
Citation 
 Permanent Link:
 http://digital.auraria.edu/AA00006083/00001
Material Information
 Title:
 Geosyntheticreinforced soil walls under multidirectional seismic shaking
 Creator:
 Lee, ZehZon
 Publication Date:
 2011
 Language:
 English
 Physical Description:
 xliii, 420 leaves : illustrations (some color) ; 28 cm
Subjects
 Subjects / Keywords:
 Geosynthetics ( lcsh )
Reinforced soils ( lcsh ) Earthquake hazard analysis ( lcsh ) Earthquake resistant design ( lcsh ) Earthquake hazard analysis ( fast ) Earthquake resistant design ( fast ) Geosynthetics ( fast ) Reinforced soils ( fast )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Bibliography:
 Includes bibliographical references (leaves 404420).
 Statement of Responsibility:
 by ZehZon Lee.
Record Information
 Source Institution:
 University of Florida
 Rights Management:
 All applicable rights reserved by the source institution and holding location.
 Resource Identifier:
 780176658 ( OCLC )
ocn780176658
 Classification:
 LD1193.E53 2011D L44 ( lcc )

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GEOSYNTHETICREINFORCED SOIL WALLS UNDER MULTIDIRECTIONAL
SEISMIC SHAKING by
ZehZon Lee
B.S., University of Colorado Denver, 1998 M.S., University of Colorado Denver, 2000
A thesis submitted to the University of Colorado Denver in partial fulfillment of the requirements for the degree of Doctor of Philosophy Civil Engineering
2011
This thesis for the Doctor of Philosophy degree by ZehZon Lee has been approved by
HonYim Ko
// ^0 //
Date
Lee, ZehZon (Ph.D., Department of Civil Engineering, College of Engineering and Applied Science)
GeosyntheticReinforced Soil Walls under Multidirectional Seismic Shaking Thesis directed by Professor NienYin Chang
ABSTRACT
This research study was conducted to determine the validity of the current seismic analysis of a freestanding simple geosyntheticreinforced soil (GRS) wall. The case histories of GRS walls during the past several large earthquakes have indicated satisfactory seismic performances and suggested that the current design methodology is adequate. Despite successful cases, several GRS wall failures have been reported. Numerical simulation of GRS wall subjected to seismic loading can thus offer the opportunity to identify the discrepancy between the current design methodology and the more rigorous finite element method (FEM) solutions.
The predictive capability of the FEM computer program LSDYNA was first validated against fullscale shaking table test walls. Material characterizations of the backfill and geosynthetic reinforcements were performed in the validation process. Material model parameters were determined from the available laboratory data. In particular, the backfill was simulated with a cap model with parameters dependent of stress level. Model calibration was also performed to fine tune the input parameters such as the viscousdamping constant, mass damping coefficient, and the soilgeosynthetic interface friction coefficient. The calibrated values along with the material characterization approaches were adopted for the subsequent parametric study. Prior to the parametric study, the extent of finite element model boundary was verified in order to minimize the boundary effect.
Results of parametric study were compared against the values determined using the Federal Highway Administration (FHWA) methodology. The parametric study results were presented as functions of peak horizontal acceleration (PHA), and the correlations of seismic performances with PHA were determined through single predictor variable regression analyses in order to indicate the trend of the calculated results. In general, the external stability analysis results predicted by FEM are higher than those determined using the FHWA methodology. However, for the internal stability analysis, FHWA methodology overestimates the reinforcement tensile load as compared to the results predicted by FEM.
Using FEM results, multivariate regression equations were developed for the various seismic performances based on multiple design parameters that are essential in the design of GRS walls. The prediction equations can provide firstorder
estimates of the performances for use in the preliminary analysis of freestanding simple GRS walls. The prediction equations are applicable for PHA greater than maximum limitation stipulated by the FHWA methodology.
This abstract accurately represents the content of the candidate's thesis. I recommend its publication.
DEDICATION
I dedicate this thesis to my parents, who gave me an appreciation of learning and taught me the values of perseverance and resolve. I also dedicate this thesis to my wife, TzeHong, my children, Alexis YingShan and Stanley HsinYu, for their unfaltering support and understanding while I was completing this thesis.
ACKNOWLEDGEMENT
I would like to thank foremost my advisor Professor NienYin Chang for providing me the freedom to manage my own work. This helped me to hone my skills to solve difficult problems independently. Professor Chang has provided continuous support throughout this study. Without his guidance and assistance, the completion of this study would not have been possible. I have benefited greatly from his extensive expertise and wisdom.
I also would like to thank the committee members, Professor HonYim Ko, Professor Brian Brady, Dr. Aziz Khan, Professor John McCartney, and Dr. Trever ShingChun Wang for their comments and suggestions on the thesis. I am indebted to Professor Ko for serving as my coadvisor, and his classes in soil mechanics had prepared me well to understand indepth topics in geotechnical engineering. I am also grateful to Professor Brady for his continuous encouragement throughout my graduate study, and his advices on working for the federal government have been beneficial. Dr. Wang has also provided much technical insight in the actual design of retaining structures.
It was a privilege to work with my classmates, Mohammad AbuHassan, Hien Manh Nghiem, and Brian Volmer. Mohammad shared his knowledge and experience on using the contact algorithm in LSDYNA. Hien was a great source of help on issues related to numerical analyses and constitutive modeling. Last but not least, discussions on engineering problems and practical issues with Brian had always been fruitful.
TABLE OF CONTENTS
Figures.....................................................................xiii
Tables.......................................................................xli
Chapter
1. Introduction..............................................................1
1.1 Problem Statement.........................................................1
1.2 Research Objectives.......................................................2
1.3 Scope of Study............................................................2
1.4 Engineering Significances.................................................3
2. Literature Review.........................................................5
2.1 Post Earthquake Investigations............................................5
2.2 Laboratory Model Tests...................................................12
2.3 Numerical Model Studies..................................................19
2.4 Seismic Design of Retaining Walls........................................26
2.4.1 MononobeOkabe Method.................................................26
2.4.2 Design and Analysis of GeosyntheticReinforced Soil
Retaining Walls......................................................30
2.4.2.1 Federal Highway Administration (FHWA) Methodology.....................31
2.4.2.2 National Concrete Masonry Association (NCMA) Methodology..............37
2.4.3 Peak Ground Acceleration Coefficients.................................47
Vll
2.4.4 Permanent Displacement Methods
.48
2.4.5 Empirical Methods........................................................52
2.5 Behavior of Geosynthetics subject to Cyclic Loads........................56
2.6 Soilreinforcement Interface Friction....................................60
3. Validation of Computer Program..............................................62
3.1 Terminology..............................................................62
3.1.1 Verification and Validation..............................................63
3.1.2 Prediction...............................................................64
3.1.3 Calibration..............................................................66
3.1.4 Validation Assessment....................................................66
3.2 Material Characterization for Computer Program Validation................67
3.2.1 Geologic Cap Model.......................................................68
3.2.1.1 Cap Model Strength Parameters..........................................71
3.2.1.2 Cap Model Hardening Parameters.........................................75
3.2.2 PlasticKinematic Model.................................................86
3.3 Validation of Computer Program with FullScale Tests........................89
3.3.1 Model Configuration......................................................89
3.3.1.1 Element Types..........................................................91
3.3.1.2 Loading and Boundary Conditions........................................91
3.3.1.3 Contact Types and Contact Details......................................96
viii
3.3.2 Model Calibration
99
3.3.2.1 ViscousDamping Constant..............................................99
3.3.2.2 Mass Damping Coefficient..............................................99
3.3.2.3 Friction Coefficient of SoilGeogrid Interface......................102
3.3.3 Response Comparison..................................................104
3.3.3.1 Wall Facing Displacement.............................................105
3.3.3.2 Backfill Surface Settlement..........................................105
3.3.3.3 Lateral Earth Pressure behind Facing Blocks.........................111
3.3.3.4 Bearing Pressure.....................................................Ill
3.3.3.5 Geogrid Reinforcement Tensile Load..................................111
3.3.3.6 Accelerations in Reinforced and Retained Soil Zones.................115
3.3.3.7 Energy Observation...................................................115
3.3.4 Quality of Prediction................................................118
3.3.5 Variability within the Measured Data.................................120
3.4 Discussion of Numerical Simulation.......................................123
4. Parametric Study........................................................125
4.1 Input Ground Motions....................................................125
4.2 Material Characterization for Parametric Study..........................126
4.2.1 Soil Characterization................................................137
4.2.1.1 Cap Model Strength Parameters in Parametric Study...................139
4.2.1.2 Cap Model Hardening Parameters in Parametric Study..................140
IX
4.2.2
Geosynthetic Reinforcement Characterization
148
4.3 Development of Model Dimensions........................................150
4.4 Parametric Study Program...............................................152
4.5 Global Stability.......................................................152
4.6 Modeling Procedure.....................................................154
5. Results of Parametric Study..............................................172
5.1 Effects of Multidirectional Shaking....................................172
5.2 Seismic Performances...................................................176
5.2.1 Effects of Wall Height.................................................178
5.2.2 Effects of Wall Batter Angle...........................................188
5.2.3 Effects of Soil Friction Angle.........................................192
5.2.4 Effects of Reinforcement Spacing.......................................197
5.2.5 Effects of Reinforcement Stiffness.....................................201
5.3 Distribution of Reinforcement Tensile Load.............................206
5.4 Soil Thrusts and Reinforcement Resultants at Distances behind
Wall Facing...........................................................222
6. Multivariate Statistical Modeling........................................241
6.1 Correlation Analysis...................................................241
6.2 Regression Analysis....................................................244
6.2.1 Prediction of Maximum Horizontal Displacement.........................245
6.2.2 Prediction of Maximum Crest Settlement................................249
x
6.2.3 Prediction of Total Driving Resultant................................253
6.2.4 Prediction of Total Overturning Moment Arm...........................257
6.2.5 Prediction of Maximum Bearing Stress.................................258
6.2.6 Prediction of Maximum Reinforcement Tensile Load.....................262
6.2.7 Prediction of Maximum Horizontal Acceleration at
Centroid of Reinforced Soil Mass..................................266
6.2.8 Prediction Equations without Peak Vertical Acceleration.............270
6.3 Design Considerations..................................................271
7. Conclusions and Recommendations for Future Studies.....................274
7.1 Conclusions............................................................275
7.2 Recommendations for Future Studies.....................................279
Appendix
A. RambergOsgood Material Model..........................................280
B. Global Stability of Models considered in Parametric Study..............288
C. Analysis of GRS Wall following FHWA Methodology........................292
D. Maximum Wall Facing Horizontal Displacement Profiles...................300
E. Maximum Wall Crest Settlement Profiles.................................312
F. Maximum Lateral Earth Stress Distributions.............................323
G. Maximum Bearing Stress Distributions...................................335
H. Maximum Reinforcement Tensile Load Profiles............................346
I. Correlations with Peak Vertical Acceleration...........................358
xi
J. Regression Analysis for Low Peak Horizontal Acceleration Range...............379
K. Data for Statistical Modeling.................................................396
L. Natural Period of GRS Walls...................................................403
References........................................................................404
xii
LIST OF FIGURES
Figure
1.1 Schematics of a Freestanding Simple GRS Wall...........................2
1.2 Scope of Research Study..................................................3
2.1 Relationships between Maximum Horizontal and Vertical Accelerations of
Northridge Earthquake (after Stewart et al. 1994)......................6
2.2 GeosyntheticReinforced Soil Wall with FHR Facing
(after Tatsuoka 1993)..................................................7
2.3 Collapse of GeosyntheticReinforced Soil Wall during 1999 ChiChi, Taiwan
Earthquake (after Ling et al. 2001)....................................8
2.4 Two Failed GeosyntheticReinforced Soil Walls during 1999 ChiChi, Taiwan
Earthquake (after Huang et al. 2003)...................................10
2.5 Failure of GeogridReinforced Soil Wall during 2001 El Salvador Earthquake
(after Koseki et al. 2006).............................................11
2.6 Failure of GeogridReinforced Soil Wall during 2001 Nisqually Earthquake
(after Walsh et al. 2001)..............................................11
2.7 The Influence of Wall Type on Wall Displacement with Irregular Base Shaking
(adapted from Watanabe et al. 2003)....................................15
2.8 Effect of Relative Density and Input Motion Frequency on the Wall
Displacement (adapted from Latha and Krishna 2008).....................15
2.9 Example of Twopart Wedge Failure Mechanism
(after Matsuo et al. 1998).............................................16
2.10 Comparison of Reinforced Soil Wall Model Before and After the Shaking Table
Test (after Anastasopoulos et al. 2010)................................16
xiii
2.11 Comparison of Reinforced Soil Wall Response at Different Harmonic Input
Motions; (i) 15 Hz Harmonic Input Motion; (ii) 30 Hz Harmonic Input Motion (after Wolfe et al. 1978)..........................................17
2.12 Comparison of Reinforced Soil Wall Responses with Different Components of
Irregular Input Motion; (i) Vertical Component Only; (ii) Horizontal Component Only; (iii) Combined Vertical and Horizontal Components (after Wolfe etal. 1978)..................................................17
2.13 Measured Amplification Factor in the Free Field
(after Siddharthan et al. 2004a).........................................18
2.14 Comparison of Observed Critical Seismic Coefficients to Calculated Ones against Overturning or Bearing Capacity Failure
(after Koseki et al. 2003)...............................................18
2.15 Equilibrium of Forces acting on the Active Wedge in M0 Analysis.........27
2.16 Equilibrium of Forces acting on the Passive Wedge in M0 Analysis........28
2.17 Effect of Direction of Inertia Forces on Total Seismic Active Thrust (modified
from Fang and Chen 1995).................................................29
2.18 Effect of Direction of Inertia Forces on Total Seismic Passive Thrust....30
2.19 Seismic External Stability of a GRS wall with Level Backfill in
FHWA Method..............................................................33
2.20 Seismic External Stability of a GRS Wall with Sloping Backfill in FHWA
Method...................................................................33
2.21 Seismic Internal Stability of a GRS Wall in FHWA Method..................37
2.22 Modes of Failure for GRS Walls in NCMA Method............................38
2.23 Seismic External Stability of a GRS Wall in NCMA Method..................40
2.24 Schematic for finding the Total Tensile Force in Each Reinforcement Layer in
NCMA Method..............................................................41
2.25 Schematic for Reinforcement Pullout Evaluation in NCMA Method............42
xiv
2.26 Schematic for Base Sliding and Internal Sliding Evaluation in
NCMA Method.............................................................43
2.27 Schematic for Facing Interface Shear Evaluation in NCMA Method..........44
2.28 Schematic for Facing Local Overturning Evaluation in NCMA Method.......45
2.29 Differences between the FHWA Method and the NCMA Method.................47
2.30 Relationship between A and kh...........................................48
2.31 Newmark's Doubleintegration Method for calculating Permanent Displacement
of a Sliding Soil Mass..................................................50
2.32 Schematic of the Multiblock Displacement Method (after Siddharthan et al.
2004b)..................................................................51
2.33 TwoPart Wedge Mechanism for Direct Sliding Analysis (after Leshchinsky
1997)...................................................................52
2.34 Comparison of Various Empirical Displacement Methods....................55
2.35 Response of HDPE Geogrid Specimens to (a) Multiincrement Cyclic Load Test and (b) Singleincrement Cyclic Load Test
(after Bathurst and Cai 1994)...........................................56
2.36 Characteristics of Cyclic Response of Geosynthetic Specimen (after Bathurst
and Cai 1994)...........................................................57
2.37 Area of Hysteresis Loops (Aur) for HDPE and PET Specimens during the Multiincrement Cyclic Loading (after Bathurst and Cai 1994)..................58
2.38 Unloadreload Stiffness (Jur) for HDPE and PET Specimens from the Multiincrement Cyclic Loading (after Bathurst and Cai 1994)..................58
2.39 Influence of Strain Rate on Monotonic Loadextension Behavior of Typical
Geosynthetic Specimens (after Bathurst and Cai 1994)....................59
xv
2.40 Interface Friction Angle between Ottawa sand and HDPE Specimens with (a) Monotonic Loading and (b) Repeated Loading on HDPE Lining (after O'Rourke
etal. 1990)............................................................60
2.41 Effect of Strain Rate on the Interface Friction Angle between Sand and Various Reinforcement (after Myles 1982)........................................61
3.1 Phases of Modeling and Simulation (after Schlesinger 1979)..............64
3.2 The Enhanced Soil Mechanics Triangle (after Anon. 1999).................65
3.3 Relationship of Validation to Prediction (modified from Oberkampf and
Trucano 2002)...........................................................65
3.4 Increasing Quality of Validation Metrics (after Oberkampf et al. 2002)..67
3.5 Schematic of Cap Model (modified from Desai and Siriwardane 1984)..70
3.6 Stress Paths achieved by Various Laboratory Tests in (a) J: ft Stress Space and (b) Deviatoric (Octahedral) Plane (modified from
Chen and Saleeb 1994)...................................................70
3.7 Comparison between Calculated and Measured Triaxial Compression Results of
Tsukuba Sand (measured data from Ling et al. 2005a).....................73
3.8 Effective Shear Strength Parameters from Drained CTC Tests for the Tsukuba
Sand....................................................................74
3.9 DruckerPrager and MohrCoulomb Failure Criteria in Deviatoric (Octahedral)
Plane with Different Matching Conditions (modified from Chen and Saleeb 1994).............................................................74
3.10 Stress Paths of CTC Tests and the Fixed Yield Surface ft of
Tsukuba Sand............................................................75
3.11 Hydrostatic Compression Curves of Chattahoochee Sand at Different Initial
Relative Densities (modified from Domaschuk and Wade 1969)..............76
3.12 Grain Size Distribution Curves for Tsukuba and Chattahoochee
Sands...................................................................77
xvi
3.13 Hyperbolic Representation of Mean Stress versus Total Volumetric Strain
Curve..................................................................78
3.14 Bestfit Transformed StressStrain Curves of Chattahoochee Sand........78
3.15 Variation of Initial Tangent Bulk Modulus K, and Asymptotic Total Volumetric Strain (ev)uit with Initial Relative Density for the
Chattahoochee Sand.....................................................79
3.16 Loading and Unloading Behavior of Monterey No. 0/30 Sand during
Hydrostatic Compression Test (modified from Goldstein 1988)............81
3.17 The Mean Stress versus Total, Elastic, and Plastic Volumetric Strain Curves for
Tsukuba Sand at Dr = 54%...............................................81
3.18 Variation of Tangent Bulk Modulus (Kt), Shear Modulus (G), and Hardening
Law Exponent (D) with Depth............................................84
3.19 Numerical Triaxial Compression Test of a Single Solid Element for the Tsukuba
Sand...................................................................85
3.20 Variation of Shape Factor R (Cap Surface Axis Ratio) with
Mean Stress............................................................86
3.21 Bilinear StressStrain Curve of the PlasticKinematic Model for Geogrid
Reinforcement..........................................................87
3.22 Numerical Tensile Load Test of a Single Shell Element for the Reinforcement
(dz = Prescribed Displacement).........................................88
3.23 Loadstrain Relationship of PET and PAV Geogrid Reinforcements.........88
3.24 Dimensions and Instrumentation of Walls 1, 2, and 3 (modified from Ling et al.
2005a).................................................................90
3.25 Finite Element Mesh and Boundary Condition of Walls 1, 2, and 3........92
3.26 Isometric View of the Finite Element Model Showing Various Parts.......93
3.27 Comparison between the Original Record (Uncorrected) and the Baseline
corrected Record.......................................................94
xvii
3.28 Loading Time Histories Applied to Models of (a) Wall 1, (b) Wall 2, and (c) Wall 3..............................................................................95
3.29 Contact Interfaces adopted in the Finite Element Model (e.g., Wall 1).98
3.30 Detail showing Geogrid Thickness and Incompatible Element
Boundary...............................................................98
3.31 Effect of Interface ViscousDamping Coefficient (VDC) on Facing Response of
Wall 1 due to Seismic Load.............................................100
3.32 Effect of Global Mass Damping Coefficient on Walltop Response of (a) Wall
1, (b) Wall 2, and (c) Wall 3 due to Transient Load....................101
3.33 Effect of Global Mass Damping Coefficient on Facing Response of Wall 1 due
to Seismic Loads......................................................102
3.34 Effect of Friction Coefficient on Facing Response of Wall 1 due to Seismic
Loads.................................................................103
3.35 Strength of Soilgeogrid Interface determined with Different Sizes of Direct
Shear Apparatus (modified from Ingold 1982)...........................104
3.36 Wall Face Peak Horizontal Displacement Comparison between the Calculated
and the Measured data for (a) Wall 1, (b) Wall 2, and (c) Wall 3.......106
3.37 Comparison of Wall 1 Face Displacement Time Histories for (a) First Shaking
and (b) Second Shaking................................................107
3.38 Comparison of Wall 2 Face Displacement Time Histories for (a) First Shaking
and (b) Second Shaking................................................108
3.39 Comparison of Wall 3 Face Displacement Time Histories for (a) First Shaking
and (b) Second Shaking................................................109
3.40 Comparison of Backfill Surface Settlement for (a) Wall 1, (b) Wall 2, and (c)
Wall 3................................................................110
3.41 Comparison of Lateral Earth Pressure behind Facing Blocks for (a) Wall 1, (b)
Wall 2, and (c) Wall 3................................................112
xviii
3.42 Comparison of Bearing Pressure for (a) Wall 1, (b) Wall 2, and (c) Wall 3........................................
113
3.43 Comparison of Geogrid Reinforcement Tensile Load for (a) Wall 1, (b) Wall 2,
and (c) Wall 3.........................................................114
3.44 Comparison of Absolute Horizontal Acceleration in the Reinforced Soil Zone
for (a) Wall 1, (b) Wall 2, and (c) Wall 3.............................116
3.45 Comparison of Absolute Horizontal Acceleration in the Retained Soil Zone for
(a) Wall 1, (b) Wall 2, and (c) Wall 3.................................117
3.46 Total Energy Time Histories of the Three Numerical Models..............118
3.47 Cumulative Weight of Performance for indicating the Prediction Capability of
LSDYNA................................................................120
3.48 Variability in the Measured and the Calculated Initial Lateral
Earth Pressures........................................................121
3.49 Variability in the Measured and the Calculated Initial Bearing
Pressures..............................................................121
3.50 Variability in the Measured and the Calculated Base Layer Reinforcement
Initial Tensile Load...................................................122
3.51 Variability in the Measured and the Calculated Initial Maximum Reinforcement
Tensile Load...........................................................122
4.1 Acceleration Time Histories of the 20 Selected Earthquake Records.......129
4.2 Response Spectra of the 20 Selected Earthquake Records..................133
4.3 Variation of Peak Vertical Acceleration with Peak Horizontal Acceleration for
the 20 Selected Earthquake Records.....................................137
4.4 Relation between Standard Penetration Resistance and Friction Angle for the
Granular Soils Considered in Parametric Study..........................138
xix
4.5 Determination of Dry Unit Weight and Soil Classification (modified from
NAVFAC 1986a)..........................................................139
4.6 Variation of Hardening Law Exponent D with Relative Density Dr..........142
4.7 Variation of Tangent Bulk Modulus Kt with Relative Density Dr...........143
4.8 Variation of Shear Modulus G with Relative Density Dr...................143
4.9 Variation of Hardening Law Coefficient W with Relative Density Dr.......144
4.10 Stressstrain Curves for ' = 32 Soil from Numerical Triaxial Tests.145
4.11 Stressstrain Curves for ' = 36 Soil from Numerical Triaxial Tests.145
4.12 Stressstrain Curves for <)>' = 40 Soil from Numerical Triaxial Tests.146
4.13 Variation of (a) Tangent Bulk Modulus Kt and (b) Shear
Modulus G with Depth...................................................147
4.14 Variation of Hardening Law Exponent D with Depth.......................148
4.15 Comparison of Tensile Load Test Results between Idealized Geogrids and
Typical Geogrids.......................................................149
4.16 Numerical Model Dimensions Adopted in the Parametric Study.............150
4.17 Effect of Lateral Boundary Extent on Wall Displacement.................151
4.18 Parametric Study Program...............................................153
4.19 Global Factor of Safety of the Baseline Model Configuration............154
4.20 Isometric View of 6 m High with 15 Wall Batter Finite
Element Model..........................................................157
4.21 Wall Dimensions and Materials for Model of H = 6 m, Sv = 0.4 m,
and co = 10...........................................................158
4.22 Finite Element Mesh for Model of H = 6 m, Sv = 0.4 m, and co = 10.....159
xx
4.23 Wall Dimensions and Materials for Model of H = 3 m, Sv = 0.4 m,
and (o=10...........................................................160
4.24 Finite Element Mesh for Model of H = 3 m, Sv 0.4 m, and co = 10.161
4.25 Wall Dimensions and Materials for Model of H = 9 m, Sv = 0.4 m,
and co = 10.........................................................162
4.26 Finite Element Mesh for Model of H = 9 m, Sv = 0.4 m, and co = 10.163
4.27 Wall Dimensions and Materials for Model of H = 6 m, Sv = 0.4 m,
and (0 = 5..........................................................164
4.28 Finite Element Mesh for Model of H = 6 m, Sv = 0.4 m, and co = 5..165
4.29 Wall Dimensions and Materials for Model of H = 6 m, Sv = 0.4 m,
and (0=15...........................................................166
4.30 Finite Element Mesh for Model of H = 6 m, Sv = 0.4 m, and co = 15.167
4.31 Wall Dimensions and Materials for Model of H = 6 m, Sv = 0.2 m,
and co = 10.........................................................168
4.32 Finite Element Mesh for Model of H = 6 m, Sv = 0.2 m, and co = 10.169
4.33 Wall Dimensions and Materials for Model of H = 6 m, Sv = 0.6 m,
and co = 10.........................................................170
4.34 Finite Element Mesh for Model of H = 6 m, Sv = 0.6 m, and co = 10.171
5.1 Vartiation of Maximum Horizontal Wall Dispalcement with Peak Horiztonal
Acceleration by the Effect of Multidirection Shaking.................173
5.2 Vartiation of Maximum Wall Crest Settlement with Peak Horiztonal
Acceleration with by Effect of Multidirection Shaking................173
5.3 Vartiation of Maximum Bearing Stress with Peak Horiztonal Acceleration by
the Effect of Multidirection Shaking.................................174
xxi
5.4 Vartiation of Maximum Reinforcement Tensile Load with Peak Horiztonal
Acceleration by the Effect of Multidirection Shaking...................174
5.5 Comparison of Percent Increase for the Effect of Multidirection Shaking.175
5.6 Correlations of Maximum Wall Displacement with PHA and Other Spectral
Accelerations..........................................................177
5.7 Effect of Wall Height on Maximum Wall Facing Horizontal
Displacement...........................................................180
5.8 Effect of Wall Height on Maximum Wall Crest Settlement.................180
5.9 Effect of Wall Height on Total Driving Resultant.......................181
5.10 Effect of Wall Height on Total Overturning Moment Arm of Total Driving
Resultant..............................................................181
5.11 Effect of Wall Height on Maximum Bearing Stress.......................182
5.12 Effect of Wall Height on Maximum Reinforcement Tensile Load...........182
5.13 Effect of Wall Height on Maximum Horizontal Acceleration at Centroid of the
Reinforced Soil Mass...................................................183
5.14 Contours of XDisplacement at End of Analysis with Northridge Earthquake
P0883 ORR090 (Model: H = 6 m, Sv = 0.4 m, to = 10, <)>' 36, T5o/o = 36 kN/m).............................................................184
5.15 Contours of XDisplacement at End of Analysis with Northridge Earthquake
P0935 TAR360 (Model: H 9 m, Sv = 0.4 m, co 10, f 36, T5o/o = 36 kN/m).............................................................185
5.16 Comparison of Centorid Horizontal Acceleration Time Histories for the
Reinforced Soil Mass and the Retained Earth with Northridge Earthquake P0883 ORR090 (Model: H = 6 m, to = 10, f = 36, Sv = 0.4 m, T5% = 36 kN/m)..................................................................187
5.17 Effect of Wall Batter Angle on Maximum Wall Facing Horizontal
Displacement...........................................................188
xxn
5.18 Effect of Wall Batter Angle on Maximum Wall Crest Settlement..........189
5.19 Effect of Wall Batter Angle on Total Driving Resultant................189
5.20 Effect of Wall Batter Angle on Total Overturning Moment Arm of Total
Driving Resultant......................................................190
5.21 Effect of Wall Batter Angle on Maximum Bearing Stress.................190
5.22 Effect of Wall Batter Angle on Maximum Reinforcement Tensile Load.....191
5.23 Effect of Wall Batter Angle on Maximum Horizontal Acceleration at Centroid
of the Reinforced Soil Mass............................................191
5.24 Effect of Soil Friction Angle on Maximum Wall Facing Horizontal
Displacement...........................................................193
5.25 Effect of Soil Friction Angle on Maximum Wall Crest Settlement........193
5.26 Effect of Soil Friction Angle on Total Driving Resultant..............194
5.27 Effect of Soil Friction Angle on Total Overturning Moment Arm of Total
Driving Resultant......................................................194
5.28 Effect of Soil Friction Angle on Maximum Bearing Stress...............195
5.29 Effect of Soil Friction Angle on Maximum Reinforcement
Tensile Load...........................................................195
5.30 Effect of Soil Friction Angle on Maximum Horizontal Acceleration at Centroid
of the Reinforced Soil Mass............................................196
5.31 Effect of Reinforcement Spacing on Maximum Wall Facing Horizontal
Displacement...........................................................197
5.32 Effect of Reinforcement Spacing on Maximum Wall Crest Settlement.....198
5.33 Effect of Reinforcement Spacing on Total Driving Resultant............198
5.34 Effect of Reinforcement Spacing on Total Overturning Moment Arm of Total
Driving Resultant......................................................199
xxiii
5.35 Effect of Reinforcement Spacing on Maximum Bearing Stress
199
5.36 Effect of Reinforcement Spacing on Maximum Reinforcement
Tensile Load.........................................................200
5.37 Effect of Reinforcement Spacing on Maximum Horizontal Acceleration at
Centroid of the Reinforced Soil Mass.................................200
5.38 Effect of Reinforcement Stiffness on Maximum Wall Facing Horizontal
Displacement.........................................................202
5.39 Effect of Reinforcement Stiffness on Maximum Wall Crest Settlement..202
5.40 Effect of Reinforcement Stiffness on Total Driving Resultant........203
5.41 Effect of Reinforcement Stiffness on Total Overturning Moment Arm of Total
Driving Resultant....................................................203
5.42 Effect of Reinforcement Stiffness on Maximum Bearing Stress.........204
5.43 Effect of Reinforcement Stiffness on Maximum Reinforcement
Tensile Load.........................................................204
5.44 Effect of Reinforcement Stiffness on Maximum Horizontal Acceleration at
Centroid of the Reinforced Soil Mass.................................205
5.45 Fictitious Reinforcement Tension Distribution: (a) Distribution along Individual
Layer (b) Contours of Maximum Reinforcement Tensile Load.............207
5.46 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b)
Contours of Maximum Reinforcement Tensile Load under Static Condition (Model: H = 6 m, Sv = 0.4 m, to = 10, <(>' = 36, Ts% = 36 kN/m)...208
5.47 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b)
Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m, Sv = 0.4 m, to = 10,
5.48 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b)
Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 3 m, Sv = 0.4 m, co = 10,
xxiv
5.49 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b)
Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 9 m, Sv = 0.4 m, to = 10, (j>' = 36, T5o/(> = 36 kN/m).211
5.50 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b)
Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m, Sv = 0.4 m, co = 5, = 36, T5o/o = 36 kN/m)........212
5.51 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b)
Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m, Sv = 0.4 m, co = 15,
5.52 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b)
Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m, Sv = 0.4 m, co = 10, f = 32, TSo/o = 36 kN/m).....214
5.53 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b)
Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m, Sv = 0.4 m, co = 10, ' = 40, T5% = 36 kN/m)....215
5.54 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b)
Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m, Sv = 0.2 m, co = 10, ' = 36, TSo/o = 36 kN/m)..216
5.55 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b)
Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m, Sv = 0.6 m, co = 10, f = 36, Ts% = 36 kN/m).......217
5.56 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b)
Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m, Sv = 0.4 m, co = 10, f = 36, T5% = 72 kN/m).......218
5.57 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b)
Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m, Sv = 0.4 m, co = 10, f = 36, T5o/o = 12 kN/m).....219
5.58 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b)
Contours of Maximum Reinforcement Tensile Load under Seismic Condition with Coalinga earthquake, P0346, HZ14000, PHA = 0.282 g (Model: H = 6 m, Sv = 0.4 m, co = 10, ' = 36, T5o/o = 36 kN/m)...................223
XXV
5.59 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b)
Contours of Maximum Reinforcement Tensile Load under Seismic Condition with Northridge earthquake, PI020, SPV270, PHA = 0.753 g (Model: H = 6 m, Sv = 0.4 m, to = 10, <()' = 36, = 36 kN/m)......................224
5.60 Static and Seismic Earth Pressure Distributions (Model: H = 6 m, co 10, (j)' 
36, Sv = 0.4 m, T5o/0 = 36 kN/m)......................................225
5.61 Magnitudes and Locations of Soil Thrusts and Reinforcement Resultants
(Model: H = 6 m, co = 10, ' = 36, Sv = 0.4 m, T5% 36 kN/m)......226
5.62 Earth Pressure Distributions for Static Thrust, Inertia Force, and Seismic Thrust
Increment per FHWA Methodology.........................................227
5.63 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b)
Moment Arms of Thrusts and Resultants (Model: H = 6 m, co = 10,
5.64 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b)
Moment Arms of Thrusts and Resultants (Model: H = 3 m, co = 10, 4>' = 36, Sv = 0.4 m, T5% = 36 kN/m)...........................................230
5.65 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b)
Moment Arms of Thrusts and Resultants (Model: H = 9 m, oo = 10, <)>' = 36, Sv = 0.4 m, T5o/o = 36 kN/m).........................................231
5.66 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b)
Moment Arms of Thrusts and Resultants (Model: FI = 6 m, co = 5, (j)' = 36, Sv = 0.4 m, T5o/o = 36 kN/m).........................................232
5.67 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b)
Moment Arms of Thrusts and Resultants (Model: H = 6 m, co = 15, (j)' = 36, Sv = 0.4 m, T5o/o = 36 kN/m).........................................233
5.68 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b)
Moment Arms of Thrusts and Resultants (Model: H = 6 m, co = 10,
XXVI
5.69 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b)
Moment Arms of Thrusts and Resultants (Model: H = 6 m, co = 10, ' = 40, Sv = 0.4 m, Ts% = 36 kN/m)..........................................235
5.70 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b)
Moment Arms of Thrusts and Resultants (Model: H = 6 m, co = 10, <()' = 36, Sv = 0.2 m, Ts% = 36 kN/m)..........................................236
5.71 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b)
Moment Arms of Thrusts and Resultants (Model: H = 6 m, to = 10, ' = 36, Sv = 0.6 m, T5o/o = 36 kN/m)........................................237
5.72 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b)
Moment Arms of Thrusts and Resultants (Model: H = 6 m, co = 10, <))' = 36, Sv = 0.4 m, T5o/0= 12 kN/m).........................................238
5.73 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b)
Moment Arms of Thrusts and Resultants (Model: H = 6 m, co = 10,
6.1 Scatter Diagrams with Various Degrees of Correlation between Two Variables
(modified from Kachigan 1991).........................................242
6.2 Comparison between Predicted Ah and the FEM Results with the Effect of Wall
Height H..............................................................245
6.3 Comparison between Predicted Ah and the FEM Results with the Effect of Wall
Batter Angle co.......................................................246
6.4 Comparison between Predicted Ah and the FEM Results with the Effect of Soil
Friction Angle '...................................................246
6.5 Comparison between Predicted Ah and the FEM Results with the Effect of
Reinforcement Spacing Sv..............................................247
6.6 Comparison between Predicted Ah and the FEM Results with the Effect of
Reinforcement Stiffness T5o/0.........................................247
6.7 Comparison between Predicted Ah and the FEM Results with the Effect of Peak
Vertical Acceleration PVA.............................................248
xxvii
6.8 Comparison between Predicted Av and the FEM Results with the Effect of Wall
Height H............................................................249
6.9 Comparison between Predicted Av and the FEM Results with the Effect of Wall
Batter Angle k>.....................................................250
6.10 Comparison between Predicted Av and the FEM Results with the Effect of Soil
Friction Angle <>'.................................................250
6.11 Comparison between Predicted Av and the FEM Results with the Effect of
Reinforcement Spacing Sv............................................251
6.12 Comparison between Predicted Av and the FEM Results with the Effect of
Reinforcement Stiffness ............................................251
6.13 Comparison between Predicted Av and the FEM Results with the Effect of Peak
Vertical Acceleration PVA...........................................252
6.14 Linear Prediction Equation for Av given Ah.........................253
6.15 Comparison between Predicted XPde and the FEM Results with the Effect of
Wall Height H.......................................................254
6.16 Comparison between Predicted XPde and the FEM Results with the Effect of
Wall Batter Angle co................................................254
6.17 Comparison between Predicted XPde and the FEM Results with the Effect of
Soil Friction Angle
6.18 Comparison between Predicted EPde and the FEM Results with the Effect of
Reinforcement Spacing Sv............................................255
6.19 Comparison between Predicted EPde and the FEM Results with the Effect of
Reinforcement Stiffness Tso/0.......................................256
6.20 Comparison between Predicted XPde and the FEM Results with the Effect of
Peak Vertical Acceleration PVA......................................256
xxviii
6.21 Normalized Total Overturning Moment Arm with Wall Height versus Peak
Horizontal Acceleration.............................................258
6.22 Comparison between Predicted qVE and the FEM Results with the Effect of Wall
Height H............................................................259
6.23 Comparison between Predicted qVF. and the FEM Results with the Effect of Wall
Batter Angle to.....................................................259
6.24 Comparison between Predicted qVE and the FEM Results with the Effect of Soil
Friction Angle (j)'.................................................260
6.25 Comparison between Predicted qV[; and the FEM Results with the Effect of
Reinforcement Spacing Sv............................................260
6.26 Comparison between Predicted qVF and the FEM Results with the Effect of
Reinforcement Stiffness T504........................................261
6.27 Comparison between Predicted qVE and the FEM Results with the Effect of Peak
Vertical Acceleration PVA...........................................261
6.28 Comparison between Predicted Ttotai and the FEM Results with the Effect of
Wall Height H.......................................................263
6.29 Comparison between Predicted Ttotai and the FEM Results with the Effect of
Wall Batter Angle co................................................263
6.30 Comparison between Predicted Ttotai and the FEM Results with the Effect of
Soil Friction Angle
6.31 Comparison between Predicted Ttotai and the FEM Results with the Effect of
Reinforcement Spacing Sv............................................264
6.32 Comparison between Predicted Ttotai and the FEM Results with the Effect of
Reinforcement Stiffness T5o/o.......................................265
6.33 Comparison between Predicted Ttotai and the FEM Results with the Effect of
Peak Vertical Acceleration PVA......................................265
6.34 Comparison between Predicted Am and the FEM Results with the Effect of Wall
Height H............................................................267
XXIX
6.35 Comparison between Predicted Am and the FEM Results with the Effect of Wall
Batter Angle co......................................................267
6.36 Comparison between Predicted Am and the FEM Results with the Effect of Soil
Friction Angle
6.37 Comparison between Predicted Am and the FEM Results with the Effect of
Reinforcement Spacing Sv.............................................268
6.38 Comparison between Predicted Am and the FEM Results with the Effect of
Reinforcement Stiffness Tso/o........................................269
6.39 Comparison between Predicted Am and the FEM Results with the Effect of Peak
Vertical Acceleration PVA............................................269
6.40 Variation of Maximum Reinforcement Tensile Load with Maximum Horizontal
Displacement for the Baseline Model..................................273
A. 1 Extended Masing Rules: (a) Variation of Shear Stress with Time; (b) Resulting StressStrain Behavior with Backbone Curve indicated by Dashed Line (after Kramer 1996).........................................................281
A.2 Definitions of (a) Secant Shear Modulus and (b) Backbone Curve (after Kramer 1996)................................................................285
A.3 Modulus Reduction and Damping Ratio Curves of Average Sand (after Seed at al. 1986)............................................................286
A.4 Example of a Best Fit Straight Line for Determining Parameters a and r (after
Ueng and Chen 1992)......................................................287
B. 1 Effect of Wall Height on Global Factor of Safety.................288
B.2 Effect of Wall Batter Angle on Global Factor of Safety.................289
B.3 Effect of Soil Friction Angle on Global Factor of Safety.................290
B.4 Effect of Reinforcement Spacing on Global Factor of Safety.................291
XXX
D.l Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m, co = 10, (j)'
= 36, Sv = 0.4 m, T5o/o 36 kN/m)...................................301
D.2 Maximum Wall Facing Horizontal Displacement Profiles (H = 3 m, co = 10, (j)'
= 36, Sv = 0.4 m, T5% = 36 kN/m)......................................302
D.3 Maximum Wall Facing Horizontal Displacement Profiles (H = 9 m, co = 10, (j)' = 36, Sv = 0.4 m, T5o/o = 36 kN/m).....................................303
D.4 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m, co = 5,
36, Sv = 0.4 m, T5% = 36 kN/m)........................................304
D.5 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m, co = 15,
D.6 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m, co = 10, ((>'
 32, Sv = 0.4 m, T5o/o = 36 kN/m)....................................306
D.7 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m, co = 10, (j)'
= 40, Sv = 0.4 m, T5% = 36 kN/m)......................................307
D.8 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m, co = 10, (j)'
= 36, Sv = 0.2 m, T5% = 36 kN/m)......................................308
D.9 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m, co = 10,
= 36, Sv = 0.6 m, T5o/o = 36 kN/m)....................................309
D.10 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m, co = 10, '
= 36, Sv = 0.4 m, T5o/o = 12 kN/m)....................................310
D. l 1 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m, to = 10,
= 36, Sv = 0.4 m, T5o/o = 72 kN/m)....................................311
E. l Maximum Wall Crest Settlement Profiles (H = 6 m, to = 10, ' = 36, Sv = 0.4
m, T5o/o = 36 kN/m)....................................................312
E.2 Maximum Wall Crest Settlement Profiles (H = 3 m, to = 10, (j)' = 36, Sv = 0.4
m, T5% = 36 kN/m)...........................................................313
XXXI
E.3 Maximum Wall Crest Settlement Profiles (H = 9 m, co = 10, (j)' = 36, Sv = 0.4
m, Ts% = 36 kN/m)................................................................314
E.4 Maximum Wall Crest Settlement Profiles (H = 6 m, to = 5, <)>' = 36, Sv = 0.4 m, T5% = 36 kN/m)...........................................................315
E.5 Maximum Wall Crest Settlement Profiles (H = 6 m, co = 15, (j)' = 36, Sv = 0.4 m, T5% = 36 kN/m)...........................................................316
E.6 Maximum Wall Crest Settlement Profiles (H = 6 m, co = 10, <(>' = 32, Sv = 0.4 m, T5% = 36kN/m)............................................................317
E.7 Maximum Wall Crest Settlement Profiles (H = 6 m, co = 10, (j)' = 40, Sv = 0.4 m, T5% = 36kN/m)............................................................318
E.8 Maximum Wall Crest Settlement Profiles (H = 6 m, co = 10, ' = 36, Sv = 0.2 m, T5o/o = 36 kN/m).........................................................319
E.9 Maximum Wall Crest Settlement Profiles (H = 6 m, co = 10, (j)' = 36, Sv = 0.6 m, Ts% = 36 kN/m)...........................................................320
E.10 Maximum Wall Crest Settlement Profiles (H = 6 m, co = 10, ' = 36, Sv = 0.4 m, T5o/o= 12 kN/m)..........................................................321
E. l 1 Maximum Wall Crest Settlement Profiles (H = 6 m, co = 10, ' = 36, Sv = 0.4
m, T5o/o = 72 kN/m)..........................................................322
F. l Maximum Lateral Earth Stress Distributions (H = 6 m, co = 10, <>' = 36, Sv =
0.4 m, T5o/o = 36 kN/m).....................................................324
F.2 Maximum Lateral Earth Stress Distributions (H = 3 m, co 10, ' = 36, Sv = 0.4 m, T5o/o = 36 kN/m).....................................................325
F.3 Maximum Lateral Earth Stress Distributions (H = 9 m, co = 10, cj) = 36, Sv = 0.4 m, T5% = 36 kN/m).......................................................326
F.4 Maximum Lateral Earth Stress Distributions (H = 6 m, co = 5, <)>' = 36, Sv = 0.4
m, T5o/o = 36 kN/m)...............................................................327
xxxn
F.5 Maximum Lateral Earth Stress Distributions (H = 6 m, co = 15,
0.4 m, T5/o = 36 kN/m)...............................................................328
F.6 Maximum Lateral Earth Stress Distributions (H = 6 m, co = 10, (j)' = 32, Sv = 0.4 m, T5% = 36 kN/m)............................................................329
F.7 Maximum Lateral Earth Stress Distributions (H = 6 m, co = 10, ' = 40, Sv = 0.4 m, T5% = 36 kN/m)............................................................330
F.8 Maximum Lateral Earth Stress Distributions (H = 6 m, co = 10,
F.9 Maximum Lateral Earth Stress Distributions (H = 6 m, oj = 10,
F.10 Maximum Lateral Earth Stress Distributions (H = 6 m, co = 10,
F. l 1 Maximum Lateral Earth Stress Distributions (H = 6 m, co = 10, (j)' = 36, Sv =
0.4 m, T5o/o = 72 kN/m).........................................................334
G. l Maximum Bearing Stress Distributions (H = 6 m, co = 10, <)>' = 36, Sv = 0.4 m,
T5o/o = 36 kN/m)................................................................335
G.2 Maximum Bearing Stress Distributions (H = 3 m, co = 10,
T5o/o = 36 kN/m)................................................................336
G.3 Maximum Bearing Stress Distributions (H = 9 m, co = 10,
G.4 Maximum Bearing Stress Distributions (H = 6 m, to = 5, (J)' = 36, Sv = 0.4 m, T5o/o = 36 kN/m)...........................................................338
G.5 Maximum Bearing Stress Distributions (H = 6 m, co = 15, ' = 36, Sv = 0.4 m, T5o/o = 36 kN/m)...........................................................339
G.6 Maximum Bearing Stress Distributions (H = 6 m, co = 10,
T5o/o = 36 kN/m)................................................................340
xxxiii
G.7 Maximum Bearing Stress Distributions (H = 6 m, co = 10, (j)' = 40, Sv = 0.4 m,
T5% = 36 kN/m)...............................................................341
G.8 Maximum Bearing Stress Distributions (H = 6 m, to 10,
T5o/o = 36 kN/m).......................................................342
G.9 Maximum Bearing Stress Distributions (H = 6 m, co = 10, ' = 36, Sv = 0.6 m, T5o/o = 36 kN/m)........................................................343
G.10 Maximum Bearing Stress Distributions (H = 6 m, co = 10, ' = 36, Sv = 0.4 m,
T5o/o = 12 kN/m)........................................................344
G. l 1 Maximum Bearing Stress Distributions (H = 6 m, co = 10,
T5% = 72 kN/m)..........................................................345
H. l Maximum Reinforcement Tensile Load Profiles (H = 6 m, co = 10, <(>' = 36, Sv
= 0.4 m, Ts% = 36 kN/m)................................................347
H.2 Maximum Reinforcement Tensile Load Profiles (H = 3 m, co = 10, <(>' = 36, Sv = 0.4 m, T5o/o = 36 kN/m)...............................................348
H.3 Maximum Reinforcement Tensile Load Profiles (H = 9 m, co = 10, <(>' = 36, Sv = 0.4 m, T5% = 36 kN/m).................................................349
H.4 Maximum Reinforcement Tensile Load Profiles (H = 6 m, co = 5, <>' = 36, Sv = 0.4 m, T5% = 36 kN/m)...................................................350
H.5 Maximum Reinforcement Tensile Load Profiles (H = 6 m, co = 15, ' = 36, Sv = 0.4 m, T5o/o = 36 kN/m)...............................................351
H.6 Maximum Reinforcement Tensile Load Profiles (H = 6 m, co = 10, (j)' = 32, Sv = 0.4 m, T5o/o = 36 kN/m)...............................................352
H.7 Maximum Reinforcement Tensile Load Profiles (H = 6 m, co = 10, ())' = 40, Sv = 0.4 m, T5o/o = 36 kN/m)...............................................353
H.8 Maximum Reinforcement Tensile Load Profiles (H = 6 m, co = 10,
= 0.2 m, T5% = 36 kN/m)......................................................354
XXXIV
H.9 Maximum Reinforcement Tensile Load Profiles (H = 6 m, a> = 10,
H.10 Maximum Reinforcement Tensile Load Profiles (H = 6 m, co = 10, ' = 36, Sv = 0.4 m, T5o/o = 12 kN/m)................................................356
H. l 1 Maximum Reinforcement Tensile Load Profiles (H = 6 m, w = 10, ' = 36, Sv
= 0.4 m, T5o/o = 72 kN/m)...............................................357
I. 1 Effect of Wall Height on Maximum Wall Facing Horizontal Displacement
(Correlated with PVA)....................................................359
1.2 Effect of Wall Height on Maximum Wall Crest Settlement
(Correlated with PVA)....................................................359
1.3 Effect of Wall Height on Total Driving Resultant
(Correlated with PVA)....................................................360
1.4 Effect of Wall Height on Total Overturning Moment Arm of Total Driving
Resultant (Correlated with PVA)..........................................360
1.5 Effect of Wall Height on Maximum Bearing Stress
(Correlated with PVA)....................................................361
1.6 Effect of Wall Height on Maximum Reinforcement Tensile Load (Correlated
with PVA)................................................................361
1.7 Effect of Wall Height on Maximum Horizontal Acceleration at Centroid of the
Reinforced Soil Mass (Correlated with PVA)...............................362
1.8 Effect of Wall Batter Angle on Maximum Wall Facing Horizontal Displacement
(Correlated with PVA)....................................................363
1.9 Effect of Wall Batter Angle on Maximum Wall Crest Settlement (Correlated
with PVA)................................................................363
1.10 Effect of Wall Batter Angle on Total Driving Resultant
(Correlated with PVA)....................................................364
1.11 Effect of Wall Batter Angle on Total Overturning Moment Arm of Total Drive
Resultant (Correlated with PVA)..........................................364
xxxv
1.12 Effect of Wall Batter Angle on Maximum Bearing Stress
(Correlated with PVA).................................................365
1.13 Effect of Wall Batter Angle on Maximum Reinforcement Tensile Load
(Correlated with PVA).................................................365
1.14 Effect of Wall Batter Angle on Maximum Horizontal Acceleration at Centroid
of the Reinforced Soil Mass (Correlated with PVA).....................366
1.15 Effect of Soil Friction Angle on Maximum Wall Facing Horizontal
Displacement (Correlated with PVA)....................................367
1.16 Effect of Soil Friction Angle on Maximum Wall Crest Settlement (Correlated
with PVA).............................................................367
1.17 Effect of Soil Friction Angle on Total Driving Resultant
(Correlated with PVA).................................................368
1.18 Effect of Soil Friction Angle on Total Overturning Moment Arm of Total Drive
Resultant (Correlated with PVA).......................................368
1.19 Effect of Soil Friction Angle on Maximum Bearing Stress (Correlated with
PVA)..................................................................369
1.20 Effect of Soil Friction Angle on Maximum Reinforcement Tensile Load
(Correlated with PVA).................................................369
1.21 Effect of Soil Friction Angle on Maximum Horizontal Acceleration at Centroid
of the Reinforced Soil Mass (Correlated with PVA).....................370
1.22 Effect of Reinforcement Spacing on Maximum Wall Facing Horizontal
Displacement (Correlated with PVA)....................................371
1.23 Effect of Reinforcement Spacing on Maximum Wall Crest Settlement
(Correlated with PVA).................................................371
1.24 Effect of Reinforcement Spacing on Total Driving Resultant (Correlated with
PVA)..................................................................372
XXXVI
1.25 Effect of Reinforcement Spacing on Total Overturning Moment Arm of Total
Drive Resultant (Correlated with PVA).................................372
1.26 Effect of Reinforcement Spacing on Maximum Bearing Stress (Correlated with
PVA)..................................................................373
1.27 Effect of Reinforcement Spacing on Maximum Reinforcement Tensile Load
(Correlated with PVA).................................................373
1.28 Effect of Reinforcement Spacing on Maximum Horizontal Acceleration at
Centroid of the Reinforced Soil Mass (Correlated with PVA)............374
/
1.29 Effect of Reinforcement Stiffness on Maximum Wall Facing Horizontal
Displacement (Correlated with PVA)....................................375
1.30 Effect of Reinforcement Stiffness on Maximum Wall Crest Settlement
(Correlated with PVA).................................................375
1.31 Effect of Reinforcement Stiffness on Total Driving Resultant (Correlated with
PVA)..................................................................376
1.32 Effect of Reinforcement Stiffness on Total Overturning Moment Arm of Total
Drive Resultant (Correlated with PVA).................................376
1.33 Effect of Reinforcement Stiffness on Maximum Bearing Stress (Correlated with
PVA)..................................................................377
1.34 Effect of Reinforcement Stiffness on Maximum Reinforcement Tensile Load
(Correlated with PVA).................................................377
1.35 Effect of Reinforcement Stiffness on Maximum Horizontal Acceleration at
Centroid of the Reinforced Soil Mass (Correlated with PVA)............378
J.l Effect of Wall Height on Maximum Wall Facing Horizontal Displacement (for PHA < 0.29 g).........................................................381
J.2 Effect of Wall Height on Maximum Wall Crest Settlement
(for PHA <0.29 g).....................................................381
J.3 Effect of Wall Height on Total Driving Resultant (for PHA < 0.29 g)....382
xxxvii
J.4 Effect of Wall Height on Maximum Bearing Stress (for PHA < 0.29 g).382
J.5 Effect of Wall Height on Maximum Reinforcement Tensile Load (for PHA < 0.29 g)...........................................................383
J.6 Effect of Wall Height on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass (for PHA < 0.29 g)...........................383
J.7 Effect of Wall Batter Angle on Maximum Wall Facing Horizontal Displacement (for PHA <0.29 g).................................................384
J.8 Effect of Wall Batter Angle on Maximum Wall Crest Settlement (for PHA <
0.29 g)..............................................................384
J.9 Effect of Wall Batter Angle on Total Driving Resultant
(for PHA <0.29 g)....................................................385
J. 10 Effect of Wall Batter Angle on Maximum Bearing Stress
(for PHA <0.29 g)....................................................385
J. 11 Effect of Wall Batter Angle on Maximum Reinforcement Tensile Load (for
PHA <0.29 g).........................................................386
J.12 Effect of Wall Batter Angle on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass (for PHA < 0.29 g).......................386
J. 13 Effect of Soil Friction Angle on Maximum Wall Facing Horizontal
Displacement (for PHA < 0.29 g)......................................387
J.14 Effect of Soil Friction Angle on Maximum Wall Crest Settlement (for PHA < 0.29 g)..............................................................387
J. 15 Effect of Soil Friction Angle on Total Driving Resultant
(for PHA <0.29 g)....................................................388
J. 16 Effect of Soil Friction Angle on Maximum Bearing Stress
(for PHA <0.29 g)....................................................388
J.17 Effect of Soil Friction Angle on Maximum Reinforcement Tensile Load (for PHA <0.29 g).........................................................389
xxxviii
J. 18 Effect of Soil Friction Angle on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass (for PHA < 0.29 g)............................389
J. 19 Effect of Reinforcement Spacing on Maximum Wall Facing Horizontal
Displacement (for PHA < 0.29 g)......................................390
J.20 Effect of Reinforcement Spacing on Maximum Wall Crest Settlement (for PHA <0.29 g).............................................................390
J.21 Effect of Reinforcement Spacing on Total Driving Resultant
(for PHA <0.29 g)....................................................391
J.22 Effect of Reinforcement Spacing on Maximum Bearing Stress
(for PHA <0.29 g)....................................................391
J.23 Effect of Reinforcement Spacing on Maximum Reinforcement Tensile Load (for PHA < 0.29 g)...................................................392
J.24 Effect of Reinforcement Spacing on Maximum Horizontal Acceleration at
Centroid of the Reinforced Soil Mass (for PHA < 0.29 g).............392
J.25 Effect of Reinforcement Stiffness on Maximum Wall Facing Horizontal
Displacement (for PHA < 0.29 g)......................................393
J.26 Effect of Reinforcement Stiffness on Maximum Wall Crest Settlement (for PHA <0.29 g).............................................................393
J.27 Effect of Reinforcement Stiffness on Total Driving Resultant
(for PHA < 0.29 g)..................................................394
J.28 Effect of Reinforcement Stiffness on Maximum Bearing Stress
(for PHA <0.29 g)..................................................394
J.29 Effect of Reinforcement Stiffness on Maximum Reinforcement Tensile Load (for PHA < 0.29 g).............................................395
J.30 Effect of Reinforcement Stiffness on Maximum Horizontal Acceleration at
Centroid of the Reinforced Soil Mass (for PHA < 0.29 g).............395
xxxix
L.l Variation of Natural Period with Wall Height and Soil Friction Angle
.403
xl
LIST OF TABLES
Table
2.1 Summary of shaking table model tests.....................................21
2.2 Summary of centrifuge model tests........................................24
2.3 Summary of computer program validation on seismic performance of
reinforced soil wall....................................................25
2.4 Comparison of recommended minimum factors of safety for
GRS walls...............................................................46
3.1 Summary of model parameters for facing block, EPS board, and geogrid
reinforcements..........................................................71
3.2 Cap model parameters for Tsukuba sand....................................71
3.3 Physical properties of Tsukuba and Chattahoochee sands (data from Ling et al.
2005a and AlHussaini 1973).............................................77
3.4 Summary of parameters for determining mean stress dependent variables D, Kt,
and G at Dr = 54%.......................................................83
3.5 Effect of hardening parameters on CTC stressstrain relationship.........86
3.6 Summary of shaking table input motions for numerical test walls..........96
3.7 Summary of LSDYNA contact interfaces defined in the
numerical model.........................................................98
3.8 Summary of calibrated parameters........................................104
3.9 Prediction quality classes (after Morgenstern 2000).....................119
3.10 Quality of numerical prediction (% error)..............................119
4.1 Ground motion parameters of the 20 earthquake records...................127
xli
4.2 Spectral parameters from response spectrum at 5% damping of the 20
earthquake records......................................................128
4.3 Correlation of relative density with standard penetration resistance (after
Kulhawy and Mayne 1990).................................................138
4.4 Physical properties of soils used in parametric study....................139
4.5 Cap model parameters for the three soils of parametric study.............146
4.6 Parameters for finding mean stress dependent variables Kt, G, and D for the
three soils of parametric study.........................................147
4.7 Plastickinematic model parameters for geogrid reinforcements............149
4.8 Summary of coefficient of friction for contact interfaces................156
4.9 Parametric study model summary...........................................156
5.1 Summary of design parameters and seismic performances....................178
6.1 Correlation matrix of independent and dependent variables................243
6.2 Change in seismic performance of GRS wall due to increase in design
parameter...............................................................270
6.3 Summary of constant and regression coefficients from the multiple regression analysis including independent variable PVA [Y = exp (bo + bi PHA + b2
PVA + b3 H + b4 co + b5 f + b6 Sv + b7 Ts%)]..................272
6.4 Summary of constant and regression coefficients from the multiple regression
analysis without independent variable PVA [Y = exp (bo + bi PHA + b2 H + b3 to + b4 ' + bs Sv + b6 T5%)]...........................272
A. 1 Material parameters for the RambergOsgood model.........................283
J. 1 Summary of constant and regression coefficients from the multiple regression
analysis using earthquake records with PHA less than 0.29 g [Y = exp (bo + bi PHA + b2 PVA + b3 H + b4 co + b5 + b6 Sv + b7 Ts%)].380
xlii
K. 1 Data for statistical modeling
397
xliii
1. Introduction
1.1 Problem Statement
Over the past several decades, numerous reinforced soil walls, either with inextensible or extensile reinforcements, have been built in the earthquake prone areas of the U.S. and abroad. The growing acceptance of reinforced soil wall over its conventional counterparts is mainly due to its cost effectiveness, which includes low material cost, short construction period, and ease of construction. The competitiveness of reinforced soil wall is even greater when the extensible geosynthetic reinforcement is employed. Case histories have indicated that when designed and constructed properly, geosyntheticreinforced soil (GRS) wall have performed well during the past several large earthquakes. Typical crosssection of a GRS wall is shown in Figure 1.1. Satisfactory seismic performances of GRS walls may have attributed to the conservatism implemented in the sateofpractice design and analysis of the structure.
To uncover the validity of the current design, seismic responses of GRS wall will need to be examined. The seismic responses can be examined by means of physical model tests or through a numerical modeling study. It is, however, uneconomical and impractical to examine the seismic responses of GRS wall by conducting a series of fullscale tests with different types of soils and reinforcements under various seismic loads. Hence, a more economical and practical approach to examine the seismic responses of GRS wall is to conduct a numerical modeling study, in which the numerical tool would need to be validated from physical model tests with well controlled conditions.
It is the interest of this research study to examine the seismic performances of freestanding simple GRS walls under real multidirectional ground motion shaking. The validated numerical tool with proven predictive capability would be used to perform a parametric study, where essential design parameters such as wall height, wall batter angle, soil friction angle, reinforcement spacing, and reinforcement stiffness would be evaluated. The results of the numerical parametric study would be compared with values determined from the current design methodology, and discrepancies between the two would be identified. The results of the numerical parametric study would also provide the data needed to develop seismic performance prediction equations. The prediction equations can assist a designer to estimate the seismic performances of GRS wall in a preliminary design setting.
1
Figure 1.1 Schematics of a Freestanding Simple GRS Wall
1.2 Research Objectives
The objectives of this research study are threefold. The first objective is to validate the numerical tool from well controlled physical model tests. The second objective is to identify discrepancies between the numerical parametric study results with values determined using the current design methodology. The third objective is to develop prediction equations for estimating seismic performances of freestanding simple GRS walls.
1.3 Scope of Study
To achieve the research study objectives, following undertakings were performed:
Review seismic performances of GRS walls in the field and from the laboratory physical model tests
Review current seismic design and analysis of GRS walls.
Characterize material parameters with laboratory test results.
Validate and calibrate the finite element method (FEM) computer program LSDYNA using fullscale shaking table wall tests.
Conduct parametric study using the validated FEM computer program.
2
Compare the numerical results with values determined from the current design methodology and identify the discrepancies between the two.
Establish seismic performance prediction equations that incorporate the essential design parameters for use in the preliminary design.
The flow chart showing the sequence of the research undertakings is depicted in Figure 1.2. The organization of the thesis follows the tasks listed above.
Figure 1.2 Scope of Research Study
1.4 Engineering Significances
The engineering significances of the research study are listed as follows:
A validated computer program can provide strong inference on its predicted results.
Although the cost of computation for numerical simulation is more expensive, the seismic performances such facing displacement and crest settlement that are otherwise not achievable in the limit equilibrium approach could be estimated.
By including the realistic earthquake shaking, nonlinear soil behavior, and the geosynthetic stiffness characteristics, numerical simulation is a better representation of GRS wall than the model analyzed using the limit equilibrium approach.
3
The hardening parameters of the cap model for describing the nonlinear granular soil behavior can be determined based on the relative density of the soil, which were derived based on laboratory stressstrain curves.
Prediction equations developed in this study are directly applicable in assisting the seismic design and analysis of freestanding simple GRS wall. The design parameters were incorporated in the prediction equations.
The established numerical modeling technique can be used to analyze more complex GRS structures.
Numerical simulation terminology (i.e., verification, validation, calibration, and prediction) used in this study adheres to those adopted by the computational mechanics community.
4
2. Literature Review
The literature review given below includes post earthquake investigations on the field performances of geosyntheticreinforced soil (GRS) retaining structures, laboratory and numerical model tests of GRS walls, and the seismic design and analysis of GRS walls. It is through the literature review that the adequacies in the current seismic design and analysis of GRS walls could be evaluated. The results of studies conducted by other researchers are used in deriving tasks to be performed in this study.
2.1 Post Earthquake Investigations
Numerous GRS structures have been built in seismically active regions in the U.S. and abroad. The post earthquake investigations on the performances of these structures can reveal the adequacy of the stateofpractice seismic design. Case histories have been reported from the 1989 Loma Prieta earthquake, 1994 Northridge earthquake, 1995 HyogoKen Nanbu, Japan earthquake, 1999 ChiChi, Taiwan earthquake, 2001 El Salvador earthquake, and 2001 Nisqually earthquake. Seismic performances of reinforced soil retaining structures from each earthquake are summarized as follows.
1989 Loma Prieta Earthquake
The performances of five geogridreinforced slopes and walls that experienced the 1989 Loma Prieta earthquake near San Francisco, California were evaluated by Collin et al. (1992). The 1989 Loma Prieta earthquake registered a Richter local magnitude of 7.1 and had duration of shaking of 10 to 15 seconds. Maximum horizontal and vertical accelerations of 0.64 g and 0.60 g, respectively, were recorded near the epicenter. Based on visual observations, the five geogrid reinforced slopes and walls experienced no signs of distress (e.g., no apparent movement, no apparent cracks, and no signs of sloughing). One of the geogrid reinforced slope was originally designed with a maximum horizontal acceleration of 0.10.2 g and had performed well with estimated site acceleration of as high as 0.4 g.
1994 Northridge Earthquake
The performances of geosyntheticreinforced soil structures that experienced the 1994 Northridge, California earthquake were reported by Sandri (1997) and White and Holtz (1997). The 1994 Northridge earthquake had a Richter local magnitude of 6.7 with strong motion duration of 10 to 15 seconds. Unlike most other earthquakes, the vertical accelerations of the Northridge earthquake were as strong as the
5
horizontal accelerations, which in turn made the Northridge earthquake destructive (see Figure 2.1). Sandri visually examined eleven geogridreinforced soil slopes and walls, and signs of distress inspected include: (1) structure alignment, (2) relative facade movements, (3) facade bulging, (4) facade cracking, (5) geogrid slippage at the wall face, (6) soil sloughing, and (7) tension crack near the slope/wall crest. The post earthquake inspection results indicated that none of the geogrid reinforced soil structures showed signs of distress. As compared to excellent seismic performances of one of the geogridreinforced soil structures, cantilever walls located nearby (within 100 m) experienced significant cracking and required major remedial measures. Similar findings regarding the seismic performances of geosyntheticreinforced soil slopes and walls located in the Greater Los Angles area were reported by White and Holtz. As indicated by White and Holtz, an unreinforced natural cut slope in a weakly cemented sand and gravel at approximate 200 m away from a geosyntheticreinforced slope had failed when subjected to strong ground shaking, while the geosynthetic reinforced slope showed no signs of distress. The post earthquake investigations have indicated that geosyntheticreinforced soil structures were able to resist a horizontal acceleration up to 2 times greater than that specified in the design.
Figure 2.1 Relationships between Maximum Horizontal and Vertical Accelerations of Northridge Earthquake (after Stewart et al. 1994)
1995 HyogoKen Nanbu (Kobe), Japan Earthquake
The performances of reinforced soil structures subjected to the 1995 HyogoKen Nanbu, Japan earthquake were reported by Tatsuoka et al. (1997). The 1995
6
HyogoKen Nanbu earthquake had a Richter local magnitude of 7.2. One of the recorded peak horizontal ground accelerations was as high as 0.8 g. Many retaining structures were located in the severely affected areas, where these structures were used to support the railway tracks of Japan Railway. The retaining structures evaluated were categorized into three groups: (1) gravity walls, (2) reinforced concrete cantilever walls, and (3) geosyntheticreinforced soil walls with fullheight rigid facing. A number of gravity walls (e.g., masonry walls and unreinforced walls) were seriously damaged, where the damaged walls were later demolished and reconstructed. Many reinforced concrete cantilever walls suffered moderate damages and were substantially repaired before reuse. The damaged gravity and reinforced concrete cantilever retaining walls suffered mostly from overturning mode of failure, and the causes of the overturning failure were due to the large inertial force and the bearing capacity failure beneath the toe of the wall. Observation from the gravity retaining wall also indicated that the orientation of the actual failure plane was steeper than the one predicted by the M0 method.
As compared to the gravity and cantilever reinforced concrete walls, geosyntheticreinforced soil walls with fullheight rigid (FHR) facing performed well and suffered only slight damages (e.g., limited amount of displacement). The geosyntheticreinforced soil walls with FHR facing were constructed by the staged construction procedures, where a relatively short length of reinforcement of 0.5H and a relatively short reinforcement spacing of 0.3 m were adopted. Key features of the FHR reinforced soil wall are shown in Figure 2.2. Superior seismic performance of the FHR reinforced soil wall may be due to the intrinsic ductility and flexibility of the GRS structure. Despite the high seismic stability, the geosyntheticreinforced soil walls with FHR facing were only constructed in Japan due to their higher cost and longer construction time than the regular segmental facing reinforced soil walls constructed elsewhere.
Figure 2.2 GeosyntheticReinforced Soil Wall with FHR Facing (after Tatsuoka 1993)
7
1999 ChiChi, Taiwan Earthquake
The performances of six geosyntheticreinforced soil walls and slopes at the vicinity of the CheLungPu fault during the 1999 ChiChi, Taiwan earthquake were evaluated by Ling et al. (2001). The devastating 1999 ChiChi, Taiwan earthquake had a Richter local magnitude of 7.3, and the rupture of ground long the CheLungPu fault was about 105 km long. The frequency of shaking ranged from 1.0 to 4.0 Hz. One of the recorded peak horizontal ground accelerations was as high as 1.0 g. All of the six geosyntheticreinforced soil structures suffered moderate to severe damages. Signs of distress included total collapse of the structure, large facade movement (in both transverse and vertical directions), facade cracking, and tension cracks at the wall crest. An example of total collapse of the geosynthetic reinforced soil wall is shown in Figure 2.3. Some of the factors contributing to the failure of the geosynthetic reinforced soil structures include: (1) unqualified onsite soil was used as backfill, (2) inadequate global stability analysis, (3) unusual large reinforcement spacing (e.g., spacing > 800 mm [32 in.]), (4) mixture of unreinforced and reinforced retaining walls within a common structure, (5) insufficient connection strength between the modular block and the reinforcement, and (6) peripheral structures such as lamppost were installed close to the modular block facing.
Figure 2.3 Collapse of GeosyntheticReinforced Soil Wall during 1999 ChiChi, Taiwan Earthquake (after Ling et al. 2001)
8
Failures of three gravity retaining walls due to the ChiChi earthquake have also been reported by Fang et al. (2003). The failures include: (1) shear failure of concrete wall (or sliding) along the construction joints, (2) excessive settlement, and
(3) overturning of the wall structure about the toe due to bearing capacity failure. These modes of failure are similar to the gravity wall failures observed in the 1995 HyokoKen Nanbu, Japan earthquake. Possible factors contributed to the failure of gravity retaining walls may be: (1) insufficient compaction of the backfill, (2) improper design of the retaining wall, and (3) excessive fault displacement. As noted by Fang et al., the evaluation of factors of safety against various modes of failure (viz., base sliding, overturning, and bearing capacity failure) should never be neglected in the seismic design of retaining structures.
A reinforced soil slope failure during the ChiChi earthquake was reported by Chen et al. (2001). The 40 m high reinforced soil slope was located at the entrance road of National ChiNan University. The reinforced soil slope was constructed in tiers with the height of each tier being 10 m. The slope had a wraparound facing and was reinforced with woven polyester geogrids at vertical spacing of 1 m. The length of reinforcement (ranged between 4 m to 13 m) was gradually reduced with the height of slope. Onsite material was used as the backfill. The failure of the reinforced slope took place several hours after the ChiChi earthquake. Backanalysis indicated that internal stability of the reinforced slope was satisfactory; however, the factor of safety against external stability is less than unity when pseudostatic acceleration exceeds 0.24 g. Failure occurred along the interface between the reinforced soil and the natural soil, which agrees with the backanalysis.
Additional field performances of four geosyntheticreinforced soil modular block walls during the ChiChi earthquake were evaluated by Huang et al. (2003).
The peak horizontal acceleration near the observed walls was approximately 0.46 g. Two collapsed reinforced soil modular block walls with a height of 3.2 m are shown in Figure 2.4. Other two walls examined were either lightly damaged or undamaged. Features of the undamaged wall included larger modular block, wellcompacted gravel backfill, and higher tensile strength geogrid reinforcement. In contrast to the observation made during the Kobe earthquake, close to the collapsed wall, a reinforced concrete cantilever retaining wall was undamaged. The two collapsed walls were reinforced with knitted polyester geogrid with vertical spacing of 0.8 m and length of approximately 0.8 of the wall height. The backfill of the collapsed walls was a loose sandysilty soil. The study had found that the facingreinforcement connection strength and the blocktoblock interface resistance along with large vertical spacing significantly affected the stability of reinforced soil walls. The failure of the two walls could also be attributed to the overly large vertical spacing of the geogrids.
9
Figure 2.4 Two Failed GeosyntheticReinforced Soil Walls during 1999 ChiChi, Taiwan Earthquake (after Huang et al. 2003)
2001 El Salvador Earthquake
On January 13, 2001, a Richter local magnitude 7.6 earthquake struck off the coast of El Salvador and was followed by a strong aftershook on February 13, 2001. The devastating earthquake caused significant damage, and hundreds of people were killed. Many people were killed due to the earthquake induced landslides. Performances of two geosyntheticreinforced soil walls during the El Salvador earthquake were reported by Race and del Cid (2001). Maximum horizontal and vertical accelerations close to the sites were 0.3 g and 0.15 g, respectively. First wall had a height of 6.5 m and was reinforced with high density polyethylene geogrid (HDPE) with vertical spacing of 1 m and length 0.6 of the wall height. Failure of the first wall was caused by the toppling of the 1.7 m of unreinforced soil placed on top of the wall (see Figure 2.5). Second wall had a height of 5.4 m and was also reinforced by HDPE geogrid with vertical spacing of 1 m and length 0.6 of the wall height. A 1.8 m masonry wall was built on top of the reinforced soil wall. Although the second wall did not collapse, it experienced some lateral movement that resulted in a negative facing batter.
2001 Nisqually Earthquake
The seismic performances of two reinforced soil wall failures during the 2001 Nisqually earthquake were evaluated by Kramer and Paulsen (2001). The moment magnitude of 6.8 earthquake occurred on February 28, 2001, and the epicenter was located 17 km northeast of Olympia, Washington. The earthquake caused significant damage in Olympia, minor damage in Tacoma, and moderate damage in Seattle. One of the walls evaluated was located in Tacoma and was named Costco Wall. The 12 batter wall has a height of 5.5 m and was reinforced with geogrid at vertical spacing of 0.8 m and reinforcement length 0.5 of the wall height. The anticipated peak ground acceleration was 0.1 g, and the wall had performed satisfactorily. The second
10
wall named Extended Stay America Wall, which was located in Tumwater, had suffered a catastrophic failure (see Figure 2.6). The failed modular block wall had a height that ranged from 2 to 4 m. Failure of the Extended Stay America Wall was attributed to: (1) inadequate compaction of the fill material, (2) weak foundation material, and (3) poor drainage condition within the wall system.
Figure 2.5 Failure of GeogridReinforced Soil Wall during 2001 El Salvador Earthquake (after Koseki et al. 2006)
Figure 2.6 Failure of GeogridReinforced Soil Wall during 2001 Nisqually Earthquake (after Walsh et al. 2001)
11
Summary of Field Case Histories
GRS retaining walls have performed well during the 1989 Loma Prieta earthquake, 1994 Northridge earthquake, and 1995 HyogoKen Nanbu, Japan earthquake. The high seismic stability of GRS retaining walls observed in the U.S. and Japan may be due to the conservatism built into the design procedure. However, on the other hand, failure of GRS retaining walls reported in 1999 ChiChi, Taiwan earthquake, 2001 El Salvador earthquake, and 2001 Nisqually earthquake indicated that inadequate design considerations and poor construction quality control can render GRS structure vulnerable to severe ground motions. Some probable causes of failure include: (1) insufficient compaction of the backfill, (2) weak foundation material, (3) large reinforcement spacing (e.g., spacing > 800 mm), and (4) additional overturning load from the addon structures.
2.2 Laboratory Model Tests
Well documented case history on the seismic performances of GRS retaining structures in the field is extremely scarce (e.g., insufficient information regarding input ground motion, material properties, boundary conditions, construction details, and preearthquake static performances). Consequently, many physical model shaking table tests were conducted to investigate the seismic performances of geosyntheticreinforced soil structures. Both the reducedscale and the fullscale models seated on laboratory shaking tables have been reported. To better simulate the stress conditions experienced by the fullscale prototype, other researchers have used dynamic centrifuge tests to examine the seismic performance of reinforced soil walls and embankments. Tables 2.1 and 2.2 summarize the laboratory model test results using shaking table and centrifuge, respectively. Note that the fullscale model is preferred over the reducedscale model, since the fullscale model is not associated with problems such as similitude and boundary effects often encountered in reducedscale model and in centrifuge testing. Implications of the shaking table test and centrifuge test results to seismic design and analysis of reinforced soil structures are summarized as follows:
(1) According to Matsuo et al. (1998), reinforced soil walls with flexible facing that have long reinforcement (i.e., L = 0.7H) are more stable than those with short reinforcement (i.e., L = 0.4H).
(2) The influence of wall type was examined by Watanabe et al. (2003). Under the same shaking condition, reinforced soil wall experienced less deformation than the gravity and cantilever walls. In addition, reinforced soil wall with partially extended reinforcements performed better than uniform reinforcement length wall (see Figure 2.7).
12
(3) Latha and Krishna (2008) observed that reinforced soil walls with flexible facing experienced larger lateral displacement than those with rigid facing. Rigid facing adds to wall seismic resistance. In addition, retaining walls with rigid facing experienced higher acceleration than reinforced soil walls with flexible facing. Latha and Krishna also indicated that seismic deformation of reinforced soil wall was inversely proportional to the initial relative density of backfill (see Figure 2.8).
(4) Koseki et al (1998) indicated that overturning failure appeared as the dominant failure mode for retaining walls with rigid facing.
(5) A rigid facing panel with restraints at the toe can greatly reduce the lateral wall movement and tensile load in the reinforcement layers (ElEman and Bathurst 2005).
(6) The reinforced zone behaves as a monolithic block, and a twopart wedge (or graben) failure mode was observed in the backfill (see Figures 2.9 and 2.10).
(7) As observed by Wolfe et al. (1978), seismically induced deformations decreased with increasing reinforcement stiffness. Seismic induced reinforcement tensile force is proportional to the acceleration amplitude. Resonance between the vertical harmonic shaking and the vertical natural frequency of the modeled wall resulted in the increase of reinforcement tensile force (see Figure 2.11). The scaled model shaking table test conducted by Wolfe et al. has a natural frequency of 20 Hz and 30 Hz for the horizontal and the vertical excitation, respectively, and the amplification in the reinforcement tensile force is observed in Figure 2.11 (ii) when input frequency matches the natural frequency in the vertical direction. The tie force was measured at a distance of 0.2H from the all base, where H is the total wall height.
(8) In the irregular shaking, the combined horizontal and vertical input motion yielded larger reinforcement tensile force and wall displacement than the one with only vertical input motion and the one with only horizontal input motion. Figure 2.12 shows the response of reinforced soil wall due to irregular shaking with (i) vertical component only, (ii) horizontal component only, and (iii) combined vertical and horizontal components. Wall response due to combined horizontal and vertical shaking is not the same as the superposition of the wall responses due to the horizontal shaking alone and the vertical shaking alone (see Figure 2.12). Test result also suggested that peak wall response was observed to be synchronized with the peak of the input motion.
(9) Acceleration attenuation was observed in the free field of the reinforced soil structure when base acceleration is greater than 0.4g. Figure 2.13 plots the amplification factor versus the peak base acceleration at various depths in
13
the free field (Siddharthan et al. 2004a). The amplification factor is defined as the ratio of the maximum acceleration in the backfill to the corresponding peak base acceleration. Attenuation is said to have occurred when the amplification factor is less than 1.
(10) As shown in Figure 2.14, for a given retaining structure, the observed critical seismic coefficient using the irregular input motion is greater than the one using the sinusoidal input motion. In other words, test results with sinusoidal input motion are more conservative than those with irregular (but more realistic) input motion. Figure 2.14 also indicates that the observed critical seismic coefficients, in general, are higher than those calculated.
(11) ElEman and Bathurst (2004) observed that acceleration was generally noted at top of reinforced soil wall with the reducedscale model, and amplification increased with peak base acceleration. Upper reinforcements provided the much needed resistance against overturning of the facing blocks. In addition, the seismic performance was affected by reinforcement stiffness rather than the ultimate tensile strength.
(12) As observed by Bathurst et al. (2002b), a wall with reinforcement connected mechanically to the facing yielded less seismic displacement than the wall with only frictional connection at the facing. Bathurst et al. also indicated that reinforcement pullout failure as predicted by pseudostatic analysis was not observed in the reducedscale model test. Toppling of facing block was more dominant in the laboratory model test, and the horizontal displacement of segmental reinforced soil wall can greatly be reduced by implementing a wall batter of 8 from vertical.
(13) Budhu and Halloum (1994) indicated that the vertical spacing to length ratio does not have a significant effect on the external seismic stability. The results suggested that the vertical spacing to length ratio should be kept small at least in the top portion of the wall to ensure seismic stability.
(14) The location of maximum tensile load remained unchanged irrespective of the different loading magnitudes (Ling et al. 2005a).
14
1 Svrchrec
! 53 1 1 ModriBackflll
20 ^
Ti ModriBackflll
a. Cantilever type(C)
b. Gravity type(G)
Sarchr**
50 ModriBackflll
20
c. Reinforcedsoil type I(R1)
. Surckrjr rTx ill gUi iiiiiil
50 HA Krinforcrtnrn
ModriBackflll
d. Rein forcedsoil type 2(R2)
Seismic coefficient, k^a^/g
Figure 2.7 The Influence of Wall Type on Wall Displacement with Irregular Base Shaking (adapted from Watanabe et al. 2003)
Figure 2.8 Effect of Relative Density and Input Motion Frequency on the Wall Displacement (adapted from Latha and Krishna 2008)
15
0.5 5 m
ISSS&fct
JDODaSSE
IQDDdS^
mnnnQgg
i
juiJLJLjfcn n'l
jaODDSn
laoQDKd
]QDDQmD
raaODD^
moaat^L
joszspQ tnnDT
= o n me nr
H i
n
Figure 2.9 Example of Twopart Wedge Failure Mechanism (after Matsuo et al. 1998)
Figure 2.10 Comparison of Reinforced Soil Wall Model Before and After the Shaking Table Test (after Anastasopoulos et al. 2010)
16
(i)
(ii)
Tie Material Fiberglass Screen Tie Length  20 In (0.51 n)
(a) Displacement at Top of Wall.
(b) Tie Force at Location 4.5 1n (O.llm) up From Base.
Tie Material Fiberglass Screen Tie Length  20 1n (0.51 m)
Figure 2.11 Comparison of Reinforced Soil Wall Response at Different Harmonic Input Motions; (i) 15 Hz Harmonic Input Motion; (ii) 30 Hz Harmonic Input Motion (after Wolfe et al. 1978)
(i)
Tie Material Fiberglass Screen Tie Length  12 1n (0.30 m)
Tie Material Fiberglass Screen Tie Length  12 1n (0.30 m)
(a) Horizontal Acceleration
(d) Wall Movement
Tie Material Fiberglass Screen Tie Length  12 In (0.30 m)
Figure 2.12 Comparison of Reinforced Soil Wall Responses with Different Components of Irregular Input Motion; (i) Vertical Component Only; (ii) Horizontal Component Only; (iii) Combined Vertical and Horizontal Components (after Wolfe et al. 1978)
17
1.30
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Peak base acceleration (g)
Figure 2.13 Measured Amplification Factor in the Free Field (after Siddharthan et al. 2004a)
Calculated critical seismic coefficient against overturning or bearing capacity failure whichever is smaller
3011
200
500
Reinforcedsoil type 1 (Rl)
50
800
500
450
Reinforcedsoil type 2 50 (R2)
500
3011
Reinforcedsoil type 3 (R3)
50 Units in mm
350
Figure 2.14 Comparison of Observed Critical Seismic Coefficients to Calculated Ones against Overturning or Bearing Capacity Failure (after Koseki et al. 2003)
18
2.3 Numerical Model Studies
Numerical investigation on seismic performance of GRS wall is more economical than the physical model tests. In addition, numerical solutions are much more rigorous than the conventional limit equilibrium approach as they satisfy forceequilibrium condition, straincompatibility condition, and constitutive material laws. Conclusions drawn from numerical investigations could either be implemented or served as supplements to the design guidelines. Both qualitative and quantitative performances could be obtained from numerical investigations. For example, the influences of reinforcement stiffness and reinforcement length on seismic response of GRS wall were examined by Bathurst and Hatami (1998) using computer program FLAC. The numerical models were 6 m high with soil and reinforcements simulated by MohrCoulomb and linear elastic models, respectively. Bathurst and Hatami showed that for the same base condition (i.e., either fixed or free to slide horizontally), the wall deflections diminish with increase reinforcement stiffness, and longer reinforcement length wall deforms less than shorter reinforcement length. Furthermore, there was no evidence of a welldefined failure surface within the reinforced soil zone. Numerical modeling performed by Rowe and Ho (1998) also suggested that the reinforcement tensile stiffness have significant effect on the deformation of GRS walls, where large deformation is associated with low reinforcement stiffness.
FLAC was also used by Vieira et al. (2006) to evaluate seismic response of geosyntheticreinforced steep slope, where the reinforcement layers were modeled using linear elastoplastic model. Program input parameters evaluated by Vieira et al. include boundary conditions (e.g., freefield and quiet boundaries) and mechanical damping implemented in FLAC, and the results indicated that higher wall response is associated with application of freefield boundary and local damping. Effect of design parameters on natural frequency of GRS structure was investigated by Hatami and Bathurst (2000) also using FLAC, and they concluded that the natural frequency is not significantly affected by the reinforcement stiffness, reinforcement length, toe restraint condition, and the strength of granular backfill.
GRS walls with complex geometry such as the tiered walls and bridge abutments under seismic loads had been analyzed numerically by Guler and Bakalci (2004) and Fakharian and Attar (2007). Computer program PLAXIS was used in the parametric study performed by Guler and Bakalci for the tiered wall, and the parameters analyzed include wall height (i.e., 4 m and 6 m), ratio of geosynthetic reinforcement length to wall height of the upper and lower wall, ratio of distance between two walls to wall height, vertical spacing of geosynthetics (i.e., 0.25 m and 0.5 m), and stiffness of geosynthetics. The soil and geosynthetic reinforcements were simulated by elastoplastic MohrCoulomb model and linear elastic model,
19
respectively. It was found that wall deformation is significantly influenced by the ratio of distance between two walls to wall height; a bigger berm is effective in reducing the amount of deformation. It was also found that reinforcement length in the lower wall contributes more to the wall deformation than the reinforcement length in the upper wall. FLAC was used by Fakharian and Attar for the seismic numerical modeling of GRS segmental bridge abutments, where the Founder/Meadows segmental bridge abutment completed by the Colorado Department of Transportation was used as their validation experiment. The Founder/Meadows segmental bridge abutment was then analyzed dynamically using artificial variable amplitude harmonic ground motion records. The numerical modeling suggested that the calculated reinforcement tensile loads are lower than those based on the current active earth pressure theory, and the seismically induced horizontal displacement is larger than the vertical displacement.
Other seismic design parameters such as soil properties and reinforcement layout were considered by Ling et al. (2005b) using the computer program DIANASWANDYNEII. The walls considered in the parametric study were 6 m high GRS wall with modular block facing. A generalized plasticity soil model and a bounding surface geosynthetic model were utilized in the study. Reinforcement length was varied between 0.23 to 0.9 of the wall height, and the spacing was varied between 0.2 m to 1.0 m. Six earthquake input motions were analyzed. The results suggested that larger lateral displacement is associated with shorter reinforcement length and larger spacing. The results also suggested that amplification of acceleration was affected by the stiffness of the backfill, where higher amplification is associated with stiffer backfill.
Validations of various numerical tools specifically for GRS structures subjected to dynamic shaking done by different researchers are summarized in Table 2.3. Close agreement between the measured data and the calculated results from the validation process indicates good promise of numerical investigation. In the validation tests performed by Ling et al. (2004) using dynamic centrifuge models, it was found that acceleration amplified with wall height for both reinforced and retained soil zones comparing to the base input, and the largest settlement occurred behind the reinforced soil zone. Burke (2004) indicated that results of finite element analysis were significantly influenced by the damping properties of the wall. Fujii et al. (2006) noted large discrepancy in maximum earth pressure and maximum response acceleration between the finite element analysis and dynamic centrifuge model test results, while good agreement was observed in lateral displacement. The numerical modeling and shaking table tests performed by ElEman et al. (2001) evaluated the significance of toe restraint condition for the reinforced soil walls and reported that the magnitude and distribution of reinforcement loads were influenced by the boundary toe condition, where the reinforcement load at the bottom layer was greater for the unrestrained case as compared to the restrained case.
20
Table 2.1 Summary of shaking table model tests
Reference Height Facing type Reinforcement Reinf. Reinf. Backfill Input base Critical Failure
(mm) length (mm) spacing (mm) motion acceleration (g) mode
Anastasopoulos et al. (2010) 375 Segmental plexiglass strip Steel wire mesh 263 30 Uniform quartz sand Sinusoidal and irregular horizontal Not reported Tow wedge
Bathurst et al. 1020 Concrete Weak geogrid 700 170 Laboratory Sinusoidal Varied from Excessive
(2002a; blocks (HDPE bird silica #40 horizontal (5 0.23 to 0.36 wall top
2002b) (vertical and 8 batter) fencing) sand (Dr = 67%) Hz) movement
Budhu and 720 Wrapfaced Woven 1330 180; Dry silica Sinusoidal 0.45 and Sliding of
Halloum (1994) polypropylene fabric 240; 360 sand horizontal 0.55 top layer
ElEmam and 1000 Fullheight Knitted PET 600;1000 185; Synthetic Sinusoidal Varied from Excessive
Bathurst rigid panel geogrid 225 olivine sand horizontal 0.42 to 0.57 wall top
(2004, 2005, 2007) facing (vertical and 10 batter) (Dr = 86%) (2.5 to 5 Hz) movement
Koga et al. 1000 Varied from Nonwoven Varied Varied Airdried Sinusoidal Varied from Excessive
(1988) wrapped faced fabrics; plastic from 320 from sand (SP, w horizontal 0.73 to 0.86 crest
to sand bag net; steel bar to 2250 100 to = 6 12%) (4 Hz) settlement
faced at slopes 45 to 90 200
Koseki et al. 500 Full height Phosphor 200;200 50 Airdried Sinusoidal 0.54; 0.61; Two
(1998, 2003); Watanabe et rigid facing bronze strips grid mix with 450 & Toyoura sand (5 Hz) and irregular 0.62; 0.68; 0.83; 0.86 wedge
al. (2003) 800;350 horizontal
Krishna and 600 Wrapfaced; Woven 420 150; Poorly Sinusoidal Not reported Not
Latha (2007); Latha and Krishna rigidfaced polypropylene (PP) geotextile 200; 300 graded sand (Dr = 37 to 87%) horizontal (1 to 3 Hz) reported
(2008)
Table 2.1 (Cont.)
Reference Height (mm) Facing type Reinforcement Reinf. length (mm) Reinf. spacing (mm) Backfill Input base motion Critical acceleration (g) Failure mode
Ling et al. (2005a) 2800 Concrete blocks (12 batter) Polyester (PET) and polyvinyl alcohol (PVA) geogrids 2050; 1680 mix with 2520 600; 400 Tsukuba sand (Dr = 5256%) Irregular horizontal and vertical Not reported Not failed
Lo Grasso et al. (2005) 350 Aluminum Lshaped section with slope = 70 Biaxial polypropylene (PP) geogrid 350 Varied from 25 to 50 Dry silica sand (Dr = 70%) Sinusoidal (4, 5, & 7 Hz) and irregular horizontal Not reported Two wedge
Matsuo et al. Varied Varied from Geogrid Varied 200 Airdried Sinusoidal Varied from Two
(1998) from 1000 to 1400 discrete panel to continuous panel reinforcement (Tit = 19.4 kN/m) from 400 to 700 Toyoura sand horiztonal (5 Hz); irregular horizontal 0.38 to 0.59 wedge
Murata et al. (1994) 2480 Full height rigid facing Geogrid reinforcement (Tult = 9.8 kN/m) 1000 150 Inagi sand Sinusoidal horizontal (3.4 Hz); irregular horizontal Stable at 0.51 g Not reported
Perez and 1219 Wrapped faced Pel Ion Varied Varied Silica sand Sinusoidal Varied from Two
Holtz (2004) at slope = 63 nonwoven fabric from 305 to 610 from 61 to 203 (Dr = 91%) horizontal (5 Hz) 0.15 to 0.29 wedge
Ramakrishnan et al. (1998) 900 Wrapped faced; segmental block Woven polypropylene (PP) geotextile 500 150 Dry silica sand Sinusoidal horizontal (3 Hz) 0.25; 0.45 Not reported
Table 2.1 (Cont.)
Reference Height (mm) Facing type Reinforcement Reinf. length (mm) Reinf. spacing (mm) Backfill Input base motion Critical acceleration (g) Failure mode
Richardson 300; 25 mm curved 3.8 mm wide Varied 38 Sand Sinusoidal Not reported Not
and Lee 380 aluminum aluminum foil from 76 to horizontal reported
(1975) sheet; 38 mm strips; 6 mm 150 (11.6 Hz)
flat aluminum wide mylar
sheet tape strips
Sabermahani 1000 Wraparound Knitted textile; 500; 700; 100; SP(D,0 = Sinusoidal Not reported Bulging
et al. (2009) wall facing nonwoven 900 200 0.16 mm; Dr horizontal and over
textile; geogrid = 47% & (0.6, 1.5, turning
84%) 2.4, and 3.34
Hz)
Sakaguchi 1500 Cement coated Geogrid Not 300 Standard Sinusoidal 0.33 Not
(1996) foam block reported laboratory horizontal reported
silica sand (4 Hz)
No. 4
Sofronie et al. 900 Rigid facing High density 360; 540 150 Leighton Sinusoidal Not reported Tilting of
(2001) polyethylene Buzzard (5 Hz) and facing
(HDPE) sand irregular
geogrid horizontal
Sugimoto et Varied Wrapped faced Tensar SS1 Not Not Niigata sand Sinusoidal Not reported Excessive
al. (1994) from at slopes reported reported horizontal crest
700 to varied from settlement
1050 34 to 79
Wolfe et al. 600 Discrete panel Mylar tape Varied 38 Uniform Sinusoidal Not reported Not
(1978) strips; from 305 fine dry horizontal & reported
fiberglass to 762 quartz sand vertical;
screen strips irregular
horizontal &
vertical
Table 2.2 Summary of centrifuge model tests
Reference Scale factor N Model height (mm) Facing type Reinforcement Reinf. length (mm) Reinf. spacing (mm) Backfill Input base motion
Howard et al. (1998) and Siddharthan et al. (2004a) 24 310 Discrete facing panel made of aluminum Galvanized steel wire mesh; ribbed steel strips Varied from 155 to 434 24 Nevada sand (Dr = 65%) Varied with sinusoidal and irregular horizontal
Ichikawa, et al. (2005) 30 150 Modular blocks; rigid panel Wire netting; steel anchor 100;160 30 Toyoura sand (Dr = 80%) Sinusoidal (2 Hz) horizontal
Izawa and Kuwano (2006) 50 150 Rigid facing panel Polycarbonate plate geogrid 90 30 Toyoura sand (Dr = 80%) Varied with sinusoidal (100 Hz) and irregular horizontal
Kutter et al. (1990) and Casey et al. (1991) 50 152.4 Discrete face panel made of aluminum Wire screen 106.7 19.1 Nevada sand; Nevada sand mixed with Yolo loam Irregular horizontal
Liu et al. (2010) 40 195 Aluminum blocks Non woven geotextile; model geogrid 140 45 Silty clay; sand Sinusoidal (2 Hz) and irregular horizontal
NovaRoessig and Sitar (1998, 2006) 48 152.4 Wrapped faced at slope = 63 Pellon TruGrid (Tult = 0.18 kN/m) 106.7 Varied from 15 to 17 Monterey #0/30 sand Sinusoidal and irregular horizontal
Sakaguchi (1996) 30 150 Light weight blocks Non woven geotextiles Varied from 50 to 150 30 Toyoura sand Sinusoidal horizontal
Takemura and Takahashi (2003) 50 150 Discrete facing panel made of aluminum Glass fiber geogrid 40; 90; 120 15; 30 Inagi sand (pd = 1.4 and 1.48 Mg/m3) Sinusoidal horizontal (100 Hz)
Table 2.3 Summary of computer program validation for seismic performance of reinforced soil wall
Reference Code (method) Facing Reinforcement Backfill Interface Input Validation Performance
model model model model motion model type examined
(element) (element) (element) (element)
Burke DIANA Linear 1D bounding Pastor Slip element Kobe Fullscale Acc.; disp.;
(2004) SWANDYNE elastic (8 & surface (3 Zienkiewicz record shaking vertical and
11 (finite 6node node bar III (8 & 6 table lateral
element) element) element) node element) pressures; reinf. load
ElEman et FLAC (finite Not given Elasticplastic Mohr Note given Sinusoida Reduced Disp.; reinf.
al. (2001) difference) (2node cable Coulomb 1 record scale load; acc.; toe
element) elasticplastic strain softening (5 Hz) shaking table condition
Fujii et al. FLIP (finite Elastic Elastic (linear Multispring Joint element Kobe Dynamic Acc.; disp.;
(2006) element) (linearbeam element) beam element) record centrifuge earth pressure
Helwany et al. (2001) DYNA3D (finite element) Not given Linear elastic (shell element) Ramberg Osgood (solid element) Penalty based interface Sinusoida 1 record Fullscale shaking table Acc.; disp.; interface load
Ling et al. (2004) DIANASWANDYNE II (finite element) Linear elastic Bounding surface Generalized plasticity soil Elastic perfectly plastic interface element Sinusoida 1 record Dynamic centrifuge Acc.; disp.; vertical and lateral pressures
Liu et al. (2011) ABAQUS (finite element) Linear elastic Elastoplastic viscoplastic bounding surface (1D bar element) DruckerPrager creep model Mohr Coulomb (thin layer element) Kobe record Dynamic centrifuge Acc.; disp.; crest settlement; reinf. strain
2.4 Seismic Design of Retaining Walls
The limit equilibrium (LE) methods have been used widely in the design of earth structures. The LE analysis offers a designer with the advantages of simple input data and useful design output information. Adequate factors of safety against potential failure modes utilizing the LE methods can often provide satisfactory performances of the earth structures. The stateofpractice in both the static and the seismic design and analysis of retaining structures also involves the LE methods. Examples of LE methods in static design of retaining structure include the conventional Rankine's and Coulombs earth pressure theories. Consequently, LE methods are extended to the seismic design and analysis of retaining structures. One of such extensions is the venerable MononobeOkabe (MO) method, which is the successor of the static Coulomb method for estimating the seismic earth pressure imparted on retaining structures. The M0 method in analyzing conventional retaining wall is first described. The influence of the direction of inertia forces on the seismic thrust is then discussed. Subsequently, current design and analysis of reinforced soil retaining wall are described. Lastly, the seismic induced permanent displacement of retaining wall and its design implications are discussed.
2.4.1 MononobeOkabe Method
Okabe (1926) and Mononobe and Matsuo (1929) formulated the basis of the pseudostatic analysis of seismic earth pressures on retaining structures that is known as the MononobeOkabe (MO) method. M0 method is similar to static Coulomb method, in which the additional pseudostatic accelerations are applied to the Coulomb active or passive wedges. Figure 2.15 shows the active wedge with pseudostatic accelerations ah = khg and av = kvg in horizontal and vertical directions, respectively, where g is the gravitational acceleration. Note that kh and kv are earthquake acceleration coefficients in the horizontal and vertical directions, respectively. Positive kh and kv are acting towards the wall and upward, respectively. The backfill considered in the M0 method is both cohesionless and unsaturated. In an active condition, the total active thrust PAe can be calculated, in a form similar to that of the static condition, as:
PAE^KMyH!(lk) (2.1)
where y = unit weight of the backfill, H = total wall height, and Kae = seismic active earth pressure coefficient. KAE is calculated as:
cos2 ((j)to ^)/[cos^ cos2 cocos(8 + co + 4)]
sin((j) + 8)sin( p ^)
Kak =
1 +
cos(5 + co + ^)cos(p co)
(2.2)
26
where (> = peak soil friction angle, 5 = soilwall interface friction angle, co = wall inclination angle, P = slope angle of backfill, and Ã‚Â£, = seismic inertia angle (Ã‚Â£ = arctan[kh/(lkv)]. The orientation of the critical failure surface for active condition from the horizontal ocae proposed by ZarrabiKashani (1979) can be expressed as:
d~a (23)
a
AE
= <> E, + tan
i
where:
a = tan(
The original M0 method implies that the total active thrust Pae would act at a location H/3 above the wall base. From experimental results, Seed and Whitman (1970) suggested that PAe be resolved into the static thrust Pa and the seismic increment APae components as:
PAE = PA + AP^ (2.4)
where PA acts at H/3 above the wall base, and APae acts at 0.6H above the wall base. Hence, the location of the total active thrust from the wall base, h, is:
h PaH/3 + APae(0.6H)
Pae
(2.5)
The magnitudes and locations of APae and PA are used in determining the factor of safety against overturning. Note that in order to get a real solution of KAe from Equation 2.2, the term Ã‚Â£. Furthermore, for horizontal backfill, P = 0; hence E, < . Since E, = tan''[kh/(lkv)], which leads to kh < (lkv)tan. The critical value of horizontal acceleration coefficient is then kh(Cr) = (lkv) tan<().
Figure 2.15 Equilibrium of Forces acting on the Active Wedge in M0 Analysis
27
In the passive earth pressure condition, the horizontal component of earthquake peak acceleration ah is directed toward the backfill as shown in Figure 2.16. The total passive thrust on the retaining wall is given by:
Ppe =KrarH1(ik.)
(2.6)
Kpe is the seismic passive earth pressure coefficient and is determined as: cos2(() + co^)/[cos 4 cos2 co cos(S (o + Ã‚Â£,)]
sin( + P ^)
K
1
(2.7)
cos(d co + ^)cos(p co)
The orientation of the critical failure surface for the passive condition from the horizontal ape is given by:
'j + f
a
PE
= 0 <> + tan
l
(2.8)
where:
f = tan( + Ã‚Â£, + p) g = tan(
j = v/g(g + hXhi + 1)
k = 1 + i(g + h)
Similar to the total active thrust, the total passive thrust Ppe can also be resolved into the static component PP and the seismic increment APpE as:
PPE = Pp + APpe (2.9)
Figure 2.16 Equilibrium of Forces acting on the Passive Wedge in M0 Analysis
28
The influence of the direction of earthquake induced inertia forces (i.e., khW and kvW) on the seismic earth pressure has been examined by Fang and Chen (1995). For the active condition, the conventional M0 method assumes the horizontal and vertical inertia forces to act upward and toward the wall, respectively (see Figure 2.15). It is a concern whether or not the assumed directions of the inertia forces would yield the maximum total seismic active thrust Pae In reality, a total of four combinations for the direction of inertia forces are possible. Figure 2.17 compares the effect of direction of inertia forces on the magnitude of the total active thrust, and the total seismic active thrust is the maximum for horizontal and vertical inertia forces to act toward the wall and downward, respectively, for kh < 0.4. Furthermore, the effect of direction of inertia forces on the total seismic passive thrust is shown in Figure 2.18 and the total seismic passive thrust is the minimum for horizontal and vertical inertia forces to act toward the wall and upward, respectively.
Figure 2.17 Effect of Direction of Inertia Forces on Total Seismic Active Thrust (modified from Fang and Chen 1995)
29
Towards backfill < kh Towards wall
Figure 2.18 Effect of Direction of Inertia Forces on Total Seismic Passive Thrust
2.4.2 Design and Analysis of GeosyntheticReinforced Soil Retaining Walls
This section describes the current design and analysis of geosyntheticreinforced soil (GRS) retaining walls subjected to earthquake loads. In North America, a widely accepted design guidelines, which includes the seismic design of GRS retaining walls, is the Federal Highway Administration (FHWA) manual put forth by Elias et al. (2001). Another seismic design method for GRS walls follows the National Concrete Masonry Association (NCMA) manual (Bathurst 1998). Design of reinforced soil slope can be found in the US Army Corps of Engineers Waterways Experiment Station publication (Leshchinsky 1997). Other design methodologies from abroad have been summarized by Zomberg and Leshchinsky (2003) and Koseki et al. (2006). Design criteria and analysis methods from the FHWA and NCMA manuals are summarized as follows. The assumptions involved in the design are also presented.
30
2.4.2.1 Federal Highway Administration (FHWA) Methodology
Limit equilibrium (LE) method is adopted in the FHWA methodology, where one can only estimate the margins of safety against collapse and cannot estimate the deformation of the structure given the external loads. In the seismic design of GRS walls, the FHWA methodology requires both the external stability and the internal stability be evaluated in addition to the static design considerations. The design peak horizontal acceleration at a site can be obtained from Division IA (AASHTO 2002) and Section 3.10 (AASHTO 2007) of the AASHTO Specifications. As specified by the FHWA methodology, the seismic design is needed whenever the peak acceleration coefficient (A) at the site being considered is greater than 0.05. The coefficient is expressed as a fraction of gravitational constant, g, and is dimensionless. The maximum limiting value of A in which the FHWA seismic design requirements are applicable is 0.29, and the FHWA methodology recommends that the seismic design of a GRS wall should be reviewed by a specialist when A at the project site exceeds 0.29.
FHWA External Stability Evaluation
In the external stability evaluation for GRS walls, three potential modes of failure considered are: (1) base sliding, (2) eccentricity, (3) bearing capacity. Taking into account the flexibility/ductility exhibited by the GRS walls, the recommended minimum seismic factors of safety with respect to the failure modes are assumed as 75 percent of the static factors of safety, and the eccentricity should be within L/3 (L = length of the reinforcement) for both soil and rock foundations. Two forces in addition to the static forces in the external stability evaluation are the horizontal inertia force (Pir) and the seismic horizontal thrust increment (APAe). APAe is exerted on the reinforced soil by the retained soil. Both AP^ and Pir are shown in Figures 2.19 and 2.20 for level and sloping backfill conditions, respectively.
The seismic external stability is evaluated in the following steps:
Select the acceleration coefficient A from Section 3 of AASHTO Division I
A.
Calculate the maximum acceleration (Am) developed within the GRS wall system.
Am = (1.45 A) A (2.
Calculate the horizontal inertia force Pjr and the seismic horizontal thrust increment APAg. The height H2 should be used in finding Pir and AP^ for sloping backfill condition (see Figure 2.20).
H = H + t^ni3':5H =M (2.
2 (l0.5 tan p) 10.5 tan p
The horizontal inertia force Pir is calculated as follows:
10)
11)
31
P.R == P,r + Pis
Pir=0.5AmYfH2H
Pis ~ 0.125 Am yf (H2)2 tanp
(2.12)
(2.13)
(2.14)
Note that Pjr is the inertial force caused by acceleration of the reinforced backfill, and PjS is the inertial force cased by acceleration of the sloping soil surcharge above the reinforced backfill. The seismic horizontal thrust increment APae is calculated using the pseudostatic MononobeOkabe method with the horizontal acceleration coefficient kh equal to Am and vertical acceleration coefficient kv equal to zero.
The total seismic earth pressure coefficient Kae is calculated following the general MononobeOkabe expression:
<)> = the peak soil angle of friction to = angle of the wall face from vertical The seismic earth pressure coefficient associated with the seismic thrust increment (APAe) is AKAe, and AKAe = Kae KA. Note that the mobilized interface friction angle 8 is assumed to be equal to P in the FHWA method. Note also that the wall batter angle to in the FHWA method is with the facing blocks inclined into the backfill, which is the opposite of the Coulomb method.
Check factors of safety against failures of base sliding, eccentricity, and bearing capacity with Pir and 50% of APAe The reduction of 50% on APae was reasoned with possible phase lag between the inertial force and the seismic thrust from the retained backfill.
(2.15)
(2.16)
where: p = backfill slope angle
Ã‚Â£, = seismic inertia angle = tan''[kh/(l kv)]
For level backfill condition (P = 0), H2 = H, PjS = 0, and Pir =Pjr.
32
Mass for Inertial Force
Mass for resisting forces
Figure 2.19 Seismic External Stability of a GRS wall with Level Backfill in FHWA Method
Mass for resisting forces
Figure 2.20 Seismic External Stability of a GRS Wall with Sloping Backfill in FHWA Method
33
As indicated by the FHWA manual, the use of full value of Am for kh in the pseudostatic MononobeOkabe method to find Pae can result in an excessively conservative design. To achieve a more economical GRS wall, a reduced kh can be used if the following conditions are met:
The wall system and any structures supported by the wall can tolerate later movement resulting from sliding of the structure.
The wall is unrestrained regarding its ability to slide, other than soil friction along its base and minimal soil passive resistance.
If the wall functions as an abutment, the top of the wall must also be unrestrained, e.g., the superstructure is supported by sliding bearings.
With the conditions listed above and provided that the GRS wall can tolerate displacements up to 250A (mm), kh may be reduced to 0.5A (i.e., kh = 0.5A).
FHWA methodology also provides an alternative method for estimating the horizontal acceleration coefficient kh in finding APAe kh can be computed as:
^A ^
k =1.66 A.
(2.17)
where d is the anticipated lateral wall displacement in mm. Noted that this equation should not be used for displacement of less than 25 mm or greater than 200 mm. FHWA manual suggests that typical anticipated lateral wall displacement in seismically active area ranges from 50 mm to 100 mm.
It is to be noted that although a trapezoidal dynamic pressure distribution was proposed by the FHWA methodology (see Figures 2.19 and 2.20), and the actual dynamic pressure distribution was not specified. The equation for determining the seismic horizontal thrust increment APae has otherwise suggested a triangular dynamic pressure (hydrostatic) distribution. For the seismic thrust to be located at 0.6H and with a trapezoidal pressure distribution, the ratio of long length (at the top) to the short length (at the bottom) of the trapezoid needs to be 4.
FHWA Internal Stability Evaluation
The internal failure of a GRS wall can occur in three ways: (1) pullout of reinforcement, (2) reinforcement rupture, and (3) connection pullout failure. To evaluate the internal stability of a GRS wall, one needs to determine the maximum developed tensile force in each reinforcement layer, the critical slip surface, and the resistance provided by the reinforcements in the resistant zone. It is assumed that the critical slip surface coincide with the locus of the maximum tensile force in each reinforcement layer Tmax, and the critical slip surface is further assumed to be linear in the case of extensible reinforcements which passes through the toe of the wall (see Figure 2.21). Also assumed is that the location and the slope of the linear critical slip surface is not affected by the seismic loads (i.e., the seismic critical slip surface is the
34
same as the one for the static condition). The critical static slip surface, following the Coulomb's active condition, is inclined at an angle aA from the horizontal as:
aA =(() + tan
i
tan(()p) + C1
(2.18)
where
C, = .yjtan((j) p)[tan(<) p)+ cot( + co)]
C2 = 1 + {tan (5 co )[tan(
As has mentioned earlier, in the FHWA methodology, the mobilized interface friction angle 8 is assumed to be equal to the backfill slope angle p (i.e., 8 = P).
The static maximum tensile force in each reinforcement Tmax is a function of horizontal stress at each reinforcement level along the critical slip surface (an) and reinforcement spacing (Sv), and Tmax is computed as:
Tmax = cth Sv (2.19)
Furthermore, the horizontal stress an is a function of the overburden stress, uniform surcharge loads, and concentrated surcharge loads. Alternatively, the tributary area from horizontal stress distribution can be used to calculate Tmax for each of the reinforcements. Note that the reinforcement spacing should not exceed 800 mm as required by the FHWA methodology.
In a seismic event, seismic loads would produce an inertial force Pi acting horizontally in addition to the static forces (see Figure 2.21). The inertial force Pi is calculated as:
P. = Am WA (2.20)
where WA is the weight of the active zone (shaded area in Figure 2.21), and Am is the maximum acceleration. Each reinforcement layer would receive additional seismic tensile force induced by the inertial force Pi. The additional seismic tensile force Tmd in each reinforcement layer is determined by proportionally distributing the Pi based on the embedment length of reinforcements in the resistant zone and is computed as follows:
T P ^ei 1md M n
IL<.
i=l
where n is number of reinforcement layers in the GRS wall. Knowing Tmax the total tensile force in each reinforcement layer Ttotai is calculated as:
(2.21)
and Tmd,
(2.22)
Ttotai is then used to evaluate the reinforcement pullout failure. Note that the factor of
safety against reinforcement pullout failure FSpo under static condition should be
greater than or equal to 1.5, and in seismic design, the factor of safety is said to be
35
75% of the static value. The total tensile force in each reinforcement layer Ttotai should not exceed the pullout resistance Pr at that layer as:
T <
total
P. R.
(0.75)FS,
(2.23)
where Rc is the coverage ratio and is often assumed to be unity for geotextiles and geomembranes. Pr is a function of embedment length Le, overburden stress, and coefficient of friction (or the frictionbearinginteraction factor). According to the FHWA methodology, the coefficient of friction between the soil and reinforcement in the seismic condition should be reduced to 80% of the static value.
In evaluating the rupture failure during seismic loading, the reinforcement is to be designed to resist both the static and seismic forces, which requires the following:
T <
S R.
(0.75)RFcrRFdRFidFS
(static component)
(2.24)
Tnd ^ (0.75)RFD RF,,, FS component) (2.25)
where Rc = coverage ratio, RFcr = creep reduction factor, RFD = durability reduction factor, RFid = installation damage factor, FS = overall factor of safety, Srs = reinforcement strength to resist static load, and Srt = reinforcement strength to resist seismic load. Note that the creep reduction factor RFcr is not applicable to Tmd, since seismic load occurs in a short time. The values of various reduction factors have been suggested in the FHWA methodology. Moreover, with both Srs and Srt known, the required ultimate strength of the geosynthetic reinforcement Tui, can be calculated as follows:
Tult=Srs+Srt (2.26)
A particular geosynthetic reinforcement can be selected based on the value of Tut.
The connection pullout failure during seismic loading is evaluated using the following conditions:
T <
< Srs CRcr
RFd FS
Tmd 0.8
RFd FS
(static component)
(seismic component)
(2.27)
(2.28)
where CRcr = connection strength reduction factor resulting from longterm testing and CRuit = connection strength reduction factor resulting from quick connection tests. Both CRcr and CRUt are to be determined using laboratory testing technique described in Appendix A of the FHWA manual. Both CRcr and CRuit are function of normal stress, which is developed by the weight of the facing units. Calculation of normal stress should be limited by the hinge height in the case of a batter wall.
36
Internal inertial force due to the weight of the backfill within the active zone. The length of reinforcement in the resistant zone of the i'th layer.
The load per unit wall width applied to each reinforcement due to static forces. The load per unit wall width applied to each reinforcement due to dynamic forces.
The total load per unit wall width applied to each layer.
T + T ,
max md
Figure 2.21 Seismic Internal Stability of a GRS Wall in FHWA Method
Similar to the reinforcement rupture failure evaluation, Srs and Srt can be back calculated from Equations 2.27 and 2.28, respectively, and the ultimate strength of the geosynthetic reinforcement Tuit can then be found. Subsequently, a particular reinforcement can be selected in the wall design based on the value of Tut. Note that for AASHTO Division 1A seismic performance categories C or higher (i.e., A >
0.19), FHWA methodology recommends that the modular block facing connection should not depend solely on the frictional resistance between the facing units and the reinforcement; rather, shear resisting devices such as shear keys or pins should be installed in the modular block facing wall.
2.4.2.2 National Concrete Masonry Association (NCMA) Methodology
Similar to the FHWA methodology, the NCMA manual utilizes the pseudostatic MononobeOkabe (MO) earth pressure theory in assessing the seismic stability of GRS walls. As indicated in the NCMA manual, it is applicable with restrictions of A < 0.4 and kv = 0. Three types of stabilities considered in the NCMA manual are external, internal, and facing stabilities. The possible modes of failure for the GRS wall are presented in Figure 2.22. Figures 2.22a to 2.22c, 2.22d to 2.22f, and 2.22g to 2.22h show the external failure modes, internal failure modes, and facing failure modes, respectively. The factors of safety in the seismic design are the same as those
37
proposed in the FHWA manual, where the minimum seismic factors of safety are 75% of the static values.
With the assumption that GRS walls are freestanding and can tolerate horizontal displacement at the base without any lateral constraints, NCMA manual adopts the same approach as those from the FHWA methodology in determining the horizontal acceleration coefficients kh for the external, internal, and facing stabilities evaluations. For external stabilities and internal sliding:
kh (ext) = 0.5 A (Figures 2.22a, b, c, and f) (2.29)
For facing stabilities and other internal stabilities:
kh(int) = (l.45A)A (Figures 2.22d, e, g, h, i, and j) (2.30)
where A can be obtained from Section 3 of AASHTO Division IA. Summarized below are the approaches for evaluating the three types of stabilities.
(a) base sliding (external failure mode)
(b) overturning (external failure mode)
(c) bearing capacity (external failure mode)
(d) pullout
(internal failure mode)
(e) tensile overstress (internal failure mode)
(f) internal sliding (internal failure mode)
(g) column shear failure (h) connection failure (i) local overturning (j) crest toppling
(facing failure mode) (facing failure mode) (facing failure mode) (facing failure mode)
Figure 2.22 Modes of Failure for GRS Walls in NCMA Method
NCMA External Stability Evaluation
Using the approach suggested by Seed and Whitman (1970), the seismic thrust Pae is comprised of the static thrust component PA and the seismic thrust increment APae component as:
38
(2.31 )(2.4 bis)
and in terms of the earth pressure coefficients as:
(l + k>)KAÃ‚Â£=KA+AKAE
(2.32)
The static and seismic increment earth pressure distributions for evaluating the external stabilities are shown in Figure 2.23. In the NCMA manual, it is assumed that positive kh and kv acts toward the wall and downward, respectively. Note that the static pressure increases linearly from top to the base, and the static thrust acts at h/3 from the base. The seismic increment pressure takes the shape of a trapezoid with higher stress at the top. The trapezoidal pressure distribution was adopted from Ebling and Morrison (1992) in the seismic design of anchored sheet pile wall, and the seismic thrust increment acts at 0.6h from the base. This trapezoidal pressure distribution is said to have indirectly accounted for the amplification of horizontal ground acceleration. The seismic active earth pressure coefficient KAe is calculated as:
where <)> = peak soil friction angle, co = wall facing column inclination, 8 = mobilized interface friction angle, P = backfill slope angle, and 9 = seismic inertia angle (0 = arctan[kh/(lkv)]). In the FHWA manual, Ã‚Â£ is used to denote seismic inertia angle instead of 0 in the NCMA manual. Note that in the above equation, the wall inclination angle co is measured from vertical and increases as the wall inclined into the backfill, whereas the expression of Kae (Equation 2.2) for gravity retaining has a co that is measured from vertical as well but increases as the wall inclined away from the backfill (i.e., the difference is the sign in front of co). Note also that Equations 2.33 and 2.16 are different by the parameters 8 and P, where in Equation 2.16 of the FHWA manual, 8 is assumed to be equal to p. In the NCMA external stability evaluations, the mobilized interface friction angle 8 is further assumed to be equal to <(> (with (f> equal to lesser of r and (pf values). Hence, the seismic thrust increment APae is calculated as:
where h is the height at the back of the reinforced soil zone (see Figure 2.23). Note that only 50% of APae is considered in the seismic analysis. The justification for the 50% reduction on APae is that the inertial forces within the reinforced soil mass would not peak at the same time as the seismic thrust increment generated by the retained backfill.
(2.33)
APae 2 AK ae Yf ^
(2.34)
39
In addition to the seismic thrust increment APae, a horizontal inertial force within the reinforced soil mass P]R is also considered in the external stability evaluations. The method in finding PiR is similar to the one proposed in the FHWA manual. The inertial force includes the mass of the facing column and the mass of the reinforced soil zone extended to a distance of 0.5H behind the wall facing, which makes PiR < kh(ext) W with W being the weight of the entire reinforced soil mass. With 50% of APae, Pir, and static forces, external stabilities of base sliding, overturning, and bearing capacity in terms of factors of safety can thus be evaluated.
Figure 2.23 Seismic External Stability of a GRS Wall in NCMA Method
NCMA Internal Stability Evaluation
The tensile force for each reinforcement layer needs to be evaluated. Figure 2.24 shows the schematic in finding the total tensile force in each layer Ttota!. Note that only the horizontal components of earth pressure coefficients are considered in the stability evaluation. The horizontal components are calculated as:
Kah =Ka cos(5co) (2.35)
KAEH = Kae cos(h to) (2.36)
AKahh = AK^ cos(5(o) (2.37)
The tributary area approach is used, where the tensile force in each layer is the pressure integrated over the vertical reinforcement spacing Sv. Ttotai is comprised of the facing column inertial force (kh(int)AWw), static component (Tmax), and the seismic increment (Fdyn) as:
40
Tt0(fli = kh (int) AWw + Tmas + Fdyn (2.38)
where AWw is the weight of the facing column within the contributory area. The contributory area approach is said to be more conservative, since the seismic increment pressure is higher toward the top.
Figure 2.24 Schematic for finding the Total Tensile Force in Each Reinforcement Layer in NCMA Method
The factor of safety against reinforcement rupture failure (or reinforcement overstressing) is evaluated by the ratio of the allowable tensile load for the reinforcement with seismic loading Ta(dyn) to the total reinforcement tensile force Ttotai as:
FS
OS
T
1 a(dyn)
T
total
(2.39)
Note that Ta(dyn) should not include the creep reduction factor [Ta(dyn) = TUt / (RFd RFid FS)], since the seismic loading is of short duration.
In the reinforcement pullout evaluation, a failure surface is first identified, and the potential failure surface would initiate from the toe of the reinforced soil mass. Equation 2.3 can be used to determine the angle of the failure surface with a difference of using opposite sign in front of to (wall inclination angle). The embedment length Le that extends beyond the potential failure surface provides the
41
pullout resistance to resist the Ttotai of the reinforcement (see Figure 2.25). The factor of safety against pullout failure FSpo at each reinforcement layer is calculated as:
FSp. = (2.40)
total
where Pr is the pullout resistance (or the anchorage capacity), which is a function of the overburden stress ctv, embedment length Le, and coefficient of soilgeosynthetic interface friction. In absence of the laboratory test data, the coefficient of soilgeosynthetic interface friction can be estimated as tan
Similar to the base sliding in the external stability evaluation, the factor of safety against internal sliding FSsii is calculated as:
pÃ‚Â§ =________________^s(z)______________ (2 41)
s" APir(z) + PAH (z) + 0.5 APaeh(z)
where Rs is the sliding resistance along a reinforcement layer at depth z, which is a function of the interface shear capacity between facing column units and the frictional resistance between the soil and the reinforcement. Note that the inertial force APir is due to the 0.5H in length of the reinforced soil mass, and only 50% of the seismic thrust increment APAeh contributes to the sliding evaluation (see Figure 2.26).
Figure 2.25 Schematic for Reinforcement Pullout Evaluation in NCMA Method
42
Figure 2.26 Schematic for Base Sliding and Internal Sliding Evaluation in NCMA Method
NCMA Facing Stability Evaluation
Four potential modes of failure associated with the facing instability are: (1) interface shear, (2) connection failure, (3) local overturning, and (4) crest toppling. In the interface shear mode of failure, the facing column is treated as a beam, and the factor of safety against the interface shear failure FSsc is expressed as:
V (z)
FS = (2.42)
Si(z)
where Vu(z) is the shear capacity of the interface between facing column units, and Sj(z) is the outofbalance shear force transmitted through facing unit interface at depth z, as shown in Figure 2.27. Sj(z) is calculated as:
S1(z)=k1(int)AW.(z)+PAH+APAEH Ã‚Â£T101>1 (2.43)
where ETtotai is the sum of the horizontal forces carried by the reinforcement layers above the interface of interest.
The factor of safety against connection failure FScs is evaluated as:
FSt!=L (2.44)
* total
where Ttotai is the tensile force within the reinforcement, and Tc is the connection capacity between the facing column units and the reinforcement. Note that Tc is to be obtained from the NCMA Test Method SRWU1 "Determination of connection
43
strength between geosynthetics and segmental concrete units" (Collin 1997), which is often referred as the "quick connection test" in the laboratory. Tc takes the same form as the MohrCoulomb failure criterion, where Tc is a function of the minimum available peak connection strength (similar to the cohesion component), the apparent interface friction angle (describing the failure envelope), and the applied normal stress.
The beam analogy used in the interface shear failure evaluation is also applicable to the local overturning mode of failure. The factor of safety against local overturning about the toe of the facing column unit FS0ti is evaluated as ratio of resisting moment to the overturning moment as:
FS M' <245) M0
The resisting moment is comprised of the moment due to the weight of the facing column units Mr and the summation for moments due to reinforcement tensile forces STt0taixy. M0 is the overturning moment (or the driving moment) due to the horizontal inertial force of the facing column, the static thrust, and the seismic thrust increment (see Figure 2.28). Note that the crest toppling mode of failure is considered as a subset to the local overturning mode of failure. The crest toppling is the local overturning that occurs at the elevation of the topmost layer of reinforcement.
It
total
Figure 2.27 Schematic for Facing Interface Shear Evaluation in NCMA Method
44
Figure 2.28 Schematic for Facing Local Overturning Evaluation in NCMA Method
Comparison between FHWA and NCMA Methods
The minimum factors of safety stipulated in the seismic design of GRS wall from the FHWA and the NCMA methods are summarized in Table 2.4. The similarities observed between the FHWA and the NCMA methods are listed as follows:
(1) The reinforced and retained soils are assumed to be cohesionless and unsaturated.
(2) The peak friction angle is assumed as the design friction angle.
(3) GRS wall is founded on competent foundation, and the global stability is not a problem.
(4) The horizontal acceleration coefficients kh remains constant and uniform in the GRS wall structure.
(5) The vertical acceleration coefficient kv is assumed to be zero.
(6) The minimum seismic factors of safety are 75% of the values considered in the statically loaded structures.
The differences observed between the two methods are listed as follows:
(1) The design methods are considered applicable when the maximum horizontal accelerations are limited to 0.29 g and 0.4 g for the FHWA manual and the NCMA manual, respectively.
(2) In finding the inertia force Pir, FHWA manual applies Am [Am = (1.45 A)A] to the reinforced soil mass of O.5H2 in length, whereas NCMA manual applies
45
kh(ext) [kh(ext) = 0.5A] to the reinforced soil mass of 0.5H in length (see Figure 2.29).
(3) In sloping backfill, the seismic thrust is found to act at O.6H2 in FHWA manual versus 0.6h in the NCMA manual (see Figure 2.29).
(4) FHWA method assumes 8 = (3 in finding Kae versus 8 (3 in the NCMA method.
(5) FHWA manual assumes the wall to be vertical when the wall inclination angle co is less than 8, whereas NCMA manual adopts the general Coulomb earth pressure theory for any wall inclination angles.
(6) In external stability analysis, FHWA method checks for the eccentricity requirement (i.e., e < L/3) and not the factor of safety against overturning, whereas NCMA manual requires the overturning and omits the eccentricity.
(7) In external stability analysis, FHWA manual requires a factor of safety of 1.9 (75% of 2.5) against the seismic bearing capacity failure versus a factor of safety of 1.5 for the NCMA manual.
(8) The coefficient of friction between soil and reinforcement in determining the pullout resistance is reduced to 80% of static value in the FHWA manual, whereas there is no reduction on the coefficient of friction in the NCMA manual.
(9) FHWA method assumes that the inclinations of static and seismic thrusts are parallel to the backslope angle (i.e., inclination = P), whereas the inclinations of static and seismic thrusts recommended by NCMA method follows the conventional Coulomb approach (i.e., inclination = 8 to).
Table 2.4 Comparison of recommended minimum factors of safety for GRS walls
Failure Modes FHWA NCMA
13 Base sliding 1.1 1.1
B S Eccentricity L/3 
* 3 PQ t/3 Overturning  1.1
Bearing capacity 1.9 1.5
13 Ã‚Â£ Reinf. Rupture 1 1
B Reinf. pullout 1.1 1.1
G C/D Internal sliding  1.1
Interface shear  1.1
CD & .5 tB Connection failure 1.1 1.1
O JD cd cd Ll. +* Local overturning  1.1
Crest toppling  1.1
46
Figure 2.29 Differences between the FHWA Method and the NCMA Method 2.4.3 Peak Ground Acceleration Coefficients
The pseudostatic M0 method depends entirely on the horizontal and vertical peak ground acceleration coefficients kh and kv, respectively. For engineering applications, kv is often assumed to be twothirds of kh (i.e., kv = 2/3k,; Newmark and Hall 1982). However, in typical seismic retaining wall design, the stateofpractice is to assume kv equal to zero, and that kh remains constant throughout the retaining structure. Furthermore, there is no consensus in determining the design value of kh. Seed (1983) suggested that kh = 0.15 be the maximum level ascribed to the limit equilibrium analyses. Bonaparte et al. (1986) suggested that kh = 0.85A, where A is the peak horizontal ground acceleration coefficient found in Section 3 of AASHTO Division IA. Whitman (1990) recommended that kh be ranged from 0.05 to 0.15. Segrestin and Bastick (1988) suggested that kh be found as:
kh =(1.45A)A (2.52)
for 0.05 < A < 0.5. Note that the above equation was incorporated in both the FHWA and NCMA manuals. Figure 2.30 shows the relation between kh and A as determined from Equation 2.52. As depicted in Figure 2.30, amplification of the peak horizontal ground acceleration is observed for A < 0.45. For A > 0.45, kh is less than A, and the maximum of the curve occurs at A = 0.725.
47
c
0
o
it
0
o
o
c
o
2
0
0
o
o
ro
o
N
o
X
0 0.2 0.4 0.6 0.8 1
Peak horizontal ground acceleration, A
Figure 2.30 Relationship between A and kh 2.4.4 Permanent Displacement Methods
In the pseudostatic seismic analysis and design of GRS walls, only the factors of safety against various modes of failure or collapse of the wall could be estimated, and wall deformation could not be estimated directly from the pseudostatic analysis. This is a common deficiency in all of the limit equilibrium analyses. The following paragraphs describe the indirect methods used to estimate the horizontal wall movements (or the timedeformation response of the wall system) to accompany the seismic stability analysis.
In Newmark's doubleintegration displacement method (Newmark 1965) applied to retaining wall structure, the total displacement is termed unsymmetrical displacement, since the permanent displacement only accumulates in one direction (outward direction). The displacement in the reverse direction (toward the backfill) requires a greater critical acceleration to overcome the passive resistance in the backfill. A passive failure in the backfill would require a force on the order of 10 times the static resistance (Richards and Elms 1979). Hence in retaining wall structure application, it is assumed the displacement of the rigidplastic block at failure toward the backfill is zero. Note that the calculation of displacement is based on the assumption that the moving mass displaces as a rigidplastic block with shear resistance mobilized along the potential sliding surface. Permanent displacement of the rigidplastic block is said to have occurred whenever the forces acting on the soil
48
mass (both static and seismic forces) overcomes the available shear resistance along the potential sliding surface.
Sliding Block Method by Cai and Bathurst
The three seismic induced sliding mechanisms in a GRS wall proposed by Cai and Bathurst (1996a) are: (1) external sliding along the base of the entire wall structure, (2) internal sliding along a reinforcement layer and through the facing column, and (3) block interface shear between facing column units. The displacements are estimated using the conventional sliding block method. The pseudostatic dynamic earth pressure used in the analysis and design of GRS wall by the NCMA method (Bathurst 1998) is also adopted in this displacement method.
Note that both kh and kv are used to calculate dynamic active forces and are assumed to remain constant through out the entire wall structure. According to Cai and Bathurst (1996a), in absence of the ground motion record, kv can be estimated as kv = 2*kh/3, and the vertical inertial force is assumed to act upward to produced the most critical factors of safety for the horizontal sliding mechanisms.
Critical accelerations associated with the three sliding mechanisms are needed in order to determine the seismic induced permanent displacements. The critical accelerations are found by setting the factor of safety equations of the three sliding mechanisms equal to unity. The factor of safety against the base sliding FSsi is:
FS =i= 1.0 (2.53)
Pir+Pah+0.5APaeh
where Rs = the frictional resisting force mobilized along the sliding boundary at the base of the wall structure, Pir = the seismic inertial force due to 0.5H in length of the reinforced soil mass, and APaeh = the seismic thrust increment. Similar to the FSsi, the factor of safety against internal sliding FSsn at depth z is:
R.(z)
FSsll =
= 1.0
(2.54)
APir (z) + PAH (z) + 0.5 AP^ (z)
The factor of safety against block interface shear between facing column units FSsc is: Vu(z) au +AWw(z)tan?iu
FS =
= 1.0
(2.55)
Si (z) k h (int) AWw (z) + P^ (z) + APaeh (z) Ã‚Â£Ttotal where Vu(z) = the peak shear capacity of the facing column interface, which is dependent of the minimum available interface shear strength (au) and the apparent peak interface friction angle between facing units (?^u), S,(z) = outofbalance shear force transmitted through facing unit interface, AWw(z) = weight of the facing column above the sliding interface, and STtotai = sum of the horizontal forces carried by the reinforcement layers above the interface of interest. Note that both kh and kv are incorporated in the factor of safety equations, and with the assumed kv and FS = 1.0, the critical acceleration in the horizontal direction ac = kcg can be back calculated.
49
The permanent displacements are assumed to accumulate each time the ground acceleration exceeds the critical acceleration, where the Newmark's doubleintegration method is used to calculate the permanent displacement. In Newmark's doubleintegration method, the potential sliding soil mass is treated as a rigidplastic monolithic mass subjected to inertial force. Figure 2.31 shows an example of the Newmarks doubleintegration from the acceleration time history to the velocity time history, and finally to the displacement time history. The shaded area in Figures 2.31 indicates deceleration. The total displacement at the wall face is determined in an accumulative manner from the wall base to the topmost layer of reinforcement. The displacement of a layer is determined as the larger of the internal sliding and the facing shear displacement.
Figure 2.31 Newmark's Doubleintegration Method for calculating Permanent Displacement of a Sliding Soil Mass
Sliding Block Method by Siddharthan et al.
Based on many seismic centrifuge test results of mechanically stabilized earth (MSE) walls, a rigidplastic multiblock computational method was developed to predict the permanent displacement of MSE wall subjected to seismic loading (Siddharthan et al. 2004b). This method is applicable to inextensible reinforcement such as the steel bar mat reinforcement. The failure mechanism is comprised of three
50
rigid blocks and possesses a bilinear failure plane; the top two blocks are rectangular, and the bottom block is triangular. There are a total of four degrees of freedom for the proposed failure mechanism, where the top two blocks only have the translational degrees of freedom, and the bottom block has both translational and rotational degrees of freedom. The length of the top blocks is assumed to be equal to the length of the reinforcement. The schematic of the multiblock method is shown in Figure 2.32. In the proposed multiblock model, the slope of the failure plane a and the thickness D* are variables. Through an iterative approach, the combination of a and D* that yields the largest displacements are considered as the permanent displacements.
The forcemassacceleration method was used to obtain the equations of motion. In the verification study (verifying the computational method with the seismic centrifuge test results), the acceleration time history in each block is calculated by multiplying the base acceleration time history by the amplification factor determined from the seismic centrifuge test results (see Figure 2.13). The displacements of the blocks are determined using the Newmark's doubleintegration method. As reported by Siddharthan et al., the calculated displacements using the proposed method were in good agreement with the measured displacements.
>\
I
D*
(Variable)
T
Figure 2.32 Schematic of the Multiblock Displacement Method (after Siddharthan et al. 2004b)
51
Sliding Block Method by Ling et al.
The twopart wedge mechanism has been used to determine the reinforcement length of the base layer of a reinforced steep slope (Leshchinsky 1997). The twopart wedge mechanism is further considered in determining the seismic induced permanent displacement of a reinforced steep slope (Ling et al. 1997). The schematic of the twopart wedge mechanism is shown in Figure 2.33. The displacement evaluation procedure is similar to the base sliding approach proposed by Cai and Bathurst (1996a), in which the reinforced soil zone is treated as a rigidplastic block. The displacement of the rigidplastic block is induced when the factor of safety against direct sliding is less than unity. The factor of safety equation is first used to determine the critical acceleration (or yield acceleration), and with a known design earthquake motion time history, Newmark's doubleintegration method is then used to find the cumulative permanent displacement of the rigidplastic block.
Figure 2.33 TwoPart Wedge Mechanism for Direct Sliding Analysis (after Leshchinsky 1997)
2.4.5 Empirical Methods
Newmark's doubleintegration method in finding the seismic induced permanent displacement requires the ground motion time history to be known. In absence of the ground motion time history, several empirical methods have been
52
developed to predict the seismic induced permanent displacement of earth structures. Newmark's sliding block theory has been used as the basis for developing the empirical methods, where the total permanent displacement determined by Newmark's doubleintegration method is correlated with input ground motion parameters such as peak ground acceleration (kmg), peak ground velocity (vm), and critical acceleration ratio (kc/km). This section describes the empirical methods of Newmark (1965), Richards and Elms (1979), Whitman and Liao (1984), and Cai and Bathurst (1996b).
Empirical Method by Newmark
Four U.S. west coast earthquakes with peak acceleration ranging from 0.178 g to 0.32 g were selected and normalized by Newmark (1965) with each having a peak acceleration of 0.5 g and a peak velocity of 762 mm/s (30 in./s). These normalized earthquake motions were used to determine the displacement of sliding of rigidplastic mass in attempt to simulate the block sliding of an embankment or earth dam. The standardized maximum seismic induced displacements were plotted against the critical acceleration ratio (kc/km). Both the critical acceleration ratio and the peak ground velocity were used in the prediction of the standardized displacement. In the case of unsymmetrical displacement, the upper bound limits of the standardized displacement are:
d. =
6vm
2kcg
dr =
(for kc/km <0.16)
(for kc/km >0.16)
(2.56)
(2.57)
2kcg kc
where ds = standardized displacement (m), vm = peak ground velocity (m/s), g = gravitational acceleration (9.81 m/s2), km = peak ground acceleration coefficient, and kc = critical ground acceleration coefficient. Furthermore, as proposed by Franklin and Chang (1977), the standardized displacement ds can be converted to the actual permanent displacement d by:
, , 0.86 v"
d = d
(2.58)
where ds and d are in unit of meter.
Empirical Method by Richards and Elms
Richards and Elms (1979) fitted an upper bound curve to the standardized displacement results integrated by Franklin and Chang (1977) for more than 196 strong motion records. The upper bound is given by the expression:
53
2 /
d = 0.087
kmg
_k
V^m J
>4
(2.59)
where d = total permanent displacement (m), vm = peak ground velocity (m/s), g = gravitational acceleration (9.81 m/s2), km = peak ground acceleration coefficient, and kc = critical ground acceleration coefficient. Since the above expression is an upper bound curve, the predicted permanent displacement may be conservative as compared to other empirical methods.
As suggested by Richards and Elms, an alternative seismic design approach for the retaining wall is to consider its allowable permanent displacement. In this alternative design approach, an allowable permanent displacement is estimated first based on the function of the retaining wall (e.g., 50 to 100 mm for typical retaining walls). Using the displacement equation above and with prescribed peak acceleration coefficient km and peak velocity vm, the critical acceleration coefficient kc is back calculated. The seismic thrust using the M0 method along with back calculated kc can then be determined. The wall dimensions are to be sized based on satisfactory factors of safety against various modes of failure (e.g., base sliding) using the seismic thrust just determined.
Empirical Method by Whitman and Liao
Whitman and Liao (1984) performed the regression analysis based on the standardized displacement data performed by Franklin and Chang (1977) and proposed the following equation to predict the mean displacement of a sliding gravity retaining wall:
(
\
d = 37
exp
9.4Ã‚Â£k
(2.60)
m J
where d = total permanent displacement (m), vm = peak ground velocity (m/s), g = gravitational acceleration (9.81 m/s2), km = peak ground acceleration coefficient, and kc = critical ground acceleration coefficient. Whitman (1990) listed the potential errors and uncertainties associated with the permanent displacement prediction based on the sliding block method and hence the empirical method. Examples of uncertainties and errors include: (1) unpredictable details of ground motion (e.g., frequency content, duration, and directionality), (2) material parameters (e.g., friction angle of backfill and interface friction angle between backfill and wall), and (3) model errors (e.g., deformability of backfill and wall tilting).
Empirical Method by Cai and Bathurst
More recently, Cai and Bathurst (1996b) have reexamined the permanent displacement results ofNewmark (1965) and Franklin and Chang (1977) and proposed a mean upper bound curve for predicting the permanent displacement as:
54
(
V
, 0.38
d = 35
g
exp
 6.91 k
(2.61)
where d = total permanent displacement (m), vm = peak ground velocity (m/s), g = gravitational acceleration (9.81 m/s2), km = peak ground acceleration coefficient, and kc = critical ground acceleration coefficient. This mean upper bound curve is meant to reduce the conservatism involved with the upper bound of Richards and Elms (1979) and to avoid potential underestimate of displacement from the mean curve of Whitman and Liao (1984). Figure 2.34 compares the permanent displacement estimated by the various empirical methods with the assumptions of peak velocity vm = 762 mm/s (30 in./s) and peak ground acceleration coefficient km = 0.5 (the same as those used by Newmark 1965).
Figure 2.34 Comparison of Various Empirical Displacement Methods
55
2.5 Behavior of Geosynthetics subject to Cyclic Loads
The loaddeformation behavior of geosynethetics subject to cyclic loads is dependent of the loading frequencies and the load amplitudes. Bathurst and Cai (1994) performed a series of inisolation cyclic loadextension tests on high density polyethylene (HDPE) and polyester (PET) geogrid specimens, in which the specimens were tested at five different loading frequencies (e.g., 0.1, 0.5, 1.0, 2.0, and
3.5 Hz) and over a range of load amplitudes. Figure 2.35 shows a typical loaddeformation curve of the HDPE geogrid specimens under multiincrement and singleincrement cyclic loadings. There were five load amplitudes applied in the multiincrement cyclic loading, where the amplitudes ranged from about 20% to 80% of the ultimate strength and that each load amplitude was applied for 10 cycles. For the singleincrement cyclic loading, a single load amplitude at approximately 80% of the ultimate strength was applied for 10 cycles.
Strain e {%)
(a) multiincrement cyclic load lest
Figure 2.35 Response of HDPE Geogrid Specimens to (a) Multiincrement Cyclic Load Test and (b) Singleincrement Cyclic Load Test (after Bathurst and Cai 1994)
Some qualitative characteristics of a cyclic loaddeformation curve are illustrated in Figure 2.36. Figure 2.36 identifies the parameters that can be used to
56
characterize the loaddeformation curve as a function of strain. A nonlinear hysteresis loaddeformation loop for each unloadreload cycle (sUr, Tur) is defined by the average unloadreload stiffness (Jur) of the unloadreload cycle and the contained area (Aur). A loaddeformation cap (or envelope) that is tangent to the peak response of the initial unloadreload cycle at each loading stage can be quantified by an initial tangent stiffness (Jj) and secant stiffness values at selected strain values (e.g., JseC2 and
Jsec5)
average unloadreload stiffness Jur
/
Ã‚Â£ (%)
Figure 2.36 Characteristics of Cyclic Response of Geosynthetic Specimen (after Bathurst and Cai 1994)
The area of a hysteresis loop (Aur) was found to be strongly influenced by the strain level and the frequency of loading. As shown in Figure 2.37, the area Aur increases with the strain level and decreases with increasing frequency. Note that below 0.5% strain (with load amplitude at approximately 12% of the ultimate strength) of the HDPE geogrid and 0.8% strain (with load amplitude at approximately 15% of the ultimate strength) of the PET geogrid, the specimens behaved in a linear elastic manner with fully recoverable strain (i.e., the area of the hysteresis loop is nearly zero). Figure 2.38 shows the average unloadreload stiffness Jur versus strain relationships for different load amplitude and frequencies. For the HDPE specimen, Jur is influenced by the strain level but is essentially independent of the frequency of load above 0.5 Hz. At frequencies greater than 0.5 Hz and strains greater than about 2%, Jur of HDPE specimen decreases with increasing strain (see Figure 2.38a). The Jur of PET specimen, on the other hand, is relatively insensitive to the frequency of loading but varies with the strain level. At strain level of grater than about 3%, Jur increases with increasing strain (see Figure 2.38b). Similar hysteresis behavior of
57

Full Text 
Ling, H.I., Cardany, C.P., Sun, L.X., and Hashimoto, H. (2000). "Finite element study of a geosyntheticreinforced soil retaining wall with concreteblock facing," Geosynthetics International, Vol. 7, No. 3, pp. 163188.
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GEOSYNTHETICREINFORCED SOIL WALLS UNDER MULTIDIRECTIONAL SEISMIC SHAKING by ZehZon Lee B.S. University of Colorado Denver 1998 M.S. University of Colorado Denver 2000 A thesis submitted to the University of Colorado Denver in partial fulfillment of the requirements for the degree of Doctor of Philosophy Civil Engineering 2011
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This thesis for the Doctor of Philosophy degree by ZehZon Lee has been approved by Aziz Khan 1/U ")...eI II Date
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Lee ZehZon (ph.D. Department of Civil Engineering College of Engineering and Applied Science) GeosyntheticReinforced Soil Walls under Multidirectional Seismic Shaking Thesis directed by Professor NienYin Chang ABSTRACT This research study was conducted to determine the validity of the current seismic analysis of a freestanding simple geosyntheticreinforced soil (GRS) wall. The case histories of GRS walls during the past several large earthquakes have indicated satisfactory seismic performances and suggested that the current design methodology is adequate Despite successful cases several GRS wall failures have been reported Numerical simulation of GRS wall subjected to seismic loading can thus offer the opportunity to identify the discrepancy between the current design methodology and the more rigorous finite e l ement method (FEM) solutions. The predictive capability of the FEM computer program LSD YNA was first validated against fullscale shaking table t est walls. Material characterizations of the backfill and geosynthetic reinforcements were performed in the validation process Material model parameters were determined from the available laboratory data In particular the backfill was simulated with a cap model with parameters dependent of stress level. Model calibration was also performed to fine tune the input parameters such as the viscousdamping constant mass damping coefficient and the soil geosynthetic interface friction coefficient. The calibrated values along with the material characterization approaches were adopted for the subsequent parametric study. Prior to the parametric study the extent of finite element model boundary was verified in order to minimize the boundary effect. Results of parametric study were compared against the values determined using the Federal Highway Administration (FHWA) methodology. The parametric study results were presented as functions of peak horizontal acceleration (PHA) and the correlations of seismic performances with PHA were determined through single predictor variable regression analyses in order to indicate the trend of the calculated results. In general the external stability analysis results predicted by FEM are higher than those determined using the FHW A methodology. However for the internal stability analysis, FHW A methodology overestimates the reinforcement tensile load as compared to the results predicted by FEM. Using FEM results multivariate regression equations were developed for the various seismic performances based on multiple design parameters that are essential in the design of GRS walls. The prediction equations can provide firstorder
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estimates of the performances for use in the preliminary analysis of freestanding simple GRS walls. The prediction equations are applicable for PHA greater than maximum limitation stipulated by the FHW A methodology. This abstract accurately represents the content of the candidate's thesis. I recommend its publication. Signed
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DEDICATION I dedicate this thesis to my parents who gave me an appreciation of learning and taught me the values of perseverance and resolve. I also dedicate this thesis to my wife, TzeHong, my children Alexis YingShan and Stanley HsinYu for their unfaltering support and understanding while I was completing this thesis.
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ACKNOWLEDGEMENT I would like to thank foremost my advisor Professor NienYin Chang for providing me the freedom to manage my own work This helped me to hone my skills to solve difficult problems independently. Professor Chang has provided continuous support throughout this study. Without his guidance and assistance the completion of this study would not have been possible. I have benefited greatly from his extensive expertise and wisdom. I also would like to thank the committee members Professor HonYim Ko Professor Brian Brady Dr. Aziz Khan Professor John McCartney and Dr. Trever ShingChun Wang for their comments and suggestions on the thesis. I am indebted to Professor Ko for serving as my coadvisor and his classes in soil mechanics had prepared me well to understand indepth topics in geotechnical engineering. I am also grateful to Professor Brady for his continuous encouragement throughout my graduate study and his advices on working for the federal government have been beneficial. Dr. Wang has also provided much technical insight in the actual design of retaining structures. It was a privilege to work with my classmates Mohammad AbuHassan Hien Manh Nghiem and Brian Volmer. Mohammad shared his knowledge and experience on using the contact algorithm in LSDYNA. Hien was a great source of help on issues related to numerical analyses and constitutive modeling. Last but not least discussions on engineering problems and practical issues with Brian had always been fruitful.
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TABLE OF CONTENTS Fig ures ....... .... . ...... ....... ................ .... . ...... . . . . ........ . .......... . .... . ... ... ....... ......... ... xiii Tables ....... .... . .... . . . .................... . .... .... .... .... . . . .... . . ..... ... .... ..... . .... ....... . ............. xli Chap t e r 1. Introduction .... ................. . ............ ................. .... . ..... . ..... . . .... . ......... ... .. ........ .... l 1.1 Probl em Sta tem ent ....... . ....... .......................... ...... .......... ....... .......... . . ..... . .... 1 1 2 Research Objecti ves ................ .......... . . .... . ........................................................ 2 1 3 Scope of Study ...... ... . ... . . ........ ...... . ....... .... . . . ................ . ..... ...... ...... . ...... 2 1.4 Engineer in g Significances ... .... . . . .... . . . ..... ..... ..... ...... . . .... . . .... . . . . ............. 3 2 L iteratur e R ev ie w .... . ....... . . ... . ........ .... ................... . ... ... . ................... . . ..... . ..... 5 2.1 Post Eart hquake Investigations . .... . . . . . . ...... .... .......... . .... .... . ....... . . ... . . . 5 2 2 Laboratory Model Tests . . ..... ..... . ....... ........... ....... . .... .... . ..... . ... ...... .... ..... . . ... 12 2.3 Numerical Model Studies ........ . . ..... .............. ... .... . ........ ...... ....... ... .. ... .... . . 19 2.4 Seismic Design of Retaining Walls . ...... .... . ........ . . .... . .................................. 26 2.4.1 MononobeOkabe Method . ....... . . ...... .... . ............... . ................ .... ..... . ..... . . 26 2.4.2 D esign and Analysis of G eosynt heticR einforce d Soil Retaining Walls . . ............. ............................. . ........ ................. . ..... ........ .... 30 2.4.2. 1 Federa l Highway Admini stratio n (FHW A) Methodology ........ ............. . ... 31 2.4.2.2 Natio nal Concrete Masonry Assoc iation (NCMA) Methodology ... . ..... . . 37 2.4 3 P eak Ground Acceleration Coefficie nt s . . . . . .... . .... .... .... .... . . .... . ..... . ..... . .47 Vll
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2.4.4 Permanent Displacement Methods ... ........ ................... ................... .... ...... .48 2.4.5 Empirical Methods ......... ........................... . . . ... ............................. .... ....... ... 52 2.5 Behavior of Geosynthetics subject to Cyclic Loads ... . ........... ............... ..... . ... 56 2.6 Soilreinforcement Interface Friction ...... ........ . ........ ...... .... . . .... .......... ....... 60 3. Validation of Computer Program ............... ..... ....................................................... 62 3.1 Terminology .................................................. ....................................... ........... .... 62 3.1.1 Verification and Validation .............................................................................. 63 3.1.2 Prediction ..................... ................ ......................... ......... .................................. 64 3.1.3 Calibration ........................................................................................................ 66 3.1.4 Validation Assessment.. ................................................................................... 66 3.2 Material Characterization for Computer Program Validation ............................ 67 3.2.1 Geologic Cap Model ........................ ...... ... ............ ...... ................ ..... ............. 68 3.2.1. 1 Cap Model Strength Parameters .......... ..... .......... ......................................... 71 3.2.1.2 Cap Model Hardening Parameters ................................................................ 75 3.2.2 PlasticKinematic Model .............................................. ................................... 86 3.3 Validation of Computer Program with FullScale Tests .............. ....................... 89 3.3.1 Model Configuration .......................... ............................... ............................... 89 3.3.1. 1 Element Types ...... .... .......... ....... ......... ....... .......................... ........................ 91 3.3.1.2 Loading and Boundary Conditions ...................... ................................ ......... 91 3.3 .1.3 Contact Types and Contact Details ... ..................... .... ........ ........................ 96 3.3.2 Model Calibration .................................................... ....... ....... .......................... 99 Vlll
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3.3.2.1 ViscousDamping Constant ........ .................... . . ...................... ........ ..... ..... 99 3.3.2 2 Mass Damping Coefficient ........................................... .................. ....... .... 99 3.3. 2.3 Friction Coefficient ofSoilGeogrid Interface .......................................... .10 2 3.3.3 Response Comparison ......... .... . .......... .......... . . .... ..... ........ ........... .... ..... .... 104 3.3.3. 1 Wall Facing Displacement.. ..... ................... . . . . ....... ........................... . . 105 3.3.3.2 B ac kfill Surface Settlement ..... . .......... ..... ...... .... .... ............... .............. .... 105 3.3.3 3 Lateral Earth Pressure behind Faci n g Blocks ... . . . ..................... .............. 111 3.3.3.4 Bearing Pressure ........... .................... ...... ....... .. ........... .. .. .......................... 111 3.3.3.5 Geogrid Reinforcement Tensile Load ................ . ....... ....... .......... . ........... 111 3.3.3.6 Accelerations in Reinforc e d and Retained Soi l Zones ..... ...... ........... . . ..... 115 3.3.3. 7 E n e rgy Observation . ........... ........ ....... ....... . ..... . . ..... . ... ...... . ....... ............ 115 3.3.4 Quality of Prediction ... ....... ............. . .... .... . .............. .............. . .... . ......... .. 118 3.3.5 Variability within the Measured D ata .... ........ ..... .... ..... . .... ............. ..... ...... 1 20 3.4 Discussion of Numerical Simulation . ...... . ...... .................... . .... ..... ...... ......... 123 4. Parametric Study .... ........ .... . ................................... ..... .... ..... ..... .......................... 125 4.1 Input Ground Motions ............................................... . ............. ... ... . .... . ...... .... 125 4.2 Material Characterization for Parametric Study ..... . . .............. .......... . . ........ .126 4.2.1 Soil Characterization . . ................... ........ ...... ..... . .... . .... ........... ...... ..... ........ 137 4.2.1.1 Cap Model Strength Parameters in Parametric Study ......... . . .... . .......... . . 139 4.2.1.2 Cap Model Hardening Parameters in Parametric Study ............ .......... ...... 140 I X
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4 2.2 Geosynthetic Reinforcement Characterization . . . . . ....... .............. . . . ...... . 148 4.3 Development of Model Dimensions ....................... .................... . . .......... ..... 150 4.4 Parametric Study Program ..... .... . . ........ .... ..... .... ..... . . ...... ...................... . .... 152 4.5 Global Stability ...... ..... ...................... . . .... .................. ...... .......... ..... ............. 152 4.6 Modeling Procedure .... ..... .... ........ . . . ...... . ............................................ ... ... . . 154 5 Results of Parametric Study ............... ....... ..... .. ......... . ...... ...... . . .... ............. ..... 172 5 1 Effects of Multidirectional Shaking .................... . . . ....... . .... ......... .......... ....... 172 5.2 Seismic Performances .............. . ............... ..................... .... ..... ..... ..... ........ .... 176 5.2.1 Effects of Wall Height . . . . ...... .. ... . .... . .... . . ................................ ............. 178 5.2.2 Effects of Wall Batter Angle ............................... ...... . . ........ ..... ..... .... ....... . 188 5.2.3 Effects of Soil Friction Angle ............ ...... .... . ..... . ...... .............. ..... .... ......... . 192 5.2.4 Effects of Reinforcement Spacing . ...... ...... ........ ........ .......... ..... ............ . . 197 5.2.5 Effects of Reinforcement Stiffness ........ . .... . . ............................................ 201 5 3 Distribution of Reinforcement Tensile Load .......... ........ ...... . . . . ............ . . 206 5.4 Soil Thrusts and Reinforcement Resultants at Distances behind Wall Facing ....... ..... ....................... . . . .... ...... ........ .... . .............. .... . ..... . . 222 6. Multivariate Statistical Modeling . .... . ........ ...... ..... ................ .............. ..... . .241 6 1 Correlation Anal y sis ................... .... . ..... . .... . . . ........ .... . . ..... . ...... . . ........ .241 6.2 Regression Analysis ........................................ .... .... . .... .... ............. .... . ..... . . 244 6.2.1 Prediction of Maximum Horizontal Displacement.. ..... .. ... . ..... .......... ..... . . 245 6.2.2 Prediction of Maximum Crest Settlement.. ..... . .... ..... .... ..... . . .......... ........ . 249 x
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6 2.3 Prediction of Total Driving Resultant.. ..... ......... ...... .. ......... ........ .. ....... .. ...... 253 6.2.4 Prediction of Total Overturning Moment Arrn ...................... ...... ............ ..... 257 6.2 5 Prediction of Maximum Bearing Stress .... .................. .................. ...... .. .. .... 258 6.2 6 Prediction of Maximum Reinforcement Tensile Load .... .. .... ................ ..... 262 6.2.7 Prediction of Maximum Horizontal Acceleration at Centroid of Reinforced Soil Mass ....... ...... ..... ...... .. .. ...... ............ ...... .... ..... 266 6.2 8 Prediction Equations without Peak Vertical Acceleration ............. .. ....... ..... 270 6 3 Design Considerations .. ....... ................... ................ ..... ...... .. .. ............. ..... .. .... .271 7. Conclusions and Recommendations for Future Studies ................. .. ...... ..... .. .... .274 7.1 Conclusions ...... ....... .. .. ...... ................ .. ... ...... ............ .... .. .......... ...... .. ............. 275 7.2 Recommendations for Future Studies ..... ............. ............ .... ...... ............... .. .. .. 279 Appendix A. RambergOsgood Material Model . ...... ..... .............. .. ..... .... ........ ................. .. .... .280 B. Global Stability of Models considered in Parametric Study ... ...... .. ..... .. .. .. .. ..... .288 C. Analysis of GRS Wall following FHWA Methodology ... .. ..... ..... .. ......... ..... .... .29 2 D Maximum Wall Facing Horizontal Displacement Profiles ..... ..................... ...... 300 E. Maximum Wall Crest Settlement Profiles ...... ..... .. .. ..... ................. ........... ....... 312 F. Maximum Lateral Earth Stress Distributions ...... .. .. .. ............. ....... .. ....... ............. 323 G Maximum Bearing Stress Distributions ....... ............ .. ............. ...... .. .... .. .... .... .. ... 335 H. Maximum Reinforcement Tensile Load Profiles ............. .. .... ...... .. .... ......... .. ..... 346 I. Correlations with Peak Vertical Acceleration ........... .. .... ........... .... .. ....... ..... ...... 358 Xl
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J. Regression Analysis for Low Peak Horizontal Acceleration Range ................... 379 K. Data for Statistical Modeling .. .. .. ......................................................................... 396 L. Natural Period of GRS Walls ...... ........................................................................ .403 References ........ . ........ .... . . ....... ..... ...... ..... ...... ....... .. ....... ............. ............ .... . . ........ 404 XlI
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LIST OF FIGURES Figure 1.1 Schematics of a Freestanding Simple GRS Wall .... ............... .... ... ............. .. .... .2 1.2 Scope of Research Study .................................... ... ...... ........... ..................... ...... 3 2.1 Relationships between Maximum Horizontal and Vertical Accelerations of Northridge Earthquake (after Stewart et al. 1994) ..... ................. ...... ............ .. ...... 6 2.2 GeosyntheticReinforced Soil Wall with FHR Facing (after Tatsuoka 1993) ........................ .. ...... ...... .. ..... .. ............. ........... ....... .......... 7 2.3 Collapse of GeosyntheticReinforced Soil Wall during 1999 ChiChi Taiwan Earthquake (after Ling et al. 2001) .... .. ..... .. ......... ................................... ........... 8 2.4 Two Failed GeosyntheticReinforced Soil Walls durin g 1999 ChiChi Taiwan Earthquake (after Huang et al. 2003) ............ ..... ....... .. .. .... .. .... .................... ..... 10 2.5 Failure of GeogridReinforced Soil Wall during 2001 EI Salvador Earthquake (after Koseki et al. 2006) ....... .......... ......................... .. .. .... ................. ........... .... .11 2.6 Failure of GeogridReinforced Soil Wall during 2001 Nisqually Earthquake (after Walsh et al. 2001) ........................ ............. .. ................... .. .......... .. .. .. ....... 11 2 7 The Influence of Wall Type on Wall Displacement with Irregular Base Shaking (adapted from Watanabe et al. 2003) ...... ............... ..................... ...................... .15 2.8 Effect of Relati ve Density and Input Motion Frequency on the Wall Displacement (adapted from Latha and Krishna 2008) .... .. ............ ........... ..... .. 15 2.9 Example of Twopart Wedge Failure Mechanism (after Matsuo et al. 1998) .. .... .. ..... .. ..... . .. .. .... .. ..... ........ .. ...... .......... .. ................. .16 2.10 Comparison of Reinforced Soil Wall Model Before and After the Shaking Table Test (after Anastasopoulos et al. 2010) ... ..... .. ....................... .......... ....... .. ...... .. .16 Xlll
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2.11 Comparison of Reinforced Soil Wall Response at Different Harmonic Input Motions; (i) 15 Hz Harmonic Input Motion ; (ii) 30 Hz Harmonic Input Motion (after Wolfe et al. 1978) . ...... ... ... .. ... . .... . . .... . ..... ...... . . ................... . . . . . .... 17 2 .12 Comparison of Reinforced Soil Wall Responses with Different Components of Irregular Input Motion ; (i) Vertical Component Only; (ii) Horizontal Component Only ; (iii) Combined Vertical and Horizontal Components (after Wolfe et al. 1978 ) ...... . ............... ... .............. ............ ...... . ................... .............. 17 2.13 Measured Amplification Factor in the Free Field (after Siddharthan et al. 2004a) ......... ..... ... ............. ......... . ..... . .................. . ... 18 2.14 Comparison of Observed Critical Seismic Coefficients to Calculated Ones against Overturning or Bearing Capacity Failure (after Koseki et al. 2003) . . ...... . ...... . . .... .... . ... .... . . ... ........... . .... ... ........ . 18 2.15 Equilibrium of Forces acting on the Active Wedge in MO Analysis .... . ......... 27 2.16 Equilibrium of Forces acting on the Passive Wedge in MO Analysis ... . ... . . . 28 2.17 Effect of Direction ofInertia Forces on Total Seismic Active Thrust (modified from Fang and Chen 1995) . ... .... . . . ...... . ...... ... . ....... . .... . ... . ............ ..... .... .29 2.18 Effect of Direction ofInertia Forces on Total Seismic Passive Thrust ............ . 30 2 .19 Seismic External Stability of a GRS wall with Level Backfill in FHWA Method . ...... . . . . ..... .... . . . ... ...... ....... . . ...... . . ....... . ..... ....... . ........ . 33 2.20 Seismic External Stability of a GRS Wall with Sloping Backfill in FHWA Method .... .... ... ........ ....................... .......... ...... . ....... ...... . ....... ... ..... ............... 33 2 .21 Seismic Internal Stability of a GRS Wall in FHW A Method .......... . ................ 37 2.22 Modes of Fa ilure for GRS Walls in NCMA Method ... ............ .......... .......... . . 38 2.23 Seismic External Stability of a GRS Wall in NCMA Method ........ .......... ..... . .40 2.24 Schematic for finding the Total Tensile Force in Each Reinforcement Layer in NCMA Method .................. ...... .......... . ............. ........ ... . . . . ..... ...... ............... 41 2 25 Schematic for Reinforcement Pullout Evaluation in NCMA Method ... ..... ..... .42 XIV
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2.26 Schematic for Base Sliding and Internal Sliding Evaluation in NCMA Method ...... .... . . . ........... ............. ........ . ..... ....... .... . . . ....... ... ........ ... 43 2.27 Schematic for Facing Interface Shear Evaluation in NCMA Method ................ 44 2 .28 Schematic for Facing Loca l Overturning Evaluation in NCMA Method .......... .45 2 29 Differences between the FHWA Method and the NCMA Method ................... .47 2 .30 Relationship between A and kh .............................................................. .... ...... .. .48 2.31 Newmark's Doubleintegration Method for calculating Permanent Displacement of a Sliding Soil Mass ............................................... ......................................... 50 2.32 Schematic of the Multib l ock Displacement Method (after Siddharthan et al. 2004b ) ........ ......................... ............... ........................ ............................. .. ........ 51 2 .33 TwoPart Wedge Mechanism for Direct Sliding Analysis (after Leshchinsky 1997) ... . . . ... ........ . . ..... . ...... ...... ...... ...... ............. ............. ........ ............. . 5 2 2.34 Comparison of Various E mpirical Displacement Methods ................................ 55 2.35 Response ofHDPE Geogrid Specimens to (a ) Multiincrement Cyclic Load Test and (b) Singleincrement Cyclic Load Test (after Bathurst and Cai 1994) .... .... ........................ ............ .... ................ .............. 56 2.36 Characteristics of Cyclic Response of Geosynthetic Specimen (after Bathurst and Cai 1994) .. ................ ................. ........ .................... ............. .. .. ..................... 57 2 37 Area of Hysteresis Loops (Aur) for HDPE and PET Specimens during the Multiincrement Cyclic Loading (after Bathurst and Cai 1994 ) .. .................. .... ........... 5 8 2.38 Unloadreload Stiffness (Jur) for HDPE and P E T Specimens from the Multiincrement Cyclic Loading (after Bathurst and Cai 1994 ) ................................... 58 2.39 Influence of Strain Rate on Monotonic Load extension Behavior of Ty pical Geos y nthetic Specimens (after Bathurst and Cai 1994) ...... .............................. 59 xv
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2.40 Interface Friction Angle betw een Ottawa sand and HDPE Specimens with (a) Monotonic Loading and (b) Repeated Loading on HDPE Lining (after O'Rourke et al. 1990) .. ... .... ..... ............. ...................... .... ..... ...... ..... .... . ....... .... ...... . ....... 60 2.41 Effect of Strain Rate on the Interface Friction Angle between Sand and Various Reinforcement (after My les 1982) .......... ............. . . .... . ..... . ..... ........... . ..... . 61 3.1 Phases of Modeling and Simulation (after Schlesinger 1979) ..... ...... ....... ......... 64 3 2 The Enhanced Soil Mechanics Triangle (after Anon 1999) .... .......... .... . . ..... . 65 3.3 Relationship of Validation to Prediction (modified from Oberkampf and Trucano 2002) .. ... . . ...... ..... . . .... . ..... . . .... . ........... . ..... ....... . .... . . ...... ............ 65 3.4 Increasing Quality of Validation Metrics (after Oberkampf et al. 2002) .... . ..... 67 3.5 Schematic of Cap Model (modified from Desai and Siriwardane 1984) ... . .... . 70 3.6 Stress Paths achieved by Various Laboratory Tests in (a) .J];: II Stress Space and (b) Deviatoric (Octahedral) Plane (modified from Chen and Saleeb 1994) ..... ..... .... .... . . ...... .... ..... . . . .... ........ . . . ...... . .... . . .... 70 3.7 Comparison between Calculated and Measured Triaxial Compression Results of Tsukuba Sand (measured data from Ling et al. 2005a) . . . ......... ....... ............. 73 3 8 Effective Shear Strength Parameters from Drained CTC Tests for the Tsukuba Sand . ....... . . ....... . .... .... .... ..... ...... .... ....... ........................ ..... ..... . . .... . . .... ..... . 74 3 9 DruckerPrager and MohrCoulomb Failure Criteria in Deviatoric (Octahedral) Plane with Different Matching Conditions (modified from Chen and Saleeb 1994) ............. . .... ... ........... . ...... . .... . ........... . ........... ....... . . ....... ........... . . ..... 74 3.1 0 Stress Paths of CTC Tests and the Fixed Yield Surface fl of Tsukuba Sand .... . .......... ..... ...... ............ .......... .... . ............. . . . . . . .... .... . . ..... 75 3 .11 Hydrostatic Compression Curves of Chattahoochee Sand at Different Initial Relative Densities (modified from Domaschuk and Wade 1969) ..... . . . .... ..... 76 3.12 Grain Size Distribution Curves for Tsukuba and Chattahoochee Sands ..................... ......... ...... ..... . ..... . ....... .... . ....... . .... .............. ............ . . . . 77 XVI
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3 .13 Hyperbolic Repr ese ntation of Mean Stress versus Total Volumetric Strain Curve ............ . .... ........ ... ..... ............. . ................. . . . ......... ....... . . .... ... ..... ..... 78 3.14 Bestfit Transformed StressStrain Curves of Chattahoochee Sand .... ..... . ....... 78 3.15 Variation of Initial Tange nt Bulk Modulus Ki and Asymptotic Total Volumetric Strain (Ev)ult with Initial Relati ve Density for the Chattahoochee Sand ........................ . . .......... ..... ....... .... . ...... ....... .... . . ........... 79 3.16 Loading and Unloading Behavior of Monterey No 0 /30 Sand durin g Hydrostatic Compression Test ( modified from Goldstein 1988) ............. ........ 81 3 .17 The Mean Stress versus Total E lastic and Plastic Volum et ric Strain Curves for Tsukuba Sand at D r = 54% . . ...... . .......... . ........................ ........... .............. ...... 81 3.18 Variation of Tangent Bulk Modulus (KD, Shear Modulus (G), and Hardening Law Expo nent ( D ) with Depth ........... ...... . .... ..... ..... ..... .......... ............. ..... ....... 8 4 3 .19 Numerical Triaxial Compression Test of a Single Solid E lement for the Tsukuba Sand ... ................ . ........ .... ..... ................ . . ... .. .. ...... . ....... ........ ... ..... ...... ..... ..... 85 3 20 Variation of Shape Fact or R (Cap Surface Axis Ratio ) wit h Mean Stress ... . .... ....... .... ... ...... ... ........ ...................... .... .................... ........ 86 3 2 1 Bilinear StressStrain Curve of th e PlasticKinematic Model for Geo gri d Reinforcement . .... . . . . ........... ..... ............................. ...... .... . ....................... 8 7 3.22 Numerical Tensile Load Test ofa Single Shell E lement for the Reinforc e ment (dz = Pr escr ibed Displacement ) ........ ... . .... ..... . ..... .... .... . ........ ........... .......... 88 3.23 Loadstrain Relationship of P ET and PAV Geogrid R einfo rc e ment s . ..... ........ 88 3 24 Dimensions and Instrumentation o f Walls 1,2, and 3 (modified from Ling et a1. 2005a) ...... .................... . ...... . .... ... .......... . . ..... . ..... ...... ............. . ....... ... .. ........ 90 3.25 Finite E lement Mesh and Boundar y Condition of Wall s 1 2, and 3 .................. 92 3.26 Isometric View of the Finite E l eme nt Model Showing Various Part s .... ............ 93 3 2 7 Comparison betw een the Original R e cord (U ncorrected ) and the Ba se linecorrected Record . . .... ..... .... ....... .... ... . ......................... ........ . ........ .... . ..... ....... 94 XVll
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3.28 Loading Time Histories Applied to Models o f (a) Wall 1 (b) Wall 2 and (c) Wall 3 ....... . ........ .... ..... . .... ........... . . ...... ........ ......... . . ...... . . ............. . .......... 95 3 29 Contact Interfaces adopted in the Finite Element Model (e g., Wall 1 ) ............. 98 3.30 Detail showing Geogrid Thickness and Incompatible Element Boundary .... . .... ...... . ............ .... . . . ...... ...... ..... . ............ . ............ ...... . .... . . 98 3.31 Effect ofInterface ViscousDamping Coefficient (VDC) on Facing Response of Wall 1 due to Seismic Load ...... .. ............................................ ...... ...... .. ...... ...... 1 00 3 32 Effect of Global Mass Damping Coefficient on Walltop Response of (a) Wall 1 (b) Wall 2 and (c ) Wall 3 due to Transient Load .. ...................................... .101 3.33 Effect of Global Mass Damping Coefficient on Facing Response of Wall 1 due to Seismic Loads .......................... ........ ........... ................... .............................. 1 02 3.34 Effect of Friction Coefficient on Facing Response of Wall 1 due to Seismic Loads . . .... . . .... . . .... ..... . ............... . . . .... . .... . ..... ... ... ........ . . ...... .... . . . . 103 3.35 Strength ofSoilgeogrid Interface determined with Different Sizes o f Direct Shear Apparatus (modified from Ingold 1982) ................................ ........ ........ .1 04 3.36 Wall Face Peak Horizontal Displacement Comparison between the Calculated and the Measured data for (a) Wall 1, (b) Wall 2 and (c) Wall 3 .................... 106 3.37 Comparison of Wall 1 Face Displacement Time Histories for (a) First Shaking and (b) Second Shaking .......................................................................... .... ...... 1 07 3.38 Comparison of Wall 2 Face Displacement Time Histories for (a) First Shaking and (b) Second Shaking .............................................. .... ........................ .... ...... 1 08 3.39 Comparison of Wall 3 Face Displacement Time Histories for (a) First Shaking and (b) Second Shaking ........................ ............ ................................................ 1 09 3.40 Comparison of Backfill Surface Settlement for (a) Wall 1 (b) Wall 2 and (c) Wall 3 .............................................................................. ........ .......... .... .... ....... 110 3.41 Comparison of Lateral Earth Pressure behind Facing Blocks for (a) Wall 1 (b ) Wall 2, and (c) Wall 3 . . . . .... . . .... . . ...... .... ......... ..................... .... . ..... . ..... 112 XVlll
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3.42 Comparison of Bearing Pressure for (a) Wall 1 (b) Wall 2 and (c) Wall 3 .................... ........ ................. . ...... . . . ............ ...... ..... ...... . . ...... 113 3.43 Comparison of Geogrid Reinforcement Tensile Load for (a) Wall 1 (b) Wall 2 and (c) Wall 3 .... .......... ................ .............. ........................................................ 114 3.44 Comparison of Absolute Horizontal Acceleration in the Reinforced Soil Zone for (a) Wall 1 (b) Wall 2, and (c) Wall 3 .......................................... ............... 116 3.45 Comparison of Absolute Horizontal Acceleration in the Retained Soil Zone for (a) Wall 1 (b) Wall 2 and (c) Wall 3 .... .......... ................................................. 117 3.46 Total Energy Time Histories of the Three Numerical Models ............ ...... ...... 118 3.47 Cumulative Weight of Performance for indicating the Prediction Capability of LSDYNA .... ..... .... ..... . ........... .................. ....... ......... ............. .... ......... . ... . 120 3.48 Variability in the Measured and the Calculated Initial Lateral Earth Pressures ..... ... ... .... ................ ............. . ..... ......... . ...... .... ..... . ..... ........... 121 3.49 Variability in the Measured and the Calculated Initial Bearing Pressures ........................................................................................................... 121 3.50 Variability in the Measured and the Calculated Base Layer Reinforcement Initial Tensile Load ..................................... ...... ......... .... ................................... 122 3.51 Variability in the Measured and the Calculated Initial Maximum Reinforcement Tensile Load ......................................... .... ............ ........................................... 122 4.1 Acceleration Time Histories of the 20 Selected Earthquake Records .............. 129 4.2 Response Spectra of the 20 Selected Earthquake Records ............................... 133 4.3 Variation of Peak Vertical Acceleration with Peak Horizontal Acceleration for the 20 Selected Earthquake Records ................................ .... ............................. 137 4.4 Relation between Standard Penetration Resistance and Fricti on Angle for the Granular Soils Considered in Parametric Study ............................................... 138 XIX
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4.5 Determination of Dry Unit Weight and Soil Classification (modified from NAVFAC 1986a) . ..... ......... ...... .... . ... . . ..... ... ........................ ................ ...... 139 4 6 Variation of Hardening Law Exponent D with Relative Density Dr ............... .142 4.7 Variation of Tangent Bulk Modulus Kt with Relative Density D r . .... ...... ... ... 143 4.8 Variation of Shear Modulus G with Relative Density Dr ............ . ...... ....... .... 143 4.9 Variation of Hardening Law Coefficient W with Relative Density D r . ... ... . 144 4.10 Stressstrain Curves for = 32 Soil from Numerical Triaxial Tests .......... .... 145 4 .11 Stressstrain Curves for = 36 Soil from Numerical Triaxial Tests . . ..... . ... 145 4.12 Stressstrain Curves for = 40 Soil from Numerical Triaxial Tests . . .... . .... 146 4.13 Variation of (a) Tangent Bulk Modulus Kt and (b) Shear Modulus G with Depth .............. ..... ...... ......... ..... . . . .... .... . ... . .... . ...... ....... ... . 147 4 .14 Variation of Hardening Law Exponent D with Depth .... ...... . ... .... ..... ... ....... 148 4 .15 Comparison of Tensile Load Test Results between Idealized Geogrids and Typical Geogrids ........ . . .... .... .... . ..... .... . .... .......... . .... ...... . ........ . ........ . . .... 149 4.16 Numerical Model Dimensions Adopted in the Parametric Study ....... ...... ....... 150 4.17 Effect of Lateral Boundary Extent on Wall Displacement .... .... .... ....... ....... .... 151 4.18 Parametric Study Program ................................ ............... .............. .... . ..... . . . 153 4.19 Global Factor of Safety ofthe Baseline Model Configuration ... ......... ...... . . .154 4.20 Isometric View of 6 m High with 15 Wall Batter Finite Element Model ..... . ..... . . ..... ...... . .... . . .... . . .... . ... ...... .... ........ ... . .... . ..... . . 157 4.21 Wall Dimensions and Materials for Model ofH = 6 m Sy = 0.4 m and 0) = 10 ... . . .... . ......... ... . . ...... ...... ...... .... . ................... ...... ... ...... ..... . . 158 4.22 Finite Element Mesh for Model ofH = 6 m Sy = 0.4 m and 0) = 10 ............ .159 xx
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4.23 Wall Dimensions and Materials for Model o f H = 3 m S v = 0.4 m and (0 = 10 .... ............... . . ... .... ... .... ...... . ...... ..... ........ .... ...... . . .......... . . . . 160 4.24 Finite E lement Mesh for Model ofH = 3 m S v = 0.4 m and (0 = 10 .... . ..... 161 4.25 Wall Dimensions and Materials for Model o f H = 9 m S v = 0.4 m and (0 = 10 ............... .... ...... .. .. ... .......... ....... ... ... ..................... ....... ......... . ..... 162 4.26 Finite E lement M e sh for Model ofH = 9 m S v = 0.4 m and (0 = 10 ... . . ..... 163 4.27 Wall Dimensions and Materials for Model ofH = 6 m S v = 0.4 m and (0 = 5 . . ...... ..... . ...... . .......... . .................. ... . . . . ... ....... .......................... 164 4 .28 Finit e E lement M e sh for Model ofH = 6 m S v = 0.4 m and (0 = 5 . .... ...... .. 165 4.29 Wall Dimension s and Materials for Model o f H = 6 m S v = 0.4 m and (0 = 15 ........ ..... ........ .... . ..... ........ . .... ... ..... . ........ ...... ............... . . . ..... 166 4.30 Finite E lement Mesh for Model ofH = 6 m S v = 0.4 m and (0 = 15 . . . . . . .167 4.31 Wall Dimension s and Material s for Model o f H = 6 m Sv = 0.2 m and (0 = 10 . . .... .......... ... . .... . .......... . .... . . . . ........ . .... . ........ . ...... . ..... . 168 4 .32 F init e E l e ment Mesh for Mod e l of H = 6 m S v = 0 2 m and (0 = 10 .... . . ..... 169 4.3 3 Wall Dimensions and Materials for Model of H = 6 m S v = 0 6 m and (0 = 10 . .... . .............. . ....... . ... . . .... .............. ........ .... ..... ...... . .... . ..... . 170 4.34 Finit e E lement Mesh for Model ofH = 6 m S v = 0.6 m and (0 = 10 ........... .171 5.1 Vartiation of Ma x imum Horizontal Wall Dispalcement w ith Peak Hori z tonal Accel e ration b y the Effect o f Multidirection Shaking .... ...... ... . ...... .... . . ..... .17 3 5.2 Vartiation of M ax imum Wall Crest Settlement with Peak Horizton a l Acceleration with b y E ffect o f Multidirection Shakin g ......... ..... . .... ..... .... . 173 5 3 Varti a tion of Ma x imum Bearing Stress with Peak Hori z tonal Accel e ration by the Effect of Mult i direction Shaking ... . .... .......... ... .... . ........ ... ...... .... ......... 174 XXI
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5.4 Vartiation of Maximum Reinforcement Tensile Load with Peak Horiztonal Acceleration by the Effect of Multidirection Shaking . . . . .... . . .... . ........... . .174 5 5 Comparison of Percent Increase for the Effect of Multidirection Shaking ... . .17 5 5 6 Correlations of Maximum Wall Displacement with PHA and Other Spectral Accelerations .... ...... ..... . ........... ........ ...... ... ... .. ......... .... . .................. ............ 177 5.7 Effect of Wall Height on Maximum Wall Facing Horizontal Displacement. . .... ........ ... ........... . ......... . . . .... .... . ..... ...... ..... .... ........ ...... . . ... 180 5.8 Effect of Wall Height on Maximum Wall Crest Settlement.. ........ .... . .... . . . .180 5.9 Effect of Wall Height on Total Driving Resultant.. . . ..... . . . . ... . . .... . ....... ... 181 5.10 Effect of Wall Height on Total Overturning Moment Arm of Total Driving Resultant . .... . ..... ................................................................... . ...... .... . .......... 181 5.11 Effect of Wall Height on Maximum Bearing Stress ......... ...................... ..... . . 182 5.12 Effect of Wall Height on Maximum Reinforcement Tensile Load ... . ........ ..... 182 5.13 Effect of Wall Height on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass ... ... . . .... . ........ ...... . ........... .... . . ...... . . . ......... . . . ... 183 5 .14 Contours of XDisplacement at End of Analysis with Northridge Earthquake P0883 ORR090 (Model : H = 6 m Sy = 0.4 m (0 = 10, = 36 Ts% = 36 kN/m) ................ ...... ................... ............ ...... ......... ...... .............. ...... .......... 184 5.15 Contours of XDisplacement at End of Analysis with Northridge Earthquake P0935 TAR360 (Model: H = 9 m Sy = 0.4 m (0 = 10, = 36 Ts% = 36 kN/m) . .............. ...... ......................................... ..... ............ . .......... .............. ..... 185 5.16 Comparison of Centorid Horizontal Acceleration Time Histories for the Reinforced Soil Mass and the Retained Earth with Northridge Earthquake P0883 ORR090 (Model : H = 6 m (0 = 10, = 36 S y = 0.4 m Ts% = 36 kN/m) .......... ...... ..... ........ . .... . .... . . ...... ............ ....... ...... . ........ ..... ...... ............ 187 5.17 Effect of Wall Batter Angle on Maximum Wall Facing Horizontal Displacement. . .... ... . .... . .... . ...... ............. ...................... .... . . ... .................. 188 XXll
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5.18 Effect of Wall Batter Angle on Maximum Wall Crest Settlement.. ... ..... ..... .189 5 .19 Effect of Wall Batter Angle on Total Driving Resultant.. .............. .................. 189 5 20 Effect of Wall Batter Angle on Total Overturning Moment Arm of Total Driving Resultant .................. ............................................................................ 190 5 .21 Effect of Wall Batter Angle on Maximum Bearing Stress ............ ........ .......... .190 5.22 Effect of Wall Batter Angle on Maximum Reinforcement Tensile Load ... ..... 191 5.23 Effect of Wall Batter Angle on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass .................................................................... .......... 191 5.24 Effect of Soil Friction Angle on Maximum Wall Facing Horizontal Displacement. ............ .... . . . .... . .... . ...... ... . ....... . ..... ....... . . . .... . ..... ..... ..... 193 5.25 Effect of Soil Friction Angle on Maximum Wall Crest Settlement.. ................ 193 5.26 Effect of Soil Friction Angle on Total Driving Resultant.. .............................. .194 5.27 E ffect of Soil Friction Angle on Total Overturning Moment Arm of Total Driving Resultant ................. .................... ... .................................. ................... 194 5.28 Effect of Soil Friction Angle on Maximum Bearing Stress .............................. 195 5.29 Effect of Soil Friction Angle on Maximum Reinforcement Tensile Load ................................................................................................. ..... 195 5 30 Effect of Soil Friction Angle on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass .......... .. .......... .... .................................................... 196 5.31 Effect of Reinforcement Spacing on Maximum Wall Facing Horizontal Displacement. ...... ...... . ..... ...... ..... ...... ..... ............ .... . . ........ ....... ......... . .... . .... 197 5 32 Effect of Reinforcement Spacing on Maximum Wall Crest Settlement.. ......... 198 5 .33 Effect of Reinforcement Spacing on Total Driving Resultant.. ........ ...... ........ .. 198 5.34 Effect of Reinforcement Spacing on Total Overturning Moment Arm of Total Driving Resultant . ...... ... . ........ .... .......... . . ...... . ........ ..... .............. ..... .......... 199 XXlll
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5.35 Effect of Reinforcement Spacing on Maximum Bearing Stress .... . ..... .......... .199 5 36 Effect of Reinforcement Spacing on Maximum Reinforcement Tensile Load ........... ... ..... ....... . .... ........ .... . ..... .... ........ . ................ . . ...... ..... ... 200 5.37 Effect of Reinforcement Spacing on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass . . ............ . . . . . .... ............ ....... .......... 200 5.38 Effect of Reinforcement Stiffness on Maximum Wall Facing Horizontal Displacement. ... ...... . . ....... . ................ . . .... . ........ . ..... ........ ...... . ... ............... 202 5.3 9 Effect of Reinforcement Stiffness on Maximum Wall Crest Settlement.. ...... . 202 5.40 Effect of Reinforcement Stiffness on Total Driving Resultant.. .... . . .... . ....... . 203 5.41 Effect of Reinforcement Stiffness on Total Overturning Moment Arm of Total Driving Resultant ...... ....... .... . . .......... ............... . . ...... .... . .......... . .............. . 203 5.42 Effect of Reinforcement Stiffness on Maximum Bearing Stress ........... . . ..... . 204 5.43 Effect of Reinforcement Stiffness on Maximum Reinforcement Tensile Load ............ . ......... .... ........ ..... .... .... ...... ......... . ....... .......... . . .... . ....... 204 5.44 Effect of Reinforcement Stiffness on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass . ........ ...... ..... . . ... ....... ................. ...... .205 5.45 Fictitious Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load ...... . .... . ..... .207 5.46 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Static Condition (Model: H = 6 m Sy = 0.4 m co = 10, = 36, Ts% = 36 kN / m) ....... ...... ...... 208 5.47 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m Sy = 0.4 m co = 10, = 36 Ts% = 36 kN / m) ... ...... ... ........ 209 5.48 Reinforcement Tension Distribution : (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 3 m Sy = 0.4 m co = 10, = 36, Ts% = 36 kN / m) ............. ..... .210 XXIV
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5.49 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b ) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 9 m Sy = 0.4 m co = 10 = 36, Ts% = 36 kN / m) ................... .211 5.50 Reinforcement Tension Distribution : (a) Distribution along Individual Layer (b ) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m Sy = 0.4 m co = 5 = 36, Ts% = 36 kN/m) . . ........ ....... 212 5.51 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b ) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m Sy = 0.4 m co = 15, = 36, Ts% = 36 kN/m) ...... . . ........ 213 5.52 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b ) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m Sy = 0.4 m co = 10 = 32, Ts% = 36 kN/m) ...... . ....... ..... 214 5.53 Reinforcement Tension Distribution: (a) Distribution along Individu a l Layer (b ) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m S y = 0.4 m co = 10, = 40 Ts% = 36 kN /m) ..... ....... ...... .215 5.54 Reinforcement Tension Distribution : (a) Distribution along Individual Layer (b ) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model : H = 6 m Sy = 0 2 m co = 10 = 36, Ts% = 36 kN/m) ..... ........ ....... 216 5.55 Reinforcement Tension Distribution : (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m Sy = 0 6 m co = 10 = 36 Ts% = 36 kN / m) ................... .217 5.56 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b) Contours of Max imum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m Sy = 0.4 m co = 10 = 36, Ts% = 72 kN/ m) ................. . 218 5.57 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m Sy = 0.4 m co = 10, = 36, Ts% = 12 kN/m) ............... .... 219 5 58 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition with Coalinga earthquake P0346 HZ14000 PHA = 0 .2 82 g (Model : H = 6 m S y = 0.4 m co = 10 = 36 Ts% = 36 kN/ m) ..................... ...... ............. . .... .223 xxv
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5 59 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition with Northr idg e earthquake P1020 SPV270 PHA = 0.753 g (Model: H = 6 m S y = 0.4 m co = 10, = 36 Ts% = 36 kN/m) .............. .................................. .224 5.60 Static and Seismic Earth Pressure Distributions (Model : H = 6 m, co = 10, = 36 S y = 0 4 m Ts% = 36 kN/m) ...................................... ................................ 225 5.61 Magnitudes and Locations of Soil Thrusts and Reinforcement Resultants (Model : H = 6 m co = 10, = 36 Sy = 0.4 m Ts% = 36 kN/m) ................... 226 5.62 Earth Pressure Distributions for Static Thrust Inertia Force and Seismic Thrust Increment per FHW A Methodology ............................ ..................................... 227 5.63 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Arms of Thrusts and Resultants (Mode l : H = 6 m co = 10, = 36 S y = 0.4 m Ts% = 36 kN/m) ................................................ .. .. .. .. ...... .......... .......... 229 5.64 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Arms of Thrusts and Resultants (Model : H = 3 m co = 10, = 36, S y = 0.4 m Ts% = 36 kN/m) .................................... ...... .... .................................... 230 5.65 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Arms of Thrusts and Resultants (Model: H = 9 m co = 10, = 36 S y = 0.4 m Ts% = 36 kN/m) .................................................................................. 2 3 1 5.66 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Arms of Thrusts and Resultants (Model : H = 6 m, co = 5 = 36 S y = 0.4 m Ts% = 36 kN/m) .. .. .. .................... .... .................................................... 232 5.67 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Arms of Thrusts and Resultants (Model : H = 6 m co = 15, = 36 S y = 0.4 m Ts% = 36 kN/m) .................................................................................. 23 3 5 68 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Arms of Thrusts and Resultants (Model: H = 6 m co = 10, = 32 S y = 0 4 m T s% = 36 kN/m) .................................................................................. 234 XXV i
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5 69 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Arms of Thrusts and Resultants (Mode l : H = 6 m (J) = 10, = 40, S y = 0.4 m, Ts% = 36 kN/ m) . . . ... .. ... .................. ................................... .... ...... ... 235 5.70 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Arms of Thrusts and Resultants (Model: H = 6 m (J) = 10, = 36 S y = 0.2 m Ts% = 36 kN/ m) ...... ........... ............. .................. ..... ..... .... .... ........... 236 5 .71 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Arms of Thrusts and Resultants (Model: H = 6 m, (J) = 10, = 36 S y = 0 6 m Ts% = 36 kN/ m) .... .......... ...... . . ...................... ..... . .... . ..... .... . . . . . 237 5.72 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Arms of Thrusts and Resultants (Model: H = 6 m (J) = 10, = 36 S y = 0 4 m Ts% = 12 kN/ m) ....... . .... ........ . . . ......... ....... ........ .... .... .... ..... ........ 238 5.73 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Arms of Thrusts and Resultants (Model : H = 6 m (J) = 10, = 36 S y = 0.4 m Ts% = 72 kN/ m) . .... . . .... . . ...... ........... . ..... .... . ........ . .... . . ... ......... 239 6.1 Scatter Diagrams with Various Degrees of Correlation between Two Variables (modified from Kachigan 1991) ..... . ............ ..... . ...... ...... ................ ............ .242 6.2 Comparison between Predicted and the FEM Results with the Effect of Wall Height H ............ . ...... ............... .......... . .... ......................... ............ . ..... . ..... 245 6.3 Comparison between Predicted and the FEM Results with the Effect of Wall Batter Angle (J) .. .. . ........ ... . ..... .... 246 6.4 Comparison between Predicted and the FEM Results with the Effect of Soil Friction Angle .... ... . . .... . .... . .... ... ... .... ... .. ..... ... ... . .......... .... ......... ........... 246 6.5 Comparison between Predicted and the FEM Results with the E ffect of Reinforcement Spacing Sy . .... . ....... ......... ............ . . ...... .... ......... ..... ...... . . ... .247 6.6 Comparison between Predicted and the FEM Results with the E ffect of Reinforcement Stiffness Ts% ...... ... .... ..... . .... ......... ..... . .... . ......... .... . ..... . ... .247 6.7 Comparison between Predicted and the FEM Results with the Effect of Peak Vertical Acceleration PV A ..................... .... . .............. ........ ..... ........ ........ . .... 248 XXVll
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6.8 Comparison between Predicted and the FEM Results with the Effect of Wall Height H .......... ...... .. ...... .. .. .... .. .... ..... .. .. .... ........... ... ... .... ....... ...... .. .... .. ...... .. 249 6.9 Comparison between Predicted and the FEM Results with the Effect of Wall Batter Angle co ............ .. .. .... .... ....... .................... .......... ... ..... ... .. .. .... ........ .. ...... 250 6.10 Comparison between Predicted and the FEM Results with the Effect of Soil Friction Angle ............ .. .... .. ..... .. .. .... ... .. ........... ........ ............ .. ...... ..... .. .. .. .. .. 250 6 .11 Comparison between Predicted and the FEM Results with the Effect of Reinforcement Spacing Sy ............. ...... .. ..... .. .... ..... ............................ ............ .251 6.12 Comparison between Predicted and the FEM Results with the Effect of Reinforcement Stiffness Ts% ............................................. .. .... .. .. .... .. ............ .251 6 .13 Comparison between Predicted and the FEM Results with the Effect of Peak Vertical Acceleration PV A ................................... .. ........ .. ....... .. .............. .. .. .. .. 252 6.14 Linear Prediction Equat ion for given ........ . ........ .............................. .253 6 .15 Comparison between Predicted LP D E and the FEM Results with the E ffect of Wall Height H .... ..... .......... .... .. .... ..... ...... .............. ................. ................ ...... 254 6.16 Comparison between Predicted LP D E and the FEM Results with the Effect of Wall Batter Angle co ....... .. .............. ............... ...... ...... .. ..... ............................ ... 254 6 .17 Comparison between Predicted LPDE and the FEM Results with the Effect of Soil Friction Angle ......... .... ................. .... .. ........ ..... .. ....... ....... .. .. ....... ....... ... .255 6.18 Comparison between Predicted LP D E and the FEM Results with the Effect of Reinforcement Spacing Sy .. .. .... ........... ...... ............... ..... .. .. .... .. .. ...... .............. 255 6.19 Comparison between Predicted LP D E and the FEM Results with the Effect of Reinforcement Stiffness Ts% ........... .. ..... ..... . ..... .. .... ....................................... .256 6.20 Comparison between Predicted LP D E and the FEM Results with the Effect of Peak Vertical Acceleration PV A ... ...... ...... ............... .... .... . .... .... .......... . ...... .256 xxv III
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6.21 Normalized Total Overturning Moment Arm with Wall Height versus Peak Horizontal Acceleration ................... .. .. .......................... .. .... ...... ...... .............. .258 6.22 Comparison between Predicted qvE and the FEM Results with the Effect of Wall Height H ................ ....... .......... ........ ..... .. .. .......... ....... . ...... ............ ........ ..... .... 259 6.23 Comparison between Predicted qvE and the FEM Results with the Effect of Wall Batter Angle ill .......... . . ........... ........ . . ................ . . ............... . . ............... . . . 259 6.24 Comparison between Predicted qvE and the FEM Results with the Effect of Soil Friction Angle ...... .. ......... ... ......... .. .................... .. .................... ...... .. ........... 260 6.25 Comparison between Predicted qvE and the FEM Results with the Effect of Reinforcement Spacing Sv .... .. .. .. ........... ........ ..... ......... ......... .... .. .. ..... ........... .260 6.26 Comparison between Predicted qvE and the FEM Results with the Effect of Reinforcement Stiffness T 5% . ......................................... ...... ........ . .... . ...... .... 261 6 27 Comparison between Predicted qvE and the FEM Results with the Effect of Peak Vertical Acceleration PV A ....... .. ................... .. ..................... ...... ................. .261 6.28 Comparison between Predicted Ttotal and the FEM Results with the Effec t of Wall Height H .... .. ...... ......... .. .......... ...................... ........ .................... .......... 263 6.29 Comparison between Predicted Tto t a l and the FEM Results with the Effec t of Wall Batter Angle ill ................................................................. ..................... . 263 6.30 Comparison between Predicted Tto t a l and the FEM Results with the Effect of Soil Friction Angle .... ......... ... ........ .. .. .................. .. .. .......... .. ................ ....... .264 6.31 Comparison between Predicted Ttotal and the FEM Results with the Effect of Reinforcement Spacing Sv .... .. .. .. .. .. ...... .... ....... .... ..... .. ..... ....... ......................... 264 6.32 Comparison between Predicted Ttotal and the FEM Results with the Effec t of Reinforcement Stiffness T 5 % ............. ................. ....... ..................................... .265 6.33 Comparison between Predicted Ttotal and the FEM Results with the Effect of Peak Vertical Acceleration PV A .............. ................ ................ ....... ............... .265 6.34 Comparison between Predicted Am and the FEM Results with the Effect of Wall Height H .. .... .... ............ .. ............. .. .. .... ....... .. ..... .. .. ..... .. ..... .... .. .... ....... .... .. ... ... .. 267 XXIX
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6.35 Comparison between Predicted Am and the FEM Results with the Effect of Wall Batter Angle co .... . . .......... ..... .... . ....................... ......... ................................. 267 6 36 Comparison between Predicted Am and the FEM Results with the Effect of Soil Friction Angle . . . .............. ...... ..... ... .... ......... ............ ...... . .... . ...... ...... . ... 268 6.37 Comparison between Predicted Am and the FEM Results with the Effect of Reinforcement Spacing Sv ..... ............ .................. .......... .............. . .... ....... ..... 268 6.38 Comparison between Predicted Am and the FEM Results with the Effe ct of Reinforcement Stiffness T 5% ......... ....... ..... .............. . ........... . ................... . .269 6.39 Comparison between Predicted Am and the FEM Results with the Effect of Peak Vertical Acceleration PV A . .......... .... . ..... ...... . . ............................ . .... . ..... . .269 6.40 Variation of Maximum Reinforcement Tensile Load with Maximum Horizontal Displacement for the Baseline ModeL .... ....... . . ..... ...... ...... . . .... . .... . ..... . .273 Al Extended Masing Rules : (a) Variation of Shear Stress with Time; (b) Resulting StressStrain Behavior with Backbone Curve indicated by Dashed Line (after Kramer 1996) ... .................. . .... . .............. .... ..... . . . .... . . .... . .... . .... . . . . . 281 A2 Definitions of (a) Secant Shear Modulus and (b) Backbone Curve (after Kramer 1996) . .... . . . ................... . . . .... . . . ........ . . .... . ..... . .... . .... . . ..... ....... . . . . . 285 A3 Modulus Reduction and Damping Ratio Curves of Average Sand (after Seed at al. 1986) .... ... ........ ..... . ...... . ....... ... . . ............ . ...... ....... ...... ... . .... . .... ......... 286 A.4 Example of a Best Fit Straight Line for Determining Parameters a and r (after Ueng and Chen 1992) .......... . ...... . .......... ..................... . ............ ....... ....... . 287 B.1 Effect of Wall Height on Global Factor of Safety .... . .... ...... . . . .... . .... . ..... . . 288 B.2 Effect of Wall Batter Angle on Global Factor of Safety . . .... . . .... . .... . ..... . .289 B.3 Effect of Soil Friction Angle on Global Factor of Safety ...... . . ...... .... ... ...... .290 B.4 Effect of Reinforcement Spacing on Global Factor of Safety ... .... . ...... . ....... 291 xxx
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D.1 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m ro = 10 = 36 S y = 0.4 m T5% = 36 kN / m) ... ...... ... .................... ............... .. .. .. ..... ...... .. 301 D.2 Maximum Wall Facing Horizontal Displacement Profiles (H = 3 m, ro = 10 = 36 S y = 0.4 m T5% = 36 kN / m) .......... .... .... ........ .. .. ............. ...... .... ............ 302 D.3 Maximum Wall Facing Horizontal Displacement Profiles (H = 9 m ro = 10, = 36 Sy = 0.4 m T5% = 36 kN/m) .... ...... .... ................................................... 303 D.4 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m ro = 5 = 36 S y = 0.4 m T s% = 36 kN/m) ..................................... .......... .............. .... ..... 304 D.5 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m ro = 15, = 36 Sy = 0.4 m T5% = 36 kN / m) ...... .................................... .......... ............... 305 D.6 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m ro = 10 = 32 S y = 0.4 m T5% = 36 kN / m) ................................................................... 306 D.7 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m ro = 10, = 40 Sy = 0.4 m T5% = 36 kN / m) .................. .... ..... ...................................... 307 D.8 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m ro = 10 = 36 S y = 0.2 m T5% = 36 kN / m) ................................................................... 308 D 9 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m ro = 10 = 36 Sy = 0.6 m T5% = 36 kN / m) .. ..................... .... ........... ............................. 309 D.10 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m ro = 10 = 36 S y = 0 4 m T5% = 12 kN / m) ................................................................. .. 310 D.11 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m, ro = 10 = 36 Sy = 0 4 m T5% = 72 kN/m) .................................... ..................... ........ 311 E.1 Maximum Wall Crest Settlement Profiles (H = 6 m, ro = 10, = 36 Sy = 0.4 m T5% = 36 kN/m) .................................... .... ........................ ............................ 312 E.2 Maximum Wall Crest Settlement Profiles (H = 3 m, ro = 10 = 36 Sy = 0.4 m, T5% = 36 kN/m) .................................... .. ... .... .................... .... ...... ........ ....... 313 XXXI
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E.3 Maximum Wall Crest Settlement Profiles ( H = 9 m co = 10 = 36 S y = 0.4 m T s%=36kN/m) . ...... ... ............. .......... .... ... .......... ...... ... ... ...... ... . . ... . 314 E .4 Maximum Wall Crest Settlement Profiles (H = 6 m co = 5 = 36 S y = 0.4 m Ts% = 3 6 kN/ m ) ......... . . . .... .... ....... ...... .... . . . . .... . . .... . . . . . ... ...... . . .... 315 E.5 Maximum Wall Cres t Settlement Profiles ( H = 6 m co = 15, = 36 S y = 0.4 m Ts% = 36 kN/ m ) ...... ... ... . .... . ...... ... ...... . ........ . ...... ... .... ..... . ... . ...... ......... 316 E .6 Maximum Wall Cres t Settlement Profiles ( H = 6 m co = 10, = 3 2, S y = 0.4 m T s%=36kN/ m ) ... ... .... . ... .... . ...... . . ........ .... . ... . ...... . .... ... ... . . .... . . ... . 317 E .7 Maximum Wall Crest Settlement Profiles (H = 6 m co = 10 = 40 S y = 0.4 m Ts% = 36 kN/ m ) . . ....... . ...... ...... .......... .... ... . ...... ... ..... .... . . . .... . . . . ..... 318 E.8 Maximum Wall Crest Settlement Profiles (H = 6 m co = 10, = 36 S y = 0 .2 m Ts% = 36 kN/ m ) ..... ..... ............. ...... ............. ..... ...... . .... . . . . . . .... . . . .... 319 E. 9 Maximum Wall Cre st Settlem e nt Profiles ( H = 6 m co = 10 = 3 6 S y = 0.6 m Ts% = 36 kN/ m ) ............... ...... ... .... ... . .... . . .... . . ... .... ... . . .... ... ... . ...... . . ... . 3 2 0 E .IO Maximum Wall Cres t Settlement Profiles (H = 6 m co = 10, = 36 S y = 0.4 m Ts% = 12 kN/ m ) ............................... . .... .... ... . . ... .... ... ........ . . ... ...... ... ... . 3 2 1 E.11 Maximum Wall Cre st Settlement Profiles (H = 6 m co = 10 = 3 6 S y = 0.4 m Ts% = 72 kN/ m ) ... . . ... .... . . ...... ...... . .... . . . .... . . . ...... ... .... .............. . ... . 3 22 F.l Maximum Lateral E arth Stress Distributions ( H = 6 m co = 10 = 3 6 S y = 0.4 m Ts% = 36 kN/m) ... ..... .... . ...... ... . ...... ...... . . ... ...... . . .......... .......... ... . 3 2 4 F .2 Maximum Lateral E arth Stre s s Distributions (H = 3 m co = 10, = 3 6 Sy = 0 4 m Ts% = 36 kN/m) ... ... ...... .... . . ... .... ...... ..... ........... ... . . . ........ . . . ... 3 2 5 F 3 Maximum Lateral E arth Stress Distributions (H = 9 m co = 10, = 3 6 S y = 0.4 m Ts% = 36 kN/m) .... ... ... .... ... .... ... . ...... ...... . . ...... . . ...... ... . . ... ...... . ... ... . 3 2 6 F .4 Maximum Lateral E arth Stres s Distributions ( H = 6 m co = 5 = 3 6 S y = 0.4 m Ts% = 36 kN/ m ) ... ...... ... ...... ...... ... .... . ...... . ...... . .... . . ... . . ...... ..... ....... 3 2 7 xxx 11
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F.5 Maximum Lateral Earth Stress Distribution s (H = 6 m ro = 15, = 36 Sy = 0 4 m Ts% = 36 kN/m) . . ... . .... . ... .... .... ........... . . . .... . . .... . . . .... ....... .. ... . . 328 F.6 Maximum Lateral Ea rth Stress Distribution s (H = 6 m, ro = 10, = 32 Sy = 0.4 m, Ts% = 36 kN/m) .... . . ... .... ........ . .... .... . . .............. .... ..... .... ...... ... .... 329 F 7 Maximum Lateral Earth Stress Distributions (H = 6 m ro = 10 = 40 Sy = 0.4 m Ts% = 36 kN/m) . . . . . . .... . . .... ... .... . ...... ... . . ...... . .... ..... ...... ...... .... 330 F.8 Max imum Lateral Eart h Stress Distributions (H = 6 m ro = 10, = 36 Sy = 0.2 m Ts% = 36 kN/m) . ....... . . .... . . ... .......... ... ................. ... . . ... . . .... . . ..... . 331 F.9 Ma ximum La t eral Earth Stress Distributions (H = 6 m ro = 10 = 36 Sy = 0 6 m Ts% = 36 kN/m) . . . . ..... ...... ..... .... . .... .............. . . ..... ........ ........ ..... 332 F I0 Maximum Lateral Eart h Stress Distributions (H = 6 m ro = 10 = 36 Sy = 0 .4 m Ts% = 1 2 kN/m) ... . ... ... . ......... . . .... . ... . . . ... ....... . ........ ... . .... ... ... ... 333 F.ll Max imum Lateral Eart h Stress Distribution s (H = 6 m ro = 10, = 36 Sy = 0 4 m Ts% = 7 2 kN/m) .............................................. . . .... . ............ ... . . ... ....... 334 G.l Maximum Bearin g Stress Di stributio ns (H = 6 m ro = 10, = 36 Sy = 0.4 m Ts% = 36 kN/m) ... ... . ................ ...... ... ...... ...... . ....... . ... ... .... ...... . . ... ... ... . 335 G.2 Maximum Bear ing Stress Di strib utions (H = 3 m, ro = 10 = 36 S y = 0.4 m Ts% = 36 kN/m) ... ... . ...... ....... .... ... . . ...... . .... . . ...... .... ... . . . . ... ...... . . . . 336 G 3 Maximum Bearin g Stress Dist rib utions (H = 9 m ro = 10, = 36 Sy = 0.4 m, Ts% = 36 kN/m) .... . . . . . . . . . .... ..... ....... . .... .......... . ....... . . . .... . . .... ... ......... 337 G.4 Maximum Bearin g Stress Di stributio ns (H = 6 m ro = 5 = 36 S y = 0.4 m Ts% = 36 kN/m) . .... . . ... ... ... . ...... . ...... . .... . ... .... ... ... .... ... .......... . .... . . . ... . 338 G.5 Maximum Bear ing Stress Distributions (H = 6 m ro = 15 = 36 Sy = 0.4 m Ts% = 36 kN/m) . .... . . . . . . ..... .... .......... .... . .... ... . ...... ... ..... .. ... . . ...... . . ...... 339 G.6 Maximum Bearin g Stress Distributions (H = 6 m ro = 10 = 32 Sy = 0.4 m Ts% = 36 kN/m) ... ... .... ....... ... ...... . . ....... . ...... ... ...... . ...... ............... ... . . ..... 340 XXXlll
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G.7 Maximum Bearing Stress Distributions (H = 6 m co = 10 = 40, S y = 0.4 m Ts% = 36 kN/m) .......... ....................................................................................... 341 G.8 Maximum Bearing Stress Distributions (H = 6 m co = 10 = 36 S y = 0.2 m Ts% = 36 kN/m) .... . . . . .......... . .... ... ... . . . .... ... . . ... .. ... ...... . ... .... . ...... ........ 342 G.9 Maximum Bearing Stress Distributions (H = 6 m co = 10 = 36 S y = 0.6 m Ts% = 36 kN/m) ................................................................................................. 34 3 G.lO Maximum Bearing Stress Distributions (H = 6 m co = 10 = 36 S y = 0.4 m Ts% = 12 kN/m) ................................................................................................. 344 G.ll Maximum Bearing Stress Distributions (H = 6 m co = 10 = 36, S y = 0.4 m Ts% = 72 kN/m) .... .. ... ....................................................................... ................. 345 H.l Maximum Reinforcement Tensile Load Profiles (H = 6 m co = 10 = 36 S y = 0.4 m T s% = 36 kN/m) .................................................................. ................ 347 H.2 Maximum Reinforcement Tensile Load Profiles (H = 3 m co = 10 = 36 S y = 0 4 m T s% = 36 kN/m) .................... .... .......... ................................................ 348 H.3 Maximum Reinforcement Tensile Load Profiles (H = 9 m co = 10 = 36 S y = 0.4 m T s% = 36 kN/m) ...................................................... ............................ 349 H.4 Maximum Reinforcement Tensile Load Profiles (H = 6 m co = 5 = 36 S y = 0 4 m Ts% = 36 kN/m) ................ .. .. .............. ....................................... ............. 350 H.5 Maximum Reinforcement Tensile Load Profiles (H = 6 m co = 15, = 36 S y = 0 4 m Ts% = 36 kN/m) ................................................ .................................. 351 H 6 Maximum Reinforcement Tensile Load Profiles (H = 6 m co = 10 = 32 S y = 0 4 m Ts% = 36 kN/m) ...... ...... .................................................... .................. 352 H .7 Maximum Reinforcement Tensile Load Profiles (H = 6 m co = 10 = 40, S y = 0.4 m Ts% = 36 kN/m) ............ ...... .... ...................... ...................................... 353 H 8 Maximum Reinforcement Tensile Load Profiles (H = 6 m co = 10 = 36 S y = 0.2 m Ts% = 36 kN/m) .......................................... ............................ ...... .. .. .. 354 X XXIV
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H.9 Maximum Reinforcement Tensile Load Profiles (H = 6 m, (j) = 10 = 36 Sy = 0.6 m Ts% = 36 kN/m) ........ ...................... .................................. ..... ...... ... 355 H.I0 Maximum Reinforcement Tensile Load Profiles (H = 6 m (j) = 10, = 36 Sy = 0 4 m Ts% = 12 kN / m) . ....... .............. ...... ..... . . .......... .... ..... . .... ............... 356 H.l1 Maximum Reinforcement Tensile Load Profiles (H = 6 m (j) = 10, = 36 S y = 0.4 m Ts% = 72 kN/ m) . . .... . .... . ..... ...... ......... . . .... ..... .... ......... .............. . 357 1.1 Effect of Wall Height on Maximum Wall Facing Horizontal Displacement (Correlated with PV A) .... . . ...... ... ........ . . ...... ..... . . ... ............. . . .... . ...... . ..... 359 1.2 Effect of Wall Height on Maximum Wall Crest Settlement (Correlated with PV A) ...... ........... ........ . ..... .... ... . . . . . .... .......... ....... ............. 359 1.3 Effect of Wall Height on Total Driving Resultant (Correlated with PV A) ....... .... . .......... .... ...... . ............ ................ ...... .......... . 360 1.4 Effect of Wall Height on Total Overturning Moment Arm of Total Driving Resultant (Correlated with PVA) ............. .... .......................................... . . .... 360 1.5 Effect of Wall Height on Maximum Bearing Stress (Correlated with PV A) ... ........ ...... .............. ........ . ............. ............ ...... ........ 361 1.6 Effect of Wall Height on Maximum Reinforcement Tensile Load (Correlated with PV A) .... ... . .... . . . . . ... ........... ................ ..... ..... ..... .... ...... . ...... ......... .... 361 I. 7 Effect of Wall Height on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass (Correlated with PYA) ................ ........................ ......... 362 1.8 Effect of Wall Batter Angle on Maximum Wall Facing Horizontal Displacement (Correlated with PV A) ..... ....... ............... ....... ...... . ............ . ....... .................... 363 1.9 Effect of Wall Batter Angle on Maximum Wall Crest Settlement (Correlated with PV A) ....... .......... ............ . .......... ....... ......... ..... ... .. ...... ... ..... .... . ....... ..... 363 UO Effect of Wall Batter Angle on Total Driving Resultant (Correlated with PV A) ................................. .... . . . . . ............... . . .... . ............ 364 1.11 Effect of Wall Batter Angle on Total Overturning Moment Arm of Total Drive Resultant (Correlated with PV A) ............. ..... ...... .......... ......................... ......... 364 xxxv
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1.12 Effect of Wall Batter Angle on Maximum Bearing Stress (Correlated with PVA) ....... . . .... . ........ ...... .... .............. . .............. ...... ..... . . 365 1.13 Effect of Wall Batter Angle on Maximum Reinforcement Tensile Load (Correlated with PV A) ....... . .... ..... .... . . .... ....... ..... . ................ .... . . ........ . . . 365 1.14 Effect of Wall Batter Angle on Maximum Horizontal Acceleration at Centroid of the R einf orced Soil Mass (Correlated with PYA) ..... ...... . . ...... ...... ..... . ... 366 1.15 Effec t of Soil Friction Angle on Maximum Wall Facing Horizontal Displacement (Correlated with PV A) . . . . .... ...... ......... ...... ....................... .... 367 1.16 Effect of Soil Friction Angle on Maximum Wall Crest Settlement (Corre lated with PV A) ......... .... . ... .......... . .... ...... ............... ..... . ........ ..... ...... . . .... ..... ..... 367 1.17 Effect of Soil Friction Angle on Total Driving Resultant (Correlated with PVA) ......... . .... ........ ........ ...... ................ . .... .... .......... ........ .368 1.18 Effect of Soil Friction Angle on Total Overturning Moment Arm of Tota l Driv e Resultant (Correlated with PVA ) . .... . ....... .... . ..... . . ...... ........ ..... . ..... .......... . 368 1.19 Effect of Soil Friction Angle on Maximum Bearing Stress (Correlated with PYA) .... ....... . ...... . .............. ...... ...... ...... . ...... . . . . ............... . . . ........ . . ... 369 1.20 Effe ct of Soil Friction Angle on Maximum Reinforcement Tensile Load (Correlated with PV A) ......... .... . . .... . ...... . .... . ..... . ............ .......... ........... . . . 369 1.21 Effect of Soil Friction Angle on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass (Correlated with PV A) ................... . . ...... . ......... 3 70 1.22 Effect of Reinforcement Spacing on Maximum Wall Facing Horizontal Displacement (Correlated with PVA) . ........... . . .......... ...... . . .......... ....... ..... 371 1.23 Effect of Reinforcement Spacing on Maximum Wall Crest Settlement (Correlated with PV A) ............... .... . ...... ........ ........ ........... ..... ........... . .... ... ... 371 1.24 Effect of Reinforcement Spacing on Total Dri v ing Resultant (Correlated with PVA ) ...... ..... . ... ...... . ........... ............. . . ............... . .......... ...... . . . ............ .... 372 XXXVI
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1.25 Effect of Reinforcement Spacing on Total Overturning Moment Arm of Total Drive Resultant (Corre lated with PV A) ..... ..... .... ..... ........ ........... ..... .... ....... .. .... 372 1.26 Effect o f Reinforcement Spacing on Maximum Bearing Stress (Correlated with PYA) ... ............... ...... ........ ..... ............... ...... ..... .. .. ...... ...... . ..... .. ...... .... . .. ...... 373 1.27 Effect of Reinforcement Spacing on Maximum Reinforcement Tensile Load (Correlated with PVA) ................................................................................ ...... 373 1.28 Effect of Reinforcement Spacing on Maximum Horizontal Acceleration at Centroid ofthe Reinforced Soil Mass (Correlated with PYA) ......................... 374 / 1.29 Effect of Reinforcement Stiffness on Maximum Wall Facing Horizontal Displacement (Correlated with PV A) .................................. .... ......................... 375 1.30 Effect of Reinforcement Stiffness on Maximum Wall Crest Settlement (Correlated with PYA) ... ....... .... ................. ........ .. .. .. .. .... .......... ...... .. .... ........ 375 1.31 Effect of Reinforcement Stiffness on Total Driving Resultant (Correlated with PYA) .... ..... ........ .... ...... ........... .... ... .. .. .. .... ...... ...... ..... .......... .. ............. .. ..... .. .... 376 1.32 Effect of Reinforcement Stiffness on Total Overturning Moment Arm of Total Drive Resultant (Correlated with PVA) .......... ........ ................ ...... .... ................ 376 1.33 Effect of Reinforcement Stiffness on Maximum Bearing Stress (Correlated with PV A) .... ....................... .............. ..... .................................................................. 377 1.34 Effect o f Reinforcement Stiffness on Maximum Reinforcement Tensile Load (Correlated with PV A) .... ..... ...... ................ ...................................................... 377 1.35 Effect of Reinforcement Stiffness on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass (Correlated with PV A) .... .... .......... .. .. .. .3 78 1 1 Effect of Wall Height on Maximum Wall Facing Horizontal Displacement (for PHA 0.29 g) ....... ............ ........ ............ .... ..... ............................... ................... 381 1.2 Effect of Wall Height on Maximum Wall Crest Settlement (for PHA 0.29 g) ..................... .... ............................................ ....................... 381 1.3 Effect o f Wall Height on Total Driving Resultant (for PHA 0.29 g) ............ 382 XXXVll
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1.4 Effect of Wall Height on Maximum Bearing Stress (for PHA 0 29 g) . ....... 382 1.5 Effect of Wall Height on Maximum Reinforcement Tensile Load (for PHA 0.29 g) .... ...... . . .... . . ....... ............ . ................. . ................... ... ............... ....... 383 1.6 Effect of Wall Height on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass (for PHA 0.29 g) ............... ........ .... . . ............ ........ . 383 1 7 Effect of Wall Batter Angle on Maximum Wall Facing Horizontal Displacement (for PHA 0.29 g) ........... . . . ...... ....... ........ ...... . ..... ...... ................ ........ ......... 384 1 8 Effect of Wall Batter Angle on Maximum Wall Crest Settlement (for PHA 0.29 g) ..... . ............ .... . ......... . ........... .... ...... ............ . . .... ....... ............. ........... 384 1.9 Effect of Wall Batter Angle on Total Driving Resultant (for PHA 0.29 g) ....... ... ..... . .... . . ..... ........ ...... . . .......... .................. ...... .... .... 385 1.10 Effect of Wall Batter Angle on Maximum Bearing Stress (for PHA 0.29 g) ...... ... ..... . ........ ..... ...... ........ . .......... . ...... .......... .............. . 385 1.11 Effect of Wall Batter Angle on Maximum Reinforcement Tensile Load (for PHA 0 29 g) ....... ....... ... .... .................................... ...... .... ........................... . 386 1 .12 Effect of Wall Batter Angle on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass (for PHA 0.29 g) . ..... .... . .... . . ........... .......... 386 1.13 Effect of Soil Friction Angle on Maximum Wall Facing Horizontal Displacement (for PHA 0 29 g) . ................. . . . ........ .......... . .... . . .... . .... ... 387 1.14 Effect of Soil Friction Angle on Maximum Wall Crest Settlement (for PHA 0 29 g) .... . .... ......... .... . . ..... . ......... ..... ........... .... . ........... ............................ . . 387 1.15 Effect of Soil Friction Angle on Total Driving Resultant (for PHA 0 .29 g) ... . ..... ....... ............ ........... . ............... ..... ...... . .... ...... . . . 388 1.16 Effect of Soil Friction Angle on Maximum Bearing Stress (for PHA 0.29 g) . ......... .... ............ . ..... ................................... ...... ...... . ... 388 1.17 Effect of Soil Friction Angle on Maximum Reinforcement Tensile Load (for PHA 0 29 g) ........ . .............. ... . .... . .... . ........ ............ .... . . ..... ...... ............. 389 XXXVlll
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1.18 Effect of Soil Friction Angle on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass (for PHA 0 29 g) .............................................. 389 1.19 Effect of Reinforcement Spacing on Maximum Wall Facing Horizontal Displacement (for PHA 0.29 g) .................................................................... 390 1 20 Effect of Reinforcement Spacing on Maximum Wall Crest Settlement (for PHA 0 29 g ) ............................................................................................................ 390 1.21 Effect of Reinforcement Spacing on Total Driving Resultant (for PHA 0.29 g ) ... ................................................................................ ........ 391 1.22 Effect of Reinforcement Spacing on Maximum Bearing Stre s s (for PHA 0.29 g) ...................................................................................... .. .. .. 391 1.2 3 Effect of Reinforcement Spacing on Maximum Reinforcement Tensile Load (for PHA 0.29 g ) ..... ......................................... .... ............ .................... ........ 39 2 1 24 Effect of Reinforcement Spacing on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass (for PHA 0.29 g) .............................. .392 1.25 Effect of Reinforcement Stiffness on Maximum Wall Facing Horizontal Displacement (for PHA 0.29 g) .. .. .. ........ ........ .. ............................................. 393 1.26 Effect of Reinforcement Stiffness on Maximum Wall Crest Settlement (for PHA 0.29 g) ..................................................................... ....................................... 393 1.27 Effect of Reinforcement Stiffness on Total Driving Resultant (for PHA 0.29 g) ......... ...... .................................. ........... .... ............................ 394 1.28 Effect of Reinforcement Stiffness on Maximum Bearing Stress (for PHA 0 29 g) ............................................. .... .... .................. .... ................. 394 1.2 9 Effect of Reinforcement Stiffness on Maximum Reinforcement Tensile Load (for PHA 0 29 g) ... .............. ........ ............... ................................. .... ......... .... 395 1.30 Effect of Reinforcement Stiffness on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass (for PHA 0 29 g) ........................ .. ..... 395 XXXIX
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L.l Variation of Natural Period with Wall Height and Soil Friction Angle ..... ... .403 x l
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LIST OF TABL E S Table 2.1 Summary of shaking table model tests ...... ..................... ........ . ....................... 21 2 2 Summary of centrifuge model tests .... . .... . ...... . ......... ......... ......... ...... . . ......... 24 2.3 Summary of computer program validation on seismic performance of reinforced soil wall ......... ........ .... . ...... ........... . . . . ...... .... ..... ........... .......... . 25 2.4 Comparison of recommended minimum factors of safety for GRS walls ... ....... ......................... . .... ........ . ...... . ......... .......... ...... ...... ......... ... 46 3 1 Summary of model parameters for facing block EPS board and geogrid reinforcements . . . ........ ......... . . .... . .... . .... . . .... . ......................... . .... ..... ....... 71 3.2 Cap model parameters for Tsukuba sand . .... .... . . ....... .... . ...... .......... . . ..... . . 71 3.3 Physical properties of Tsukuba and Chattahoochee sands (data from Ling et al. 2005a and AlHussaini 1973) .... ...... . ............ ....... ....... ............. ... ... .... ..... . ..... 77 3.4 Summary of parameters for determining mean stress dependent variables D Kt, and G at Dr = 54 % ........ ....................... .......... .............. .... . . .......... . ...... . . .... . 83 3.5 Effect of hardening parameters on CTC stressstrain relationship . . .... ..... ..... . 86 3.6 Summary of shaking table input motions for numerical test walls . ...... ....... .... 96 3.7 Summary of LSDYNA contact interfaces defined in the numerical model. ..... .............................. . .... . ........... . . .... . .... . .................... . . 98 3.8 Summary of cal i brated parameters ......... ............ ........ .... . .... . ..... . . .............. 104 3 9 Prediction quality classes (after Morgenstern 2000) . ........ .......... . . ..... .......... 119 3.10 Quality of numerical prediction (% error) .... . .... . ..... ......... ..... . ..... ...... . . ..... . 119 4.1 Ground motion parameters of the 2 0 earthquake records .... . ......... . ......... ..... 1 2 7 xli
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4 .2 Spectral parameters from response spectrum at 5% damping of the 20 earthquake records .... ................ ... ...... ...... ...... . . . .... . .... . .......... . .............. 128 4.3 Correlation of relative density with standard penetration resistance (after Kulhawy and Mayne 1990) . .... . ............ .... . ...... ... .......... ........ .... . . ............ 138 4.4 Physical propertie s of soils used in parametric study ... . .............. ............. ... ... 139 4.5 Cap model parameters for the three soils of parametric study ... ... .... ........ ... .. .. 146 4.6 Parameters for finding mean stress dependent variables Kr, G and 0 for the three soils of parametric study ... ..... . .... . . ...... .... . . .......... ..................... ........ 147 4.7 Plastickinematic model parameters for geogrid reinforcements .... ... ........ . . .149 4.8 Summary of coe ffic ient of friction for contact interfaces .... . ...... ... . ... ........... 156 4.9 Parametric study model summary .......... ...... ...... . ................ ......................... 156 5.1 Summary of design parameters and seismic performance s ........ ... .................. 178 6.1 Correlation matri x of independent and dependent variables ..... ...... .... . ... ... .243 6.2 Change in seismic performance of GRS wall due to increase in design parameter ........... . ... ..... ... ...... ...... . . ............. . ..... ........ . .... ... . ...................... 270 6.3 Summary of constant and regression coefficients from the multiple regression analysis including independent variable PV A [Y = exp (bo + bl PHA + b2 PYA + b 3 H + b4 co + b s + b6 Sy + b7 T s%)] .... . .... ... ................. ....... 272 6.4 Summary of constant and regression coe ffic ients from the multiple regression analysis without independent variable PV A [Y = exp ( b o + bl PHA + H + b 3 co + b4 + b s S y + b6 Ts%)] ... .... . .... . ................. ........ . .... . . ... ......... .272 A.I Material parameters for the RambergOsgood model ... . ........ ... ........ ... ... ... .283 J l Summary of constant and regression coefficients from the multiple regression analysis using earthquake records with PHA less than 0.29 g [Y = exp (b o + bl PHA + b2 PYA + b 3 H + b4 co + b s + b6 Sy + b7 T s%)] ...... ................ 380 xlii
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K.l Data for statistical modeling ........ . ...... ..... . . ........ ... . .... . .... . . ... . .... . . ... . . 397 x liii
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1. Introduction 1.1 Problem Statement Over the past several decades numerous reinforced soil walls, either with inextensible or extensile reinforcements have been built in the earthquake prone areas of the u s and abroad. The growing acceptance of reinforced soil wall over its conventional counterparts is mainly due to its cost effectiveness which includes low material cost short construction period and ease of construction The competitiveness of reinforced soil wall is even greater when the extensible geosynthetic reinforcement is employed. Case histories have indicated that when designed and constructed properly geosyntheticreinforced soil (GRS) wall have performed well during the past several large earthquakes. Typical crosssection of a GRS wall is shown in Figure 1.1. Satisfactory seismic performances of GRS walls may have attributed to the conservatism implemented in the sateofpractice design and analysis of the structure. To uncover the validity of the current design, seismic responses ofGRS wall will need to be examined The seismic responses can be examined by means of physica l model tests or through a numerical modeling study It is, however uneconomical and impractica l to examine the seismic responses of GRS wall by conducting a series of fullsca l e tests with different types of soils and reinforcements under various seismic loads. Hence a more economical and practical approach to examine the seismic responses of GRS wall is to conduct a numerical modeling study in which the numerical tool wo uld need to be validated from physical model tests with well controlled conditions It is the interest of this research study to examine the seismic performances of freestanding simple GRS walls under real multidirectional ground motion shaking. The validated numerical tool with proven predictive capability would be used to perform a parametric study where essentia l design parameters such as wall height wall batter angle soil friction angle reinforcement spacing and reinforcement stiffness would be evaluated. T h e results of the numerical parametric study would be compared with values determined from the current design methodology and discrepancies between the two would be identified. The results of the numerical parametric study would also provide the data needed to develop seismic performance predic tion equat ions. The prediction eq uat ions can assist a designer to estimate the seismic performances of GRS wall in a preliminary design setting 1
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Cap Geosynthetic Reinforcement Free Draining Material Modular Block s Backfill I I I I I I I I I Retained I Ea rth I I Leveling Pad Drainage Collection Pipe J Limits Figure 1.1 Schematics of a Freestanding Simple GRS Wall 1.2 Research Objectives The objectives oftms research study are threefold. The first objecti ve is to validate the numerical tool from well controlled physical model tests. The second objective is to identify discrepancies between the numerical parametric study results with values determined using the current design methodology. The third objective is to develop prediction equations for estimating seismic performances of freestanding simple GRS walls 1.3 Scope of Study To achieve the research study objectives following undertakings were performed: Re view seismic performances of GRS walls in the field and from the laboratory physical model tests Review current seismic design and analysis of GRS walls. Characterize material parameters with laboratory test results. Validate and calibrate the finite e lement method (FEM) computer program LSDYNA using fullscale shaking table wall tests. Conduct parametric study usin g the validated FEM computer program. 2
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Compare the numerical results with values determined from the current design methodology and identify the discrepancies between the two. Establish seismic performance prediction equations that incorporate the essential design parameters for use in the preliminary design. The flow chart showing the sequence of the research undertakings is depicted in Figure l.2. The organization ofthe thesis follows the tasks listed above. I Establish research objectives I I Perform literature review I I Material characterization I .. Computer program validation using fullscale shaking table tests I Perform parametric study I ... Compare numerical results with current design methodology I Develop prediction equations I Figure l.2 Scope of Research Study 1.4 Engineering Significances The engineering significances of the research study are listed as follows: A validated computer program can provide strong inference on its predicted results. Although the cost of computation for numerical simulation is more expensive the seismic performances such facing displacement and crest settlement that are otherwise not achievable in the limit equilibrium approach could be estimated. By including the realistic earthquake shaking, nonlinear soil behavior and the geosynthetic stiffness characteristics, numerical simulation is a better representation of GRS wall than the model analyzed using the limit equilibrium approach. 3
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The hardening parameters of the cap model for describing the nonlinear granular soil behavior can be determined based on the relative density of the soil which were derived based on laboratory stressstrain curves. Prediction equations developed in this study are directly applicable in assisting the seismic design and analysis of freestanding simple GRS wall. The design parameters were incorporated in the prediction equations The established numerical modeling technique can be used to anal yze more complex GRS structures. Numerical simulation terminology (i. e., verification, validation, calibration and prediction) used in this study adheres to those adopted by the computational mechanics community 4
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2. Literature Review The literature review given below includes post earthquake investigations on the field performances of geosyntheticreinforced soil (GRS) retaining structures laboratory and numerical model tests of GRS walls and the seismic design and analysis of GRS walls. It is through the literature review that the adequacies in the current seismic design and analysis of GRS walls could be evaluated. The results of studies conducted by other researchers are used in deriving tasks to be performed in this study. 2.1 Post Earthquake Investigations Numerous GRS structures have been built in seismically active regions in the u .s. and abroad The post earthquake investigations on the performances of these structures can reveal the adequacy of the stateofpractice seismic design. Case histories have been reported from the 1989 Lorna Prieta earthquake 1994 Northridge earthquake 1995 HyogoKen Nanbu Japan earthquake 1999 ChiChi Taiwan earthquake 2001 EI Salvador earthquake and 2001 Nisqually earthquake. Seismic performances of reinforced soil retaining structures from each earthquake are summarized as follows. 1989 Lorna Prieta Earthquake The performances of five geogridreinforced slopes and walls that experienced the 1989 Lorna Prieta earthquake near San Francisco Californi a were evaluated by Collin et aI. (1992). The 1989 Lorna Prieta earthquake registered a Richter local magnitude of 7.1 and had duration of shaking of 10 to 15 seconds. Maximum horizontal and vertical accelerations of 0 64 g and 0.60 g respectively were recorded near the epicenter. Based on visual observations the five geogrid reinforced slopes and walls experienced no signs of distress (e .g., no apparent movement no apparent cracks and no signs of sloughing). One of the geogrid reinforced slope was originally designed with a maximum horizontal acceleration of 0 10.2 g and had performed well with estimated site acceleration of as high as 0.4 g. 1994 Northridge Earthquake The performances of geosyntheticreinforced soil structures that experienced the 1994 Northridge California earthquake were reported by Sandri (1997) and White and Holtz (1997). The 1994 Northridge earthquake had a Richter local magnitude of 6 7 with strong motion duration of 10 to 15 seconds. Unlike most other earthquakes the vertical accelerations of the Northridge earthquake were as strong as the 5
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horizontal accelerations which in turn made the Northridge earthquake destructive (see Figure 2.1). Sandri visually examined eleven geogridreinforced soil slopes and walls, and signs of distress inspected include: (1) structure alignment (2) relative facade movements (3) facade bulging (4) facade cracking, (5) geogrid slippage at the wall face (6) soil sloughing and (7) tension crack near the slope / wall crest. The post earthquake inspection results indicated that none of the geogrid reinforced soil structures showed signs of distress. As compared to excellent seismic performances of one of the geogridreinforced soil structures cantilever walls located nearby (w ithin 100 m) experienced significant cracking and required major remedial measures Similar findings regarding the seismic performances of geosynthetic reinforced soil slopes and walls located in the Greater Los Angles area were reported by White and Holtz As indicated by White and Holtz an unreinforced natural cut slope in a weakly cemented sand and gravel at approximate 200 m away from a geosy ntheticreinforced slope had failed when subjected to strong ground shaking, while the geosynthetic reinforced slope showed no signs of distress The post earthq uake investigations have indicated that geosyntheticreinforced soil structures were able to resist a horizontal acceleration up to 2 times greater than that specified in the design. 0 8 ... Fr ... fieldroclcllte o FrHfiold"';l alto o o o o o o o 0 2 0 4 0 6 0 8 1 0 Maximum hor i zontal 9 Figure 2.1 Relationships between Maximum Horizontal and Vertical Accelerations of Northridge Earthquake (after Stewart et al. 1994) 1995 HyogoKen Nanbu (Kobe), Japan Earthquake The performances of reinforced soil structures subjected to the 1995 Hyogo Ken Nanbu, Japan earthquake were reported by Tatsuoka et al. (1997) The 1995 6
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HyogoKen Nanbu earthquake had a Richter local magnitude of7.2. One of the recorded peak horizontal ground accelerations was as high as 0.8 g. Many retaining structures were located in the severely affected areas where these structures were used to support the railway tracks of Japan Railway. The retaining structures e valuated were categorized into three groups: (1) g ravity walls ( 2) reinforced concrete cantilever walls and (3) geosyntheticreinforced soil walls with fullheight rigid facing A number of gravity walls (e.g. masonry walls and unreinforced walls ) were seriou s ly damaged where the damaged walls were later demolished and reconstructed Many reinforced concrete cantilever walls suffered moderate damages and were substantially repaired before reuse. The damaged gravity and rein f orced concrete cantilever retaining walls suffered mostly from overturning mode of failure and the causes o f the overturning failur e were due to the large inertial force and the bearing capacity failure beneath the toe of the wall. Observation from the gra v ity retaining wall also indicated that the orientation of the actual failure plane was steepe r than the one predicted by the MO method As compared to the gravity and cantilever reinforced concrete walls g eosyntheticreinforced soil walls with fullheight rigid (FHR ) facing performed well and suffered only slight damages (e g., limited amount of displacement). T he g eosyntheticreinforced soil walls with FHR facing were constructed by th e staged construction procedures where a relatively short length of reinforcement o f O.SH and a relatively short reinforc e ment spacing of 0.3 m were adopted. Key featur e s of the FHR reinforced soil wall are shown in Figure 2 .2. Superior seismic performance of the FHR reinforced soil wall may be due to the intrinsic ductility and flexibility of the GRS structure Despite the high seismic stability the geosyntheticreinforced soil walls with FHR facing w e re only constructed in Japan due to their higher cost and longer construction time than the regular segmental facing reinforced soil w alls constructed elsewhere. R C Faci n g Reinfo r cement forC.J C.J. +IfD rain age Anchor E l e m ent Constructio n J o int (C.J ) Geo t ex t i l e or Geogrid Sandbag o r Gabio n Filled with Grave l Figure 2 2 GeosyntheticReinforced Soil Wall with FHR Facing (after Tatsuoka 1993) 7
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1999 ChiChi, Taiwan Earthquake The performances of six geosyntheticreinforced soil walls and slopes at the vicinity of the CheLun g Pu fault during the 1999 ChiChi Taiwan earthquake were evaluated by Ling et al. ( 2001) The devastating 1999 ChiChi Taiwan earthquake had a Richter local magnitude of7. 3 and the rupture of ground long the CheLungPu fault was about 105 krn long The frequency of shaking ranged from 1 0 to 4.0 Hz. One of the recorded peak horizontal ground accelerations was as high as 1.0 g All of the six geosyntheticreinforced soil structures suffered moderate to severe damages Signs of distress included total collapse of the structure large facade movement (in both transverse and vertical directions) facade cracking and tension cracks at the wall crest. An example of total collapse of the geosynthetic reinforced soil wall is shown in Figure 2.3. Some of the factors contributing to the failure of the geosynthetic reinforced soil structures include: (1) unqualified onsite soil was used as backfill (2) inadequate global stability analysis (3) unusual large reinforcement spacing (e.g., spacing > 800 mm [32 in.]) (4) mixture ofunreinforced and reinforced retaining walls within a common structure (5) insufficient connection strength between the modular block and the reinforcement and (6) peripheral structures such as lamppost were installed close to the modular block facing. Figure 2.3 Collapse of GeosyntheticReinforced Soil Wall during 1999 ChiChi Taiwan Earthquake (after Ling et al. 2001) 8
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Failures of three gravity retaining walls due to the ChiChi earthquake have also been reported by Fang et al. (2003). The failures include: (1) shear failure of concrete wall (or sliding) along the construction joints (2) excessive settlement and (3) overturning of the wall structure about the toe due to bearing capacity failure. These modes of failure are similar to the gravity wall failures observed in the 1995 HyokoKen Nanbu Japan earthquake. Possible factors contributed to the failure of gravity retaining walls may be: (1) insufficient compaction of the backfill (2) improper design of the retaining wall and (3) excessive fault displacement. As noted by Fang et aI., the evaluation of factors of safety against various modes of failure (viz. base sliding overturning and bearing capacity failure) should never be neglected in the seismic design of retaining structures. A reinforced soil slope failure during the ChiChi earthquake was reported by Chen et al. (2001). The 40 m high reinforced soil slope was located at the entrance road of National ChiNan University. The reinforced soil slope was constructed in tiers with the height of each tier being 10m. The slope had a wraparound facing and was reinforced with woven polyester geogrids at vertical spacing of 1 m. The length of reinforcement (ranged between 4 m to 13 m) was gradually reduced with the height of slope. Onsite material was used as the backfill. The failure of the reinforced slope took place several hours after the ChiChi earthquake. Backanalysis indicated that internal stability of the reinforced slope was satisfactory; however, the factor of safety against external stability is less than unity when pseudostatic acceleration exceeds 0.24 g. Failure occurred along the interface between the reinforced soil and the natural soil which agrees with the backanalysis Additional field performances of four geosyntheticreinforced soil modular block walls during the ChiChi earthquake were evaluated by Huang et al. (2003) The peak horizontal acceleration near the observed walls was approximately 0.46 g Two collapsed reinforced soil modular block walls with a height of 3.2 m are shown in Figure 2.4. Other two walls examined were either lightly damaged or undamaged. Features of the undamaged wall included larger modular block wellcompacted gravel backfill and higher tensile strength geogrid reinforcement. In contrast to the observation made during the Kobe earthquake, close to the collapsed wall a reinforced concrete cantilever retaining wall was undamaged. The two collapsed walls were reinforced with knitted polyester geogrid with vertical spacing of 0.8 m and length of approximately 0.8 of the wall height. The backfill of the collapsed walls was a loose sandysilty soil. The study had found that the facingreinforcement connection strength and the blocktoblock interface resistance along with large vertical spacing significantly affected the stability of reinforced soil walls. The failure of the two walls could also be attributed to the overly large vertical spacing of the geogrids 9
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Figure 2.4 Two Failed GeosyntheticReinforced Soil Walls during 1999 ChiChi, Taiwan Earthquake (after Huang et al. 2003) 2001 EI Salvador Earthquake On January 13, 2001 a Richter local magnitude 7 6 earthquake struck off the coast ofEl Salvador and was followed by a strong aftershook on February 13,2001. The devastating earthquake caused significant damage and hundreds of people were killed. Many people were killed due to the earthquake induced landslides Performances of two geosyntheticreinforced soil walls during the El Salvador earthquake were reported by Race and del Cid (200 1). Maximum horizontal and vertical accelerations close to the sites were 0 3 g and 0.15 g, respectively First wall had a height of 6.5 m and was reinforced with high density polyethylene geogri d (H DPE) with vertical spacing of 1 m and length 0.6 of the wall height. Failure ofthe first wall was caused by the toppling of the 1.7 m ofunreinforced soil placed on top of the wall (see Figure 2.5). Second wall had a height of 5.4 m and was also reinforced by HDPE geogrid with vertical spacing of 1 m and length 0.6 of the wall height. A 1.8 m masonry wall was built on top of the reinforced soil wall. Although the second wall did not collapse, it experienced some lateral movement that resulted in a negative facing batter. 2001 Nisqually Earthquake The seismic performances of two reinforced soil wall failures during the 200 1 Nisqually earthquake were evaluated by Kramer and Paulsen (20 0 1). The moment magnitude of 6 8 earthquake occurred on February 28 2001, and the epicenter was located 17 km northeast of Olympia Washington. The earthquake caused significant damage in Olympia minor damage in Tacoma, and moderate damage in Seattle. One of the walls evaluated was located in Tacoma and was named Costco Wall. The 12 batter wall has a height of 5.5 m and was reinforced with geogrid at vertical spacing of 0.8 m and reinforcement length 0.5 of the wall height. The anticipated peak ground acceleration was 0.1 g, and the wall had performed satisfactorily. The second 10
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wall named Extended Stay America Wall which was located in Tumwater had suffered a catastrophic failure (see Figure 2.6). The failed modular block wall had a height that ranged from 2 to 4 m Failure ofthe Extended Stay America Wall was attributed to: (1) inadequate compaction of the fill material (2) weak foundation material and (3) poor drainage condition within the wall system. Figure 2.5 Failure of GeogridReinforced Soil Wall during 2001 EI Salvador E arthquake (after Koseki et al. 2006) F igure 2 6 Failure of GeogridReinforced Soil Wall during 200 1 Nisqually E arthquake (after Walsh et al. 2001) 11
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Summary of Field Case Histories GRS retaining walls have performed well during the 1989 Lorna Prieta earthquake 1994 Northridge earthquake and 1995 HyogoKen Nanbu Japan e arthquake. The high seismic stability of GRS retaining walls observed in the U.S and Japan may be due to the conser v atism built into the design procedure. However on the other hand failure of GRS retaining walls reported in 1 999 ChiCh i, Taiwan earthquake 2001 EI Salvador earthquake and 2001 Nisqually earthquake indicated that inadequate design considerations and poor construction quality control can render GRS structure vulnerable to severe ground motions Some probable causes of failure include: (1) insufficient compaction of the backfill (2) weak foundation material (3) large reinforcement spacing (e .g., spacing > 800 mm) and (4) additional overturnin g load from the addon structures. 2.2 Laboratory Model Tests Well documented case history on the seismic performances of GRS retaining structures in the field is extremely scarce (e.g., insufficient information re g arding input ground motion material properties boundary conditions construction details and preearthquake static performances) Consequently many physical model shaking table tests were conducted to investigate the seismic performance s of geosyntheticreinforced soil structures. Both the reducedscale and the fullscale models seated on laboratory shaking tables have been reported To better simulate the stress conditions experienced by the fullscale prototype other researchers have used dynamic centrifuge tests to examine the seismic performance of reinforced soil walls and embankments Tables 2.1 and 2.2 summarize the laboratory model test results using shaking table and centrifuge respectively Note that the fullscale model is preferred over the reducedscale model since the fullscale model is not associated with problems such as similitude and boundary effects often encountered in reducedscale model and in centrifuge testing Implications ofthe shaking table test and centrifuge test results to seismic design and analysis o f reinforced soil structures are summarized as follows: (1) According to Matsuo et al. (1998) reinforced soil walls with fle x ible facing that have long reinforcement (i.e., L = 0 7H) are more stable than those with short reinforcement (i.e. L = O.4H). (2) The influence of wall type was examined by Watanabe et al. (2003). Under the same shaking condition reinforced soil wall experienced less deformation than the gravity and cantilever walls. In addition reinforced soil wall with partially extended reinforcements performed better than uniform reinforcement length wall (see Figure 2.7). 12
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(3) Latha and Krishna (2008) observed that reinforced soil walls with flexible facing experienced larger lateral displacement than those with rigid facing. Rigid facing adds to wall seismic resistance. In addition retaining walls with rigid facing experienced higher acceleration than reinforced soil walls with flexible facing. Latha and Krishna also indicated that seismic deformation of reinforced soil wall was inversely proportional to the initial relative density of backfill (see Figure 2 8). (4) Koseki et al (1998) indicated that overturning failure appeared as the dominant failure mode for retaining walls with rigid facing. (5) A rigid facing panel with restraints at the toe can greatly reduce the lateral wall movement and tensile load in the reinforcement layers (EIEman and Bathurst 2005) (6) The reinforced zone behaves as a monolithic block and a twopart wedge (or graben) failure mode was observed in the backfill (see Figures 2 9 and 2.10). (7) As observed by Wolfe et al. (1978), seismically induced deformations decreased with increasing reinforcement stiffness Seismic induced reinforcement tensile force is proportional to the acceleration amplitude Resonance between the vertical harmonic shaking and the vertical natural frequency of the modeled wall resulted in the increase of reinforcement tensile force (see Figure 2 11). The scaled model shaking table test conducted by Wolfe et al. has a natural frequency of20 Hz and 30 Hz for the horizontal and the vertical excitation respectively and the amplification in the reinforcement tensile force is observed in Figure 2.11 (ii) when input frequency matches the natural frequency in the vertical direction. The tie force was measured at a distance of O.2H from the all base where H is the total wall height. (8) In the irregular shaking, the combined horizontal and vertical input motion yielded larger reinforcement tensile force and wall displacement than the one with only vertical input motion and the one with only horizontal input motion. Figure 2.12 shows the response of reinforced soil wall due to irregular shaking with (i) vertical component only, (ii) horizontal component only, and (iii) combined vertical and horizontal components. Wall response due to combined horizontal and vertical shaking is not the same as the superposition of the wall responses due to the horizontal shaking alone and the vertical shaking alone (see Figure 2.12). Test result also suggested that peak wall response was observed to be synchronized with the peak of the input motion. (9) Acceleration attenuation was observed in the free field of the reinforced soil structure when base acceleration is greater than O.4g. Figure 2.13 plots the amplification factor versus the peak base acceleration at various depths in 13
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the free field (Siddharthan et al. 2004a). The amplification factor is defined as the ratio of the maximum acceleration in the backfill to the corresponding peak base acceleration. Attenuation is said to have occurred when the amplification factor is less than 1. (10) As shown in Figure 2.14 for a given retaining structure the observed critical seismic coefficient using the irregular input motion is greater than the one using the sinusoidal input motion. In other words test results with sinusoidal input motion are more conservative than those with irregular (but more realistic) input motion. Figure 2.14 also indicates that the observed critical seismic coefficients in general are higher than those calculated (11) EIEman and Bathurst (2004) observed that acceleration was generally noted at top of reinforced soil wall with the reducedscale model and amplification increased with peak base acceleration. Upper reinforcements provided the much needed resistance against overturning of the facing blocks In addition the seismic performance was affected by reinforcement stiffness rather than the ultimate tensile strength (12) As observed by Bathurst et al (2002b) a wall with reinforcement connected mechanically to the facing yielded less seismic displacement than the wall with only frictional connection at the facing. Bathurst et al. also indicated that reinforcement pullout failure as predicted by pseudostatic analysis was not observed in the reducedscale model test. Toppling of facing block was more dominant in the laboratory model test and the horizontal displacement of segmental reinforced soil wall can greatly be reduced by implementing a wall batter of g o from vertical. (13) Budhu and Halloum (1994) indicated that the vertical spacing to length ratio does not have a significant effect on the external seismic stability The results suggested that the vertical spacing to length ratio should be kept small at least in the top portion of the wall to ensure seismic stability (14) The location of maximum tensile load remained unchanged irrespective of the different loading magnitudes (Ling et al. 2005a) 14
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110 110 53 1 rSO I MIHltf8ac1dl1l (D.tO% ) 140s.wc. .. b Gravity type(G) 10 Reinfor.edsoil type I(RI) d Reinforred soil type 2(Rl) E ..sao . }:70 c 60 Q) E 50 fl .!l! 40 a. :5 30 20 10 o d4E5cm . I Subsoil I I y I I c","'tv 111'. I / I Cantilever type II ./ 1 / ""I ',j ... / "'=.= V l/k l I I V Reinforced soil1.fi 0 0 0 1 0 2 0 3 0.4 0 5 0 6 0 7 0 8 0 9 1 0 1 1 1 2 1 3 Seismi c coefficient. k" =a...J Q Figure 2 7 The Influence of Wall Type on Wall Displacement with Irregular Base Shaking (adapted from Watanabe et al. 2003) 5 '" C .S! :; ;. .!! 60 50 40 30 RD Test No F rsg 20 __ OW T6 2 Hz __ OW T7 3 H z 5 H z RD 10 T es t N o Q'OWT9 2 H z Q' OWTIO AOW II o 10 20 30 40 50 60 Horizontal di splacement, mm Figure 2.8 Effect of Relative Density and Input Motion Frequency on the Wall Displacement (adapted from Latha and Krishna 2008) 15
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/ / Figure 2.9 Example of Twopart Wedge Failure Mechanism (after Matsuo et al. 1998) Figure 2.10 Comparison of Reinforced Soil Wall Model Before and After the Shaking Table Test (after Anastasopoulos et al. 2010) 16
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(i) .004 u .5 002 002 <5 .004 E .004 u u L u .004 u j (a) Displacement at Top of Wall 1 Horizontal and Vertical Input Horizontal Only Input 3. Vertical Only Input .. I ... .) 'V .(1 / \ i i (i r 0 1 sec. r I \ J / \ I "'/ \ j \t .II \// 1.::: { ( (b) Tie Force at location 4.5 in (D.ll m) up From 8ase Tie Haterlal Fiberglass Screen Tie Length 20 In (0.51 0) arnax 0.2 9 (ii) i .008 .004 J 0.0 004 008 .Q 008 .. 00 4 u L U 0 0 u .004 .008 (a) Displacement It Top of Wall. 1. Horizontal and Vertical Input 2 Horizonta l Only Input 3. Vertical Only Input I" 0.10 sec al (b ) Tfe Force a t location 4.5 in ( 0 .11m) up fran Base Tie Material Fiberglass Screen Tie Length 20 In (0 .51 m) aNX 0 2 9 F igure 2_11 Comparison of Reinforced Soil Wall Response at Different Harmonic Input Motions; (i) 15 Hz Harmonic Input Motion; (ii) 30 Hz Harmonic Input Motion (after Wolfe et aL 1978 ) (i) h ) Horhontal Ac.celer&tlon 0.0/_________ 0 (b) VertiUl,l.cceleration HHsured Pel\ Tie Force 0 .01 l b 0 .01___________ 0 2 I DefonMtfon len thin 0 ,001 in (0,0l .) ( d ) W.lI Move.n t TIe ""ud., fiberglns Screen fle length 12 i n (0.30.) ( ii ) (.) ""',,.,,' """mtloo 0 D.D f; 1.0 : 0.2 1 0.0 ( b ) Vertlul AccelerUion ( e l IlynMtc ne Force T1e Miterhl flbtrg1us ScrHfl Tfe length 12 1n ( 0 10 II) (iii) (.) ""","", "'ctlmtl,. 0 0.0 (b) VtrtiCiI Acceltntton a 1.0 0'1. ;: O DrV_ (c) Oynatc TIe force .:; D.' i _,,:. Cd) 11,11 Movement TIe ""ted., Fiberglass Screen TIe length 1 2 in (0.30.) Figure 2.12 Comparison of Reinforced Soil Wall Responses with Differ ent Components of Irregular Input Motion; (i) Vertical Component Only; (ii) Horizontal Component Only; (iii) Combined Vertical and Horizontal Components (after Wolfe et al. 1978) 17
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1.30 1.20 & Ii .... 1.10 0 11 (.(\'<, i ... A t D =1.I4m AtD= 3.54m AtD= 5 .94m '0
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2.3 Numerical Model Studies Numerical investigation on seismic performance of GRS wall is more economical than the physical model tests In addition numerical solutions are much more rigorous than the conventional limit equilibrium approach as they satisfy force equilibrium condition straincompatibility condition and constitutive material laws. Conclusions drawn from numerical investigations could either be implemented or served as supplements to the design guidelines. Both qualitative and quantitative performances could be obtained from numerical investigations. For example, the influences of reinforcement stiffness and reinforcement length on seismic response of GRS wall were examined by Bathurst and Hatami (1998) using computer program FLAC. The numerical models were 6 m high with soil and reinforcements simulated by MohrCoulomb and linear elastic models respectively Bathurst and Hatami showed that for the same base condition (i.e either fixed or free to slide horizontally) the wall deflections diminish with increase reinforcement stiffness and longer reinforcement length wall deforms less than shorter reinforcement length. Furthermore there was no evidence of a welldefined failure surface within the reinforced soil zone. Numerical modeling performed by Rowe and Ho (1998) also suggested that the reinforcement tensile stiffness have significant effect on the deformation of GRS walls where large deformation is associated with low reinforcement stiffness. FLAC was also used by Vieira et al. (2006) to evaluate seismic response of geosyntheticreinforced steep slope where the reinforcement layers were modeled using linear elastoplastic model. Program input parameters evaluated by Vieira et al. include boundary conditions (e.g., freefield and quiet boundaries) and mechanical damping implemented in FLAC, and the results indicated that higher wall response is associated with application of freefield boundary and local damping. Effect of design parameters on natural frequency of GRS structure was investigated by Hatami and Bathurst (2000) also using FLAC and they concluded that the natural frequency is not significantly affected by the reinforcement stiffness reinforcement length toe restraint condition and the strength of granular backfill. GRS walls with complex geometry such as the tiered walls and bridge abutments under seismic loads had been analyzed numerically by GuIer and Bakalci (2004) and Fakharian and Attar (2007). Computer program PLAXIS was used in the parametric study performed by GuIer and Bakalci for the tiered wall and the parameters analyzed include wall height (i .e., 4 m and 6 m) ratio of geosynthetic reinforcement length to wall height of the upper and lower wall ratio of distance between two walls to wall height vertical spacing of geosynthetics (i .e., 0.25 m and 0.5 m) and stiffness of geosynthetics. The soil and geosynthetic reinforcements were simulated by elastoplastic MohrCoulomb model and linear elastic model 19
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respectively. It was found that wall deformation is significantly influenced by the ratio of distance between two walls to wall height ; a bigger berm is effective in reducing the amount of deformation. It was also found that reinforcement length in the lower wall contributes more to the wall deformation than the reinforcement length in the upper wall. FLAC was used by Fakharian and Attar for the seismic numerical modeling of GRS segmental bridge abutments where the FounderlMeadows segmental bridge abutment completed by the Colorado Department of Transportation was used as their validation experiment. The Founder / Meadows segmental bridge abutment was then analyzed dynamically using artificial variable amplitude harmonic ground motion records. The numerical modeling suggested that the calculated reinforcement tensile loads are lower than those based on the current active earth pressure theory and the seismically induced horizontal displacement is l arger than the vertica l displacement. Other seismic design parameters such as soil properties and reinforcement layout were considered by Ling et al. (2005b) using the computer program DIANA SW ANDYNEII. The walls considered in the parametric study were 6 m high GRS wall w ith modular block facing A generalized plasticity soil model and a bounding surface geosynthetic model were utilized in the study. Reinforcement length was varied between 0.23 to 0 9 of the wall height and the spacing was varied between 0.2 m to 1.0 m. Six earthquake input motions were analyzed. The results suggested that larger lateral displacement is associated with shorter reinforcement length and larger spacing The results also suggested that amplification of acceleration was a f fected by the stiffness of the backfill where higher amplification is associated with stiffer backfill. Validations of various numerical tools specifically for GRS structures subjected to dynamic shaking done by different researchers are summarized in Table 2.3 Close agreement between the measured data and the calculated results from the validation process indicates good promise of numerical investigation. In the validation tests performed by Ling et al. (2004) using dynamic centrifuge models, it was found that acceleration amplified with wall height for both reinforced and retained soil zones comparing to the base input and the largest settlement occurred behind the reinforced soil zone. Burke (2004) indicated that results of finite element analysis were significantly influenced by the damping properties of the wall. Fujii et al. (2006) noted large discrepancy in maximum earth pressure and maximum response acceleration between the finite element analysis and dynamic centrifuge model test results while good agreement was observed in lateral displacement. The numerical modeling and shaking table tests performed by ElEman et al. (2001) evaluated the significance of toe restraint condition for the reinforced soil walls and reported that the magnitude and distribution of reinforcement loads were influenced by the boundary toe condition, where the reinforcement load at the bottom layer was greater for the unrestrained case as compared to the restrained case 20
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Table 2 1 Summary of shaking table model tests Reference Height Facing type Reinforcement Reinf. Re in f. Backfill Input base Critical Failure (mm) length spacing motion acceleration mode (mm) (mm) (g) Anastasopou375 Segmental Steel wire 263 30 Uniform Sinusoidal Not reported Towlos et al. plexiglass mesh quartz sand and irregular wedge (2010) striE horizontal Bathurst et al. 1020 Concrete Weak geogrid 700 170 Laboratory Sinusoidal Varied from Excessive (2002a; blocks (HDPE bird silica # 40 horizontal (5 0 23 to 0 36 wall top 2002b) (vertical and fencing) sand (D, = Hz) movement 8 batter) 67%) Budbu and 720 Wrapfaced Woven 1330 180 ; Dry silica Sinusoidal 0.45 and Sliding of Halloum polypropylene 240 ; sand horizontal 0 .55 top layer {19942 fabric 360 EIEmam and 1000 Fullheight Knitted PET 600 ; 1000 185; Synthetic Sinusoidal Varied from Excessive N Bathurst rigid panel geogrid 225 olivine sand horizontal 0.42 to 0 57 wall top (2004 2005 facing (vertical (D, = 86%) (2.5 to 5 Hz) movement 2007 2 and 10 batter) Koga et al. 1000 Varied from Nonwoven Varied Varied Airdried Sinusoidal Varied from Excessive (1988) wrapped faced fabrics ; plastic from 320 from sand (SP w horizontal 0.73 to 0 .86 crest to sand bag net ; steel bar to 2250 100 to = 6 12%) (4 Hz) settlement faced at slopes 200 45 to 90 Koseki et al. 500 Full height Phosphor200;200 50 Airdried Sinusoidal 0.54 ; 0.61 ; Two(1998,2003); rigid facing bronze strips mix with Toyoura (5 Hz) and 0.62 ; 0.68; wedge Watanabe et grid 450& sand irregular 0 83; 0.86 al. (2003) 800;350 horizontal Krishna and 600 Wrapfaced; Woven 420 150 ; Poorly Sinusoidal Not reported Not Latha (2007) ; rigidfaced polypropylene 200 ; graded sand horizontal (1 reported Latha and (PP) geotextile 300 (D, = 37 to to 3 Hz) Krishna 87%) (2008)
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Table 2 1 (Cont.) Refere n ce Height Facing type Reinforcement Reinf. Reinf. Backfill Input base Critica l Fai l ure ( mm ) len gt h spacin g motion acc e l e ration mode (mm ) (mm) ( g ) Ling e t al. 2 800 Concr ete Pol y e s t e r 2 050 ; 600 ; T s ukub a Irregular Not r e ported Not failed ( 2005a ) blocks (12 (PET) and 1680 mix 400 s and ( D r = horizontal batter ) pol y vin y l with 25 2 0 5 2 56 %) and v ertical alcohol (PV A ) g eo grids Lo Grasso e t 350 Aluminum LBiaxial 350 Varied Dr y s ilica Sinusoidal Not r e ported Two al. ( 2005 ) s haped section polyprop y len e from 25 sand ( D r = ( 4 5 & 7 wedge with s l ope = (PP) geogrid to 50 70 %) Hz) and 70 irregular horizontal Mat suo e t al. Varied Varied from G e o grid Varied 2 00 Airdri e d Sinusoidal Vari e d from TwoN ( 1998 ) from dis crete panel reinforcem e nt from 400 To y oura horiztonal ( 5 0.38 to 0 59 wedge N 1000 to to cont i nuous ( Tult = 19.4 to 700 s and Hz ); 1 4 00 pan e l kN/ m ) irre g ular hori zo ntal Murata et al. 2480 Full h e i g ht G e ogrid 1000 150 Ina g i s and Sinu s oidal St able a t Not ( 1994 ) rigid f a cing reinforc e m e nt horizontal 0 .51 g reported ( T ult= 9 8 (3.4 Hz) ; kN / m ) irre g ular horizontal P e rez and 1 219 Wr a pped fac e d P e lion Varied Varied Silica s and Sinusoida l Varied from TwoHoltz ( 200 4) at s lop e = 63 nonwov e n from 305 from 61 ( Dr=91%) horizontal ( 5 0 .15 to 0 29 wedge fabric to 610 to 203 Hz) R a m akr i s hn900 Wr a pp e d Wov e n 5 00 1 5 0 D ry s ilic a Sinu s oidal 0 25 ; 0 .4 5 Not an e t al. fac e d ; po I y prop y l e n e s and hori z ont a l (3 report e d ( 1998 ) seg m e nt a l ( PP ) ge o tex til e H z) block
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Table 2.1 (Cont.) Reference Height Facing type Reinforcement Reinf. Reinf. Backfill Input base Critical Failure (mm) length spacing motion acceleration mode (mm) (mm) (g) Richardson 300; 25 mm curved 3.8 mm wide Varied 38 Sand Sinusoidal Not reported Not and Lee 380 aluminum aluminum foil from 76 to horizontal reported (1975) sheet; 38 mm strips ; 6 mm 150 (11.6 Hz) flat aluminum wide mylar sheet taEe striEs Sabermahani 1000 Wraparound Knitted textile ; 500; 700 ; 100 ; SP (DIO = Sinusoidal Not reported Bulging et aL (2009) wall facing non woven 900 200 0 .16 mm ; Dr horizontal and overtextile ; geogrid =47%& (0.6,1.5 turning 84%) 2.4, and 3.34 Hz) Sakaguchi 1500 Cement coated Geogrid Not 300 Standard Sinusoidal 0.33 Not N (1996) foam block reported laboratory horizontal reported w silica sand (4 Hz) No 4 Sofronie et aL 900 Rigid facing High density 360;540 150 Leighton Sinusoidal Not reported Tilting of (2001) polyethylene Buzzard (5 Hz) and facing (HDPE) sand irregular geogrid horizontal Sugimoto et Varied Wrapped faced Tensar SSI Not Not Niigata sand Sinusoidal Not reported Excessive al.(1994) from at slopes reported reported horizontal crest 700 to varied from settlement 1050 34 to 79 Wolfe et al. 600 Discrete panel Mylar tape Varied 38 Uniform Sinusoidal Not reported Not ( 1978) strips; from 305 fine dry horizontal & reported fiber g lass to 762 quartz sand vertical; screen strips irregular horizontal & vertical
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Table 2.2 Summary of centrifuge model tests Reference Scale factor Model Facing type Reinforcement Reinf. Reinf. Backfill Input base motion N height length spacing (mm) (mm) (mm) Howard et al. 24 310 Discrete facing Galvanized Varied 24 Nevada sand Varied with (1998) and panel made of steel wire from 155 (Dr = 65%) sinusoidal and Siddharthan aluminum mesh; ribbed to 434 irregular horizontal et al. (2004a) steel strips Ichikawa et 30 150 Modular Wire netting; 100 ; 160 30 Toyoura sand Sinusoidal (2 Hz) al. (2005) blocks; rigid steel anchor (Dr = 80%) horizontal panel Izawa and 50 150 Rigid facing Polycarbonate 90 30 Toyoura sand Varied with Kuwano panel plate geogrid (Dr = 80%) sinusoidal (100 Hz) (2006) and irregular horizontal N Kutter et al. 50 152.4 Discrete face Wire screen 106.7 19.1 Nevada sand; Irregular horizontal (1990) and panel made of Nevada sand Casey et al. aluminum mixed with (1991) Yolo loam Liu et al. 40 195 Aluminum Nonwoven 140 45 Silty clay ; Sinusoidal (2 Hz) (2010) blocks geotextile ; sand and irregular model geogrid horizontal NovaRoessig 48 152.4 Wrapped faced Pelion Tru106.7 Varied Monterey Sinusoidal and and Sitar at slope = 63 Grid (Tul! = from 15 to # 0 / 30 sand irregular horizontal (1998 2006) 0.18 kN/ m) 17 Sakaguchi 30 150 Light weight Nonwoven Varied 30 Toyoura sand Sinusoidal (\996) blocks geotextiles from 50 to horizontal 150 Takemura and 50 150 Discrete facing Glass fiber 40 ; 90 ; 15; 30 Inagi sand (P d Sinusoidal Takahashi panel made of geogrid 120 = 1.4 and 1.48 horizontal (100 Hz) (2003) aluminum Mglm3 )
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Table 2.3 Summary of computer program validation for seismic performance of reinforced soil wall Reference Code (method) Facing Reinforcement Backfill Interface [nput Validation Performance model model model model motion model type examined (element) (element) (element) (e lement) Burke DlANALinear ID bounding PastorSlip element Kobe Full scale Acc. ; disp.; (2004) SWANDYNE elastic (8& surface (3Zienkiewicz record shaking vertical and II (fmite 6node node bar III (8& 6table lateral element) element) element) node pressures ; reinf element) load EIEman et FLAC (fmite Not given Elasticplastic MohrNote given Sinusoida Reduced Disp.; reinf. al. (2001) difference) (2node cable Coulomb I record scale load ; acc. ; toe element) elastic(5 Hz) shaking condition plastic strain table softening Fujii et al. FLIP (finite Elastic Elastic (IinearMultispring Joint element Kobe Dynamic Acc.; disp.; N (2006) element) (linear beam beam element) record centrifuge earth pressure VI element) Helwanyet DYNA3D Not given Linear elastic RambergPenalty based Sinusoida Fullscale Acc.; disp.; al. (2001) (finite (shell element) Osgood interface I record shaking interface load element) (solid table element) Ling et al. DIANALinear Bounding Generalized Elastic Sinusoida Dynamic Acc. ; disp.; (2004) SWAN DYNE elastic surface plasticity perfectly I record centrifuge vertical and II (fmite soil plastic lateral element) interface pressures element Liu et al. ABAQUS Linear Elastoplastic DruckerMohrKobe Dynamic Acc.; disp.; (2011 ) (finite elastic viscoplastic Prager creep Coulomb (thin record centrifuge crest element) bounding model layer element) settlement ; surface ()D reinf. strain bar element)
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2.4 Seismic Design of Retaining Walls The limit equilibrium (LE) methods have been used widel y in the design of earth structures. The L E analysis offers a designer with the advantages of s imple input data and useful d e sign output information. Adequate factors of safety against potential failure modes utilizing the L E methods can often provide satisfactory performances of the earth structures. The stateofpractice in both the static and the seismic design and anal y sis of retaining structures also involves the LE methods Ex amples of L E methods in static desi g n of retaining structure include th e conventional Rankine's a nd Coulomb's earth pressure theories. Consequently L E methods are extended to the seismic design and analysis of retaining structures. One of such extensions is the venerable MononobeOkabe (MO) method which is the s uccessor o f the static Coulomb method for estimating the seismic earth pr e ssur e imparted on retaining structures The MO method in analyzing conventional retaining wall is first described. The influence of the direction of inertia forces on the seismic thrust is then discussed. Subsequently current design and analysis of reinforced soil retainin g wall are described Lastly the seismic induced p e rmanent displacement of retaining wall and its design implications are di s cussed 2.4.1 MononobeOkabe Method Okabe (1926) and Mononobe and Matsuo (1929) formulated the basis o f the pseudostatic analysis o f seismic earth pressures on retaining s tructures that is known as the MononobeOkabe (MO) method MO method is similar to static Coulomb method in which the additional pseudostatic accelerations are applied to the Coulomb active or passive wedges Figure 2.15 shows the active wedge with pseudo static accelerations ah = kh g and a v = k v g in horizontal and vertical directions respectively where g is the gravitational acceleration. Note that kh and k v are earthquake acceleration coefficients in the horizontal and vertical directions respectively. Positive k h and kv are acting towards the wall and upward respectively. The backfill considered in the MO method is both cohesionless and unsaturated. In an active condition the total active thrust P AE can be calculated in a form similar to that of the static condition as : PAE = .!.KAEyH2(IkJ (2.1 ) 2 where Y = unit weight of the backfill H = total wall height and KAE = sei s mic active earth pressure coefficient. KAE is calculated as: K AE = cos2 (0 S ) /[COsS cos2 (Ocos(o + + S)] [ 1 sin + 0 sin p S ] + cos(o + (0 + S )cos(P (0) (2.2 ) 26
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where = peak soil friction angle 0 = soilwall interface friction angle (0 = wall inclination angle = slope angle of backfill and = seismic inertia angle = arctan[k h /( 1k y ) ] The orientation of the critical failure surface for active condition from the horizontal aAE proposed by ZarrabiKashani (1979) can be expressed as: where : [ d a] aAE = + tan I e a = b = (0) d = (a + b)(bc + 1) e=1+c(a+b) The original MO method implies that the total active thrust P AE would act at a location Hl3 above the wall base. From experimental results Seed and Whitman (1970) suggested that P AE be resolved into the static thrust P A and the seismic increment L1P AE components as: (2.3) P AE = P A + L1P AE (2.4) where P A acts at Hl3 above the wall base and L1P AE acts at 0 6H above the wall base. Hence the location of the total active thrust from the wall base h is: h = PAH / 3+L1PAE(0.6H) PAE (2 5) The magnitudes and locations of AE and P A are used in determining the factor of safety against overturning. Note that in order to get a real solution of KAE f rom E quation 2.2 the term in the denominator needs to be greater than zero hence the limiting slope of the backfill is Furthermore for horizontal backfill = 0 ; hence Since = tan I [k h / (1k y)], which leads to kh The critical v alue of horizontal acceleration coefficient is then kh(cr ) = F Figure 2.15 Equi librium of Forces acting on the Active Wedge in MO Analysis 27
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In the passive earth pressure condition the horizontal component of earthquake peak acceleration ah is directed toward the backfill as shown in Figure 2.16. The total passive thrust on the retaining wall is given by : PpE YH2(Ikv ) (2 6) 2 K p E is the seismic passive earth pressure coefficient and is determined as: K p E = (2 .7 ) [ 1 sin + 8 sin + S ] cos(8 00 + (0) T he orientation o f the critical failure surface for the passive condition from the horizontal a.P E is given by: where: I [j + f] a.pE = e + tan k (2.8 ) f = tan + S + ) g = + S ) h = + 00 s ) i =tan(8+soo) j = Jg(g + hXhi + 1) k = 1 + i(g + h) Similar to the total active thrust the total passive thrust PPE can also be resolved into the static component P p and the seismic increment as: P p E = P p + (2 9 ) w Figure 2 16 Equilibrium of Forces acting on the Passive Wedge in MO Analysis 28
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The influence of the direction of earthquake induced inertia forces ( i .e., kh W and k v W) on the seismic earth pressure has been examined by Fang and Chen (1995). For the active condition the conventional MO method assumes the horizontal and vertical inertia forces to act upward and toward the wall respectively (see F igure 2 .15). It is a concern whether or not the assumed directions of the inertia forces would yield the maximum total seismic active thrust P AE. In reality a total of four combinations for the direction of inertia forces are possible Figure 2.17 compares the e f fect of direction o f inertia forces on the magnitude of the total active thrust and the total seismic active thrust is the maximum for horizontal and vertical inertia forces to act toward the wall and downward respectively for kh < 0.4 Furthermore the effect of direction of inertia forces on the total seismic passive thrust is shown in Figure 2 .18 and the total seismic passive thrust is the minimum for horizontal and vertical inertia forces to act toward the wall and upward respectively. 0 EI 0.4 1 kv = 0 2 (t) 0 kv = 0 2 (l..) 0 8 """ 0 6 0 2 N I . w 0... N 0.4 o 0 2 Towards backfill +kh Towards wall 0.4 Figure 2 .17 Effect of Direction of Inertia Forces on Total Seismic Active Thrust (modified from Fang and Chen 1995) 29 0 6
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12 A ky = 0.2 (t) eky = 0 10 EI ky = 0 2 (l..) .......... 4 N I ?........ w a. 0... N 0.4 0.2 0 0 2 0.4 Towards backfill kh Towards wall Figure 2.18 Effect of Direction of Inertia Forces on Total Seismic Passive Thrust 2.4.2 Design and Analysis of GeosyntheticReinforced Soil Retaining Walls 0 6 This section describes the current design and analysis of geosynthetic reinforced soil (GRS) retaining walls subjected to earthquake loads. In North America, a widely accepted design guidelines which includes the seismic design of GRS retaining walls is the Federal Highway Administration (FHWA) manual put forth by Elias et al. (2001). Another seismic design method for GRS walls follows the National Concrete Masonry Association (NCMA) manual (Bathurst 1998). Design of reinforced soil slope can be found in the US Army Corps of Engineers Waterways Experiment Station publication (Leshchinsky 1997). Other design methodologies from abroad have been surnrnarized by Zomberg and Leshchinsky (2003) and Koseki et al. (2006). Design criteria and analysis methods from the FHW A and NCMA manuals are surnrnarized as follows. The assumptions involved in the design are also presented. 30
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2.4.2.1 Federal Highway Administration (FHW A) Methodology Limit equilibrium (LE) method is adopted in the FHWA methodology where one can only estimate the margins of safety against collapse and cannot estimate the deformation of the structure given the external loads. In the seismic design of GRS walls, the FHW A methodology requires both the externa l stability and the internal stability be evaluated in addition to the static design considerations. The design peak horizontal acceleration at a site can be obtained from Division IA (AASHTO 2002) and Section 3.10 (AASHTO 2007) of the AASHTO Specifications As specified by the FHWA methodology the seismic design is needed whenever the peak acceleration coefficient (A) at the site being considered is greater than 0.05. The coefficient is expressed as a fraction of gravitational constant g, and is dimensionless. The maximum limitin g value of A in which the FHW A seismic design requirements are applicable is 0.29 and the FHWA methodology recommends that the seismic design of a GRS wall should be reviewed by a specialist when A at the project site exceeds 0.29. FHW A External Stability Evaluation In the external stability evaluation for GRS walls three potential modes of failure considered are : (1) base sliding (2) eccentricity (3) bearing capacity Taking into account the flexibility / ductility exhibited by the GRS walls, the recommended minimum seismic factors of safety with respect to the failure modes are assumed as 75 percent of the static factors of safety, and the eccentricity should be within L/3 (L = length of the reinforcement) for both soil and rock foundations. Two forces in addition to the static forces in the external stability evaluation are the horizontal inertia force (PrR) and the seismic horizontal thrust increment AE). AE is exerted on the reinforced soil by the retained soil. Both AE and P1R are shown in Figures 2.19 and 2.20 for level and sloping backfill conditions, respectively. The seismic external sta bili ty is eva luat ed in the following steps: Select the acceleration coefficient A from Section 3 of AASHTO Division 1A. Calculate the maximum acceleration (Am) developed within the GRS wall system Am = (1.45 A) A (2.10) Calculate the horizontal inertia force P1R and the seismic horizontal thrust increment AE The height H 2 should be used in finding PIR and AE for sloping backfill condition (see Figure 2 .2 0) H H H 2 + (2.11 ) The horizontal inertia force PIR is calculated as follows : 31
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P'R = Pir + Pis Pir = 0.5Am Y r H 2 H Pis = 0.125A m Yr (H2Y tanp (2.12) (2.13) (2.14) Note that Pir is the inertial force caused by acceleration of the reinforced backfill and Pis is the inertial force cased by acceleration of the sloping soil surcharge above the reinforced backfill. The seismic horizontal thrust increment AE is calculated using the pseudostatic MononobeOkabe method with the horizontal acceleration coefficient kh equal to Am and vertical acceleration coefficient k v equal to zero. 1 2 AE ="2 AE Y r H (2.15) The total seismic earth pressure coefficient KAE is calculated following the general MononobeOkabe expression: KAE = [ 1 sin + p sin p s 1 + cos(p 0) + S)cos(O) + p) where: p = backfill slope angle S = seismic inertia angle = tan I [kh/(l kv)] = the peak soil angle of friction 0) = angle of the wall face from vertical (2.16) The seismic earth pressure coefficient associated with the seismic thrust increment AE) is and = KAE KA Note that the mobilized interface friction angle 8 is assumed to be equal to p in the FHW A method. Note also that the wall batter angle 0) in the FHW A method is with the facing blocks inclined into the backfill which is the opposite of the Coulomb method. Check factors of safety against failures of base sliding eccentricity and bearing capacity with P[R and 50% of AE. The reduction of 50% on AE was reasoned with possible phase lag between the inertial force and the seismic thrust from the retained backfill. For level backfill condition (P = 0) H 2 = H Pis = 0 and P[R =Pir. 32
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Mass for Inertial Force Mass for resisting forces Figure 2.19 Seismic E xternal Stability of a GRS wall with Le v el Backfill in FHW A Method H O 5H2 Mass for resisting forces Retained Backfi 11 +r '1r K.r F H W A meth o d ass u m es Ii = 13. Figure 2 20 Seismic Ex ternal Stability o f a GRS Wall with Sloping Backfill in FHWAMethod 33
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As indicated by the FHW A manual the use of full value of Am for kh in the pseudostatic MononobeOkabe method to fmd P AE can result in an excessively conservative design. To achieve a more economical GRS wall a reduced kh can be used if the following conditions are met: The wall system and any structures supported by the wall can tolerate later movement resulting from sliding of the structure. The wall is unrestrained regarding its ability to slide other than soil friction along its base and minimal soil passive resistance. If the wall functions as an abutment the top of the wall must also be unrestrained e g., the superstructure is supported by sliding bearings. With the conditions listed above and provided that the GRS wall can tolerate displacements up to 250A (mm) kh may be reduced to 0 5 A (i.e. kh = OSA). FHW A methodology also provides an alternative method for estimating the horizontal acceleration coefficient kh in finding L1P AE kh can be computed as: ( A ) 0 .25 k h = 1.66Am dm (2 .17) where d is the anticipated lateral wall displacement in mm. Noted that this equation should not be used for displacement of less than 25 mm or greater than 200 mm. FHW A manual suggests that typical anticipated lateral wall displacement in seismically active area ranges from 50 mm to 100 mm. It is to be noted that although a trapezoidal dynamic pressure distribution was proposed by the FHWA methodology (see Figures 2. 19 and 2.20) and the actual dynamic pressure distribution was not specified The equation for determining the seismic horizontal thrust increment AE has otherwise suggested a triangular dynamic pressure (hydrostatic) distribution For the seismic thrust to be located at 0.6H and with a trapezoidal pressure distribution the ratio oflong length ( at the top ) to the short length (at the bottom) of the trape z oid needs to be 4. FHW A Internal Stability Evaluation The internal failure of a GRS wall can occur in three ways : (1) pullout of reinforcement (2) reinforcement rupture and (3) connection pullout failure To evaluate the internal stability of a GRS wall one needs to determine the maximum developed tensile force in each reinforcement layer the critical slip surface and the resistance provided b y the reinforcements in the r e sistant zone. It is assumed that the critical slip surface coincide with the locus of the maximum tensile force in each reinforcement layer T max, and the critical slip surface is further assumed to be linear in the case of extensible reinforcements which passes through the toe of the wall (see Figure 2.21 ) Also assumed is that the location and the slope of the linear critical slip surface is not affected by the seismic loads (i.e ., the seismic critical slip surface is the 34
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same as the one for the static condition). The critical static slip surface, following the Coulomb's active condition is inclined at an angle UA from the horizontal as: where uA ='f'+tan C2 C I = p p )+ + 0) m1 + tan(8 0) + 0) )] C2 = 1 + {tan(8 0) p) + + O))ll (2.18) As has mentioned earlier in the FHWA methodology the mobilized interface friction angle 8 is assumed to be equal to the backfill slope angle p (i.e., 8 = P). The static maximum tensile force in each reinforcement T rna,,, is a function of horizontal stress at each reinforcement level along the critical slip surface (O'H) and reinforcement spacing (Sv), and T max is computed as : T max = 0' H S v (2.19) Furthermore the horizontal stress O'H is a function of the overburden stress uniform surcharge loads and concentrated surcharge loads. Alternatively the tributary area from horizontal stress distribution can be used to calculate T max for each of the reinforcements. Note that the reinforcement spacing should not exceed 800 mm as required by the FHWA methodo l ogy. In a seismic event seismic loads would produce an inertial force PI acting horizontally in addition to the static forces (see Figure 2.21) The inertial force PI is calculated as: PI = Am WA (2.20) where W A is the weight of the active zone (shaded area in Figure 2.21) and Am is the maximum acceleration. Each reinforcement layer would receive additional seismic tensile force induced by the inertial force PI. The additional seismic tensile force Tmd in each reinforcement layer is determined by proportionally distributing the PI based on the embedment length of reinforcements in the resistant zone and is computed as follows: T = P L e ; md I n (2.21) ILe ; ; = 1 where n is number of reinforcement layers in the GRS wall. Knowing T max and T md, the total tensile force in each reinforcement layer Ttota l is calculated as: Tto t a = Tmax + Tmd (2. 22) Ttotal is then used to evaluate the reinforcement pullout failure. Note that the factor of safety against reinforcement pullout failure FS p o under static condition should be greater than or equal to 1.5, and in seismic design the factor of safety is said to be 35
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75% of the static value. The total tensile force in each reinforcement layer Ttotal should not exceed the pullout resistance P r at that layer as: T < P, Re t o tal (0.75)FSpo (2.23 ) where Re is the coverage ratio and is often assumed to be unity for geotextiles and geomembranes. P r is a function of embedment length Le overburden stress and coefficient of friction (or the frictionbearinginteraction factor). According to the FHW A methodology the coefficient of friction between the soil and reinforcement in the seismic condition should be reduced to 80% of the static value. In evaluating the rupture failure during seismic loading the reinforcement is to be designed to resist both the static and seismic forces which requires the following: T < S,s R e (static component) (2.24) max (0.75)RFC R RFo RFlO FS (seismic component) (2.25) where Re = coverage ratio RFC R = creep reduction factor RFo = durability reduction factor, RFIO = installation damage factor FS = overall factor of safety Srs = reinforcement strength to resist static load and Srt = reinforcement strength to resist seismic load Note that the creep reduction factor RFC R is not applicable to T md, since seismic load occurs in a short time. The values of various reduction factors have been suggested in the FHWA methodology. Moreover with both Srs and Srt known the required ultimate strength of the geosynthetic reinforcement T ult can be calculated as follows : Tult = S,s + Srt (2 26 ) A particular geosynthetic reinforcement can be selected based on the value of T ult. The connection pullout failure during seismic loading is evaluated using the following conditions : (static component) (2.27) (seismic component) (2.28) where CRer = connection strength reduction factor resulting from longterm testing and CRuit = connection strength reduction factor resulting from quick connection tests. Both CR e r and CRuit are to be determined using laboratory testing technique described in Appendix A of the FHW A manual. Both CRer and CRuit are function of normal stress which is developed by the weight of the facing units Calculation of normal stress should be limited by the hinge height in the case of a batter wall. 36
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JL7''++i i (i' th Laye r ) PI I nterna l i n ertial force due to th e wei g h t of t h e backfill wi thin the ac t ive zone Lei The l e n g th of reinforceme n t i n t he resi s t ant zone of the i' t h l aye r T max = The load pe r unit wall wid t h applied to eac h r ei n forcemen t d u e t o s t atic f orce s T md = The load pe r unit wall wid t h applied to each rei n forcement due t o dynamic for ces T,o,al = The to t a l load per unit wall width app l ied t o each la yer. T1o,al = Tmax + Tmd Extensib l e Rei nfor ceme n ts F igure 2.21 Seismic Internal Stability of a GRS Wall in FHW A Method Similar to the r e inforcement rupture failure evaluation Srs and Srt can be back calcu l ated from Eq u ations 2.27 and 2 2 8 respective l y and the ultimate str e ngth of th e geosynthetic reinforcem e nt Tuil can then be found Subsequently a particular reinforcement can be selected in the wall design based on the v alue of T uil' Note that for AASHTO Division 1A seismic performance categories C or higher (i. e., A > 0 19) FHW A methodolo g y recomm e nds that the modular block facing connection should not depend solely on the frictional resistance between the facing units and the reinforcement ; rather shear resisting devices such a s shear key s or pins should b e insta ll ed in the modular block facing wall. 2.4.2. 2 Na tional C o nc r e t e Masonry Ass oc iatio n (NeMA) Me thodology Similar to the FHWA methodology the NCMA manual utilizes the pseudo static MononobeOkab e (MO) earth pressure theory in assessing the seismic stability of GRS walls As indicated in the NCMA manual it is applicable with restrictions o f A s 0.4 and k v = O. Three types of stabilities considered in th e NCMA manua l ar e external internal and fa cing stabilities The possible modes o f failur e for the GRS wall are presented in Figure 2.22. Figures 2 22a to 2 22c 2 22d to 2 .22f, and 2 22g to 2.22h show the external failure modes internal failure modes and facing f ailure modes respectively. The factors of safety in the seismic design are the same as those 37
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proposed in the FHW A manual where the minimum seismic factors of safety are 75% of the static values With the assumption that GRS walls are freestanding and can tolerate horizontal displacement at the base without any lateral constraints NCMA manual adopts the same approach as those from the FHW A methodology in determining the horizontal acceleration coefficients kh for the external, internal and facing stabi l ities evaluat ions. For external stabilities and internal sliding: k h (ext) = 0.5 A (Figures 2 22a b c and f) (2. 29) For facing stabilities and other internal stabi lities : kh (int) = (1.45 A) A (Figures 2.22d e g, h i and j) (2.30 ) where A can be obtained from Section 3 of AASHTO Division IA. Summarized below are the approaches for evaluating the three types of stabilities. (a) ba se s l idin g ( b ) overturnin g ( c) b ea rin g ca pa c ity (ex t erna l failur e m ode) (ex t e rnal fail ure m ode) (ex t erna l fai lu re m o d e) ( d ) pull o ut (e) t e n sile overstress (I) i n ternal s l i din g (interna l failur e m ode) (interna l failure mo d e) ( int ernal failur e m o d e) (g) co lumn s h e ar failure ( h ) co nne ctio n failur e ( i ) l o c a l ove rturnin g (j) c r es t topplin g (facing failu r e m ode) (facing failur e mo d e) (facing failur e m o d e) (faci n g failur e m ode) Figure 2.22 Modes of Failure for GRS Walls in NCMA Method NCMA External Stability Evaluation Using the approach suggested by Seed and Whitman (1970), the seismic thrust P AE is comprised of the static thrust component P A and the seismic thrust increment AE component as : 38
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PAE=PA+L'lPAE and in terms of the earth pressure coefficients as: (1 kJKAE = K A + L'lKAE (2.31)(2.4 bis) (2. 32) The static and seismic increment earth pressure distributions for evaluating the external stabilities are shown in Figure 2.23. In the NCMA manual it is assumed that positive kh and ky acts toward the wall and downward respectively. Note that the static pressure increases linearly from top to the base and the static thrust acts at hl3 from the base. The seismic increment pressure takes the shape of a trapezoid with higher stress at the top The trapezoidal pressure distribution was adopted from Ebling and Morrison (1992) in the seismic design of anchored sheet pile wall and the seismic thrust increment acts at 0.6h from the base. This trapezoidal pressure distribution is said to have indirectly accounted for the amplification of horizontal ground acceleration. The seismic active earth pressure coefficient KAE is calculated as: (2.33) where = peak soil friction angle, ()) = wall facing column inclination, () = mobilized interface friction angle p = backfill slope angle and 8 = seismic inertia angle (8 = arctan[k h / (1ky)]). In the FHWA manual is used to denote seismic inertia angle instead of 8 in the NCMA manual. Note that in the above equation the wall inclination angle ()) is measured from vertical and increases as the wall inclined into the backfill whereas the expression of KAE (Equation 2.2) for gravity retaining has a ()) that is measured from vertical as well but increases as the wall inclined away from the backfill (i.e. the difference is the sign in front of ())). Note also that Equations 2 .33 and 2.16 are different by the parameters () and p where in Equation 2.16 ofthe FHWA manual () is assumed to be equal to p. In the NCMA external stability evaluations the mobilized interface friction angle () is further assumed to be equal to (with equal to lesser of and values) Hence the seismic thrust increment L'lP AE is calculated as: 1 2 L'lP AE = L1K AE Y f h (2.34) 2 where h is the height at the back of the reinforced soil zone (see Figure 2.23). Note that only 50% of L'lP AE is considered in the seismic analysis. The justification for the 50% reduction on L'lP AE is that the inertial forces within the reinforced soil mass would not peak at the same time as the seismic thrust increment generated by the retained backfill. 39
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In addition to the seismic thrust increment AE, a horizontal inertial force within the reinforced soil mass PIR is also considered in the external stability evaluations The method in finding PIR is similar to the one proposed in the FHW A manual. The inertial force includes the mass of the facing column and the mass of the reinforced soil zone extended to a distance of 0.5H behind the wall facing which makes P[R < k h ( ext) W with W being the weight of the entire reinforced soil mass. With 50% of AE, P[R, and static forces external stabilities of base sliding overturning and bearin g capacity in terms of factors of safety can thus be evaluated. h H O 6 h ="''(Mass for resisting forces) Static component Seismic increment Figure 2.23 Seismic External Stability of a GRS Wall in NCMA Method NCMA Internal Stability Evaluation The tensile force for each reinforcement layer needs to be evaluated Figure 2.24 shows the schematic in finding the total tensile force in each layer Tto t al. Note that only the horizontal components of earth pressure coefficients are considered in the stability evaluation. The horizontal components are calculated as: K AH = K A cos( 8 (J) ) K AEH = K AE cos( 8 (J) ) AEH = AE cos( 8 (J) ) (2.35) (2.36) (2.37) The tributary area approach is used where the tensile force in each layer is the pressure integrated over the vertical reinforcement spacing Sy. Ttotal is comprised of the facing column inertial force (kh(int) W w), static component (T max), and the seismic increment (Fdyn) as: 40
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Ttotal = +Tmax +Fdyn (2.38) where W w is the weight of the facing column within the contributory area. The contributory area approach is said to be more conservative since the seismic increment pressure is higher toward the top z H Figure 2.24 Schematic for finding the Total Tensile Force in Each Reinforcement Layer in NCMA Method The factor of safety against reinforcement rupture failure (or reinforcement overstressing) is evaluated by the ratio of the allowable tensile load for the reinforcement with seismic loading T a(dyn ) to the total reinforcement tensile force Ttotal as: T FS = a(dyn) (2.39) os T total Note that Ta( d y n ) should not include the creep reduction factor [Ta(dyn) = Tu1t / (RFD RFID FS)] since the seismic loading is of short duration. In the reinforcement pullout evaluation, a failure surface is first identified and the potential failure surface would initiate from the toe of the reinforced soil mass. Equation 2.3 can be used to determine the angle of the failure surface with a difference of using opposite sign in front of (() (wall inclination angle). The embedment length Le that extends beyond the potential failure surface provides the 41
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pullout resistance to resist the Ttotal of the reinforcement (see Figure 2.25) The factor of safety against pullout failure FS p o at each reinforcement layer is calculated as: FS =l (2.40) po T,o a l where P r is the pullout resistance (or the anchorage capacity), which is a function of the overburden stress crv embedment length Le and coefficient of soilgeosynthetic interface friction. In absence of the laboratory test data the coefficient of soil geosynthetic interface friction can be estimated as Note that from the repeated pullout testing Bathurst and Alfaro (1997) concluded that the pullout resistance Pr should not be reduced by 20% as what has specified in the FHW A manual. Similar to the base sliding in the external stability evaluation the factor of safety against internal sliding FS sii is calculated as: FS r = R s (z) (2.41) s I MIR (z) + P AH (z) + 0 5 AEH (z) where R s is the sliding resistance along a reinforcement layer at depth z, which is a function of the interface shear capacity between facing column units and the frictional resistance between the soil and the reinforcement. Note that the inertial force is due to the 0.5H in length of the reinforced soil mass and only 50% of the seismic thrust increment AEH contributes to the sliding evaluation (see Figure 2.26). H Figure 2.25 Schematic for Reinforcement Pullout Evaluation in NCMA Method 42
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H (Mass for resisting forces) Figure 2.26 Schematic for Base Sliding and Internal Sliding Evaluation in NCMA Method NCMA Facing Sta b ility Eva l uation Four potential modes of failure associated with the facing instability are: (1) interface shear (2) connection failure (3) local overturning and (4) crest toppling. In the interface shear mode of failure the facing column is treated as a beam and the factor of safety against the interface shear failure FSsc is expressed as : FS = V u (z) (2.42 ) sc Si (z) where V u(z) is the shear capacity of the interface between facing column units and Si(Z) is the outofbalance shear force transmitted through facing unit interface at depth z as shown in Figure 2 27. S i(Z) is calculated as : S i(Z)= LTt o t a l (2.43 ) where LTt o t a l is the sum of the hori z ontal forces carried by the reinforcement layers above the interface of interest. The factor of safety a g ainst connection failure FScs is evaluated as : FS =l (2.44) cs T t otal where Tto t a l is the tensile force within the reinforcement and T c is the connection capacity between the facing column units and the reinforcement. Note that T c is to b e obtained from the NCMA Test Method SR WUl "Determination of connection 43
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strength between geosynthetics and segmental concrete units" (Collin 1997) which is often referred as the "quick connection test" in the laboratory T c takes the same form as the MohrCoulomb failure criterion where T c is a function of the minimum available peak connection strength (similar to the cohesion component) the apparent interface friction angle (describing the failure envelope) and the applied normal stress. The beam analogy used in the interface shear failure evaluation is also applicable to the local overturning mode of failure. The factor of safety against local overturning about the toe of the facing column unit FSotl is evaluated as ratio of resisting moment to the overturning moment as: FS = M r + LTtotal x y ( 2.45 ) otl M o The resisting moment is comprised of the moment due to the weight of the facing column units Mr and the summation for moments due to reinforcement tensile forces LTto t a l xy. M o is the overturning moment (or the driving moment) due to the horizontal inertial force of the facing column the static thrust and the sei s mic thrust increment (see Figure 2.28) Note that the crest toppling mode offailure is considered as a subset to the local overturning mode of failure The crest toppling is the local overturning that occurs at the elevation of the topmost layer of reinforcement. z H Vu(z) FSs c = S;(z) KAHY,H Y,H S ; (z) = kh(int) Ww(z) + P AH + AEH ETtotaJ Figure 2.27 Schematic for Facing Interface Shear Evaluation in NCMA Method 44
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z z "2 H KAHY,H O .26KAEHY,H Mo= [kh (int) W .. (z) 1 1 + ( P AH 1) + (6P AEH Ydyn) Figure 2.28 Schematic for Facing Local Overturning Evaluation in NCMA Method Comparison between FHW A and NCMA Methods The minimum factors of safety stipulated in the seismic design of GRS wall from the FHW A and the NCMA methods are summarized in Table 2A. The similarities observed between the FHW A and the NCMA methods are listed as follows: (1) The reinforced and retained soils are assumed to be cohesionless and unsaturated. (2) The peak: friction angle is assumed as the design friction angle. (3) GRS wall is founded on competent foundation and the global stability is not a problem. (4) The horizontal acceleration coefficients kh remains constant and uniform in the GRS wall structure. (5) The vertical acceleration coefficient k v is assumed to be zero (6) The minimum seismic factors of safety are 75% of the values considered in the statically loaded structures. The differences observed between the two methods are listed as follows : (1) The design methods are considered applicable when the maximum horizontal accelerations are limited to 0.29 g and OA g for the FHW A manual and the NCMA manual respectively. (2) In finding the inertia force PIR FHWA manual applies Am [Am = (lA5 A)A] to the reinforced soil mass of 0.5H 2 in l e ngth whereas NCMA manual applies 45
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kh(ext) [kh(ext) = 0 5A] to the reinforced soil mass ofO.5H in length (see Figure 2.29). (3) In sloping backfill the seismic thrust is found to act at 0 .6H2 in FHWA manual versus 0.6 h in the NCMA manual (see Figure 2.29). (4) FHWA method assumes () = P in finding KAE versus () :j:. P in the NCMA method (5) FHWA manual assumes the wall to be vertical when the wall inclination angle 00 is less than 8, whereas NCMA manual adopts the general Coulomb earth pressure theory for any wall inclination angles (6) In external stability analysis FHWA method checks for the eccentricity requirement (i.e. e ::; Ll3) and not the factor of safety against overturning, whereas NCMA manual requires the overturning and omits the eccentricity (7) In external stability analysis FHWA manual requires a factor of safety of 1.9 (75% of 2.5) against the seismic bearing capacity failure versus a factor of safety of 1.5 for the NCMA manual. (8) The coefficient of friction between soil and reinforcement in determining the pullout resistance is reduced to 80% of static value in the FHW A manual whereas there is no reduction on the coefficient of friction in the NCMA manual. (9) FHW A method assumes that the inclinations of static and seismic thrusts are parallel to the backslope angle (i.e. inclination = P) whereas the inclinations of static and seismic thrusts recommended by NCMA method follows the conventional Coulomb approach (i.e., inclination = () 00). Table 2.4 Comparison of recommended minimum factors of safety for GRS walls Failure Modes FHWA NCMA Base sliding 1.1 1.1 e:..:= Eccentricity Ll3 ..... (1).0 Overturning 1.1 X ro Bearing capacity 1.9 1.5 Reinf. Rupture 1 e:..:= Reinf pullout 1.1 1.1 s:: ..... Internal sliding 1.1 ....... (/) onO Interface shear 1.1 s=:..::= Connection failure 1.1 1.1 ot"""'l .....t (').0 Local overturning 1.1 ro ro ..... (/) Crest toppling 1.1 46
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h H H L (a) FHWA Method ,h H L (b) CMA M ethod Figure 2 29 Differences between the FHW A Method and the NCMA Method 2.4.3 Peak Ground Acceleration Coefficients The pseudostatic MO method depends entirely on the horizontal and vertical peak ground acceleration coefficients kh and ky respectively. For engineering applications ky is often assumed to be twothirds ofkh (i.e. ky = 2/3kh ; Newmark and Hall 1982) However in typical seismic retaining wall design the stateofpractice is to assume ky equal to zero and that kh remains constant throughout the retaining structure. Furthermore there is no consensus in determining the design value of k h Seed (1983) suggested that kh = 0.15 be the maximum level ascribed to the limit equilibrium analyses. Bonaparte et al. (1986) suggested that kh = 0 85A where A is the peak horizontal ground acceleration coefficient found in Section 3 of AASHTO Division IA. Whitman (1990) recommended that kh be ranged from 0.05 to 0.15. Segrestin and Bastick (1988) suggested that kh be found as: k h = (l.45 A) A (2.5 2) for 0 .05 A 0.5. Note that the above equation was incorporated in both the FHWA and NCMA manuals. Figure 2 30 shows the relation between kh and A as determined from Equation 2.52. As depicted in Figure 2.30 amplification of the peak horizontal ground acceleration is observed for A < 0.45. For A > 0.45 kh is less than A and the maximum of the curve occurs at A = 0.725. 47
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/ / .c .><: 0 8 .... / c: QJ '0 / lE QJ 0 / u 0 6 c: / 0 .. QJ Qj u 0.4 u ro ro c 0 N 0 2 I V ..... V / / / / / o o 0 2 0.4 0 6 0 8 Peak horizontal ground acceleration A Figure 2.30 Relationship between A and kh 2.4.4 Permanent Displacement Methods In the pseudostatic seismic analysis and design of GRS walls, only the factors of safety against various modes of failure or collapse of the wall could be estimated and wall deformation could not be estimated directly from the pseudostatic analysis. This is a common deficiency in all of the limit equilibrium analyses. The following paragraphs describe the indirect methods used to estimate the horizontal wall movements (or the timedeformation response of the wall system) to accompany the seismic stability analysis In Newmark's doubleintegration displacement method (Newmark 1965) applied to retaining wall structure the total displacement is termed unsymmetrical displacement since the permanent displacement only accumulates in one direction (outward direction) The displacement in the reverse direction (toward the backfill) requires a greater critical acceleration to overcome the passive resistance in the backfill. A passive failure in the backfill would require a force on the order of 10 times the static resistance (Richards and Elms 1979). Hence in retaining wall structure application it is assumed the displacement of the rigidplastic block at failure toward the backfill is zero Note that the calculation of displacement is based on the assumption that the moving mass displaces as a rigidplastic block with shear resistance mobilized along the potential sliding surface. Permanent displacement of the rigidplastic block is said to have occurred whenever the forces acting on the soil 48
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mass (both static and seismic forces) overcomes the available shear resistance along the potential sliding surface Sliding Block Method by Cai and Bathurst The three seismic induced sliding mechanisms in a GRS wall proposed by Cai and Bathurst (1996a) are: (1) external sliding along the base of the entire wall structure (2) internal sliding along a reinforcement layer and through the facing column and (3) block interface shear between facing column units. The displacements are estimated using the conventional sliding block method. The pseudostatic dynamic earth pressure used in the analysis and design of GRS wall by the NCMA method (Bathurst 1998) is also adopted in this displacement method. Note that both kh and k v are used to calculate dynamic active forces and are assumed to remain constant through out the entire wall structure According to Cai and Bathurst (1996a) in absence of the ground motion record k v can be estimated as kv = 2kh/ 3 and the vertical inertial force is assumed to act upward to produced the most critical factors of safety for the horizontal sliding mechanisms. Critical accelerations associated with the three s lidin g mechanisms are needed in order to determine the seismic induced permanent displacements. The critical accelerations are found by setting the factor of safety equations of the three sliding mechanisms equal to unity. The factor of safety against the base s lidin g FS s l is : FS 1= R s = 1.0 (2. 53) s P1R +PAH +0.5M>AEH where R s = the frictional resisting force mobilized along the sliding boundary at the base of the wall structure P[R = the seismic inertial force due to OSH in length of the reinforced soil mass and AEH = the seismic thrust increment. Similar to the FS sl, the factor of safety against internal sliding FS sii at depth z is: FS sii = R s (z) = 1.0 (2. 54) (z) + P AH (z) + AEH (z) The factor of safety against block interface shear between facing column units FSsc is : FS = Vu(z) = =1.0 (2.55) s c S i (z) k h (int) W w (z) + P A H (z) + AEH (z) LTt o tai where V u(z) = the peak shear capacity of the facing column interface which is dependent of the minimum available interface shear strength (au) and the apparent peak interface friction angle between facing units (Au), Si(Z) = outofbalance shear force transmitted through facing unit interface W w(z) = weight of the facing column above the sliding interface and LT1 01aJ = sum of the horizontal forces carried by the reinforcement layers above the interface of interest. Note that both kh and kv are incorporated in the factor of safety equations and with the assumed kv and FS = 1.0 the critical acceleration in the horizontal direction ac = kcg can be back calculated. 49
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The permanent displacements are assumed to accumulate each time the ground acceleration exceeds the critical acceleration, where the Newmark's double integration method is used to calculate the permanent displacement. In Newmark's doubleintegration method the potential sliding soil mass is treated as a rigidplastic monolithic mass subjected to inertial force. Figure 2.31 shows an example of the Newmark's doubleintegration from the acceleration time history to the velocity time history and finally to the displacement time history. The shaded area in Figures 2.31 indicates deceleration The total displacement at the wall face is determined in an accumulative manner fr om the wall base to the topmost layer of reinforcement. The displacement of a layer is determined as the larger of the internal sliding and the facing shear displacement. 
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rigid blocks and possesses a bilinear failure plane ; the top two blocks are r e ctangular and the bottom block is triangular. There are a total of four degrees of freedom for the proposed failure mechanism where the top two blocks only have the translational degrees of freedom and the bottom block has both translational and rotational degrees of freedom. The length of the top blocks is assumed to be equal to the length of the reinforcement. The schematic of the multi block method is shown in Figure 2 32 In the proposed multiblock model the slope of the failure plane a and the thickness D are variables Through an iterative approach the combination of a and D* that yields the largest displacements are considered as the permanent displacements. The forcemassacceleration method was used to obtain the equations of motion In the verification study (verifying the computational method with the seismic centrifuge test results) the acceleration time history in each block is calculated by multiplying the base acceleration time history by the amplification factor determined from the seismic centrifuge test results (see Figure 2.13) The displacements of the blocks are determined using the Newmark's doubleintegration method. As reported by Siddharthan et al. the calculated displacements using the proposed method were in good agreement with the measured displacements BLOCK I BLOCK II H a (Variable) D* (Variable) Figure 2.32 Schematic of the Multiblock Displacement Method (after Siddharthan et al. 2004b ) 51
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Sliding Block Method by Ling et al. The twopart wedge mechanism has been used to determine the reinforcement length of the base layer of a reinforced steep slope (Leshchinsky 1997) The twopart wedge mechanism is further considered in determining the seismic induced permanent displacement of a reinforced steep slope (Ling et al. 1997). The schematic of the twopart wedge mechanism is shown in Figure 2.33 The displacement evaluation procedure is similar to the base sliding approach proposed by Cai and Bathurst (1996a) in which the reinforced soil zone is treated as a rigidplastic block. The displacement of the rigidplastic block is induced when the factor of safety against direct sliding is less than unity The factor of safety equation is first used to determine the critical acceleration (or yield acceleration) and with a known design earthquake motion time history Newmark's doubleintegration method is then used to find the cumulative permanent displacement of the rigidplastic block. Wedge B Wedge A FS d s = factor of safety a g ainst direct s lidin g FS = TB d s P co s 8 Figure 2 .33 TwoPart Wedge Mechanism for Direct Sliding Analysis (after Leshchinsky 1997) 2.4.5 Empirical Methods Newmark's doubleintegration method in finding the seismic induced permanent displacement requires the ground motion time history to be known. In absence of the ground motion time history several empirical methods have been 52
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developed to predict the seismic induced permanent displacement of earth structures. Newmark's sliding block theory has been used as the basis for developing the empirical methods where the total permanent displacement determined by Newmark's doubleintegration method is correlated with input ground motion parameters such as peak ground acceleration (km'g) peak ground velocity (v m ), and critical acceleration ratio (kJkm ). This section describes the empirical methods of Newmark (1965), Richards and Elms (1979) Whitman and Liao (1984) and Cai and Bathurst (1996b). Empirical Method by Newmark Four U.S. west coast earthquakes with peak acceleration ranging from 0.178 g to 0 .32 g were selected and normalized by Newmark (1965) with each having a peak acceleration of 0.5 g and a peak velocity of 762 mrn/s (30 in.ls). These normalized earthquake motions were used to determine the displacement of sliding of rigid plastic mass in attempt to simulate the block sliding of an embankment or earth dam The standardized maximum seismic induced displacements were plotted against the critical acceleration ratio (kJkm). Both the critical acceleration ratio and the peak ground velocity were used in the prediction of the standardized displacement. In the case of unsymmetrical displacement the upper bound limits of the standardized displacement are: 2 d s 2kc g (for kJkm < 0.16) (2.56) 2 d = (for k Jkm > 0.16) (2.57) s 2kc g k c where ds = standardized displacement (m), Vm = peak ground velocity (m/s), g = gravitational acceleration (9.81 m/s2 ) km = peak ground acceleration coe fficie nt and kc = critical ground acceleration coefficient. Furthermore, as proposed by Franklin and Chang (1977), the standardized displacement ds can be converted to the actual permanent displacement d by : 2 d = ds 0.8:V m (2.58) m where ds and d are in unit of meter. Empirical Method by Richards and Elms Richards and Elms (1979) fitted an upper bound curve to the standardized displacement results integrated by Franklin and Chang (1977) for more than 196 strong motion records. The upper bound is given by the expression: 53
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(2.59) where d = total permanent disp l acement (m) Vm = peak ground vel ocity (m/s) g = gravitational acceleration (9 .81 mls2 ) km = peak ground acceleration coefficient and k c = crit ical ground acceleration coefficient. Since the above expression is an upper bound curve the predicted permanent displacement may be conservative as compared to other empirica l methods. As suggested by Richards and Elms an alternative seismic design approach for the retaining wall is to consider its allowable permanent displacement. In this alternative design approach an allowable permanent displacement is estimated first based on the function of the retaining wall (e.g ., 50 to 100 mm for typical retaining walls). Using the d i splacement equation above and with prescribed peak acceleration coefficient km and peak ve l ocity vm the critical acceleration coefficient k c is back calculated The seismic thrust using the MO method along with back calculated k c can then be determined The wall dimensions are to be sized based on satisfactory factors of safety against various modes of failure (e.g base sliding) using the seismic thrust just determined. Em p ir ic a l Met hod b y W hi t m a n and L i a o Whitman and Liao (1984) performed the regression analysis based on the standardized displacement data performed by Franklin and Chang (1977 ) and proposed the following equation to predict the mean displacement of a sliding gravity retaining wall: d = 37 v m 2 exp((2.60 ) km g km where d = total permanent displacement (m) Vm = peak ground velocity (m/s) g = gravitational acceleration (9.81 m/s2 ) km = peak ground acceleration coe f ficient and k c = critical ground acceleration coefficient. Whitman (1990) listed the potential errors and uncertainties associated with the permanent displacement prediction based on the sliding block method and hence the empirical method. E xamples of uncertainties and errors include : (1) unpredictable details of ground motion (e g ., frequency content duration and directiona l ity) (2) material parameters (e .g., friction angle of backfill and interface friction ang l e between backfill and wall) and (3) model errors (e g. deformabi l ity of backfill and wall ti l ting) Empir ic a l Met hod b y Cai and B a t h urst More recently Cai and Bathurst (1996b) have reexamined the permanent displacement results of Newmark (1965) and Franklin and Chang (1977) and proposed a mean upper bound curve for predicting the permanent displacement as: 54
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d = 35 v .m 2 exp(6.91 k c J 0.38 km g km km (2.61 ) where d = total permanent displacement (m) VOl = peak ground velocity (m/s) g = gravitational acceleration (9.81 m/s2 ) kOl = peak ground acceleration coefficient and k c = critical ground acceleration coefficient. This mean upper bound curve is meant to reduce the conservatism involved with the upper bound of Richards and Elms (1979) and to avoid potential underestimate of displacement from the mean curve of Whitman and Liao (1984). Figure 2.34 compares the permanent displacement estimated by the various empirical methods with the assumptions of peak velocity VOl = 762 mm/s (30 in ./ s) and peak ground acceleration coefficient kOl = 0.5 ( the same as those used by Newmark 1965). 1 00 ,.... .... E '' 10 .... s::
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2.5 Behavior of Geosynthetics subject to Cyclic Loads The loaddeformation behavior of geosynethetics subject to cyclic loads is dependent of the loading frequencies and the load amplitudes. Bathurst and Cai (1994) performed a series of inisolation cyclic loadextension tests on high density polyethylene (HDPE) and polyester (PET) geogrid specimens in which the specimens were tested at five different loading freq uencies (e .g. 0.1, 0.5 1.0 2.0 and 3.5 Hz) and over a range ofload amplitudes Figure 2.35 shows a typical load deformation curve of the HDPE geogrid specimens under multiincrement and single increment cyclic loadings. There were five load amplitudes applied in the multi increment cyclic loading where the amplitudes ranged from about 20% to 80 % of the ultimate strength and that each load amplitude was applied for 10 cycles For the singleincrement cyclic loading a single load amplitude at approximately 80% of the ultimate strength was applied for 10 cycles. 70 60 E Z 50 .... U
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characterize the loaddeformation curve as a function of strain A nonlinear hysteresis loaddeformation loop for each unloadreload cycle (cur, Tur) is defined by the average unloadreload stiffness (Jur) of the unloadreload cycle and the contained area (Aur). A loadd eformation cap (or envelope) that is tangent to the peak response of the initial unloadreload cycle at each loading stage can be quantified by an initial tangent stiffness (Ji) and secant stiffness values at selected strain values (e.g. Jscc2 and JsecS)' average unloadreload stiffness Jur I I initial stiffness / Jsec5 of loadstrain cap J; Js c2 / T o 2 C (%) I I Cur 5 loadstrain cap area of hysteresis loop .Avr Figure 2.36 Characteristics of Cyclic Response of Geosynthetic Specimen (after Bathurst and Cai 1994) The area of a hysteresis loop (Aur) was found to be strongly influenced by the strain level and the frequency of loading. As shown in Figure 2.37 the area Aur increases with the strain level and decreases with increasing frequency Note that below 0.5% strain (with load amplitude at approximately 12% of the ultimate strength) of the HDPE geogrid and 0 .8% strain (with load amplitude at approximately 15% of the ultimate strength) of the PET geogrid the specimens behaved in a linear elastic manner with fully recoverable strain (i.e ., the area of the hysteresis loop is nearly zero). Figure 2.38 shows the average unloadreload stiffness Jur versus strain relationships for different load amplitude and frequencies. For the HDPE specimen Jur is influenced by the strain level but is essentially independent of the frequency of load above 0.5 Hz. At frequencies greater than 0.5 Hz and strains greater than about 2%, Jur ofHDPE specimen decreases with increasing strain (see Figure 2 38a). The JUT of PET specimen on the other hand is relatively insensitive to the frequency of loading but varies with the strain level. At strain level of grater than about 3%, Jur increases with increasing strain (see Figure 2.38b) Similar hysteresis behavior of 57
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geosynthetics subject to cyclic loading has also been reported by Moraci and Montanelli (1997). Au, (kN/m)3 HOPE o 2 3 4 5 6 7 8 Strain ur (%) (a) HDPE specimens Au, (kN/m) 3 o PET mulll increment cyclic load tests o singleIncrement cyclic load tests Au, I I I e (%l Eur 0 1 Hz 2 3 4 5 6 7 8 9 10 11 12 Strain Cur (%) (b) PET specimens Figure 2.37 Area of Hysteresis Loops (Aur) for HDPE and PET Specimens during the Multiincrement Cyclic Loading (after Bathurst and Cai 1994 ) cyclic loa d l esls o singlelncremenl cyclic l oad lests 4500r, HOPE X 0 1 Hz 0 5 Hz + 1.0 Hz 2 0 Hz 3 5 Hz (0. 1 Hzr o 1 2 3 4 5 678 Strain Cur (%) ( a) HDPE specimens PET E (%) o 1 2 3 4 5 6 7 8 9 10 1 1 12 Strain Cur (%) (b) PET specimens Figure 2.38 Unloadreload Stiffness (Jur) for HDPE and PET Specimens from the Multiincrement Cyclic Loading (after Bathurst and Cai 1994 ) 58
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In addition to the inisolation cyclic loadextension tests Bathurst and Cai (1994) also conducted inisolation monotonic loadextension tests on the HDPE and P ET geogrid specimen with various strain rates The results of monotonic load extension test at various strain rates are shown in Figure 2.39. As indicated in F igure 2.39 the HDPE specimen is more sensitive to the strain rate then the PET specimen, and that the secant stiffness increases with increasing strain rate. Note that the monotonic loadextension test has commonly referred to as the ASTM D 4595 Wide Width Strip Test (Standard Test Method for Tensile Properties of Geote xti les b y the WideWidth Strip Method) where the strain rate is specified at 10% strain/min The loaddeformation relationship from the widewidth strip test particularly the ultimate strength, is often used in the design of reinforced soil structures T (kN / m) eSlin.ted from inltill cycte 01 0 1 Hz single load amplitude cyclic lest estimated from Initial cycle of 1 0 Hz single load amplilude cyclic teet 1 % stra in/min PET 1050% strain/m in" o 2 3 4 5 6 7 8 9 10 11 12 strain t (%) Figure 2.39 Influence of Strain Rate on Monotonic Loadextension Behavior of Typical Geosynthetic Specimens (after Bathurst and Cai 1994) The loaddeformation relationship obtained from the index test of ASTM 0 4595 is generally not applicable to the response of geosynthetics subject to cyclic loading and cyclic responses of different geosynthetic materials can be markedly different. However, as shown in Figure 2.39, higher strain rate yields higher reinforcement stiffness, and since rates of loading under seismic excitation are greater 59
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than those employed in the standard laboratory tensile tests (e.g ASTM D 4595) thus the seismic design of reinforced soil structures based on the reinforcement properties found using the standard laboratory tensile tests would consider to be conservative. Furthermore at small strains (0.5 % to 0.8%) the area of hysteresis loop found in cyclic loading is nearly zero hence the geosynthetic materials is said to exhibit a linear elastic behavior at small strains 2.6 Soilreinforcement Interface Friction The soilreinforcement interface friction can be determined by using either direct shear test or a pullout test. It has been shown that the strain rate and the number of repeated loading have no significant effect on the soilreinforcement interface friction when a granular soil is employed (Myles 1982 and O'Rourke et al. 1990) Figure 2.40 shows the variation of interface friction angle under repeated loading utilizing standard direct shear test apparatus and only small variation was observed. Figure 2.41 shows the interface friction angles between sand and various geotextiles at different strain rates also using standard direct shear device where the interface friction angle was found to be insensiti v e to the strain rate. Hence the monotonic loading standard direction shear test is considered to be appropriate for determining the soilreinforcement friction angle in the design of a geosynthetic reinforced soil structure. Normal Sfre.. IT I kPa (a) 40'.. .., y,. 16. 3 17. 0 !i 1 30 20 20 It Number 01 Ropeoltd Loodinql (b) Figure 2.40 Interface Friction Angle between Ottawa sand and HDPE Specimens with (a) Monotonic Loading and (b) Repeated Loading on HDPE Lining ( after O'Rourke et al. 1990) 60
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kN/m2 200 kN/ml 200 40 SAND/SAND NORMAL STRESS SAND/HEAVY\\EIGHT WOVEN 380 kN/rnZ 200 SAND/NEEDLEPUNCHED kN/m2 200 NORMAL S TRESS SAN O/LICHT WEIGHT WOVEN 4 40 NORflAL STRESS 40 NORl1AL STRESS ANGlES OF FRICTIOII BETWEEN SAND GEOTEXTI LES STEEL SAND LEIGHTON BUZZARD SAND 0.85 0.60 11111 (US SIEVE 20 30) VOID RATIO UNDER TEST 0.60 0.65 ioo kN/m2 kN/m2 200 kN/ml 200 SAIID I HEAT (JONDED NORl1AL STRESS SAND/POLISHED STEEL 40 tlORI1AL STRESS RATES Of STRAIN A 10 .... /mln o 75 .... /ml n C1 75 mm/mln II Figure 2.41 Effect of Strain Rate on the Interface Friction Angle between Sand and Various R einforcement (after Myles 1982) 61
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3. Validation of Computer Program This chapter presents the results of numerical simulation of threefullscale geosynthetic reinforced soil walls that were seismically loaded by a shaking table. The numerical simulations were intended to validate the finite element computer program LSOYNA. Terms associated with the numerical simulation are first defmed Material model parameters were determined from the available laboratory data. Particularly, the backfill was simulated with a cap model with parameters dependent of stress level. Hardening parameters of cap model were determined from hyperbolic relation derived from the relevant hydrostatic compression tests. A discussion on the calibration of modeling parameters is presented. Responses compared include: (1) maximum wall displacement (2) maximum backfill settlement (3) maximum lateral earth pressure (4) maximum bearing pressure (5) maximum reinforcement tensile load (6) absolute maximum acceleration in reinforced soil zone, and (7) absolute maximum acceleration in retained soil zone Quality of simulations was also evaluated and is discussed. 3.1 Terminology More than ever numerical simulation of a physical model or process is performed for all disciplines of engineering on a routine basis The popularity of numerical simulation nowadays could be attributed to easy accessibility of computer programs and the ability to include model details. Numerical simulation fulfills engineering enquiries such as to make quantitative predictions to compare alternatives to identify governing parameters, design limitations, and modes of failure. In addition numerical simulation draws great attention since it (1) provides indepth understanding of physical process, (2) is more economical model than a physical model (3) allows for parametric study (4) involves no safety concern for personnel performing the simulation (5) contains known solutions in the domain of interest, and (6) allows for different boundary and initial conditions that are not easily achievable in a physical model (Krahn 2004a). It is noted in the geotechnical engineering literature that "verification" and "validation" are sometimes interchangeable. However particularly for numerical simulation distinction between the two needs to be made in order to conform to the simulation community for an effective communication. The terms of verification validation prediction and calibration are defined explicitly in this section. 62
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3.1.1 Verification and Validation Potential use of the results from numerical simulation especially for decision making or engineering design is related to the confidence that one might have in the simulation. An approach to establish the confidence or the credibility of the numerical simulation is through the verification and validation (V & V) process. The American Institute of Aeronautics and Astronautics (AIAA 1998) has defined verification as "The process of determining that a model implementation accurately represents the developer's conceptual description of the model and the solution to the model." and validation as "The process of determining the degree to which a model is an accurate representation of the real world from the perspective of the intended uses of the modeL" More concisely Oberkampf et al. (2002) have defined verification as "the assessment of the accuracy of the solution to a computational model" and validation as "the assessment of the accuracy of a computational simulation by comparison with experimental data." Oberkampf and his associates further indicated that "In verification the relationship of the simulation to the real world is not an issue. In validation the relationship between computation and the real world i.e ., experimental data is the issue." The process of V & V proposed by Schlesinger (1979) is shown graphically in Figure 3.1, in which the reality can be thought of as the experimental measurements or observed response of a structure. The reality is interrelated to the conceptual model and the computerized model. The conceptual model is composed of mathematical representations including equations and data of physical process of interest. Some examples of conceptual model are: (1) partial differential equations (PDE's) for conservation of mass momentum and energy (2) initial and boundary conditions of the PDE's and (3) constitutive models for materials (Oberkampf et al. 2002). The computerized model on the other hand is a computer program or code complied based on the conceptual model. As shown in Figure 3 1 verification is performed between the conceptual model and the computerized model whereas validation relates the outcomes of computerized model to the experimental measurements. According to these definitions verification is the responsibility of computer code developer while validation is to be performed by the computer code user (e .g., an analyst or engineer). Examples of verification activity include numerical algorithm verification software quality assurance and numerical error estimation ; on the other hand conducting validation experiments and performing confidence assessment are examples of validation activity (Oberkampf et al. 2002). Therefore as the computer code users this study concerns only the validation activities. 63
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REA LITY < ............ I Analysis M o d e l qualification I ................. : Computer simulation I / / / / Model validation I I I I ,///// COMPUTERIZED MODEL Model verification Figure 3 1 Phases of Modeling and Simulation (after Schlesinger 1979) v &V process is needed in order for a computer program to establish its predictive capability and this process requires interaction between the experimental and the computational activities The interaction concept has also been recognized in geotechnical engineering as is evident by the soil mechanics triangle illustrated in Figure 3.2 where three aspects of soil mechanics (i .e., ground profile, soil behavior and modeling) suggested by Burland (1996) are interlinked. Note that the soil mechanics triangle concurs with the phases of modeling and simulation as depicted in Figure 3.1 in the way that modeling is interlinked with reality where reality in geotechnical engineering is represented by soil behavior and ground profile. Burland has emphasized that modeling and reality (e.g. experiments) in the soil mechanics triangle must be linked closely in order to achieve advancement. A meaningful simulation is founded on continuous interactions with the reality and improvement of simulation can thus be achieved through continuous cycles of interactions. Note that the completion or sufficiency ofV&V process is often vaguely defined since it depends on practical issues such as financial constraint and the intended uses of the model. 3.1.2 Prediction Types of prediction in geotechnical engineering has been defined by Lambe (1973) as either being a Class A (before event) Class B (during event) or Class C (after event) prediction. A more stringent definition of prediction pertaining to numerical simulation which is adopted in this study is given by AlAA (1998) as "use of a computational model to foretell the state of a physical system under conditions for which the computational model has not been validated." The prediction definition by AlAA coincides with Class A prediction suggested by Lambe. The usage of prediction by AIAA is less general as it excludes the precedent comparison of computation results with the experimental data (i.e. validation). The 64
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relationship between validation and prediction is delineated in Figure 3.3 where the prediction is linked by the dashed lines and validation by the solid lines. Note that prediction as defined by AlAA could not indicate the accuracy of a complex system that has not been validated ; rather the accuracy could only be inferred based on the previous quantitative comparison Lab/field testing observation measurement Genesis/geology experience Site investigation soil description Idealization followed by evaluation Conceptual physical or analytical modeling Figure 3.2 The Enhanced Soil Mechanics Triangle (after Anon. 1999) PREDICTION Exp erimental Out comes Di fTerences Between Computation and Experiment Figure 3.3 Relationship of Validation to Prediction (modified from Oberkampf and Trucano 2002) 65
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3.1.3 Calibration Often times a great amount of effort could be allocated in model "calibration." As defined by AIAA (1998) model calibration is the explicit fine tuning of the unknown parameters in the computational model to achieve some level of agreement with the experimental data. With such definition model calibration is then equivalent to the Class C prediction suggested by Lambe (1973), which is an after event activity. In addition calibration allows the identification of those input parameters that could significantly a f fect the result of numerical simulation. Stated differently the sensitivity of the parameters is evaluated in the calibration process A successful calibration could hence establish appropriate values of the parameters when making future prediction. 3.1.4 Validation Assessment Validation of a computer program essentially involves the comparison between the calculated result and the experimental data. Oberkampf et al. ( 2002) have suggested various "validation metrics" to quantitatively compare the computational results with the experimental data in which the word "metric" implies "measure." Ex amples of validation metric with progression of confidence are as shown in Figure 3.4. The most primitive metric (see Figure 3.4a) hence with the least confidence is the viewgraph comparison where experimental data and calculation are presented side by side At the other end of spectrum high quality metric would (see Figure 3.4f) account for both uncertainty and error involved in computationexperiment activities It should be noted that although high quality metric is desirable it is very difficult to produce especially in geotechnical engineering modeling that involves great extent of uncertainties. The metric applicable to this study falls in the deterministic category (i.e. Figure 3.4b) since experimental uncertainties and computational errors could not be evaluated quantitatively As has been described by Starfield and Cundall (1988 ), geotechnical modeling falls into the class of "datalimited problems in which the relevant data are unavailable or cannot easily be obtained. Starfield and Cundall have suggested that it may be impossible to validate a geotechnical model due to the inherent complexity. Starfield and Cundall further indicate that the expectation of what geotechnical modeling can achieve should be different than the other braches of mechanics that possess the wellposed problems (e g., aerospace and structural mechanics ) Improvement from a datalimited problem to a wellposed problem can be achieved through the "adaptive modeling which is a process that requires continuous 66
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interaction between the measured data and the calcu l ated results simi l ar to the cycles p r esen t ed in Fig u res 3.1 an d 3 .2. experiment computation (a) Viewgraph Norm + experiment + computation 51 c: 8. (I) e input (d) Numerical Error experiment .... computation Ir input (b) Deterministic + experiment +computation CI) r (I) c:: 8. Xl "input (e) Nondeterministic Computation I experiment .... computation CI) yYri (I) c: 8. Xl "input (c) Experimental Uncertainty "E CI) E c 8. an c:: 0 @ input E :::J a. E 0 0 (f) Quantitative Comparison Figure 3.4 Increasing Quality of Validation Metrics (after Oberkampfet al. 2002) 3.2 M at er i a l C h a r acterizat ion for C omput e r Pr o gra m Va lid a tion LSDYNA is a general purpose nonlinear threedimensional finite element computer code (Hallquist 1998) LSDYNA can be used to analyze large deformation dynamic response of solids and structures. The code utilizes explicit central difference method to integrate the equations of motion in time The available elements include 4node tetrahedron and 8node solid elements 2node beam elements 3and 4node shell e l ements 8node thick she ll elements and ridge bodies LSDYNA includes many builtin material models to simulate a wide range of material behaviors including elasticity plasticity composites thermal effects and rate dependence. Also included in LSDYNA is the sophisticated contact interface capability The contactimpact algorithm can solve difficult contact problems such as frictional sliding with closure and separation single surface contact arbitrary mechanical interactions between independent bodies and in draw beads in metal 67
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stamping applications. The companion preprocessor etaIVPG (ETA 2004) was utilized in this study to generate input files of LSDYNA. In essence validation of a computer program requires the comparison between the calculated results and the measured data from a physical model test. In geotechnical earthquake engineering the commonly performed physical model tests include dynamic centrifuge test, reducedscale and fullscale shaking table tests As mentioned earlier there are known limitations within the dynamic centrifuge test such as particle size effect and boundary restrictions. It is also known that since soil behavior is stress dependent the reducedscale shaking table test in 1 g gravitational field may not reflect the true behavior of the fullscale prototype. Consequently the measured performance of the reducedscale model may not be directly applicable to the fullscale prototype The shaking table tests performed by Ling et al. (2005a) were selected as the validation tests in this study since they are fullscale models and that detail descriptions of backfill reinforcement and interface properties were provided. Materials considered in the numerical modeling include concrete facing block expanded polystyrene (EPS) board geogrid reinforcement and backfill. Concrete facing blocks and EPS boards were simulated as linear elastic materials and their properties are presented in Table 3 .1. The facing block was assigned with typical concrete elastic properties. The density p of EPS board of 15 kg/m3 was based on ASTM C578 Type I material. The maximum shear modulus of EPS is approximately 2.76 MPa. With an assumed Poisson's ratio v of 0.4 the Young's modulus E ofEPS was calculated to be 7 72 MPa. Geogrid reinforcement and soil are simulated by the plastickinematic model and the geologic cap model respectively. Note the RambergOsgood material model (i. e., a nonlinear elastic model) could also be used to simulate soil. Since the failure envelope could not be defined RambergOsgood model was not used in this study For historic reference though the description of RambergOsgood model is given in Appendix A. Derivation of the material model parameters based on material behaviors for geologic cap model and plastickinematic model are presented as follows. 3.2.1 Geologic Cap Model The backfill is represented by the geologic cap model. The cap model is situated in the A: II stress space where II is the first invariant of the stress tenor and h is the second invariant of the deviatoric stress tensor. I I and h are defined by Equations 3.1 and 3.2 respectively as: I I = 0xx + 0yy + 0zz = 0I + 02 + 03 (3.1 ) 1 2 = ![cOxx _oyy) 2 +(oyy Ozz) 2 +(Ozz (3.2) 6 68
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The schematic of the cap model is shown in Figure 3.5 in which the cap model is comprised of a fixed yield surface fl and a yield cap f 2 The fixed yield surface fl is considered as the failure surface where region above the failure surface is not permissible. Associated flow rule is implemented in the cap model in which the plastic strain increment vector is normal to the yield surface and that the plastic deformation is associated with the yield surface of the material (i.e. the yield surface coincides with the potential surface). The expression for fl' originally adopted by DiMaggio and Sandler (1971) and later modified by Sture et al. (1979) is given as : f = A +yeP't 81, u =0 (3.3 ) where u p and y are the material parameters. The yield cap f 2 is a moving yield surface. The moving yield cap follows the shape of an ellipse and is represented by: f 2 = R 2 J 2 + (I C)2 = R 2 b2 (3.4) where R is termed the shape factor and is the ratio of the major axis to minor axis of the ellipse and R b = (X C). Note that X is the value ofll at the intersection ofthe yield cap and the IIaxis C is the value ofll at the center of the ellipse and b is the value of A when II = C (see Figure 3 5) X is a hardening parameter that controls the change in size of the moving yield surface and the magnitude of the plastic deformation and X depends on the plastic volumetric strain through: X = )+X D W 0 (3. 5 ) where D W and X o are the material parameters. Note that W characteri z es the ultimate plastic volumetric strain D denotes the total volumetric plastic strain rate and Xo determines the initiation of volumetric plastic deformation under h y drostatic loading conditions (Zaman et al. 1982). X o can also be thought as the preconsolidation hydrostatic pressure. The stress state inside the yield surfaces is considered to exhibit elastic behavior. The cap model parameters with their physical significance are summarized in Table 3.2. Note that the descriptions of cap model along with other geologic constitutive models under general loading conditions have been summarized by Ko and Sture (1981) The stress paths needed to fully describe the yield surfaces of the cap model are delineated in Figure 3 6. Although it is desirable to use different stress paths to probe the yield surfaces laboratory tests other than the conventional triaxial compression test (CTC) are seldom performed ; as is the case for the fullscale shaking table test where only the CTC test results are available. The determination of cap model parameters is discussed in the following sections. Due to insufficient laboratory test results some assumptions were hence introduced in the process of determining the cap model parameters 69
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a (a y) / / / / ",," /:;/ ; / / / / ,/;' EP _/ Transition Initial cap c f, (Fixed yield surface) f 2 (Elliptical yield cap) I a = R b x Figure 3.5 Schematic of Cap Model (modified from Desai and Siriwardane 1984) Triaxial compression (TC) } Conventional triaxial compression (CTC) Lode angle = 0 Reduced triaxial compression (RTC) Triaxial extension (TE) } Conventional triaxial extension (CTE) Lode angle = 60 Reduced triaxial extension (RTE) Simple shear (SS): Lode angle = 30 TC TE SS HC (a) 'crl Lode angle ''TE SS (b) /' /' /' /' /' /' Figure 3 6 Stress Paths achieved by Various Laboratory Tests in (a) .j'i';: II Stress Space and (b) Deviatoric (Octahedral) Plane (modified from Chen and Saleeb 1994) 70
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Table 3.1 Summary of model parameters for facing block EPS board and geogrid reinforcements Material Facing block EPS Board PET PAY reinforcement reinforcement Material Linearelastic Linearelastic PlasticPlasticmodel kinematic kinematic Parameters p = 2320 kg/m3 p = 15 kg/m3 p = 1030 kg/m3 p = 1030 kg/m3 E =250Pa E = 7 72 MPa E = 433 MPa E = 464 MPa v = 0.15 v =O.4 v = 0.3 v = 0.3 cry = 4.33 MPa cry = 4.64 MPa Et = 162 5 MPa Et = 304 MPa Table 3.2 Cap model parameters for Tsukuba sand Parameter Mass density p BuLk modulus K[ Shear modulus G Failure envelope parameter a Failure envelope linear coefficient e Failure envelope exponential coef. y Failure envelope exponent p Shape factor R Hardening law exponent 0 Hardening law coefficient W Hardening law exponent X o Value 1596 2 kglm3 Vary with p (see Fig. 3 .18) Vary with p (see Fig 3.18) 4 .844 kPa 0 2872 o kPa o (kPar' Vary with p (see Fig. 3.20) Vary with p (see Fig. 3 .18) 0 0342 o kPa 3.2.1.1 Cap Model Strength Parameters Physical significance Elastic properties Parameters for defining the fixed yield surface fl. Parameters for definin g the moving yield cap f2 and the hardening function The sand used in the shaking table test (i.e., Tsukuba sand) was subjected to a series of CTC drain tests. The laboratory stressstrain curves under three confining pressures presented by Ling et al. (2005a) are shown in Figure 3.7. The peak deviatoric stress and the corresponding confining pressure were then used to construct the Mohr circle at failure (see Figure 3.8). The MohrCoulomb fai lure envelope based on the three Mohr circles at failure indicates a drained friction angle of 36.7 and a drained cohesion c' of 4.19 kPa. Note that the friction angle reported by Ling et al. (2005a) is 38, which is slightly higher than the based on the bestfit failure enve lope is considered to be a design value. For the purpose of numerical simulation, it is thought that the shear strength parameters based on the bestfit failure envelope is more representative of the actual backfill placed in the shaking table test 71
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than the design value The design friction angle along with zero cohesion on the other hand would be better suited in a routine design It was assumed that y in Equation 3.3 is zero so the curved transition portion coincides with the linear portion of the fixed yield surface. It was further assumed that P in Equation 3.3 is also zero so the fixed yield surface at high stress range remains linear. With y and P being zero Equation 3.3 is reduced to : f) = A 81) a = 0 (3.6) Parameters a and 8 now become the A intercept and the slope of the assumed linear fixed yield surface respectively. Note that Equation 3.6 takes the same form as the DruckerPrager (also known as the extended von Mises) failure criterion which requires only two parameters. With apexes of the DruckerPrager and Mohr Coulomb surfaces matching on the space diagonal (hydrostatic axis) and depending on the meridian chosen the two shear strength parameters c' and from the Mohr Coulomb irregular hexagonal pyramid can be converted to a and 8 parameters of the DruckerPrager cone For the compressive meridian with the two failure surfaces matching at Lode angle of 0, the equations relating the two failure criteria are (Chen and Saleeb 1994) : 6 a = (3.7a) .J3 (3 sin 8= (37b) For tensile meridian with the two failure surfaces matching at Lode angle of 60, the relations between the two failure criteria are (Chen and Sleeb 1994): 6 a = (3.8a) .J3 (3 + sin 8 = (3. 8b ) .J3. (3 + sin The MohrCoulomb criterion with of 36.7 and the two DruckerPrager criteria matching the compressive and tensile meridians of the MohrCoulomb criterion in the deviatoric plane are shown in Figure 3.9. For the compressive meridian matching condition the DruckerPrager surface circumscribes the MohrCoulomb surface. Since only CTC test results are available it is assumed that a and 8 of Tsukuba sand are governed by Equations 3.7a and 3 7b of the compressive meridian matching condition. The values of a and 8 following Equations 3.7a and 3.7b are calculated as 4 84 kPa and 0.2872 respectively. Figure 3.10 shows the stress paths for the three drained CTC tests and the fixed yield surface or the failure envelope fl with parameters a and 8 as determined from above. It is observed that failure state of each CTC test lies in close proximity 72
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with the failure envelope ; hence the failure envelope just determined is thought representative of the Tsukuba sand. Note that the stress path for the CTC test in the A : II space has a slope of J3 1 3V: IH. 400 ,, 0"3 = 100 kPa ro 300 a.. C b 100 kPa w c iii 0 0 c Q) E 40 kPa ::J (5 1 > 70 kPa 100 kPa 2 0 5 10 15 20 25 Axial strain E a (%) Figure 3.7 Comparison between Calculated and Measured Triaxial Compression Results of Tsukuba Sand (measured data from Ling et al. 2005a) 73
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200 '' en en en $'= 36.7 o 50 100 150 200 250 300 350 400 450 cr' Effective normal stress (kPa) Figure 3.8 Effective Shear Strength Parameters from Drained CTC Tests for the Tsukuba Sand MohrCoulomb W = 36 .r) DruckerPrager (matching at Lode angle = 0; compressive meridian ) CJ3 DruckerPrager (matching at Lode angle = 60; tensile meridian) Figure 3.9 DruckerPrager and MohrCoulomb Failure Criteria in Deviatoric (Octahedral) Plane with Different Matching Conditions (modified from Chen and Saleeb 1994) 74
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300 ......... 200 ro a.. 6 li: 100 0.= 4 .84 o 100 200 300 400 500 600 700 11 (kPa) Figure 3.10 Stress Paths of CTC Tests and the Fixed Yield Surface fl of Tsukuba Sand 3.2.1.2 Cap Model Hardening Parameters The parameters D and Ware to be determined from the hydrostatic compression (HC) test. Since HC test results were not reported for the Tsukuba sand the Chattahoochee sand which has the same soil classification as the Tsukuba sand was used to determine D and W. According to the Unified Soil Classification System both soils are classified as poorly graded sand (SP). The HC test results of Chattahoochee sand at different initial relative densities have been reported by Domaschuk and Wade (1969) and AI Hussaini (1973) and those stressstrain curves are shown in Figure 3.11 The physical characteristics and the grain size distribution curves for Tsukuba and Chattahoochee sands are summarized in Table 3.3 and Figure 3 .12, respectively. Note that both sands are alluvial deposits and the Tsukuba sand has textural classification of fine sand whereas the Chattahoochee sand is a medium fine sand. As shown in Figure 3.11 the HC nonlinear stressstrain curves can be readily represented by hyperbolas The hyperbolic stressstrain relationships were developed herein to approximate the soil behavior under HC loading condition. The procedure to determine the hyperbolic parameters presented is similar to the procedure proposed by Duncan et al. (1980) The hyperbola relating the tota l volumetric strain E v to mean stress p can be represented by the following equation: 75
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E y = "P'l K + P I (EJasy where Ki = initial tangent bulk modulus and (Ev)asy = asymptotic total volumetric strain. As depicted in Figure 3.13 the values ofKi and (Ev)asy can readily be determined from the linear relationship of the transformed hyperbolic equation: (3.9) p 1 =K+ P E y I (Ey )asy (3.10) A linear regression curve can be drawn through the laboratory data of p/Ev plotted against p and hyperbolic parameters K i (intercept) and lI(Ev)asy (slope) can then be determined for each initial relative density Dr. Figure 3.14 shows the bestfit transformed hyperbolic curves of Chattahoochee sand and summarizes the parameters associated for each initial relative density. Note that the hyperbolic equation was formulated so that Ev is a function of p and p is the independent variable. However, to be consistent with the typical HC stressstrain curve presentation Ev is plotted along the abscissa and p is on the ordinate (see Figure 3.13). m a. a. .; 1;; c: '" Q) :; o 0 .01 0 02 0 03 0 04 0 05 Total volumetric strain E., (in fraction) Figure 3.11 Hydrostatic Compression Curves of Chattahoochee Sand at Different Initial Relati ve Densities (modified from Domaschuk and Wade 1969) 76
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S ize i n I nches U S Standard S ieve Ope n ing i n Inches SIEVE ANALYSIS I U S Standard S i eve Numbe r s #10 #40 HYDROMETER ANALYSIS T ime Readings 4 1 9 60 7 hr 25 h r 100 48" 24 12" 8 6 3 3/ 4 318" # #8 tL16 #30 #50 #100 #200 m i n min m i n m i n 15 m i n 45 m i n 90 80 I I If I 1 1 1 1 1 \ 1 I I 1 1 \ 70 0> 60 >eO> c 50 ., a. C 40 ., a. 30 H Tsukuba sand : [ 1 1 r( from Ling et al. 2005a) : I I \ : vi used in shaking table test 1 1 1 Chattahoochee sand 1 \ I (from AIHussaini 1973 ) : I "\ to determine D and W 1 1 ':\ \ 1 \ \ 1 1 20 I I I 10 0 1 1 1 \ 1 1 1 '" I I I 1220 610 305 152 75 37. 5 19 9 5 4 7 5 2 .36 1 .18. 6 3 .15 075 037 019 009 005 002 .001 20 3 2 .425 0 .01 0 005 0 .001 F i nes ( Silt Clay ) Figure 3.12 Grain Size Distribution Curves for Tsukuba and Chattahoochee Sands Table 3.3 Physical properties of Tsukuba and Chattahoochee sands (data from Ling et al. 2005a and AIHussaini 1973) Properties Tsukuba sa nd Chattahoochee sa nd (used in shaking table test) (to determine 0 and W) Specific gravity Gs 2. 67 2 .66 Maximum void ratio e max 1.2 9 1.09 Minimum void ratio emin 0 .7 8 0 59 050 (mm) 0 .2 7 0 .47 Coefficient of uniformity Cu 2 .0 2 1 Coefficient of curvature Cc 0 8 1.1 U nified soil cla ss ification SP SP 77
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(a) Real : (b) Transformed : p p E = v p p Figure 3.13 Hyperbolic Representation of Mean Stress versus Total Volumetric Strain Curve 800 ,, Dr= 18% 29 37 45 60 68 74 82 600 Iii' a. 6 a. vi II) 400 u; c '" Ql pI&., = K, + [1/(,') .. 1 p :; 0 K; ( .. l .. ( % l ( kPa ) (i n fraction ) 18 3008.6 0 .0606 200 29 3604 0 0 .0580 37 5063 1 0 0472 45 5478 3 0 0424 60 9039 0 0 .0354 68 9744 0 0 0272 0 .0240 0 .0213 o o 10000 20000 30000 40000 50000 Rat io pI&., ( kPa) Figure 3.14 Bestfit Transformed StressStrain Curves o f Chattahoochee Sand 78
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The HC nonlinear stressstrain curves vary with initial relative densities An increase in relative density will result in a steeper stressstrain curve (see Figure 3 11). The variation of hyperbolic parameters K i and (Ev)asy with initial relative density are exemplified in Figure 3.15 Note that Ki increases proportionally with Dr, whereas (Ev)asy decreases with increasing Dr. The relations ofKi and (Ev)asy to Dr can be represented by linear curves and the linear regression equations for relating the two parameters to D r with high R 2 values are shown in Figure 3.15 The established relations enables the determination ofKi and (Ev)asy given D r Using the calculated Ki and (Ev)asy together with the hyperbolic relation of Equation 3.9, the HC nonlinear stressstrain curve at a specific Dr can be defined The inferred HC stressstrain curve of Tsukuba sand used in shaking table test with initial relati ve density o f 54 % is as shown in Fi gure 3.11 'iii' a. C. UI '< 0 06 o c 3 0 04 7467 0 .0379 !!'. iil .S 4000 0 02 U K = 13776 7 x D + 27 34 R 2 = 0 975 a 2 0 40 60 80 100 Initial relative density, D (%) o 3 Figure 3.15 Variation ofInitial Tangent Bulk Modulus K i and Asymptotic Total Volumetric Strain (Ev)ult with Initial Relative Density for the Chattahoochee Sand F urthermore the loading and unloading behavior of HC stressstrain curve for a granular material is illustrated in Figure 3.16 (e.g Monterey No. 0/30 sand). The elastic component is generally recovered during the unloading path in which the test 79
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data presented by Goldstein (1988) suggests a nonlinear elastic behavior. As revealed by Figure 3.16 a single bulk modulus value may not be sufficient to describe the nonlinear elastic behavior. Based on this observation it is assumed that the bulk modulus would vary with the mean stress p. Additionally the bulk modulus is assumed to be the instantaneous slope of the stressstrain curve represented by the hyperbolic equation The tangent bulk modulus Kt can be derived by differentiating Equation 3 9 with respect to pas: dEy = 1 =K. / ( K + p J 2 or (3. 11a) dp K( I I (EJasy K ( = ( K i + P J 2 /Ki (3.lIb) (EJasy The elastic volumetric strain E : at any instant is then: (3.12) With E : from Equation 3 .12 and E y from the hyperbolic relation of Equation 3.9 the plastic volumetric strain at any given pis: EP=E Ee= P P v v v K + p i K( i I(EJasy (3.13) The total elastic and plastic curves generated using Equations 3 9 3.12 and 3.13 for Tsukuba sand at Dr = 54% are shown in Figure 3.17. Note that at low stress level the elastic contribution is more significant than the plastic contribution and the elastic contribution diminishes as the stress level increases. Kt also increases with increasing stress level. This agrees with the intuition that densely packed soi l exhibits higher stiffness than the loosely packed soil. 80
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.. a. c .. ::; 0 .004 0 008 0 012 0 016 0 .02 Total volumetric stra i n 6,. (in fractio n ) Figure 3.16 Loading and Unloading Behavior of Monterey No. 0 / 30 Sand during Hydrostatic Compression Test (modified from Goldstein 1988) (E, ) ... ( E, ) .., 600 500 400 a. m 300 ::; 200 100 a 0 .01 0 02 0 .03 0 04 0 05 Volume t ric strain (in fracti o n) Figure 3.17 The Mean Stress versus Total, Elastic and Plastic Volumetric Strain Curves for Tsukuba Sand at Dr = 54 % 81
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The hyperbolic relation of Equation 3.9 intrinsically provides an asymptotic total volumetric strain (Ev)asy, which occurs at p = 00. However the value of (Ev)asy is physically unattainable A more realistic ultimate total volumetric strain (Ev)ull was then assumed to be 9S% of the asymptotic value (i.e., (Ev)ull = 9S% (Ev)asy). This arbitrary assumption allows a tangible value of pull to be computed at th e ultimate total volumetric strain level (Ev)ull b y rearranging Equation 3.9 as : Pult = Kj (EJasy (EJult / [(EJasy (EJult] (3.14) Value of Pull can then be used to calculate the ultimate tangent bulk modulus (KDull from Equation 3 11b and the ultimate elastic volumetric strain (E:)ult from Equation 3.12: ( E e ) Pult v ult (K ) t ult (3.1S) (3.16) The calculated (KDull for Tsukuba sand is 2.99 GPa which is lower than the elastic bulk modulus of quartz sand grain that ranges from 38.0 to 47.0 GPa (Richardson et al. 2002) The calculated (KDull is deemed acceptable since Tsukuba sand is not a cemented soil. According to Zaman et al. ( 1982) most soils exhibit plastic deformation even at very low stress level hence it is reasonable to assume that Xo is negligibly small for most soils. So with X o assumed to be zero E quation 3.S can be rewritten as: 1 [ EP ) X=OLn (3.17) The value of X can be expressed in terms of mean stress pas: X=3p (3.18) B y substituting Equation 3.18 in Equation 3.17 the plastic volumetric strain can be expressed as: = W(Ie3PD ) (3.19) Thus cap model parameters D and W govern the plastic volumetric strain behavior. Equation 3.19 implies that E approaches the value of W when p is distinctly large. Consequently W is the ultimate value of at the strain level of (Ev)ult. With previously defined terms pult and (Kt)ulb W is computed as : 82 (3.20)
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Knowing the value ofW and by rearranging Equation 3.19 D at any stress level can be computed as: (3.21) where is determined using Equation 3 .13. The derivation given so far indicates that a sand with a given initial relative density would have a constant value of Wand mean stress dependent variables D and K t Furthermore by assuming a constant value of Poisson's ratio (i.e. v = 0.35) the shear modulus G as function ofKt can be computed as : G = 3 Kt (12v) 2 (1 + v) (3.22) Table 3.4 summarizes the parameters needed to determine K G and D Note that values ofK" G and D all increases with increasing mean stress. The mean stress within the backfill is assumed to be: p = (cr v + 2cr h ) / 3 (3.23) where cry and crh are vertical and horizontal stresses respectively. With the atrest earth pressure coefficient (K o = I unit weight of the backfill y and depth z the mean stress calculated as: p = (crv + 2 K o crJ/3 = y. z (3 (3.24) Figure 3.18 shows the variation ofK" G, and 0 with mean stress for a soil height of3 m which is also the height of the three shaking table test walls. Table 3.4 Summary of parameters for determining mean stress dependent variables 0 K" and G at Dr = 54% Parameters Initial relative density Dr Initial tangent bulk modulus Ki Asymptotic total volumetric strain (Ey )asy Ultimate total volumetric strain (Ey)ult Ultimate mean stress pult Ultimate tangent bulk modulus (Kt)ult Ultimate elastic volumetric strain (E: ) ult Ultimate plastic volumetric strain = W 83 Values 54% 7467 kPa 0.0379 0.0360 5381 kPa 2 99 GPa 0.0018 0.0342
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Hardening law exponent D ( MPa ) 1 o 0 .04 0 .08 0 12 o I .s:::. a. Q) o 2 3 2 4 6 8 10 Tangent bulk modulus (KJ and shear modulus (G) (MPa) Figure 3 .18 Variation of Tangent Bulk Modulus (Kt), Shear Modulus (0), and Hardening Law Exponent (D) with Depth The shape factor R was detennined using a numerical triaxial test through trialanderror procedure. The numerical triaxial test is comprised of one 8node solid element. The single e l ement model was devised using LSDYNA to predict the stressstrain behavior of Tsukuba sand under eTC loading. Figure 3.19 shows the dimension and the axisymmetric boundary conditions of the single element model. Also shown in Figure 3 19 are the stresses applied to the numerical triaxial test where 0'3 and 110' denote the confining stress and deviatoric stress respectively Using the 84
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aforementioned procedures cap model parameters were determined for each of the three confining pressures. The parameters along with R values for achieving the best fit stressstrain curves are summarized in Figure 3.19 The comparison between the calculated and the measured stressstrain curves are presented in Figure 3.7 in which good agreement is observed between the two. z
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12 10 a::: .: 0 t5 8 Q) a. ro : en 6 4 o 20 60 M 100 120 Mean stress p (kPa) Figure 3.20 Variation of Shape Factor R (Cap Surface Axis Ratio) with Mean Stress Table 3.5 Effect of hardening parameters on CTC stressstrain relationship Hardening parameter L\cr:Ea curve stiffness Volumetric strain E v R t ,j. t W t ,j. t Kt and G t t ,j. D t t ,j. 3.2.2 PlasticKinematic Model The geogrid reinforcement is simulated by the plastickinematic model in which four parameters are required to describe the model. The plastickinematic model possesses a bilinear stressstrain curve as shown in Figure 3.21. Although plastickinematic model could not fully describe the nonlinear behavior of geogrid material the bilinear aspect of the model could in part account for the strain hardening behavior observed during the tensile load test. The bilinear stressstrain approximation for geosynthetic reinforcement has also been adapted by Juran and Chirstopher (1988). 86
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cr E = Young' s modulus Et = Tangent modulus cr y = Yield stress Figure 3.21 Bilinear StressStrain Curve of the Plastic Kinematic Model for Geogrid Reinforcement To verify the materia l parameters, a single 4node shell element was devised to perform the numerical tensile load test, i n which the specimen was assigned with prescr i bed noda l displacement at the top and roller condition at the base ( see Figure 3.22) The parameters for the polyester (PET) geogrid was determined by fitting th e bilinear curve with the available laboratory tensile load test curve (see Fi g ure 3 23). Since tensile load test curve at l arge strain is not avai l ab l e for the po l yvinyl alcohol (P A V) geogrid t h e parameters were estimated based on the product information published by I n d u strial Fabric Association International (IF AI 2008) wher e this partic ul ar P A V geogrid has a tensile resistance of 2 1 kN/ m at 5% strain. F igure 3.2 3 compares the calcu l ated and the measured test results Note that the slope of the load strain curve is the product of modulus and the thickness t o f the re i nforcement (e.g., E x tor Et x t). As indicated in Figure 3.23 PA V geogr i d was assumed to be stiffer than the P E T geogrid at large strain level. Table 3 1 provides a s u mmary of material parameters for the geogrid reinforcements 87
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I I Geogrid PET PAV I N ______ + _____ E (MPa) 433.3 464 v 0.3 0 3 I cry (MPa) 4 .33 4 .64 I Et (MPa) 162 .5 304 I I x 203 2 mm Figure 3 22 Numerical Tensile Load Test of a Single Shell Element for the Reinforcement (dz = Prescribed Displacement) 40 PAV Calculated PET Calculated PET Measured (from Ling et aJ 2005a) 30 E z u ro 20 .Q .Ci5 c Q) f10 o o 2 4 6 8 10 Axial strain (%) Figure 3.23 Loadstrain Relationship of PET and P A V Geogrid Reinforcements 88
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3.3 Validation of Computer Program with FullScale Tests As defined earlier for a simulation to be meaningful and to be able to predict response with confidence validation of the computer program is required The following sections describe the undertaking to validate LSDYNA for use in predicting seismic response of GRS wall and examining the current seismic design methodology for GRS walls. It is recognized that similar to other geotechnical problems GRS walls are typically designed with the assumption of plane strain condition The three shaking table tests were also intended for plane strain condition. Numerical simulations were therefore performed to conform to these assumptions Threedimensional validation analysis is desirable but unnecessary for the purpose of this undertaking. For one reason the computation cost is more expensive in a true 3D model than the plane strain model. Furthermore, 3D analysis of GRS structure inevitably includes subtleties that are not easily characterized such as the interaction between adjacent facing blocks and these require additional assumptions and simplifications It is thought that a 3D analysis further compounds the uncertainties and would result in weak inference as compared to the predictions to be made with plane strain condition 3.3.1 Model Configuration The studies of three fullscale shaking table test walls have been reported by both Ling et al. (2005a) and Burke (2004). These test walls had segmental block facing and the backfill was reinforced with geogrids The geometry and instrumentation of the test walls are as shown in Figures 3.24. All three test walls had a height of2.8 m that overlay a 0.2 m thick foundation soil layer. A wall batter of 12 was app lied to all three walls The variations between the three walls are reinforcement spacing and reinforcement length. Walls 1 and 2 had the same reinforcement length but different reinforcement spacing. On the other hand Walls 2 and 3 had the same reinforcement spacing but different reinforcement lengths and types. The test walls were instrumented to monitor performances such as : (1) facing displacement (2) backfill surface settlement (3) lateral earth pressure behind facing blocks (4) bearing pressure beneath the wall (5) reinforcement tensile load and (6) acceleration at various parts of the wall. In the validation effort the calculated responses are compared with those that were measured. 89
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\0 o 125 100 1 301110 ['.:;;;::r;:;:::;:::==:=;:=::;:::::;.:==; Polyester (PET) 25 2 5 25 2 5 45 Geogrid ( L = 205 ) EPS Board Sv= 60 .j 4 L Backfill (Tsukuba Sand ) rfri!>e0 100 25@6 135 Instrumentation : Strain Gages B Force Transducers ( Horizontal ) c Force Transducer (V ert i cal ) Accelerometers Displacement Transducers 1""1",,1,,,,1 o 50 100 150 Scale of em c 60 60 100 135 300 125 100 1301110 Wall 2 (L = 205) 20 25 25 25 25 45 Concrete Block Polyester ( PET ) Geogrid T EPS Board Sv= 40 .j 4 L Backfill (Tsukuba Sand ) rfre
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3.3.1.1 Element Types In the finite element (FE) model all materials except the geogrid reinforcements were discretized with 8node constant stress solid elements The default 8node solid element uses one point integration with viscous hourglass control. The geogrid reinforcements were created with 4node BelytschkoTsay shell elements. The BelytschkoTsay shell element is the default s hell element formulation in LSDYNA due to its computational efficiency One integration point was assigned in the shell element formulation that allows no bending resistance and this is appropriate for the geogrid material. The thickness of the shell element is taken as the average between the rib and the junction thicknesses of geogrid. The average thicknesses of the polyester (PET) and polyvinyl alcohol (P A V) geogrids are 1.5 mm and 1.25 mm respectively. In LSDYNA a "part" is used to represent an entity or a component of the model. Hence the test wall model was created with various "parts" to reflect various components (e g., facing block backfill lift geogrid reinforcement and EPS board) During the model construction a "part" is convenient for assigning material properties element type and contact interfaces. Note that the backfill consisted of two sets of 14 parts (reinforced soil and retained soil zones) that is consistent with the number of facing blocks. Figure 3.25 shows the FE meshes of the three test walls Included in these figures are the number of parts elements and nodes for each model wall. As an example the isometric view of Wall 1 delineating various parts is as shown in Figure 3.26 3.3.1.2 Loading and Boundary Conditions The Kobe Earthquake NorthSouth and UpDown records were used as the input motions to the shaking table. Each of the three shaking table test walls was subjected to two seismic loads sequentially Walls 1 and 2 were tested for horizontal loading and Wall 3 was tested with the combined horizontal and vertical loadings. The actual base motions delivered by the shaking table have been presented in graphs by Burke (2004). These records were digitized for use in this study. Note that although Walls 1 and 2 were intended for horizontal loading only some vertical acceleration was introduced inevitably by the shaking table. Hence all numerical test walls were subjected to the combined loadings. 91
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A (node used to determ i ne (On) Model detail Wall 1 Wall 2 No of parts 50 52 No of shell elements 680 980 No of solid e l ements 2878 2998 No of nodes 7831 8673 No of node sets 4 4 No of segment sets 129 139 No of shell sets 5 7 Ini tial time step ( sec ) 1 .0051 E05 1 .0051 E05 Nodes w ith p r escribed ace l erat i on time h f stories 1""1",,1,,,,1 a 5 0 100 150 Sca l e 01 em Wall 3 52 840 2934 8621 4 139 7 1 0051 E05 Figure 3.25 F i nite Element Mes h and B oundary Condit i on of Walls 1 2 an d 3 A ( node used to determine (On) A ( node used to determ i ne (On)
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Figure 3.26 Isometric View of the Finite Element Model Showing Various Parts Baseline correction was applied to each of the digitized acceleration time histories including both the horizontal and the vertical components. Baseline correction was performed using the computer program BAP developed by the U.S. Geological Survey (Converse and Brady 1992). The differences between the uncorrected and the baselinecorrected records are shown in Figure 3.27. This particular record is the first horizontal component for Wall 2 The effect of baseline correction can be seen most markedly in the displacement record as the displacement was found by integrating the acceleration with time twice. Table 3.6 provides a summary of the shaking table input motions for the numerical test walls Note that the difference in peak acceleration between the uncorrected and baselinecorrected is insignificant. In the numerical model gravity was simulated as a body load and was applied at the outset of the analysis. This approach is known as "gravity compaction" or "gravity turnon" in geotechnical modeling The body load takes the form of a step function and was maintained at the constant gravitational acceleration (i .e., 1 g) throughout the duration of seismic loading. The effects of gravity turnon and sequential loading (or staged construction) for simulating gravity load on undrained normally consolidated slopes were reported by Smith and Hobbs (1974) where the finite element analyses showed that the two loading methods produced similar extents of yielding and horizontal displacement distribution Smith and Hobbs also indicated that the sequential loading method produced less vertical displacement than the 93
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gravity turnon method and suggested that the sequential loading method would lead to a more accurate modeling of the vertical displacement. At the present time direct comparison of gravity turnon and sequential loading methods in GRS wall application is not available in the literature. Ling et al. (2000) however showed that the static behavior of GRS wall can be simu la ted by the sequential load ing technique where good agreement was reported between the calculated horizontal displacement and the measured data but fluctuations in l ateral and vertical stresses and strains in geosynthetic layers were observed. The seismic shaking was simulated by prescribing the horizontal and vertical acceleration time histories to the boundary nodes of the model. The locations of the boundary nodes for the three numerical test walls are indicated in Figure 3.25 Since the numerical model was developed for plane strain condition all nodes were not allowed to move in the plane strain direction. The external loads applied to Walls 1 2 and 3 are as shown in Figure 3.28. Note that quiet periods of 12 seconds were added immediately following the external loads. The quiet period was added in order to damp the numerical models to a stable condition once the loading had ceased The effect of damping is discussed further in the Model Ca libr ation section .. 400 .!!! W all 2 1st Shaking c 200 0 ., a; 0 tl '" ro c: 0 Uncorrec ted Correcte d N c 0 400 I 40 0 1 0 15 20 25 i 20 u 0 a; 0 > : c 0 N c 0 I 40 E 300 0 5 1 0 15 20 25 c: Q) 200 E 1l '" a. 100 on '0 ro c: 0 0 N c 0 100 I 0 5 1 0 15 20 25 Time (sec) Figure 3.27 Comparison between the Original Record (Uncorrected) and the Baselinecorrected Record 94
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\0 VI 1200 1200 800 WoHl 800 i 400 I 400 j ] f 400 f 400 ell ell 800 1200 I I I I I I 1200 20 30 4' 50 60 7' 1200 I 1st ShakJng 2nd Shaking I 800 PHA. = 390 0 cm's' ( 0 40 g ) PHA = 862 0 uris' ( 0 88 g) 800 I 400 .uuL 400 I 400 1 400 800 .... I I I I I I 1200 1200 10 20 30 40 50 60 70 1200 I QUiet penodl 1st Shaking I QUiet period I 2nd Shaking I Quiet period I I 800 800 12, I 14, I 12. I 14. I 12. I I 400 I 400 ! 400 400 S PVA=661 crWs'( OO7g ) PYA = 146 5 emls1 (015 g ) S 800 .... > > I I I I I I 1200 10 20 30 40 50 60 70 Time (sec) (a) Wall 2 I I I I I I I I 2. 30 40 50 60 70 60 1st Shaking 2nd Shaking PHA = 3974 emil' ( 0 4, g) PHA = 852 9 em' ( 0 87 g ) 10 20 30 40 50 60 70 60 I 1st Shaking I I 2nd Shaking I,;!':: I 12. I 25. I 12. I 25. I 12. I PYA = 56 3 crnJsl (0 06 g ) PHA = 132 9 emls' (014 g) I I I I I I I I 10 20 30 40 50 60 7 0 80 nme ( sec ) (b) I ] f ell 90 I I I 90 ;:i I I S > 90 1200 Wo .. 800 400 1200 I I I I I I I I 800 400 400 .... 1200 1200 800 400 400 .... 1200 PHA = 455 1 emsl ( 0 46 g ) PHA = 814 2 cm's1 ( 0 83 g ) QUtet 12. PYA = 315 5 cm'SI ( 0 32 g ) PHA = e98 0 emls' ( 0 71 g) 10 20 30 40 50 Time ( sec ) 60 70 80 (c) 90 12. 90 Figure 3.28 Loading Time Histories Applied to Models of (a) Walll, (b) Wall 2 and (c) Wall 3
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Table 3.6 Summary of shaking table input motions for numerical test walls Peak acceleration Test Stage Component Duration before correction after correction wall (sec) ( Crn/S2 ) (crn/s2 ) 1 Horizontal 14 389.9 390.0 1 1 Vertical 14 65.4 66.1 1 2 Horizontal 14 858.2 862.0 2 Vertical 14 147.2 146 5 2 1 Horizontal 25 397.5 397.4 2 Vertical 25 55.8 56 3 2 2 Horizontal 25 854.8 852.9 2 2 Vertical 25 130.8 132.9 3 1 Horizontal 25 453 3 455.1 3 1 Vertical 25 317 9 315.5 3 2 Horizontal 29 815.5 814.2 3 2 Vertical 29 699.1 698.0 3.3.1.3 Contact Types and Contact Details The two contact types specified in thi s study were automaticsurfaceto surface and tiedsurfacetosurface. The automaticsurfacetosurface type is a penaltybased contact. In the penaltybased contact penetration of the two surfaces is resisted b y a force (represented by linear springs) proportional to the penetration distance The automaticsurfacetosurface contact type allows the contact surfaces to slide and separate. The tiedsurfacetosurface type is a constraintbased contact. In the constraintbased contact slave nodes are constrained to move with the master surface, and forces are calculated to keep the slave nodes exactly on the master surface with zero penetration Rotational degrees of freedom are not transmitted in the constraintbased contact. Furthermore the automaticsurfacetosurface contact was specified with soft constraint approach where this approach is suited for contact between two dissimilar materials. Contact interfaces the same as those observed in the physical model were specified between two distinct parts in the numerical model. The use of contact interfaces and distinct parts marks the difference between a discrete approach and a composite approach for analyzing a GRS structure. The discrete approach is the approach undertaken in this study in which the discrete components each having its own material properties are assembled to create the numerical model. The main advantage of discrete approach over the composite approach is that the material 96
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properties are more readily available for individual parts than the properties of an overall composite material. Composite approach is of practical use as the composite strengths could be used directly in the routine limit equilibrium analysis and the soil reinforcement interface property is included in the composite behavior intrinsically. The composite strengths when used in limit equilibrium analysis avoid the umealistic tensile interslice normal force and umealistic normal stress distribution along the slip surface resulted from reinforcements represented by lateral concentrated line loads Figure 3.29 shows the typical contact interfaces adopted in the numerical models The contact type for each of the contact interfaces is summarized in Table 3.7. In LSDYNA a "set" is devised to manage a group of nodes elements or facet s of elements and a "set" is used for specifying boundary conditions and contact interfaces. For example a segment set would consist of facets of elements of a part and a shell set would consist of shell elements a reinforcement layer. The contact interface requires either a pair of segment sets from the neighboring solid parts or a shell set with a segment set for the contact between a geogrid and its surrounding solid parts Since geogrid reinforcements are simulated by shell elements the thicknesses of reinforcements must be included in the mesh. E xcluding the shell thickness in the mesh would introduce initial penetration in contact interface which would result in numerical instability. Element incompatibility was hence introduced when shell thickness was considered in the model mesh. The element incompatibility occurs in the backfill at the end of geogrid layer which is also the junction between reinforced and retained zones (see Figure 3.30). The tiedsurfacetosurface contacts were used to tie the incompatible backfill elements Without the tied contact interface nonphysical behavior would occur such as intrusion of neighboring elements Although compatible mesh may be enforced by adding a thin layer of solid elements in the retained zone immediately behind the reinforcement sheet the thin elements would have extreme aspect ratios The use of thin element handicaps the computation efficiency and is prone to instability The use of thin element is thus not recommended. The contact interface is characterized by the coefficient of friction !l and the contact formulation In the numerical model the coefficient of friction was assumed to be 0.5 for all contact interfaces and the value corresponds to the estimation: (3.25) where of 36.7 is the friction angle of the backfill. This estimation is empirically adopted in the routine design of retaining wall including GRS structure when laboratory data is lacking The effect of !l on the seismic response of wall is further discussed in the Model Calibration section 97
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ConcreteConcrete GeogridConcrete SoilSoil GeogridConcrete EPSSoil ConcreteSoil ConcreteSoil ConcreteSoil ____________________________________ Figure 3.29 Contact Interfaces adopted in the Finite Element Model (e .g., Wall 1) Geogrid thickness Detail A Boundary between incompatible elements (need tied contact interfaces) Figure 3.30 Detail showing Geogrid Thickness and Incompatible E l ement Boundary Table 3.7 Summary of LSDYNA contact interfaces defined in the numerical model Materials in contact Contact type Slave segment Master segment ConcreteSoil Auto. Surface to Surface Segment set Segment set ConcreteConcrete Auto. Surface to Surface Segment set Segment set GeogridConcrete Auto. Surface to Surface Shell set Segment set GeogridSoil Auto. Surface to Surface Shell set Segment set EPSSoil Auto. Surface to Surface Segment set Segment set SoilSoil Tied Surface to Surface Segment set Segment set 98
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3.3.2 Model Calibration Calibration by definition is the "finetuning" or adjustment of parameters in order to achieve better prediction. Three parameters examined in the calibration are the interface viscousdamping constant (VDC) mass damping coefficient and coefficient of friction for soilreinforcement interface. The calibration procedure adopted in this study follows the trialanderror approach A range of values for a parameter were tested and the value that yielded the best agreement with the measured response was selected to be the calibrated value Note that the calibrated values are considered applicable only to models similar to the ones presented in this study. The calibrated values may be less reliable for models that deviate greatly from the confi g uration loadings and boundary conditions of the models established in th i s study. 3.3.2.1 ViscousDamping Constant VDC was implemented ori g inally in LSDYNA to damp out contact oscillation. It was found that it is necessary to activate VDC in the simulation of GRS wall. Without applying VDC for the contact interfaces the FE model would result in instability even at the early stage of gravity loading where reinforcements would shorten continuously in length in a unrealistic manner. As recommended b y the LSDYNA support forum a VDC value between 40 to 60 ( corresponding to 40 to 60% of critical damping) is recommended for contacts involving soft materials and 2 0 for similar materials such as metal. The effect of VDC on the wall face displacement of Wall 1 is as shown in Figure 3.31 where VDC ranging from 20 to 60 y ields similar results. It is observed that the wall face response is not sensitive to the range ofVDC e x amined hence a VDC of 40 was selected as the calibrated value 3.3.2.2 Mass Damping Coefficient Damping is a parameter that can greatly affect the result of a dynamic analysis. The mass damping coefficient was utilized in this study. The mass damping coefficient damps the lowfrequency structural modes which is considered appropriate for a GRS structure. The mass damping coefficient is related to the natural period of the structure The natural period Tn can be estimated from an undamped transient analysis. Th e gravity load that follows the form of a step function (e.g. see Figure 3.28) is considered as a transient load. The nodal displacement time histories at top of wall near the middle of the reinforced zone (e .g., see point A in Figure 3.25) due to transient load without damping are presented in Figure 3 32 for the three numerical test walls 99
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3 3,, Wall 1 1st Peak 2 2 VDC = 20 e VDC=40 VDC = 60 Measured l o I I I I I 50 40 30 20 10 o 100 80 60 40 20 o Hor i zontal displacement ( mm) Horizontal d i splacement (mm ) Figure 3 .31 Effect ofInterface ViscousDamping Coefficient (VDC) on Facing Response of Wall 1 due to Seismic Load The nodal displacement time history allows Tn of the structure to be determined. The natural circular frequency COn for each wall is then calculated as: con = 2rt/Tn (3.26) Both Tn and COn for each of the three walls are indicated in Figure 3 32 Note that the motion without mass damping coefficient does not reflect a true free vibration motion. The model was slightly damped as energy was dissipated through the use of interface VDC. The mass damping coefficient can be estimated in terms of a fraction of COn. The mass damping coefficient is applied in a global sense, where all parts are assigned with the same coefficient. The nodal displacement time histories with mass damping coefficient based on different fraction of COn are shown in Figure 3 32. It is evident from the responses that the model is overdamped for a mass damping coefficient greater than or equa l to 2'con and is underdamped for the coefficient less than 2con 100
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Wall 1 T = 0 132 s ; = 47 7 radls No damp i ng E 4 ++++1 15% .s \ 50 % "' 200 % "' '6 16 0 2 0 4 0 6 0 8 Time (sec ) (a) Wall 2 T = 0 .131 s ; "' = 48 0 radls No damp ing E 4 ++'++1 15 % "' .s \ 200 % '(j)" '6 > 12 16 0 2 0.4 0 6 0 8 Time (sec ) (b) Wall 3 Til = 0 130 5 ; Cil" = 46. 3 rad / s No dampi ng E 4 ++'++..., 15 % .s \ 50 % " 200 % 0 (j)" '6 "ii > +,+,+,i,i,ri 0 2 0 4 0 6 0 8 Time (sec ) (c) Figure 3 32 Effect of Global Mass Damping Coefficient on Walltop Response of (a) Wall 1 (b) Wall 2 and (c) Wall 3 due to Transient Load 101
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The effect of mass damping coefficient was examined for Wall 1 under seismic loads. The results are shown in Figure 3 33 which suggests that the greater the mass damping coefficient the less is the wall face displacement. Especially under high seismic load as is the case for second shaking mass damping coefficient equal to 0.15(On yielded good agreement with the measured response. It was concluded that mass damping coefficient equal to 0.15(On would represent the calibrated value. 3..3,, 2 50 Wall 1 1st Peak Ca l culated (15 % ron) e Calculated (50 % ron) ______ Calculated (200 % Measured 40 30 20 10 Horizonta l displacement (mm) 2nd Peak o 100 80 60 40 20 o Hor i zontal d i sp l acement (mm ) Figure 3.33 Effect of Global Mass Damping Coefficient on Facing Response of Wall 1 due to Seismic Loads 3.3.2.3 Friction Coefficient of SoilGeogrid Interface As reported by Ling et al. (2005a) in the shaking table tests, the friction coefficient ).l of the soilgeogrid interface as determined by the direct shear test was 0.675. In addition to the reported value three different ).l'S were tested herein for the calibration study. The values correspond to}'j X and the full (viz. 0.25 0.5 and 0.75 for = 36.70). The effect of).l on Wall 1 facing response due to seismic 102
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loads is presented in F igure 3.34. The numerical analysis indicated that a J.l between 0 .75 and 0 5 has minor effect on the facing response and J.l of 0.25 however greatly increases the wall displacement. It is observed that the J.l of 0.5 closely matches the facing response especially at larger seismic load (i.e. second shaking). Based on this observation the J.l of 0.5 was selected as the calibrated friction coefficient. Note that this observation agrees with the empirical relation of Equation 3 25. 3 r, 3 r, 2 Wa ll 1 1st Peak 50 40 30 20 10 Horizontal displacement (mm) o 240 200 160 120 80 40 0 Horizontal d i splacement (mm) Figure 3.34 Effect of Friction Coefficient on Facing Response of Wall 1 due to Seismic Loads There are many factors that could affect the value of J.l determined through laboratory tests Some of the factors include test apparatus, stress level rate of shear displacement and reinforcement extensibility (Mallick et al. 1996 ; Takasumi et al. 1991). The size effect of direct shear apparatus on the strength of soilgeogrid interface was studied by Ingold (1982), and the results are presented in Figure 3.35. The failure envelopes in the figure suggest that at low normal stress level the small (60 x 60 mm) apparatus yielded higher interface friction than the large (300 x 300 mm) apparatus and the difference subsided at higher normal stress level. The small apparatus also indicated that the interface friction decreases with increasing normal 103
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stress The large apparatus on the other hand indicated a more consistent interface friction It is evident that the laboratory determined interface friction is not unique which justifies the use of calibrated friction coefficient in the numerical simulation The calibrated parameters aforementioned are summarized in Table 3.8. 60 x 60 mm Fixed Shear (from Ingold 1982) 300 x 300 mm Fixed Shear (from Ingold 1982) 150 til a.. ... vi 100
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results of numerical simulations were compared against the measured performances reported by both Ling et al. (2005a) and Burke (2004) The performances selected for comparison include : (1) wall facing displacement (2) backfill surface settlement (3) lateral earth pressure behind facing blocks (4) bearing pressure (5) geogrid reinforcement tensile load (6) acceleration in reinforced soil zone and (7) acceleration in retained soil zone. Observation of the comparison for each performance category is discussed separately in the following sections 3.3.3.1 Wall Facing Displacement The comparison of the calculated and the measured wall face peak horizontal displacement for the three test walls is presented in Figure 3.36 The comparisons of horizontal displacement time histories are included in Figures 3 37 3.38 and 3.39 for Walls 1 2 and 3 respectively Wall facing displacement increases with increasing seismic loads. Note that the trend of horizontal displacement profile is the same between the calculated and the measured response where wall top experiences the largest displacement and wall base has the least displacement. It is observed that th e simulation of first shaking in general overestimated the wall displacement and th e prediction of second shaking is better than those of the first shaking. The simulation of the second shaking follows closely with the measured response where the calculated and the measured motions are in phase with each other (see Figures 3.37 3.38 and 3.39) When vertical component of the input acceleration is comparable to the horizontal component as is the case for Wall 3 numerical simulation tends to predict greater wall displacement. 3.3.3.2 Backfill Surface Settlement The comparison of the calculated and the measured backfill surface settlement for the three walls is presented in Figure 3.40. Similar to the prediction of wall face displacement simulation of first shaking overestimated the permanent settlement. However both the calculated and the measured results from first shaking show the similar trend of settlement profile where greater settlement took place near the wall facing and decreases in magnitude toward the back of the wall. Note that a great variability existed in the measured results of second shaking. The settlement profile is though more consistent among the numerical simulations and the greatest settlement occurs near the junction of reinforced and retained soil zones. Despite the variability of the measured results simulation of the second shaking underestimates the surface settlement of all three walls 105
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o 0\ \ 9 I Q) I I I \ I Q) I I '\ G> \ I c!> \ I \ I \ I \ I \ III c!> \ I \ I I i G> I 6 I I I I c!> I I \ I c!> I I I I Y , \ \ I c!> I I I \ I \ 1\ 6 I I I I \ I \ 6 I I I I \ 6 I \ I \ I \ I \ 1\ I I I I waU1 6 I 1st peak (calculated) \ \i _______ 2nd peak (calculated) \ \ e151 peak ( measured) \ 82nd peak (measured) 1\ I I I I Wall 2 6 I 1st peak (calculated) I ___ 2nd peak (calculated) I e151 peak (measured) I 82nd peak (measured) t '\ \ \ Wall 3 R ___ 1st peak (calculated) \ I ___ 2nd peak (calculated) \ I e151 peak (measured) I \ b B2nd peak (measured) I I I I I I I 100 80 60 40 20 100 80 60 40 20 100 80 60 40 20 Horizontal displacement (mm) Horizontal displacement (mm) Horizontal displacement (mm ) (a) (b) (c) Figure 3.36 Wall Face Peak. Horizontal Displacement Comparison between the Calculated and the Measured data for ( a) Wall 1 (b) Wall 2 and (c) Wall 3
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o ...l 20 Calculated IS 1 0 5 o 5 Wall 1 1st Shaking H = 2 8 m 10 +,,r,,,"','r,,,,.,,, 12 13 I. IS 1 6 17 18 19 20 21 22 23 2. 26 T ime ( sec ) (a) 100 80 60 .0 Calculated 20 Measured honzonta l d ispl acement at H = 2 8 m o dUring 2nd shak ing I S not complete 20 +,'l,,,,,,r,,,,,,, 80 20 80 60 H 1 .0m 4 0 : I=:=: 20 80 60 4 0 H 0 2 m 100 1 ______ ============== +,,,,,,,,,.,,,,,,,, 38 39 .0 ., 42 .3 4. .5 48 Time (sec) (b) 4 7 48 49 50 51 52 F igure 3.37 Comparison of Wall 1 Face Displacement Time Histories for (a) First Shaking and (b) Second Shaking
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o 00 20 1 5 1 0 5 Calcula ted Wall 2 1 s t Shaking H = 2 8 m 100 Wall 2 2nd Shaking SO Measured _ '0 o _____ .J' 20 5 o 10 +.r,,,'r;rr,,,,, 20 +.'r':'''''rr''''' H = 2 .Sm 201 j ; 20 ': ,,,. E _______________ I .:g '" 20 I lS 100 E 15 H=1.8m 80 &\ H=1.8m !]= I r L . 1 5 80 10 SO H= 1.0m 5 201 201 15 80 10 SO 5 __ _ H = O S m 20 1 100 I 15 8 0 10 5 .:g H 02m 13 M H T i me ( sec ) ( b ) Figure 3 .38 Comparison of Wall 2 Face Displacement Time Histories for ( a ) First Shaking and ( b ) Second Shaking
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...... o \0 H= 0 6 m 15 16 17 18 19 20 21 22 23 2. 25 26 27 28 29 T ime (sec) (a) Figure 3.39 Comparison of Wall 3 Face Displacement Time Histories for (a) First Shaking and (b) Second Shaking
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.. ..o E .s E i ,,,, 20 '0 60 , QT./gem WII' dll i u , p )J bed dunng 2nd Ihlklng Walll 60 +"+1+lltpe.k(c.lculat.d) ___ 2nd peak (calculated) e lat peak (mealufad) B 2nd peak (m ... ured ) 100 200 300 Distance from facJng block (em ) (a) E .s 8 VI 20 "(J0 '0 60 Well 2 1 ____ 1.tpeak(calculaled) 60 ++11 ______ 2nd peak (calcul ated ) 0 ht peak (mulu.ed) ) 2nd peak (mellu/ad) 100 200 300 Dlltance from faCIng block (em) (b) E .s i VI 20 ,,,,, 20 0 60 Wall I +lat peak (Calcul ated ) 60 ++1 2nd peak (calculaled) r0 ht peak (m ... ul.d) D 2nd peak (menu/.d) 100 200 300 Dlttance from facing block (em) (c) Figure 3.40 Comparison of Backfill Surface Settlement for (a) Wall 1 ( b) Wall 2 and (c) Wall 3
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3.3.3.3 Lateral Earth Pressure behind Facing Blocks The comparison between the calculated and the measured peak lateral earth pressure for the three walls is presented in Figure 3.41. Note that the lateral pressure increases with increasing seismic loads in both the calculated and the measured results. The numerical simulation also shows that the profiles due to seismic loads are near vertical for relatively small reinforcement spacing (i.e. Walls 2 and 3) The calculated and the measured pressure distribution profiles, in general are fairly irregular for all three walls ; hence no other general trend could be concluded from the pressure distribution profiles 3.3.3.4 Bearing Pressure The comparison of the calculated and the measured bearing pressure for the three walls is presented in Figure 3.42. Note that the bearing pressure increases with increasing seismic loads in both the calculated and the measured results. It is observed that in both the calculated and measured results bearing pressure remains somewhat constant in the backfill beyond the heel of facing block. It seems that bearing pressure increases with increasing reinforcement spacing, as is the case for Wall 1 and inclusion of comparable vertical component of seismic loads as is the case for Wall 3. Note also that there exists variability within the measured response especially in the region beneath the facing block. 3.3.3.5 Geogrid Reinforcement Tensile Load The comparison of the calculated and the measured peak geogrid reinforcement tensile loads is as shown in Figure 3.43. The tensile load is observed to be proportional to the magnitude of the seismic loads. Both the calculated and the measured results have shown that the reinforcement near the wall base in general experiences the highest tensile load. The rate of change in tensile load (i.e ., non uniform distribution of tensile forces) is also apparent near the wall base from both the calculated and the measured results. As compared to the measured results however the numerical simulation overestimated the tensile load in most cases The numerical simulation had predicted high tensile load near the facing block more consistently than those measured It should also be noted that the tensile load is inversely proportional to the wall displacement. It was thought that the high tensile load near the wall base is a result of both high overburden stress and restraining effect of the geogrid reinforcement. 111
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...... ...N .. a 1st peak (measured) e 2nd peak (measured) OJ 10 15 20 25 30 35 Lateral eal1h pressure (kPa) (a) 40 10 15 20 , Wan 2 Initial (calcul ated) +1s1 peak (calculated) _____ 2nd peak (calculated) $ IMial (measured) G 1st peak (measured) 8 2nd peak (measured) , i!l 25 30 35 lateral earth pressure (kPa) (b) 40 10 15 20 25 Lateral eanh pressure (kPa) (c) Wall 3 I ,.. ( calel.Uted) 11' plak (cakWlle d ) 2nd peak (calculited) Initial (rN .. wed) 1" p .. k (measured) 2nd peak (meowed) 30 35 Figure 3.41 Comparison of Lateral Earth Pressure behind Facing Blocks for ( a) Wall I ( b) Wall 2 and (c) Wall 3 40
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400 400 400 300 300 300 'l l!!I 200 ft > ...... 100 W We i 1 'iI 1'Initial (celculated ) I 1st peak ( calculated ) ____ 2nd peak ( Cllculated) I 0Initial (mealured) I 01st peak ( measured) cr 2nd peak ( measured ) I I I I .. WaU2 I ... Initial ( calculated) 1 ___ '" peak ( ealculated ) 1 __ 2nd peak ( calculated ) <>INUI (me.lured) 0<1st peak ( measured) 02nd peak (measured) h o:og_"8 j 200 ft > 100 IJ, Wall 3 +Inloal (caICYlated) q +111 peak ( Cillculated) 1 __ 2nd peak ( eaIcuiated) , ()I mtlal (me.,ufed) 0,.t peak (measured) r\q 02nd peak (measured ) , I Ap;: :l): = 6 0100 100 200 300 100 200 300 100 200 300 Distance from facing block (em) DIstance from faang block ( em ) Osstance from faang block ( em) (a) (b) (c) Figure 3.42 Comparison of Bearing Pressure for (a) Wall 1 (b) Wall 2 and (c) Wall 3
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...E 0 i 4 .. E I 6 tt Wall 1 lnitieI(meaN'1d) tll(me urld) __ 2nd (meuured) 05 1 15 Distance from facing block ( m ) (a) H=26m H = 2 0 m I 0 8 !! 4 H = 14 m E 2 e 0 6 . tt H=08m H=02 m Ini(ill(clkullt.d) _In/tilll{me.tUI.d) Wall 2 ht pllk (clkulll.dl _hl{ m .. tur.d ) 2nd I".k +2nd (mellur.d) ...... :! 1 ... ,..a"" ,.... ::::::::; IY OS 15 DIStance from faCIng block (m) (b) H=22m H = 14m H =, Om H=06m H =02m _lr\itlSl( Cllcutlt.d) 11Wt1l1{mtlllur.dl 4 WaJI3 '''{",.Itur.d) 2ndp .. k(ulculll.d) 2nd(",.lIul.d) __ H =2.2 m H = 14 m I 8 6 j 4 .... H= 1 Om C 2 oE. :orI I I I 8 5 E 6 4 i"H=06m 2 V f I I I I H=02m 05 1 15 2 5 Distance from faCing block ( m ) (c) Figure 3.43 Comparison of Geogrid Reinforcement Tensile Load for (a) Wall 1 (b) Wall 2 and (c) Wall 3
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3.3.3.6 Accelerations in Reinforced and Retained Soil Zones The comparisons of absolute peak horizontal accelerations between the calculated and the measured results in the reinforced soil zone and retained soil zone are presented in Figures 3.44 and 3.45 respectively. Acceleration amplification is observed in both the calculated and measured results. However amplification is less pronounced in the measured results especially in the reinforced soil zone. As shown in Figure 3.44 only slight amplification is observed in the reinforced soil zone; thes e results imply that the walls underwent rigid body motion which disagrees with the wall facing displacement observation. The numerical simulations indeed show that wall facing displacement is proportional to the acceleration amplification The effect of reinforcement spacing is also more noticeable in the calculated results than those measured in which high acceleration is associated with large reinforcement spacing. Findings from other numerical modeling of GRS walls subjected to seismic shaking also indicate acceleration amplification in the reinforced soil mass with amplification factor ranging from 2 (Liu et al. 2011) to 5 (Nelson and Jayasree 2010). 3.3.3.7 Energy Observation The energy data was recorded during numerical simulation. As implemented in LSDYNA the total energy is the sum of internal energy kinetic energy contact energy hourglass energy, damping energy and rigidwall energy The hourglass energy is the energy required to prevent the hourglass modes of deformation. The hourglass mode is nonphysical and anomalous and it occurs when element uses single integration point. By default the viscous hourglass control is activated when elements with one integration point are used The total energy time histories for the three numerical models are plotted in Figure 3.46. Based on the time histories, it is noted that the high wall top displacement is associated with high total energy Note also that the total energy had remained constant during the quiet periods Constant total energy implied that a static or quasistatic solution had been reached 115
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...... ...... 0\ 2 5 0 I I I I \ / I I I I ? 2 5 0 I I I I I I ? \ 2 5 0 UI r I I I ;/ / I I ? 1 5 I :c .2' r I I I I I I \ I I I I I I I i / I I I I I V I I I I I / I I I I I I I I I I I I I I I 1 5 1 5 0 5 I I I I I '( I / I 1 v I /, Wall 1 (rein f orced soil zone) I I I f/) I I '( I I I r :/1 I I Wall 2 (reinforced soil lone ) +1 st peak ( calculated) I I I I 9 1 / I I I I I A I Wall 3 (reinforced soil zone) I ____ 1 st peak (calculated) 0 5 0 5 V;", / 1st peak (calculated) __ 2nd pe a k ( c a lculated) i i 2nd peak (calculated) 01st peak (measured) /' ____ 2 n d peak (calculated) 01st peak (measured) 01st peak (measured) 02nd peak (measured ) 02nd peak ( measured) 02nd peak (measured) 0 5 I I 0 5 I I 0 5 I I 1 000 2000 3000 4000 1000 2000 3000 4000 1000 2000 3000 4000 Absolute maximum horizontal acceleration (gal) Absolute maximum horizontal acceleration (gal) Absolute maximum horizontal acceleration (gal ) (a) ( b ) ( C ) F igur e 3.44 Comparison o f A bsolut e Horizontal Acceleration in the R e inforc e d Soil Zon e f or ( a ) Wall 1 ( b ) Wall 2, and ( c ) Wall 3
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2 5 0 , a , ? J , , , 1 5 , , , , }, <;> ,/1 , , Wall 1 (retained soil zone ) j __ 1st peak ( calcu l ated ) 0 5 i 1 ___ 2nd peak ( calcu l ated ) 01st peak ( measured ) 02nd peak ( measured ) 0 5 I I 1000 2000 3000 Absolute maximum horizontal acceleration (gal) (a) 4000 so .'" :I: 2 5 1 5 0 5 0 5 0 "' I / , V , ? 2 5 0 OJ / / I i'/; / I , <;> !: , I , I I , , , 1(1 , , / '/ I , 1 5 I , '1/ , t <;> , / ,II , I WaU 2 (reta ined soi l zone) , 1st peak (calculated) 1 <;> I I , , , Wall 3 (retained soil lone ) , __ 1st peak ( calculated ) 0 5 'I I ___ 2nd peak (cal culated ) 01 st peak (measured) ;// ___ 2nd peak (calcu l ated) cr1., peak ( measured) 02nd peak (measured) 02nd peak ( measured ) I I 0 5 I I 1000 2 0 00 3 000 4 000 1 000 2000 3000 4 000 Absolute maximum h orizontal a cceleration (gal) Absolute maximum horizontal acceleration (b) (c) Figure 3.45 Comparison of Absolute Horizontal Acceleration in the Retained Soil Zone for (a) Walll, (b) WaU2, and (c) Wall 3
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4000 3000 >e> 2000 Q)
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Table 3.9 Prediction quality classes (after Morgenstern 2000) % error Qualitl:' Class within 5% Excellent within 15% Good within 25% Fair within 50% Poor >50% Bad Table 3.10 Quality of numerical prediction (% error) Performance Wall 1 Wall 2 Wall 3 1st Shaking 2nd Shaking 1st Shaking 2nd Shaking 1st Shaking 2nd Shaking Maximum wall Poor Excellent Bad Fair Bad Excellent displacement (37%) (1%) (58%) (20%) (174%) (3%) Maximum backfill Bad Bad Bad Bad Bad Poor \0 settlement (80%) (78%) (116%) (55%) (216%) (50%) Maximum lateral earth Bad Excellent Fair Poor Good Good pressure (78%) (4%) (19%) (36%) (12%) (12%) Maximum bearing Poor Poor Good Good Bad Bad pressure (39%) (39%) (12%) (12%) (58%) (59%) Maximum reinforcement Bad Poor Bad Poor Bad Excellent tensile load (282%) (36%) (144%) (47%) (118%) (5%) Reinforced soil zone Bad Bad Bad Bad Bad Bad absolute acceleration (206%) (172%) (242%) (68%) (159%) (117%) Retained soil zone Bad Good Bad Bad Bad Bad absolute acceleration (383%2 (8%2 (258%) {196%) {361%) (60%2
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20 18 16 14 12 o lIlIIr,lIIIIIrrIIIMax wall d isplacement Max backfill settlement Ma x earth pressure Max bearing pressure Ma x Ma:x, reinforced Max retained soil rei nforcement soil zone aeeei. zone acce!. tensile load Figure 3.47 Cumulative Weight of Performance for indicating the Prediction Capability of LSDYNA 3.3.5 Variability within the Measured Data Some variability's within the measured data have previously noted in the Response Comparison section. It is recognized that variability is extremely difficult to avoid especially in a large scale laboratory test since all experiments exhibit random (statistical) and bias (systematic) errors In addition it is thought that the variability encountered at the initial stage of the laboratory test might have some influences over the later collected data. In other words the variability is likely to be accumulative ; for example, high initial pressure would result in high peak pressure during the seismic shaking. Comparisons were hence made between the calculated and the measured results for the initial performances to d e lineate some o f the variability's. Figures 3.48 3.49 3.50 and 3.51 illustrate the variability of the measured initial lateral earth pressures initial bearing pressures initial tensile load within the base layer o f the reinforcement and the initial maximum tensile load for all other layers respectively. Also included in these figures are the companion numerical simulations. Note that for all the cases compared, the initial performanc es of the laboratory shaking table test are less consistent than tho se of th e numerical simulations 1 2 0
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E co iii I Measured _WaIl1 o 4 8 12 16 Lateral earth p ressure (kPa) Calculate d a Wall 1 o Wall 2 20 0 4 8 12 16 20 Lateral earth pressure (kPa) Figure 3.48 Variability in the Measured and the Calculated Initial Lateral Earth Pressures 160 .. 120 CL .. .. 80 t: .. > 40 0 160 .. 120 CL .. .. e 80 ;;; t: .. > 40 0 0 50 100 150 Distance from faci ng block (cm) Measured _+WaIl1 _WaIl2 6Wall 3 Calculated _WaIl1 o Wall 2 '" Wall 3 200 2 5 0 Figure 3.49 Variability in the Measured and the Calculated Initial Bearing Pressures 121
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Measured +Wall 1 11Wall2 C alculated ___ Wa1l1 Wall 3 0 4 0 8 1.2 1 6 Distance from facing bloc!< (m) Figure 3.50 Variability in the Measured and the Calculated Base Layer Reinforcement Initial Tensile Load 2 5 I 1 5 iii I 0 5 Measured Wall 1 . Wall 2 Wall 3 Calcul ated _
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The observed variability is an evidence of uncertainty within the measured results. Uncertainties might blur the reliability and accuracy of measured results. Cost of experimental study often inhibits the statistical study on the test reliability and accuracy. Without a rigorous statistical study, the quality of numerical prediction presented in Table 3.10 may not be realistic since the predicted results are compared against the measured results with unresolved uncertainties. It may therefore be more appropriate to term "difference percentage" instead of the "error percentage" in Table 3.10. The predictive capability of LSDYNA with the current prediction quality may appear to be less favorable. However due to the uncertainties within the measured results, it is inconclusive to state that LSDYNA is not sufficient for conducting seismic simulation of GRS structures. Furthermore the simulations presented have shown indeed prominent agreement with measured performance with high confidence such as the wall facing displacement. It is also argued that numerical simulation can conversely be used to evaluate the accuracy of experimental data. As has discussed in the Terminology section both experimental and numerical studies are needed simultaneously and continuously in order to refine the prediction capability of numerical tools. 3.4 Discussion of Numerical Simulation The discrepancy between the calculated and the measured results may be due to the reasons of: (1) imperfect material models (2) inaccurate loading conditions and (3) spatial variation. As shown in the calculated stressstrain curves (see Figures 3.7 and 3 23), the hysteretic behavior is not apprehended during the unloading reloading cycle. The hysteretic behavior accounts for energy dissipation in real material. The damping included in the numerical model is rather artificial as it was intended for computation stabi lity and is not equivalent to the hysteretic damping at least at the material l evel. Furthermore the cap model is defective in that the soil dilatancy is not properly simulated (see Figure 3.7) Note also that the cap model parameters for backfill material were developed based on conventional triaxial compression test. The shaking table test however was adhered to the plane strain condition. Behavior of soil in a plane strain condition in general, is not the same as the behavior when subjected to an axisymmetric condition The tests performed by AIHussaini (1973) indicated that the friction angle of sand obtained from plane strain test is higher than those of the triaxial compression test where the difference is approximately 1 degree for loos e sand and increases to approximately 3 degrees for dense sand Similar strength characteristics have also been observed in tests conducted by Lee (1970). In addition the geogrid reinforcement was simulated by the bilinear stressstrain curve rather than a nonlinear curve, which could also attribute to the observed discrepancy. 123
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Part of the inaccurate loading conditions may be attributed to the digitized motion which might include noises not seen in the original record since the digitization process was limited to the chart resolution. In addition to resolution uncertainty the original records were truncated ; hence the complete record was not simulated. The spatial variation concerns the geometric difference between the controlled laboratory shaking table tests and the numerical models. Although the laboratory test was tested for a plane strain condition threedimensional artifacts could not be avoided. The frictional resistance and the flexibility between adjacent drystacked modular blocks may have some contribution to the global movement of test walls and these features were not simulated in the idealized numerical models with plane strain condition It is thought that the effect of threedimensional artifacts may be more pronounced in the combined loading than the predominant horizontal loading. 124
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4. Parametric Study The validated computer program LSDYNA was utilized in the parametric study. Parametric study was performed to evaluate the effects of wall geometry material properties and reinforcement layout on the performance of GRS wall subjected seismic loading. Selection and processing of the input motions is first discussed and is followed by the material characterization for the numerical models Development of the numerical model dimensions is then presented in which a representative boundary extent was selected. Description of the parametric study program then follows the evaluation of boundary extent. This chapter also includes the evaluation of global stabilities of the proposed models and the discussion of the modeling procedure. Model geometry and finite element meshes are presented 4.1 Input Ground Motions A total of20 earthquake records were selected for the parametric study The earthquake records were obtained from the Pacific Earthquake Engineering Research Center (PE E R) Strong Motion Database (http :// peer.berkeley.edu/smcatJ) The database is maintained by PEER through the Earthquake Engineering Research Centers Program of the National Science Foundation. The database contains 1557 records of 143 earthquakes from tectonically active regions around the world The earthquake records for the parametric study were selected primarily based on the peak horizontal acceleration (PHA) as it is an essential design parameter in the seismic design of GRS walls and other earth structures. The PHA's of the selected records range from 0.114 g to 0.990 g The 20 selected records are all freefield motions with the instrum e ntation located at the ground surface. The geotechnical subsurface characteristics for the 20 records ranged from shallow 20 m thick) to deep ( > 20 m thick) soil profiles. The ground motion parameters (including duration peak acceleration peak velocity and peak displacement) of the selected earthquake records are summarized in Table 4 .1. Also included in Table 4 1 are the fault type and the moment magnitude M w of each earthquake record. The selected records were baseline corrected using the computer program BAP (Converse and Brady 1992) to eliminate possible drifts. The peak values presented in Table 4.1 are based on the baseline corrected records. Note that deconvolution of the records from the ground surface to the base of numerical model in the parametric study was not performed since the depth to the base of the model is considered shallow (i.e. < 7 m) The response spectra at 5% of critical damping of the baseline corrected records were also generated using BAP. 125
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The spectral parameters (including predominant period T p maximum spectral acceleration Sa, and spectral accelerations at natural periods of 0 2 sec., 0 5 sec. and 1 sec.) from the 5% damping response spectra of each record are summarized in Table 4.2. The acceleration time histories of the 20 earthquake records are shown in Figure 4.1. Time histories on the left hand side of Figure 4 1 are horizontal component of the record whereas the corresponding vertical components are shown on the right hand side of the figure. T he peak horizontal acce l eration (PHA) and the peak vertical acceleration (PV A) of each record are also included in Figure 4 .1. The plots of response spectra of the 20 records are shown in Figure 4.2. Similar to the time history arrangement in Figure 4 1 the horizontal and vertical components of the response spectra for the records are depicted on the left and on the right hand side of Figure 4.2 respectively. Variation of PV A with PHA for the 20 records is shown in Figure 4.3. As illustrated by Figure 4 3 most ofthe records are bounded between the 1 V: 1 Hand 1 / 3V:IH slopes. Three of the records in fact have PYA higher than PHA. Note that the fau lt type associated with each record is also indicated in Figure 4 .3. The three records having higher PV A than PHA are associated with reverse fault. 4.2 Material Characterization for Parametric Study Materials considered in the parametric study are similar to the ones in the Validation of Numerical Program section which include the facing block geosynthetic reinforcement backfill EPS and foundation material. In the parametric study the backfill and the foundation material were assumed to have the same soil properties The geologic cap model was used to sim ulate the soil. Determination of the cap model material parameters follows the same procedure presented in the Geologic Cap Model section. Geosynthetic reinforcement was simulated by the plastickinematic model which is the same model adopted in the validation section The facing block and the EPS materials were assumed to be linearelastic and their material properties are the same as those included in the PlasticKinematic Model section. Material properties of facing block and EPS are presented in Table 3 .1. Developments of material parameters for soils and geosynthetic reinforcements are presented in the following sections. 126
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...... tv ....l Tabl e 4 1 Ground mot i on param e t e r s of th e 2 0 earthqu a k e r e cord s No E arthqu ake PEE R R eco r d Date D u ratio n ID Fil e ( s ec ) I D u zce T u r k ey P I 557 1062 N 1999/ 1 1 / 1 2 30 .02 2 Oroville POliS D EBHOOO 1975108/08 1 3 .65 3 Kobe Japan P I 054 SHI090 1995/01/ 1 6 35 .03 4 Coyo t e Lake POl 49 G03 1 40 1979/08/06 26 8 1 5 5 Kobe J apan P I 054 SH IOOO 1995/ 0 1 / 1 6 35 .03 6 Coalinga P0346 HZ 1 4000 1983/05102 40.02 7 Northridge P0887 ARL090 1994/ 0 1 1 1 7 40 04 8 Lorna P r i e t a P0736 G03090 1 989 / 10/18 30 0 1 9 Lorna Prie t a P0745 CLS090 1989/10/ 1 8 30 0 1 10 No rthrid ge P0883 ORR360 1994/ 0 1 / 1 7 40 04 I I Ca p e P08 1 0 R I0360 1992/04/25 36 .04 Me n docino 12 North r idge P0883 ORR090 1994/ 0 1 / 1 7 40 04 1 3 Lorna Prie ta P0745 CLSOOO 1989/10/18 30 .01 1 4 Kob e Ja p an PI056 TAlOOO 1995/ 0 1 / 1 6 38 02 1 5 C hiC h i Taiwan PI461 TCU095N 1999/09120 60 0 1 5 1 6 Northr i dge P I 020 SPV270 1994/ 0 1/17 24 44 1 7 Northrid ge P I 00 5 RR S228 1994/ 0 1 / 1 7 14.96 1 8 Nort hr i d ge PI023 SCS I 42 1994/ 0 1 /17 40 0 1 19 C h iChi, Taiwan P I 532 WNTE 1999/09120 50 015 20 No rth ridge P0935 TAR360 1994/ 0 1 / 1 7 40 04 M w = m o m e nt m ag mtud e ; D = eplce ntr a l dIst an ce Peak Acc Hor Vert ( g) ( g) 0 1 1 4 0 093 0.168 0 073 0 2 1 2 0 059 0 229 0 1 60 0 243 0 059 0.282 0 097 0.344 0 552 0.367 0 338 0.479 0 455 0 514 0 217 0 549 0 195 0 568 0.2 1 7 0 644 0.45 5 0 694 0.433 0.7 1 2 0.255 0.753 0.467 0 838 0 852 0.897 0 586 0 958 0 3 1 1 0 990 1.048 Peak Ve l oc i ty P eak Dis p Fault Type Mw D. Hor Vert Hor. Vert. (cmls) (cmls) (c m ) (cm ) (km) 1 1.08 7 66 9 28 8 .04 S tr i k eS l i p 7 1 29 3 3 07 1.67 0 .17 0 06 Normal 4.7 7 0 27 90 6.40 7 7 1 2 57 StrikeS l i p 6 9 46 0 28 .75 5 1 8 4 86 1.26 S t rikel i p 5 7 9 6 37 76 6.40 8 8 6 2 57 StrikeS l i p 6 9 46 0 40 8 5 11.38 8 1 0 4 1 1 R eve r se 6.4 38 5 40 44 1 7 .73 15. 06 8 54 R eve r se 6.7 11.I 45 00 1 5 02 20 26 9 03 Reve r se 6 9 31.4 Ob liqu e 45.1 1 1 7 .65 11. 26 7 1 2 R eve r se 6 9 7 2 Ob l i qu e 52 .04 1 2 29 15. 52 5 22 Reve r se 6 7 40 7 41.90 1 0 54 1 9 .74 7 04 Reve r se 7 1 22 6 51.82 1 2.29 8 86 5 22 Reverse 6 7 40 7 55. 29 1 7 .65 10.62 7 .12 R eve r se 6 9 7 2 O b l iqu e 67. 8 0 34 76 30 9 1 1 1.92 St rik e S lip 6 9 38 6 49 39 22.02 26 .95 19. 20 Reve r se 7 6 95. 7 O bliqu e 84 47 33 02 1 8 70 9 .74 Reve r se 6 7 8 5 1 66 02 50 .63 28 07 11.96 Reverse 6 7 1 0.9 102 20 34 59 45 1 2 2 5 67 Reve r se 6 7 13.1 68 60 34 .16 32 08 1 6 .70 Rever s e 7 6 1 4 2 O bliqu e 77. 1 7 73.46 30.21 21.66 Reve r se 6 7 5.4
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Table 4 2 Spectral parameters from response spectrum at 5 % dampin g of the 20 e arthquake r e cords No Earthquake Peak Ace Tp Max S S at T = 0.2 sec S at T = 0 5 sec S at T = I sec Hor. Vert Hor. Vert Hor Vert Hor Vert. Hor Vert Hor. Vert ( g ) (g) (s ec ) ( sec ) (g ) (g) ( g ) ( g) ( g) (g) ( g ) (g) I Duzce 0 114 0 093 0 32 0 09 0 347 0.419 0 302 0 180 0 243 0 078 0 099 0 065 Turkey 2 Oroville 0 168 0 073 0 08 0 095 0 .681 0 268 0 140 0 110 0 066 0.028 0.021 0.003 3 Kobe Japan 0 .212 0.059 0 6 0 56 0 733 0 160 0.3 75 0 150 0 598 0.1\3 0.228 0 075 4 Coyote Lake 0.229 0 .16 0 .15 0 055 0.769 0.676 0 547 0 198 0.35\ 0.070 0.35 2 0.065 5 Kobe Jap a n 0.243 0 059 0.67 0.56 0 897 0 160 0.413 0 150 0 646 0 113 0.335 0 075 6 Coalinga 0 282 0 097 1.1 0.4 2 5 1.12 8 0.24 2 0.438 0 177 0.575 0.235 1 027 0.19 2 7 Northridge 0.344 0.552 0 245 0.12 0 923 1 783 0.667 0 808 0 610 0 314 0 5 2 7 0 158 8 Lorna Prieta 0 367 0 338 0 .215 0 05 1.475 1.136 1 3 2 6 0 309 0 704 0.233 0.379 0 148 9 Lorna Prieta 0.479 0.455 0 56 0 .215 1.417 \.42 4 1.032 1.378 1 035 0.418 0 550 0.189 10 Northridge 0 514 0 217 0 3 0.29 1.427 1 058 0.919 0 553 1 325 0 321 0 972 0 065 tv 00 11 Cape 0 549 0 .195 0.4 2 5 0 .18 2 217 0 5 2 5 1.131 0.429 1 750 0 185 0.390 0 218 Mendocino 12 Northridge 0 568 0.217 0.26 0 .2 9 1.991 1 058 1 .2 39 0 553 0 974 0 3 2 1 0 535 0 065 13 Lorna Prieta 0 644 0.455 0 3 0 .215 2 180 1.4 2 4 1.0 2 4 1.378 1.448 0.418 0 395 0.189 1 4 Kobe Japan 0.694 0.433 0.475 0 1 3 1.784 1 3 4 1 1.461 0 832 1 569 1 .051 0 90 4 0 351 15 ChiChi 0.712 0 .2 55 0 3 2 0 05 2.409 0 839 1.567 0 317 1 008 0 158 0 277 0 167 Taiwan 16 Northridge 0.753 0.467 0 635 0 .18 1.758 1.158 1.334 1.143 1.48 2 0 6 2 1 1.133 0.33 2 17 Northridge 0.838 0.852 0 74 0.075 2 050 3.246 1.485 1.716 1.775 0 947 1.834 0 369 18 Northridge 0.897 0.586 1.1 0 .11 1.691 2.464 1 378 1. 220 1.474 0.377 1.403 0.415 19 ChiChi 0 958 0 311 0 .24 5 0. 53 2 2 74 0 80 5 1 6 2 5 0.418 1.35 3 0 80 4 0.541 0.6\2 Taiwan 2 0 Northridge 0 990 1.048 0 38 0 .23 3 2 47 4 .214 2 .35 3 3 .400 1 8 22 0 794 0.507 0 249
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1000 500 o c: g 500 P0810 (RI0360) PHA= 0 549 9 1000 ''''''' IIo M 0 Time (sec) '"' 1000 .,,,,r, A ::: 500 c: :8 0 ...... P 0883 (ORR090) rCI> 500 PHA = 0 568 9 1000 ''''''' o M 0 1000 500 Time (sec) 13 Loma Prieta P0745 (CLSOOO) f .... I I c: o += CI> fi o 500 PHA = 0 644 9 ""''' I1000 ''''''' o M 0 Time (sec) 1000 ,,..,,, 500 14 Kobe Japan P1056 (TAlOOO) c: :8 0 CI> 500 PHA = 0 694 9 1000 ''''''' o 20 40 60 0 Time (sec) '"' 1000 .r,,,r, E 500 rg 0 1If'. 15 ChiChi Taiwan rP1461 (TC U095 N) CI> 500 rPHA = 0.712 9 r1000 ''''''' o 20 40 60 0 Time (sec) Figure 4.1 (Cont.) 131 (RIO UP) PVA=0.195g 20 40 60 Time (sec) (ORR UP) PVA = 0 .217 9 20 40 60 T i me (sec) (CLS UP) PVA = 0 455 9 20 40 60 Time (sec) (TAlUP) PVA = 0 433 9 20 40 60 Time (sec) (TCU095 V) "T' PVA = 0 255 9 20 40 60 Time (sec)
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1500 (fJ" 1000 6 Coalinga P0346 (HZ14000) (HZ14 UP) U 5% damping 5% damping () '" 500 Horizontal component Vertical component t5 0 Q) a. 0 1 (fJ 2 3 0 1 2 3 Period T (sec) Period T (sec) ,... III 2000 (fJ" 1500 7 Northridge P0887 (ARL090) (ARLUP) u 1000 5 % damping () '" 500 Vertical component t5 0 Q) a. 0 (fJ 1 2 3 0 1 2 3 Period T (sec) Period T (sec) ,... 2000 E. (fJ" 1500 8 Loma Prieta P0736 (G0309 0 ) (G03UP) u 1000 5 % damping 5 % damping () '" 500 Horizontal component Vert i cal component iii U 0 Q) a. 0 1 (fJ 2 3 0 1 2 3 Period T ( sec ) Period T (sec) ,... III 2000 (fJ" 1500 9 Loma Prieta P0745 (CLS 090) (CLS UP ) u 1000 5 % damping 5 % damp ing () '" Horizontal component Vert ica l component 500 t5 0 Q) a. 0 1 (fJ 2 3 0 1 2 3 Period, T (sec) Per iod, T ( sec ) 2500 (fJ" 2000 10 Northridge P0883 (ORR360) (ORR UP ) 1500 J 5 % damping 5% damping () 1000 '" Horizontal component Vert i cal component 500 t5 0 Q) a. 0 (fJ 1 2 3 0 1 2 3 Period T (sec) Period T (sec) Fig ure 4.2 (Cont.) 134
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I 2500 (/)" 2000 11. Cape Mendocino P0810 (RI0360) (RIOUP) c..i 1500 5% damping 5% damping 0 1000 Horizontal component Vertical component '" 500 ti 0 (j) a. 0 1 2 3 0 1 2 3 (/) Period T (sec) Period T (sec) N' '" 2500 (/)" 2000 12 Northridge P0883 (ORR 090) (ORRUP) c..i 1500 5% damping 5% damping 0 1000 Vertical component '" Horizontal component 500 ti 0 (j) a. 0 1 2 3 0 1 2 3 (/) Period T (sec) Period T (sec) N' 2500 .e 2000 (/)" 13 Loma Prieta P07 4 5 (CLSOOO) (CLSUP) c..i 1500 5% damping 5% damping 0 1000 Horizontal component Vertical compo nent '" 500 ti 0 (j) a. 0 1 2 3 0 1 2 3 (/) Period T (sec) Period T (sec ) E 2500 .e 2000 14 Kobe Japan P1056 (TAlOOO) (/)" (TAlUP) c..i 1500 5% damping 5% damping 0 1000 Horizon tal component Vertical component '" 500 ti 0 (j) a. 0 1 2 3 0 1 2 3 (/) Period T (sec) Period T (sec ) N' '" E 2500 .e 2000 (/)" 15. ChiChi Taiwan, P1461 (TCU095N) (TCU095V) c..i 1500 5% damping 5 % damping 0 1000 Vertical component '" Horizontal component 500 ti 0 (j) a. 0 1 2 3 0 1 2 3 (/) Period T (sec) Period T (sec ) Fig ure 4.2 (Cont.) 135
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2500 2000 16 Northridge P1020 (SPV27 0 ) (SPVUP) en" 0 1500 5 % damping 5 % damp ing 0 1000 Horizontal c o mponent Vertical co m p o nent '" 500 (3 0 <1> a. 0 1 2 3 0 1 2 3 en Period T (sec) Period T (sec ) r:;(f) 4000 en" 3000 17 Northridge P1005 ( RRS228) ( RRS UP ) 0 2000 5 % damping 5 % damping 0 Hor i zontal component Vert i cal c omponent '" 1000 (3 0 <1> a. 0 1 2 3 0 1 2 3 en P eri od T ( sec) Per iod, T (sec ) r:;(f) 2500 en" 2000 1 8 Northridge P 1 0 2 3 ( SCS142) ( SCS UP ) 0 1500 5 % damping 0 1000 Vert i cal component '" 5 % damp ing 500 Horizontal comp o nent (3 0 <1> a. 0 1 2 3 0 1 2 3 en Period T (sec) Period T (sec ) r:;2500 en" 2000 19 Ch iChi Taiwa n P 1 532 (WNT E) (WNT V) 0 1500 5% d ampin g 5 % damp i ng 0 1000 Vert i cal component '" Hor i zontal component 500 (3 0 <1> a. 0 1 2 3 0 1 2 3 en Period T (sec) Per iod, T (sec ) 4000 en" 3000 20 Northridge P0935 (TAR36 0 ) ( TAR UP) 0 2000 5 % d am p in g 5 % damp ing 0 Ve rti cal component '" Horizo ntal co mponent 1000 (3 <1> a. 2 3 0 1 2 3 en Period T ( sec) Per iod, T ( sec ) Figure 4.2 ( Cent.) 136
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1 2 1 0 a.. 0 8 c o :;:; Q) 0 6 () C1l t Q) 0.4 > C1l Q) a.. 0 2 Normal fault Strikeslip fault Reverse fault / ;J1 Reverse oblique fault 20. / / / / / / / / 1 /' / /' / / /' / /," / /' / /' 7 { g 134'/18 / /'. / /'" /' /' / /' "'" // //'/' ",, __ / /' / 10. ;;. __ 1 / ........ ___ 11 / /'/' /') .!...... __ 3 0 0 ""'''_'l._''_''_'''' 0 0 0 2 0.4 0 6 0.8 1 0 1 .2 Peak horizontal acceleration PHA (g) Figure 4.3 Variation of Peak Vertical Acceleration with Peak Horizontal Acceleration for the 20 Selected Earthquake Records 4.2.1 Soil Characterization Granular soils were considered in the parametric study. The friction angle is of significant importance in describing the behavior of granular material thus it was used as the primar y paramet er for developing other ph ysica l properties The uncorrected standard penetration resistance N in units of blows per 0 3 m was determined based on fr iction angle using the relationship proposed by P eck et al. (1974) as shown in Figure 4.4 in which three values of were preselected (viz., 32, 36, and 40). The relative densities Dr of the three soils were determined based on the standard penetration resistance following the simple correlation compiled by Kulhawy and Mayne (1990) where ranges of Dr values were correlated to the ranges ofN values as shown in Table 4.3 Linear interpolation was performed to calculate Dr values based on N for the three soils. As depicted in Figure 4.5 the combinations of friction angle and relative den sity were used to develop the dry unit weig ht Yd and to identify the soil classification for the three soils based on design chart published by NA VF AC (1986a). The moisture contents w for the three soils were selected from the typical compacted material properties presented by NAVFAC (1986b) based on soil classification The moist unit weight Y or the moist density p can be calculated 137
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given the dry unit weight and moisture content. Note that unit weight y here is not the same as the cap model exponential coefficient y as described in the Cap Model Strength Parameters section. Physical properties of the three soils are summarized in Table 4.4. The relative density could be expressed in terms of percent or relative compaction RC according to the relation suggested by Lee and Singh (1971) as: RC = 80 + 0.2xD r (4 .1) where Dr is expressed as percent. Values of RC for the three soils are included in Table 4.4. Note that even though the $' = 32 soil has a RC of 89%, this study does not suggest soil compaction be done at RC of 89% where the norm of practice is a RC of95%. Values ofRC in Table 4.4 merely reflect the probable range of values based on published relationships. 60 ,,,,.r, z 40 c g V) 'in c Q 20 di c Q) a. "E ro "0 c ro U5 Pecketal. (1974) o 28 32 36 40 44 Angle of internal fr iction, (degrees) Figure 4.4 Relation between Standard Penetration Resistance and Friction Angle for the Granular Soils Considered in Parametric Study Table 4.3 Correlation of relative density with standard penetration resistance (after Kulhawy and Mayne 1990) N value (blows / 0.3 m) o to 4 4 to 10 10 to 30 30 to 50 > 50 Relative density Very loose Loose Medium Dense Very dense 138 Dr (%) o to 15 15 to 35 35 to 65 65 to 85 85 to 100
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Dry unit weight, y d (Ib/ft 3) 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 ........ 40 0 :e. Co t5 35 :E co c .... a> C '+30 0 0) c 25 13 14 15 16 17 18 19 20 21 22 23 24 Dry unit weight Y d (kN/m 3) Figure 4.5 Determination of Dry Unit Weight and Soil Classification (modified from NA VF AC 1986a) Table 4.4 Physical properties of soils used in parametric study N Dr Yd USCS W Y P RC { 2 (blows / 0.3m) (%) {kN/ m32 (%) { kN / m32 {kg/m32 (%) 32 16 44 16.2 SP 16. 5 18.9 1926.6 89 36 30 65 18. 5 SW 12. 5 20 8 2120.3 93 40 45 80 16.2 GP 12.5 23.5 2395.5 96 4.2.1.1 Cap Model Strength Parameters in Parametric Study Two strength parameters required by the geologic cap model are a and e which govern the location of the fixed yield surface f, (see Figure 3.5). Adopting the DruckerPrager failure criterion and matching the compressive meridian of the Mohr139
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Coulomb criterion (see Figure 3.9) the strength parameters a and 8 can be related to c' and through: 6 a = (4.2a)(3 .7a bis) .J3 (3 sin 8 = (4.2b)(3.7bbis) .J3 (3 sin A small value of c' equal to 5 kN/m2 (0.7 Ib/in2 ) was assumed for all three soils in the parametric study. The small value of c' was added to provide numerical stability during finite element calculations. As has been described in the Cap Model Strength Parameters section cap model failure envelope exponential coefficient y and exponent p were assumed to be zero. Note that the exponential coefficient y here is not the same as the moist unit weight y The cap model strength parameters for the three soils are summarized in Table 4.5. 4.2.1.2 Cap Model Hardening Parameters in Parametric Study Cap model hardening parameters has been described in the Geologic Cap Model section. The hardening parameters were developed based on the assumed relative densities Dr of the three soils. Determination of the hardening parameters is presented individually as follows: (1) Initial tangent bulk modulus Ki (see Figure 3 15): Ki = 13776 7 x Dr + 27.34 (4.3) where Ki is expressed in kN/ m 2 and Dr in decimal fraction. (2) Asymptotic total volumetric strain (Ev)asy (see Figure 3.15): (Ev)asy = 0.0655 x Dr + 0.0733 (4.4) where both (Ev)asy and Dr are expressed in decimal fraction. (3) Ultimate total volumetric strain (Ev)ult: (Ev)ull = 95% x (Ev)asy (4.5) where both (Ev)ult and (Ev)asy are expressed in decimal fraction (4) Ultimate mean stress, pult: PUl l = Ki (EJasy (EJult / [(EJa s y (EJult] (4 6)(3.14 bis) where PUll has the same units as Ki and is expressed in kN/ m2 (5) Ultimate tangent bulk modulus (Kt)ult: (Kl)ult = [ K i + Pult J2/Ki (4.7)(3.15 bis) (EJa s y where (KDult has the units of stress (i.e. kN / m 2 MN / m2, or GN/ m 2). (6) Ultimate elastic volumetric strain )ult: 140
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( E e ) Pult v ult (K ) t ult (4.8)(3.16 bis) where (E: ) ult is expressed in decimal fraction. (7) Hardening law coefficient W : W = = (EJult (E:)ult = 0 .95 (EJasy Pult (Kt)ult (4.9)(3.20 bis) where is the ultimate plastic volumetric strain and both W and are expressed in decimal fraction (8) Mean stress p: (9) (10) p = (cr v + 2cr h )/3 (4 10)(3 .23 bis) where cry and crh are vertical and horizontal stresses, respectively and are expressed in kN/m2 Horizontal stress crh is calculated based on the atrest earth pressure coefficient Ko (i e. K o = 1 Tangent bu l k modulus Kt : K t =(Ki + P J 2 /Ki (EJasy (4.11 )(3 .11 b bis) where Kt has the units of stress (i e., kN/ m2 ) and varies with mean stress p Shear modulus G : G = 3 Kt (12v) 2 (1 + v) (4.12)(3.22 bis) where v is the Poisson's ratio and is assumed to be 0.35. G has the units of stress (i.e., kN/ m2 ) and varies with mean stress p (11) Plastic volumetric strain : EP=E _Ee = P P v v v K + p i Kt i I(EJasy (4.13)(3 .13 bis) where both Ey and E : are total and e lastic volumetric strains, respectively is expressed in decimal fraction. (12) Hardening law exponent D: o In(l/(3 p) (4 14)(3.21 bis) where D has the inverse units of stress (i.e., m2/kN or m2/MN) and varies with mean stress p. (13) Shape factor or cap axis ratio R (see Figure 3.20): R = 0 0833 x p + 11.3333 (0 ::; p ::; 70 kN/m2 ) (4. 15a) 141
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R = 0 0167 x p + 6 6667 (p > 70 kN/m2 ) (4.15b) where R is dimensionless and varies with mean stress p in units of kN/ m2 The lowest R value is assumed to be 0 .05, which corresponds to p = 396 2 kN/ m2 Variation of hardening law exponent D with initial relative density Dr at different mean stresses p is shown in Figure 4.6 Due to the nature of Equation 4.14 value of D increases when Dr is less than 56% resulting in an upward concave curve. This pattern contradicts the realistic soil behavior where D should increase with increasing Dr. With this argument the minimum value of D is assumed to be limited to Dr of 56% The dashed portion of the curve indicates this assertion. The variations of tangent bulk modulus Kt shear modulus G and hardening law coefficient W (or ultimate plastic volumetric strain with initial relative density Dr are shown in Figures 4.7 4.8 and 4 9 respectively. Note that parameters D Kt, and G all increase with increasing mean stress p. r0o.. 1 2 .,,y., 130 kPa 100 kPa 0 8 o c Q) c 70 kPa 8. x 0 6 +++j+,HIH Q) OJ C 40 kPa p = 10 kPa 0.4 "E ro I O 2 +==:=:=:=:=:==i==:=:=:=:=:=:=:==l==:=:=:=:=:====J::=====ILILj ++1 o +,+,+,j,+.1 o 20 40 60 80 100 Init i al relative dens i ty D r (%) Figure 4 6 Variation of Hardening Law Exponent D with Relative Density Dr 142
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80 130 kPa 60 ",:::l "S "8 40 E '" "S .0 C Ql Cl c 20 o o 20 40 60 80 1 00 In i tial relative densi ty 0 ( % ) Figure 4 7 Variation of Tangent Bulk Modulus Kt with Relative Density Dr 25 130 kPa 20 100 kPa Z 70 kPa 15 c.9 40 kPa vf :::l "S "8 p = 10 kPa E 10 ffi Ql .r::: (/) 5 o o 20 40 60 80 100 Init i al relative density, 0 ( % ) Figure 4 8 Variation of Shear Modulus G with Relative Density Dr 143
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0 08 0 06 c (1) 0 IE (1) 8 0 .04 0> c: c: (1) 'E 0 .02 o I o ........ I I I I 20 40 60 80 100 Initial relative density D r (%) Figure 4.9 Variation of Hardening Law Coefficient W with Relative Density Dr The stressstrain curves were generated from numerical triaxial test utilizing a single element model under axisymmetric loading condition (see Figure 3 19). The single element was first subjected to hydrostatic compression pressure 0'3 (or mean stress p) and subsequently subjected to the application of deviatoric stress The loading condition was consistent with the consolidateddrained conventional triaxial compression test. Figures 4.10 4.11 and 4.12 show the stressstrain curves under various mean stresses for the three soil friction angles of 32,36, and 40, respectively. Stiffer stressstrain response is observed for the soil with = 40 and during shearing the = 40 soil also contracts less in volumetric strain than the = 32 soil. The pattern is consistent with realistic soil behavior with the notion that stiffer response is associated with the high strength soil. The cap model hardening parameters for the three soils are summarized in Table 4.5 The parameters needed to determine mean stress dependent variables such as Kt G and D for the three soils are summarized in Table 4 6 Variation of mean stress dependent variables Kt and G with depth is shown in Figure 4.13. Variation of hardening law exponent D with depth for the three soil friction angles is shown in Figure 4 .14. As indicated in Figures 4.13 and 4.14 nonlinear response is observed with increasing depth for the mean stress dependent variables. 144
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500 4 00 ., Cl. b 130 kPa = 32 (D = 44 %) 2 5 Fi g ure 4 .10 Stressstrain Curves for = 32 Soil from Numerical Tria x i a l Tests 500 4 00 130 kPa ., Cl. b = 36 (D, = 65%) 2 5 F i g ur e 4 .11 Stresss tr ain Curves for = 36 Soil from Numerical Tri ax i a l T e st s 145
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4 00 Iii' a. b <1300 .; g 2 00 .9 ro 5 ., o 100 5 10 15 130 k P a 1 00 kPa 70 kPa 40 kPa p = 10 kPa 20 2 5 __ , __ __ __ __ l 0 5 of 1 ., u 1 5 E ::l :g 2 = 40 ( D = 80%) p = 1 0kPa 40kPa 70 kPa 1 00 kP a 130 kP a 2 5 Figure 4.12 Stressstrain Curves for = 40 Soil from Numerical Triaxial Tests Table 4 5 Cap model parameters for the three soils of parametric study Parameter = 32 = 36 = 40 Relative density Dr (%) 44 65 80 Mass density p (kg/m3 ) 1926,6 2120.3 2395,5 Bulk modulus Kt (MPa) (See Fig 4.13) Vary with p Vary with p Vary with p Shear modulus G (MPa) (See Fig. 4.13) Vary with p Vary with p Vary with p Failure envelope parameter a. (kPa) 5.9 5.8 5.6 Failure envelope linear coefficient 8 0.2477 0.2814 0 3149 Failure envelope exponential coef., y 0 0 0 Failure envelope exponent f3 (kPayl 0 0 0 Shape factor R (See Fig 3.20) Vary with p Vary with p Vary with p Hardening law exponent 0 (See Fig 4 14) Vary with p Vary with p Vary with p Hardening law coefficient W 0.0401 0 0277 0.0189 Hardening law exponent Xa (kPa) 0 0 0 146
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Table 4.6 Parameters for finding mean stress dependent variables K(, G and D for the three soils of parametric study Parameter = 32 = 36 = 40 Relative density Dr (%) 44 65 80 Initial tangent bulk modulus Ki (kPa) 6089 8982 11049 Asymptotic total volumetric strain (Ev )asy 0.0445 0 0307 0.0209 Ultimate total volumetric strain (Ev)ul( 0.0423 0.0292 0.0199 Ultimate tangent bulk modu l us (Kt)ult (MPa) 2436 3593 4419 Ultimate elastic volumetric strain (E: ) ull 0 0021 0.0015 0.001 Ultimate plastic volumetric strain = W 0.0401 0 0277 0.0189 <1>' = 36 <1>' = 40 4 4 I I 5 a. a. Q) Q) 0 0 8 8 5 1 0 15 20 2 5 3 0 2 4 6 8 1 0 Tangent bul k modulus K, (MN / m 2 ) Shea r modulus G (MN / m 2 ) (a) (b) Figure 4 .13 Variation of (a) Tangent Bulk Modulus K( and (b) Shear Modulus G with Depth 147
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+$' = 32 $'=36 $' = 40 4 8 o 0 1 0 2 0 3 0.4 0 5 Har den ing l aw exponent 0 ( MPa )' Figure 4.14 Variation of Hardening Law Exponent D with Depth 4.2.2 Geosynthetic Reinforcement Characterization Three geogrid reinforcements were devised for the parametric study. The tensile loadstrain responses of the three geogrids were estimated based on results of various geogrid tensile load tests as shown in Figure 4.15, and the three geogrids were termed accordingly based on their stiffness characteristics. The high strength geogrid exhibits the near upperbound tensile loadstrain response, whereas the low strength geogrid reflects closely the lowerbound tensile loadstrain response. The medium strength geogrid is viewed as having the average loadstrain response The three geogrids were assumed to have a constant thickness of 1 5 mm The geogrids were simulated by the plastickinematic model which describes a bilinear stressstrain response. Description of the bilinear stressstrain model is depicted in Figure 3 21. The slope of tensile loadstrain curve is the product of modulus (e.g. Young's modulus E or tangent modulus Et ) and thickness of the geogrid. The tensile load is expressed in units of force per unit width of the reinforcement. Inversely modulus was calculated by dividing the slope of the tensile loadstrain curve by the geogrid thickness. Similarly the yield stress cry was found by dividing the yield tensile load 148
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by the thickness of geogrid. The model parameters for the three geogrids are summarized in Table 4.7 It should be noted that no failure criterion is implemented in the plastickinematic model thus a geogrid rupture failure cannot occur. 200 0 Bathurst & Cai (1994) High stiffness x Cazzuffi et al. (1993) T 5 % = 72 kN/m Hirakawa et al. (2003) 160 t', Lee et al. (2002) + Ling et al. (2005a) E Z 120 C Polyacrylate (1 O%/min) Medium stiffness "0 t',t',t',t', T 5 %=36kN/m ro .2 w iii HOPE (10%/min) 00 80 f PET (10%/min) 40 o 5 10 15 20 25 Strain (%) Figure 4.15 Comparison of Tensile Load Test Results between Idealized Geogrids and Typical Geogrids Table 4.7 Plastickinematic model parameters for geogrid reinforcements Parameter High stiffness Medium stiffness Low stiffness Density p (kg/m3 ) 1030 1030 1030 Young's modulus E (MPa) 1155.6 533 3 160 Poisson's ratio v 0.3 0 3 0.3 Yield stress cry (MPa) 34.7 2l.3 8 Tangent modulus Et (MPa2 656.4 266 7 35. 6 149
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4.3 Development of Model Dimensions Freestanding simple geosyntheticreinforced soil wall geometry was considered in the parametric study. The models in the parametric study were developed based on dimensions defined in Figure 4.16. A uniform reinforcement length at 0.7 times the wall height was assumed for all models. The length of the retained earth behind the reinforced soil mass was extended beyond the theoretical static passive slip plane Similarly the length of the foundation soil in front of the wall was also extended beyond the static passive slip plane emanating from the lower left boundary of the model. The extend of the boundaries alongside the model was adopted in order to minimize interference between the model boundary and the potential formation of slip surfaces. The depth of the foundation soil was selected to be 0.75 of the wall height as it is the minimum boring depth required for a subsurface investigation (Arman et al. 1997) It was further assumed that competent bedrock underlies the foundation soil. This minimal foundation soil depth conforms the assumption that the wall is situated in a competent site where problematic foundation soils that might lose their strengths when subjected to seismic loads do not exist (e.g. liquefiable soils). The groundwater table was assumed to be below the bottom of the foundation soil and it does not a f fect the seismic performance of the wall. In the parametric study granular soil was assumed as the backfill for the reinforced soil mass. In addition it was assumed that onsite material was used to construct the wall ; hence all soils including the reinforced soil retained earth and the foundation soil would exhibit similar behavior. Only one soil type was specified in each numerical wall model. 12 7H '1 .,/ H ./ Reinforced so i l H IH ./ "H./ H . ./ H H H I ./ H Facing block _____ H Retained earth tH 9\ cy. R H tj 45 12 H 1 6SH ,/ Reinforcement 1 ./ , ./ <;\\9'3' (L = O .7H) O 7SH Foundat ion so i l j .J'l>.' 45 Figure 4.16 Numerical Model Dimensions Adopted in the Parametric Stud y 150
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The effect of lateral extent of the vertical side boundaries on the performance of the wall was evaluated The evaluation was conducted on a 6 m high 10 wall facing batter, 0.4 m reinforcement spacing, medium stiffness (Ts% = 36 kN/m) reinforcement and = 36 wall. The depth of foundation soil was kept constant at 0.75 of the wall height while the extent of the vertical side boundaries varied from 0 5 to 1.5 times of the proposed extent. The three models were subjected to the same Northridge earthquake shaking (i.e. No. 12, P0883 ORR090) with peak horizontal and vertical accelerations of 0.568 g and 0 217 g respectively The calculated maximum horizontal displacements are presented in Figure 4.17. As indicated in the figure the model with 0.5 times the proposed extent experienced the least amount of horizontal displacement and the models of the proposed extent and 1 5 times the proposed extent experienced larger horizontal displacements. However the variation in horizontal displacement is minimal between the proposed extent and that of the 1.5 times the proposed extent. The results suggested that the interference of vertical side boundaries to the performance of the model diminishes with the distance beyond the proposed extent of the vertical side boundaries The proposed boundary extent was thus considered adequate and was adopted in the parametric study. g I :E '" iii .c 4 2 Proposed boundary extent 1 5 Proposed extent o 100 200 300 400 Maximum horizontal d isplacemen t (mm) Figure 4 .17 Effect of Lateral Boundary Extent on Wall Displacement 151
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4.4 Parametric Study Program The purposes of the parametric study was to examine the parameters that could affect the seismic responses of geosyntheticreinforced soil walls and to determine the extent of influence a particular parameter might have on the seismic responses. The parameters considered include: (1) wall height H (2) wall batter angle 0), (3) soil friction angle (4) reinforcement spacing Sy, and (5) reinforcement stiffness T 5%. These parameters are basic features and components in the construction of a GRS wall which are also considered as fustorder factors in the design of such wall system. The parametric study program is schematically shown in Figure 4 18. A total of 11 models were developed for the parametric study. In the parametric study a baseline case was first established and gauged for subsequent analyses. For each of the parameters examined the baseline case was encompassed by a lower value and a higher value The seismic performance relationships were then estimated based on three sets of data. Note that the selected parameter values are typical of freestanding GRS wall having simple wall geometry Generic facing block dimension was used where the width and the height of block were conveniently assumed to be 0.3 m and 0.2 m respectively. For all 11 models a uniform reinforcement length of 0.7 times the wall height was specified as the ratio of 0 7 is stipulated in the FHWA design guidelines (Elias et al. 2001) Although the ratio of 0.7 may not yield the most cost effective wall system it is speculated that a reduction of the ratio could possibly result in wall instability and lead to adverse performances. 4.5 Global Stability Global sta bility (or overall stability) analysis was performed using limit equi librium commercial code SLOPE / W (Krahn 2004b) for each of the cases proposed in the parametric study The global stability analysis is similar to a deep seated stability analysis where all the structural components are included in the model but are not contributed to the analysis since the slip surface passes behind the reinforced soil mass and through the foundation soil. In other words the slip surface is not intercepted by the reinforcements in the global sta bili ty. As such the reinforcement tensile strength and the soilreinforcement interface characteristics do not affect the factor of safety calculation in the globa l stability analysis. Stated differently the effect of reinforcement stiffness could not be evaluate d in the global stability analysis. The stability of the proposed model configuration can be evaluated at first by means of global stability through limit eq uilibrium method. In the case when the globa l factor of safety is less than unity it is then evident that the proposed model 152
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configuration would not be stab le in a more rigorous computation such as the finite element analysis. On the other hand if the g lob al factor of safety is greater than unity then the initial stability of the model would be assured in the rigorous computation. This first screening analysis could be done rather rapidly and at a much lower cost than the rigorous finite element analysis. In addition globa l stability analysis verifies whether the proposed model dimensions are sufficiently large and that the slip surfaces are not interrupted by the model boundaries. The method of slices in conjunction with circular slip surface was used in the global stability analysis Specifically the MorgensternPrice method utilizing the halfsine interslice function was used. The circular slip surfaces were specified using the SLOPE/W entry and exit method in which the trial slip surfaces would enter and exit at preassigned loc ations along the ground surfaces The entry and exit locations were selected to cover wide range of trial surfaces including potential deepseated failures. The critical slip surface associated with the minimum factor of safety for the baseline case is shown in F igur e 4.19 and results of all other models are included in Appendix B. The results indicate that all of the proposed models have global factor of safety greater than 1.5 which is higher than the minimum value of 1.3 required by FHWA design guidelines. Note that the critical slip surfaces are all loc ated within the model boundary. The results suggest that the dimensions of the model (i.e. lateral extends and foundation depth) are sufficient with respect to global stability. Effects of wall height H : . 3 m .9m Effects of wall batter angle CJ): . 5 15 Baseline case : Wall height = 6 m Effects of soil friction angle Ijl': Wall batter angle = 10 Soil friction angle = 36 32 Reinforcement spacing = 0.4 m 40 Reinforcement stiffness = medium (T5% = 36 kN/m) Effects of reinforcement spacing Sy: . 0 2 m 0 6 m Effects of reinforcement stiffness T 5%: . Low stiffness (T5% = 12 kN/m) High stiffness (T5% = 72 kN/m) Figure 4.18 Parametric Study Program 153
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H = 6 m Sv = 0.4 m v.aJ1 batter = 10", soil lriet i", = 36 I2 c 0 ., III > Q) 0 [jJ 2 L 12 10 10 12 14 16 18 2D Figure 4.19 Global Factor of Safety of the Baseline Model Configuration 4.6 Modeling Procedure The modeling procedure for the parametric study is identical to the procedure described in Validation of Computer Program with FullScale Tests section. Same as the aforementioned validation test the parametric numerical models were analyzed under plane strain condition Two types of elements were used to discreti ze the numerical models. The facing blocks reinforced soil retained earth, and foundation soil were discretized with 8node constant stress solid elements The geogrid reinforcements were discretized with 4node BelytschkoTsay shell elements, and one integration point was assigned in the shell element that allows no bending resistance. With the plane strain condition, all nodes were restrained in the direction of plane strain The numerical model can be thought of as an extremely large scale shaking table test. Similar to a shaking table test buffer material that has the same material properties of expanded polystyrene (E PS) was placed in the front and at the back of the model. The EPS material was also modeled with the solid elements. Parts were defined in LSDYNA to represent components of the numerical model. During the model construction a "part" is convenient for assigning material properties element type and contact interfaces. As an example the isometric view of finite element mesh of the 6 m high wall with 150 wall batter and a reinforcement spacing of 0.4 m is shown in Figure 4.20. Note that a "part" is designated by a specific color. In the numerical model, gravity was simulated as a body load and was applied at the onset of the analysis. The body load takes the form of a step function and was maintained at the constant gravitation acceleration throughout the duration of seismic 154
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loading. Note that a quiet period of 12 seconds was sustained after the application of gravity load before the beginning of seismic loading to damp the model to reach the stable condition. The seismic loading was simulated by prescribing the horizontal and vertical acceleration time histories to the vertical side boundary nodes and nodes at the base of the model. Another quiet period of 12 seconds was added at the end of seismic loading to again bring the model to the stable condition. Contact interfaces were specified between two distinct parts within the model. The contact type for each ofthe contact interfaces is summarized in Table 3.7. As has described earlier by including the thickness of geogrid reinforcement in the model element incompatibility occurs in the backfill at the end of geogrid layer (e.g. see Figure 3 30). The tiedsurfacetosurface contacts were used to tie the incompatible backfill elements. The contact interface is defined by the coefficient of friction j.l and j.l for contacts with soil was estimated using Equation 3.25 based on friction angle of the soil. Values of j.l adopted in the parametric study are summarized in Table 4 .8. Mass damping coefficient (MDC) was determined for each of the parametric study cases following the procedure described in the Mass Damping Coefficient section From the model calibration a MDC equal to 15% of the natural circular frequency ron has been found to be in agreement with the measured results of full scale tests. The values of MDC along with the wall height H wall facing batter angle ro, soil friction angle reinforcement spacing Sy, reinforcement stiffness at 5% strain T 5%, numbers of shell elements solid elements nodes and natural period Tn for the parametric study numerical models are summarized in Table 4.9. The model geometry and the locations where performances were recorded for the baseline case are shown in Figure 4.21. The corresponding finite element mesh of the baseline case is shown in Figure 4.22. Model geometries and meshes of the 3 m and 9 m wall height wall models are shown in Figures 4.23 4.24 4.25 and 4.26 Model geometries and meshes of the 5 and 15 wall facing batter angle models are shown in Figures 4.27 4 28, 4.29 and 4 30. Model geometries and meshes of the 0.2 m and 0.6 m geogrid reinforcement spacing models are shown in Figures 4.31 4.32 4.33 and 4.34. The baseline case geometry and finite element mesh was used in studying the parameters of soil friction angle and reinforcement stiffness. Note that embedment depth at the toe of the wall was not considered The minimum wall embedment depth of 0.5 m is suggested in the FHW A design guidelines. Stability of the wall model is expected to increase if the embedment depth is incorporated in the model. The results without considering the embedment depth would thus be more conservative. 155
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Table 4 8 Summary of coefficient of friction for contact interfaces Contact 32 36 40 ConcreteSoil 0.42 0.48 0.56 ConcreteConcrete 0.5 0.5 0.5 GeogridConcrete 0.5 0.5 0.5 GeogridSoil 0.42 0.48 0.56 EPSSoil 0.42 0.48 0.56 Table 4 9 Parametric study model summary H co
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z \lx ETA/POST Figure 4.20 Isometric View of6 m High with 15 Wall Batter Finite E lement Model
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Vl 00 0 3 mj I j j I 3 9 m 12 m Concrete Mode l : H = 6 m S v = 0 4 m OJ = 10 facing Reinforced soil Ji 1 0 ... r! ,/""'" Mon i tored performance : .H .. Geosynthetic Horizontal displacement ....rt reinforcement 6m H . Retained earth j Vert i cal d i splacement ....rt .: (l = 4 2 m ) Bearing st r ess H Reinforcement load f{ 0.4 m Horizontal stress H Horizontal acceleration ....a . 02mlt/ . t i 00 0 . 0 0 9 9 m VEPS f0 2 m 4 5 m Foundat i on soi l 27. 515 m Figure 4.21 Wall Dimensions and Materials for Model ofH = 6 m S v = 0.4 m and ill = 100 I 0 .2mEPS"""'
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Model : H = 6 m Sv= 0 4 m (j) = 1 0 Mode l Summary No of parts 161 No of shell elements 1020 No of solid elements 3080 No. of nodes 8999 t 1: i f =F"lg f 1= r= .!: f f t f t E t t f I t I _=1 t t +f ! E =t:: r;, $ rt f t t == ....f 'F I t f t f t L f I $ t I + t'+ Figure 4 .22 Finite E lement Mesh for Model ofH = 6 m S y = 0.4 m and ill = 10
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0'\ o Model: H = 3 m Sv= 0.4 m co = 10 Monitored performance : 6m Horizontal displacement Vertical displacement D Bearing stress Reinforcement load 3m Horizontal stress Horizontal acceleration 4.95 m 2 25 m 0 1 m Concrete facing block ;L,4." Reinforced soi l Foundation soil 0.4 m Geosynthetic rei nforcement (L = 2 1 m) 13 74 m Figure 4 .23 Wall D imens i ons an d Materials for Mo d el ofH = 3 m S y = 0.4 m and ill = 10 Retained earth EPS 0 1 m
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Model : H = 3 m Sv= 0.4 m ro = 100 Model Summary No of parts 59 No of shell elements 364 r No. of solid elements 1198 t No. of nodes 3585 Figure 4 24 Finite Element Mesh for Model of H = 3 m, S v = 0.4 m and 0) = 10
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0'\ N Model: H 9 m Sv 0 4 m ,6l 10 Moni tored performance: 6m 1785m Reinforced soil . Horizontal displacement Vertical displacement Bearing stress Reinforcement load Horizontal stress Horizontal acce.fation 9m .J/ . 02 ml,/:j,.Hr. _________ : T : / EPS 0 .3m 6 .75m F oundation soU 39 7 9 m Geosynthetic reinforcement (LS 3 m ) Retained earth Figure 4.25 Wall Dimensions and Materials for Model ofH = 9 m S v = 0.4 m and 0) 10 0 3 m iEPS""'
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...... 0\ w Model: H 9 m S 0 4 m Q) 10 ModeiSunvnary N o of parts 1 67 No. of she elements 1760 No or solid elements 5454 No. of nodes 15556 1 I f I I Figure 4 26 Finite E lement Mesh for Model of H = 9 m S y = 0.4 m and (0 = 10 I I
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0 3 mj I 39m I 12m I Concrete Model : H = 6 m Sv= 0.4 m (j) = 5 0 facing block '" Rei nforced soil 5 0 .; Monitored performance : ., Geosynthetic Horizontal d i splacement reinforcement 6m Retained earth I Vert i cal displacement (L = 4 2 m) 0 Bearing stress Reinforcement load 0.4 m Horizontal stress Hor i zonta l accele rat ion ... 0 2 Jj I I 00 0 0 0 0 0 0 9 .9m / EPS 0 .2m 4 .5m 0 .2mFoundat ion soil EPS27 0075 m Figure 4.27 Wall Dimensions and Materials for Model ofH = 6 m, Sv = 0.4 m and 0) = 5 0
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Model : H = 6 m S v= 0 4 m <0 = 5 j: t Model Summary j: f r i E f No of parts 113 No of shell e l ements 1 020 No of solid elements 3084 No of nodes 9011 t + 1: + + + Figure 4.28 Finite Element Mesh for Model of H = 6 m S y = 0.4 m and co = 5 0
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...0\ 0\ 0 3 m I 39 m I 12m Concrete : Model: H = S m S v= 0.4 m ro = 15 facing b lock "if . Reinf orced soil .}/ 15 ....([ /' Monitored performance : VJ/ ..Geosynthetic Horizont a l d isplacement reinforcement Sm . Retained earth I Vertical disp l acement ....([ (L = 4 2 m) 0 Bear ing stress ...H Reinforcement load )/ 0.4 m Horizontal stress ...ri . Horizontal acceleration ...H .'T 0 .2mljJ 1 I 00 0 0 0 0 0 0 9 9 m VEPS f0 2 m 4 5 m Foundation soil 28 0515 m Figure 4.29 Wall Dimensions and Materials for Model ofH = 6 m Sy = 0.4 m and co = 15 I 0 .2mEPS
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Model : H = 6 m S v= 0.4 m OJ = 15 Model Summary No of parts 113 No of shell elements 1020 No of solid elements 3084 No of nodes 9011 + Figure 4.30 Finite Element Mesh for Model ofH = 6 m S v = 0.4 m and co = 150
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...... 0\ 00 0 3 m I 39m I 12m Concrete Mode l : H 6 m S v 0 2 m CJ) = 1 0 facing block Reinf orced soil 10' Mon i tored performance : r Geosynthetic Horizontal d ispl acement reinforcement 6m I Vert ical d i splacement (l4 2 m ) Reta i ned earth Bear ing stress Re i nforcement load 0 2 m Horizonta l stress Hor izontal accelerat ion 0 .2ml I I ... . . : 9 9 m / EPS 0 .2m 4 5 m Foundation soil 27 515 m Figure 4,31 Wall Dimensions and Materials for Model ofH = 6 m S y = 0.2 m and (0 = 10 I 0 .2m
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Model: H = 6 m S.= 0 2 m C!l = 10 tt r= Mode l Summary No of parts 127 No of shell elements 1972 f No of solid ele m ents 36 72 No of nodes 1 2 2 1 9 'l f +Figure 4.32 Finite Element Mesh for Model of H = 6 m S y = 0.2 m and 0) = 100
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Model: H = S m S v= o s m Ul = 10' Monitored performance : Horizontal d i splacement I Vertical d i splacement Bea ring stress Re inf orcement l oad Hor iz ontal stress Horizonta l accelerat ion Sm 0 3 m 39m Conc r ete fI facing bloc k I!...rl. R .......... ...::... 10' ..tI .1=/"1+: 1=/ R 0 2 m lJ=t O S m 10 I 00 0 ________ _____ +__ l/EPS 10 .2 m 4 5 m Foundation soil 27 515 m Reinforced soil Geosynthet i c reinforcem ent (L = 4 2 m) 12m Reta ined earth Figure 4,33 Wall Dimensions and Materials for Model ofH = 6 m S y = 0 6 m and ill = 10 0 2 m I
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Model : H = 6 m S v= 0 6 m ro = 10 Model Summary No o f parts 108 No of shell elements 680 No of solid elements 2844 No,ofnodes 7804 Figure 4,34 Finite Element Mesh for Model of H = 6 m Sy = 0.6 m, and co = 100
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5. Results of Parametric Study Results of the parametric study are presented in this chapter. The effect of multidirectional loading (i.e. combination of horizontal and vertical shaking) is first discussed. Results o f the numerical model tests specified in the parametric study program were compared to the FHWA (2001 ) methodology. Discrepancies between the LSDYNA (herein referred to as finite element method FEM) results and FHWA predictions were identified and discussed Regression analysis involvin g multiple design parameters or predictor variables is presented in Chapter 6. 5.1 Effects of Multidirectional Shaking Conventional seismic design of earth structures typically considers the horizontal component of the seismic loading and frequently ignores the vertical component in order to simplify the problem and this is the case in the current seismic design of GRS walls A study was performed to examine the effect of multidirectional seismic loading on GRS wall where the baseline case model was shaken: (1) with only the horizontal component and (2) with the combined hori z ont a l and vertical components of the 20 earthquake records Four seismic performances including horizontal wall facing displacement wall crest settlement bearing stress and reinforcement tensile load with respect to the peak horizontal acceleration (PHA ) were compared for the effect of multidirectional shaking and the results are shown i n Figures 5 1 5.2 5.3 and 5.4. The results indicate that multidirectional seismic loadings (or combined shaking) yielded greater wall face displacement crest settlement foundation bearing stress and reinforcement tensile load than the loadings with only the horizontal component. Based on the facing displacement comparison (see Figure 5.1) the displacements begin to deviate from one another at PHA of approximately 0.3 g. The threshold value coincides the maximum acceleration coefficient (A) of 0.29 specified in the FHWA methodology where the design method is considered applicable with A less than 0 29 The results of facing displacement imply that the effect of multidirectional shaking is insignificant for earthquakes with PHA less than 0.29 g. Hence with respect to horizontal facing displacement the FHW A methodology of neglecting the vertical component is considered sufficient for PHA less than 0 29 g. 172
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1800 1800 1400 1200 1000 800 600 400 200 w. :0 2 a W 0 71M / j / W /;' k"O c 0 4 0 6 0 8 Peak horizontaJ acceleration. PHA (g) 1 2 Combined shaking o Honzontalshakingonly Expon. ( Comb ined shaking) Expon. (HorIZontal shaking only) Figure 5.1 Vartiation of Maximum Horizontal Wall Dispalcement with Peak Horiztonal Acceleration b y the Effect of Multidirection Shaking 200 180 180 140 E Eo i 120 f! S 100 E 80 E 60 40 20 c / W 0 .7804 / /: W0702 'I. /1 / 0 '" .. ,.tJ 0 e j::/ 0 .// 0 o 0 2 0 4 0 6 0 8 Pea.k horizontal acceleration PHA (g) 1.2 Comb ined shaking o Horizontal shaking only Expon ( Comb ined shaking) Expon. (Horizontlltshaking only ) Figure 5.2 Vartiation of Maximum Wall Crest Settlement with Peak Horiztonal Acceleration with by Effect of Multidirection Shaking 173
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.. .. l!!. 7 00 600 soo 1 00 R" 0 .n15 0 k c /r: 0 2 / '/ c ,. II iii 0 .. 0 0 2 0.4 0 6 0 8 Peak horizontal acceleration PHA (g) u Rl06941 1 2 Combined shaking o Horizontll shaking only Expon (Combined shaking ) Expon (Honzontallhakin g o nly) Figure 5.3 Vartiation of Maximum Bearing Stress with Peak Horiztonal Acceleration by the Effect of Multidirection Shaking a j c E ,. ..... 0 2 0 4 0 6 0 8 1 2 Peak horizonta l acceleration PHA. (g) Combined shaking o Honzontll shaklng anty Expon (Comb ined shaklng ) Expo" (Horizontal shakI n g only) Figure 5.4 Vartiation of Maximum Reinforcement Tensile Load with Peak Horiztonal Acceleration by the Effect of Multidirection Shaking 174
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The effect of combined shaking as compared to the horizontal shaking alone was also evaluated quantitatively by means of percent increase. The percent increase is calculated as: X b d Xh' J 0/0 Increase = com m e onzonta X 1000/0 Xhorizon t a J (5.1) where X c ombine d = seismic performance due to combined shaking and Xhorizontal = seismic performance due to horizontal shaking only. The average percent increase from horizontal shaking only to combined shaking for horizontal wall displacement crest settlement bearing stress and reinforcement tensile load are 12.6% 26.0% 3 2%, and 5.7% respectively. The overall average percent increase for all performances was found to be 11.9% The effect of combined shaking in terms of percent increase relative to horizontal shaking only is shown in Figure 5.5. The percent increase suggests that horizontal wall displacement and crest settlement are more susceptible to combined shaking than bearing stress and reinforcement tensile load These increases are not negligible ; hence for the seismic design of GRS wall to be conservative it may be prudent to include the vertical component and to utilize th e combined shaking 3 0 0 ., .. .. 0 2 5 0 +_...____j '" :li g .. s i 150 tj 'i! '" :li ] 1 50+_____j ; .... Max hOlllontal d i sp M ax aest settle men t M ax be a r i ng_tress + M ax ,etnf lenllt.load ();oar_II ave,ege e + ; .E ; ,. :;;:""1=", .. : r t Overa a verag e 11. 9 % 0 ... + I .. 0 2 0 4... 0 6 t 0 8 1 1 2 .. Peak horizontal a cceleration PHA (g) Figure 5 5 Comparison of Percent Increase for the Effect of Multidirection Shaking 175
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5.2 Seismic Performances Seven seismic performanc es evaluated from the results of the parametric stud y include: (1) maximum wall facing horizontal displacement ( units : mrn) (2) maximum wall crest settlement (units: mrn) (3) total driving resultant (units: kN /m) (4) location of overturning moment arm for the total driving resultant Z max ( unit s : m ) (5) maximum bearing stress q v E (units : kPa ) (6) maximum reinforcement tensile load Tt otaJ (units: kN / m) (7) maximum hor iz ontal acceleration at centroid of the reinforced soil mass Am (units: g) In particular the total driving resultant was found by summing the hori z ontal force s exerted behind the reinforced soil mass where the individual force was found by multiplying the horizontal stress by the tributary width The location of overturnin g moment arm for the total driving resultant from the base of the wall was found by dividing the sum of overturning moments due to individual horizontal forces by the sum of horizontal forces The locations where the displacements and stresses were recorded in the numerical models for post processing are shown in Figures 4 .21, 4 23 4. 2 5 4.27 4.29 4.31 and 4 .33 for the various cases considered in the parametric study. All of the seismic performances are correlated to the PHA's of the earthquak e records. Comparison of correlations of maximum wall displacement with other accelerations such as spectral accelerations at di f ferent periods from th e response spectra of 5% damping is shown in Figure 5 6 It appears that the spectral acceleration S a at natural period o f 0 5second has a slightly higher R 2 v alue for the correlation than the R 2 v alue of the PHA correlation Howe v er it should be noted that spectral accelerations are seldom available and. PHA on the other hand is used directly in the design and can be obtained from the map of horizontal acc e leration included in Division IA (AASHTO 2002) and Section 3 .10 (AASHTO 2007) ofthe AASHTO Specifications FHW A design methodology requires that both external and internal stabilitie s of the GRS wall be evaluated For the external stability analysis values such as the total driving resultant location o f overturning moment arm for the total driving resultant and the maximum bearing pressure are calculated whereas the maximum reinforcement tensile load is calculated for the internal stability. The FHW A anal y sis values allow the factors of safety be determined by comparing the values to the various limit states U nder the seismic condition all the analysis values are functions of the maximum horizontal acceleration coefficient or the design acceleration coefficient Am, which is calculated as: 176
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Am = (1.45 A) A ( 5 .2)(2 .10 bis ) where A is the peak: acceleration coefficient ; the peak: acceleration coefficient A is PHA normalized by the gravitational acceleration. Am is assumed to act through the centroid of the reinforced soil mass. Note that Equation 5 2 results in a downward concave curve with a maximum Am at A = 0 725 and that Am is greater than A for A < 0.45 and l e ss than A for A 0.45 (see Figure 2.35). 25 00 R'0690 2 00 0 I I I I x . I I I R'O 718 / I I 5 00 I / 1 I R' 0 228 1 / / / / R' ",' / / .".. { . ;/ . x x ,.,' I/Y" Y )( >tit :.,J' ';' ... x x _ .. .... _ ..p.J,."..)( 0 5 1.5 A cce l eratio n (g) I I 2 5 PHA 02aecSa .. OSsecS. X 1 O.ec Sa Expon (PHA) Expon (02 sec Sa) . Expon (05 sec Sa) exPO" (1 0 sec Sa) Figure 5.6 Correlations of Maximum Wall Displacement with PHA and Other Spectral Accelerations Per FHW A methodology the total driving resultant under seismic consideration is comprised of the static active thrust inertia force due to reinforced soil mass and the seismic active thrust increment (see Figure 2 19). The location o f overturning moment arm for the total driving resultant is found by dividing the sum of overturning moments due to the three aforementioned horizontal forces by the total driving resultant. The static thrust inertia force and the seismic active thrust increment act at the distances 1 /3, 112, and 0 6 of the wall height from the wall base respectively. The maximum bearing pressure is calculated using the effective foundation width method The maximum reinforced tensile load is found by adding th e static maximum t e nsile force with the dynamic tensile load induced by the inertia force where the internal inertial force is assumed to be due to weight of the backfill within the active z one. The analysis of the baseline case following the FHW A design guidelines is presented in Appendix C. The aforementioned analysis values would be compared to the seismic performances calculated by FEM. The comparison verifies 177
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the applicability of the FHW A analysis values and assumptions made thereof. Note that the two seismic performances that cannot be verified using the FHW A limit equilibrium methodology are the wall facing displacement and the wall crest settlement. The seismic performances of maximum wall facing horizontal displacement maximum wall crest sett l ement maximum lateral stress behind the reinforced soil mass maximum bearing stress beneath the reinforced soil mass and maximum reinforcement tensile load shown in profi l es are presented in Appendices D E F G and H respectively The seismic performances were presented based on design parameters proposed in the parametric study (see Figure 4.18). The seven design parameters include peak horizontal and vertical acce l erations wall height wall batter ang l e soil friction angle reinforcement spacing and reinforcement stiffness This allows comparison of performance be made within the same design parameter. In addition, all comparisons were made against the baseline case. Various design parameters and seismic performances are summarized in Table 5.1. Following sections present the comparisons of seismic performances based on the design parameters. In this chapter the FEM results were corre l ated with PHA as it is the primary seismic design parameter. Comparisons of seismic performances correlated with PV A are included in Appendix I. Comparisons between the FEM results and the FHWA analysis values with PHA less than 0.29 g are provided and discussed in Appendix J. The data for the statistical analysis is provided in Appendix K. Table 5.1 Summary of design parameters and seismic performances Design Parameter Seismic Performance Peak horizontal acceleration Max. horizontal displacement PHA Max. crest settlement Peak vertical acceleration PV A Total driving resultant LPO E Wall height H Total overturning moment arm Zmax Wall batter angle co Max bearing stress qvE Soil friction angle $' Max. reinforcement tensile load Ttotal Reinforcement spacing, S v Max. horizontal acceleration at centroid of Reinforcement stiffness T 5 % reinforced soil mass Am 5.2.1 Effects o f Wall Hei ght Three wa ll heights of 3 m 6 m and 9 m were evaluated in the parametric study. The seismic performances of maximum horizontal displacement maximum crest settlement total driving resultant total overturning moment arm maximum bearing stress maximum reinforcement tensile load and maximum horizontal acceleration at the centroid of reinforced soil mass for the three wall heights are 178
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compared in Figures 5 7 5.8 5 9 5.10 5.11 5 .12, and 5.13 respectively. Results from the FEM calculation were correlated with the PHA's of the earthquake records through single predictor regression analysis where PHA's are the independent x value and the seismic performances are the dependent yvalue The equations from the regression analysis along with the R 2 values are included in these figures Exponential functions were considered for all of the seismic performances except the total overturning moment. As a better fit quadratic equation (polynomial of second degree) was considered for the location of overturning moment arm of the total driving resultant (see Figure 5.10). As indicated by these figures it is apparent that all seismic performances increase in magnitude with increasing wall height in both the FEM results and the FHW A analysis values. Due to the assumption in finding the maximum horizontal acceleration coefficient Am, the FHW A analysis values reach maximum at PHA = 0 725 g and decrease thereafter resulting in concave downward curves. The correlations established using the exponential function do not pass through the origin since a zero intercept violates the exponential function. Use of the exponential function inevitably introduces finite amount of response even at very low values of PHA. However the exponential correlations were considered representative of the FEM results within the range of conditions evaluated in the parametric study. Note that nonlinear response is expected given nonlinear material behavior contact interface interaction and complex loading pattern simulated in the numerical model. As indicated by the correlation curves in Figures 5 7 5.8 5.9 5.11 and 5.12 the rates of increasing response for the 9 m wall are higher than those of the 3 m wall. This suggests that tall walls are more susceptible to higher PHA than the small walls Horizontal displacement contours at the end of the analysis for the 6 m wall (PHA = 0.568 g) and the 9 m wall (PHA = 0.990 g) are presented in Figures 5.14 and 5.15 respectively. At a modest PHA of 0 568 g the integrity of the 6 m wall was not compromised. Top portion of the wall experienced greater horizontal displacement than the bottom portion of the wall. In addition the reinforced soil mass as a whole translated in the horizontal direction and tended to separate from the retained earth However toppling of top wall facing blocks was imminent for the 9 m wall at the end of earthquake shaking with a PHA of 0.990 g. The 9 m wall would have failed in toppling of facing blocks if the reinforcements were absent. Note that failure wedges can be seen in the displacement contours of Figures 5 .14 and 5.15 and the pattern is consistent with the twopart wedge failure observed in the physical model test (e.g. see Figures 2.9 and 2.10). 179
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2500 E .s 2000
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1600 y = 310 58e"32e" R 2 = 0 5871 E Z 1200 w c c.. w 'i 2 800 Y = 137 35e 6527, 'S R 2 = 0 5974 '" Ol c 5 c '0 'iii (5 I400 Y = 36 3 5e 7932> R 2 = 0 6 7 58 o 0 2 0.4 0 6 0 8 1 2 Pea k horizonta l accelerat i on PHA ( g ) Figure 5.9 Effect of Wall Height on Total Driving Resultant 5 I 4 N E n; C 3 Ql E o E Cl c E 2 :::l t::: Ql > o 'iii (5 Iy = 3 .0323x2 + 3 .5978x + 3 .4295 R 2 = 0 6041 y = 2 .3548x2 + 2 .6073x + 2 .346 R 2 = 0 .5969 J.y = 1.2486x2 + 1 .5342x + 1.0705 R 2 = 0 .7558 o o 0 2 0.4 0 6 Q8 1 2 Peak horizontal acceleration, PHA ( 9 ) H =9m H=6m .. H=3m E x pon (H = 9 m ) E x pon (H = 6 m ) Expon (H = 3 m) FHWA Limitation H =9m( FHWA ) H = 6 m (FHWA ) H = 3 m (FHWA ) H=9m H=6m ... H=3m Poly. (H = 9 m ) Poly. (H = 6 m ) Poly. (H = 3 m ) FHWA Limitation H = 9 m (FHWA) H = 6 m (FHWA) H = 3 m (FHWA) Figure 5 .10 Effect of Wall Height on Total Overturning Moment Arm of Total Driving Resultant 181
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r0o.. '!! au) V> !!! U; Ol C c '" Q) .0 E :::> E x '" 1200 800 .. 400 .... y = 316 06e 029910 R 2 = 0 3009 y = 200 24e"55A, .. R 2 = 0 686 4 .... ........ y = 76 203e' 15 '0, R 2 = 0 796 o o 0 2 0 4 0 6 0 8 1 2 Peak horizontal acceleration PHA (9) Figure 5 ,11 Effect of Wall Height on Maximum Bearing Stress 60 E y = 11. 504e '9'21< Z R 2 = 0 6956 t 0 '" 40 I .Q I I Y = 6 7123e'5645' iii +c R 2 = 0 7354 2 C Q) E y = 3 3408e'173, S c 20 R 2 = 0 8098 E :::> E A x '" A o o 0 2 0 4 0 6 0 8 1 2 Peak horizontal acceleration PHA (9) H =9m H =6m H =3m Expon (H = 9 m) Expon (H = 6 m ) Expon (H = 3 m ) FHWA Limitat i on H =9m(FHWA) H = 6 m (FHWA) H = 3 m (FHWA) H =9m H=6m H =3m Expon (H = 9 m ) E x pon (H = 6 m ) Expon ( H = 3 m ) FHWA Limitat i on H = 9 m (FHWA) H =6m(FHWA) H = 3 m (FHWA ) Figure 5 .12 Effect of Wall Height on Maximum Reinforcement Tensile Load 182
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4 o 0 2 0.4 0 6 .. y = 0 2921e20958, R 2 = 0 6342 0 8 Peak horizontal accelerat ion, PHA (9) y = 0 2805e2 0815 R 2 = 0 7636 y = 0 290ge 9622>< R 2 = 0 7535 1 2 H=9m H=6m .. H=3m Expon (H = 9 m) E x pon (H = 6 m ) Expon (H = 3 m) FHWA Limitation H = 9 m (FHWA) lOH = 6 m (FHWA ) +H = 3 m (FHWA) Figure 5 ,13 Effect of Wall Height on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass 183
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lSOYNA USER INPUT STEP 1162 TIME: 64 000008 COMPONENT: Xdisplacement 12 069748 5.279461 22.628670 39. 977879 57.327087 74 676300 Figure 5.14 Contours of XDisplacement at End of Analysis with Northridge E arthquake P0883 ORR090 (Model: H = 6 m S y = 0.4 m ill = 10, = 36 Ts% = 36 kN/ m )
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LS.oYNA USER STEP 116 2 TIlE: 64 000015 COMPOtENT: X disptacemenf Figure 5.15 Contours of XDisplacement at End of Analysis with Northridge Earthquake P0935 TAR360 (Model: H = 9 m S y = 0.4 m (J) = 10 = 36, Ts% = 36 kN/m) The FEM calculated total driving resultants are greater than the FHW A analysis values and the deviation is more pronounced for wall heights greater than 3 m (see Figure 5.9). The results indicate that the current FHWA methodology might not be conservative especially at higher PHA (i.e., greater than 0 6 g) Successful field cases suggest that the FHW A methodology may be sufficient in dimensioning the wall geometry and the reinforcement layout. However FEM results could be considered in the design when PHA is greater than the FHW A limitation of 0 29 g. The FEM calculated locations of total overturning moment arm are in general agreement with the FHWA analysis values (see Figure 5 10). The FEM results are slightly higher than the FHW A analysis values under static condition and the deviation is more pronounced for wall heights greater than 3 m. The correlation agrees with the FHW A methodology in that the location of resultant application decreases with increasing PHA for PHA greater than approximately 0.6 g The location of total driving resultant application varied from 1/3 (static condition) to 1/2 185
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(PHA 0.6 g) of the wall height. Note that the FEM correlations of total driving resultant and total overturning moment arm were anchored to the static values Bearing stress increases with increasing wall height (see Figure 5.11). Close agreement between the FEM results and FHW A analysis values is observed for the wall height of 3 m. For wall heights of 6 m and 9 m FEM results are greater than the FHW A analysis values under both the static condition and the seismic condition. The same argument for the total driving result also applied for the bearing stress where FEM results could be adopted in the design when PHA is greater than the FHW A limitation of 0.29 g. At least the equations from the single predictor regression analysis can provide a check to the FHW A methodology in establishing upper bound values. Note that the FEM correlations for the bearing stress were anchored to the static values. Unlike the other seismic performances reinforcement tensile loads calculated by FEM were lower than the FHWA analysis values for PHA up to 0.8 g (see Figure 5 12). In other words FHWA methodology provides a conservative estimate for the reinforcement tensile load. Similar findings have been reported by Allen et al. (2003) and Bathurst et al. (2005) in field case histories, where reinforcement tensile loads estimated based on measured strains were higher than the FHW A predictions. The reinforcement tensile load is used to determine factors of safety against reinforcement rupture and pullout and higher value of tensile load in turn requires higher tensile strength reinforcement. FEM results could be served as the lower bound values and the results also suggest that higher tensile strength reinforcement may not be necessary. Use ofthe FEM results to estimate reinforcement tensile load could potentially reduce the cost of the GRS wall without compromising the performance integrity of the structure. Note that the FEM correlations for the reinforcement tensile load were anchored to the static values. Comparison of maximum acceleration at centroid of reinforced soil mass computed by FEM and FHWA methodology is shown in Figure 5.13 where FEM results are considerably higher than the FHWA analysis values. Note that the maximum acceleration at the centroid based on the FHW A methodology does not depend on the wall height. FEM results also suggest that the maximum acceleration is independent of the wall height. Although the FEM results are scattered especially at higher PHA the correlations from the single predictor regression analysis for different wall heights are in general agreement at PHA less than 0.6 g. The acceleration time histories and the Fourier amplitude spectra of the time histories at the centroid of the reinforced soil mass and at the point of equa l elevation in the retained earth (see Figure 4.21) under the Northridge Earthquake (No 12, P0883 ORR090) are compared in Figure 5.16. As shown in the figure some outof phase oscillation between the two locations was observed in the first 5 seconds into the earthquake record and inphase waves were observed thereafter. In particular, the peaks of the two time histories were synchronized. Both time histories exhibit the 186
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same predominant frequency and similar frequency content. Note that this observation does not agree with the assumption stated in the FHWA methodology, where outofphase movement is assumed between the reinforced soil mass and retained earth. Note also that acceleration at the centorid of reinforced soil mass was amplified by a factor of roughly 2 as compared to the base input motion and this agrees with acceleration amplification factors found by other researchers which ranged from 2 (Liu et al. 2011) to 5 (Nelson and Jayasree 2010). 0 8 : c: 0 0 4 :; Q) u 0 u '" (ij c 0 0.4 N Reinforced soil .s:: "0 Retained earth e 0. 8 c Q) () 12 16 20 24 28 32 36 40 44 48 52 Time (sec) (a) 1000 '0 800 Q) In Reinforced soil E Q) 600 Reinforced soil (smoothed) Retained earth "0 Retained earth (smoothed) Ii E 400 '" Qj .;:: :::l 0 200 LL 0 0 2 4 6 8 10 Frequency (Hz) (b) Figure 5 16 Comparison of (a) Centorid Horizontal Acceleration Time Histories and (b) Fourier Amplitude Spectra for the Reinforced Soil Mass and the Retained Earth with Northridge Earthquake P0883 ORR090 (Model: H = 6 m S y = 0.4 m co = 100, = 360 Ts% = 36 kN/ m) 187
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5.2.2 Effects of Wall Batter Angle Three wall batter angles of 50, 100 and 150 were evaluated in the parametric study. The seismic perfonnances of maximum horizontal displacement maximum crest settlement total driving resultant total overturning moment arm maximum bearing stress maximum reinforcement tensile load and the maximum horizontal acceleration at centroid of reinforced soil mass for the three wall batter angles are compared in Figures 5.17 5.18, 5.19, 5.20, 5.21 5.22, and 5.23 respectively. Results from the FEM calculation were correlated with the PHA's of the earthquake records through single predictor regression analysis where PHA's are the indep endent x value and the seismic perfonnances are the dependent yvalue. The equations from the regression analysis along with the R 2 values are included in these figures. Exponential functions were considered for all of the seismic perfonnances except the total overturning moment. Quadratic equation (polynomial of second degree) was considered for the location of overturnjng moment arm of the total dri ving re s ultant (see Figure 5 20). 2000 E .s 1600
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350 300 E .s 250
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3 5 3 I 2 5 N g '" c: 2 CIl E 0 E '" 1 5 c E :> t:: CIl > 0 iii 15 f0 5 0 0 0 2 y = 2 .9904x2 + 3.4427x + 2 145 R = 0 .5723 ... y = 2 3548x + 2 .6073x + 2 346 :. =t= R = 0 5969 t y = 2 .1659x2 + 2 5197x + 2 .2618 R 2 = 0 .6198 OA 0 6 Peak horizontal acceleration PHA ( 9 ) 1 2 00 = 15 00 = 10 .. 00 = 5 Poly (00 = 15 ) Poly (00 = 10 ) Poly (00 = 5 ) FHWA Limitat i on 00 = 15 (FHWA) 00 = 10 (FHWA) 00 = 5 (FHWA) Figure 5,20 E ffect of Wall Batter Angle on Total Overturning Moment Arm of Total Driving Resultant 900 800 300 E x 111 :E 200 100 a a Figure 5.21 0 2 I I / I '\ 0 4 0 6 0 8 Peak horizontal accel e rat i o n PHA ( 9 ) y = 218 22e '.75, R = 0 6869 .. Y = 200 24e'155<, R = 0 6864 . y = 145 3e 41", R = 0 611 7 1 2 00 = 1 5 00 = 1 0 .. 00 = 5 Expon (00 = 1 5 ) Expon (00 = 10 ) E x pon (00 = 5 ) I FHWA Lim i ta t i o n 00 = 15 ( FHWA ) 11100 = 1 0 ( FHWA ) +00 = 5 ( FHWA ) E ffect of Wall Batter Angle on Maximum Bearing Stress 190
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50 : z 40 s 0 '" .Q 30 iii c C Q) E Q) 20 S c E ::J 10 E x '" o 0 2 y = 9 3745e '25." R 2 = 0 8506 y = 6 934ge .. R 2 = 0 .7782 y = 6 7123e'56", R 2 = 0 7354 0 4 0 8 1 2 Peak horizontal acceleration PHA (9) w = 15 w = 10 w = 5 Expon ( w = 15 ) Expon ( w = 10 ) Expon ( w = 5 ) FHWA Limitation w = 15 (F HWA ) +w = 10 (F HWA ) +w = 5 (FHWA ) Figure 5.22 Effect of Wall Batter Angle on Maximum Reinforcement Tensile Load 2 5 l y = 0 3071e'09'" y = 0 3098e'''o" ui VI R 2 = 0 7193 R 2 = 0 7536 '" E 0 2 .. Y = 0 290ge 9622>< VI R 2 = 0 7535 0 w = 15 w = 10 S c .. .. w = 5 1.5 '0 Expon ( w = 15) 0 E xpon ( w = 10 ) e c t Expon ( w = 5 ) .. FHWA Limitation iii w = 15 ( FHWA ) Qi 8 '" w = 10 (F HWA ) S 0 5 +w = 5 (FHWA) c 0 N g .I:: >i '" 0 0 0 2 0.4 0 6 0 8 1 2 Peak horizontal acceleration PHA ( 9 ) Figure 5.23 Effect of Wall Batter Angle on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass As indicated by these figures all seismic performances increase in magnitude with decreasing wall batter angle in both the FEM results and the FHWA analysis 191
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values. In other words GRS wall tends to be less stable with decreasing wall batter angle (e.g., near vertical wall facing) Based on the FEM results the total driving resultant is not significantly influenced by the wall batter angle especially at PHA greater than 0.5 g (see Figure 5.19). Maximum bearing stresses determined using FHW A methodology for the batter angle 5 wall are higher than the FEM results which is not observed for the 10 and 15 walls (see Figure 5 21). Per FHWA guidelines a GRS wall is analyzed as a vertical wall when wall batter is less than 8, and based on the comparison bearing stresses calculated using FHW A methodolog y for a near vertical facing GRS wall are more conservative than FEM results. The maximum reinforcement tensile loads calculated by FEM are lower than those found using FHWA methodology and the reverse is observed when PHA is greater than 0.8 g (see Figure 5.22). The maximum horizontal accelerations at centroid of reinforced soil mass calculated by FEM are higher than those proposed by FHW A methodolog y (see Figure 5 23). Although the accelerations are different both FEM results and FHWA analysis values show that the accelerations are not greatly affected by the wall batter angle 5.2.3 Effects of Soil Friction Angle Three soil friction angles of 32, 36 and 40 were evaluated in the parametric study The seismic performances of maximum horizontal displacement maximum crest settlement total driving resultant total overturning moment arm maximum bearing stress maximum reinforcement tensile load and maximum horizontal acceleration at centorid of reinforced soil mass for the three soil friction angles are compared in Figures 5.24 5 .25, 5 26 5 27 5 28 5 29 and 5.3 0 respectively. Results from the FEM calculation were correlated with the PHA's of the earthquake records through single predictor regression analysis where PHA's are the independent xvalue and the seismic performances are the dependent yvalue. The equations from the regression analysis along with the R 2 values are provided in these figures Exponential functions were considered for all of the seismic performances except the total overturning moment. Quadratic equation (polynomial of second degree) was considered for the location of overturning moment arm of the total driving resultant (see Figure 5 27). The figures indicate that the magnitudes of seismic performances increase with decreasing friction angle as the stiffness of the soil is directly proportional to friction angle (see Figure 4.1 0 4 .11, and 4.12 for soil behavior) with the exception in total driving resultant and bearing stress. The calculated displacements and settlements are in agreement with the notion that a GRS wall built using higher friction angle backfill would be more stable than that of a lower friction angle backfill (see Figures 5.24 and 5.25). Pressures imparted on a stiffer reinforced soil mass 192
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would be higher than a softer reinforced soil mass and this is reflected in the calculated total driving resultant and bearing stress (see Figures 5.26 and 5.28). 2000 .. E .s 1600 y = 31. 345e378 7 ' = 40 ) If) i5 Y = 19 .764e3971". Expon I>' = 36 ) R 2 = 0 7415 c Expon I>' = 32 ) 0 800 N FHWA Limitat i on .<:: E .. .. :l t E x 400 '" o 0 2 0.4 0 6 0 8 1 2 Peak horizontal accelerat i on PHA (9) Figure 5 24 Effect of Soil Friction Angle on Maximum Wall Facing Horizontal Displacement 300 250 E .s ' = 40 ) Expon I>' = 36 ) Expon I>' = 32 ) FHWA Limitat ion Effect of Soil Friction Angle on Maximum Wall Crest Settlement 193
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800 E 600 Z C Il. c' .'!l "S 400 (/) Cl c ;;: c "0 200 0 t0 0 Figure 5.26 3 5 3 g 2 5 N E C 2 Q) E 0 E Cl 1 5 c 2 :; t: Q) > 0 iii (5 t0 5 0 0 Y = 156 55e'632>< R 2 = 0 .635 8 Y = 137 35e'6527, R 2 = 0 5974 Y = 126 94e 5347, R 2 = 0 5642 $ = 40 $' = 3 6 ... $ = 3 2 ... ... E x pon W = 40 ) Expon W = 3 6 ) E xpon. W = 3 2 ) FHWA Limi tat ion ... $' = 40 ( FHWA ) $ = 36' ( FHWA ) +$' = 32' ( FHWA ) 0 2 0.4 0 6 0 8 1 2 Pea k horizontal a ccel e r at ion, PHA ( 9 ) E ffect o f Soil Friction Angle on T otal Driving Resultant 0 2 y = 2 1159x 2 + 2.471 x + 2 3532 R2 = 0.4899 ... y = 2 0444x 2 + 2 3746x + 2 3358 R 2 = 0 5302 y = 2 3548x 2 + 2 6073x + 2 346 R 2 = 0 5969 0.4 0 6 Peak horizontal acceleration PHA ( g) 1 2 $ = 40 $ = 36 ... $' = 32 Poly ($' = 40 ) Po l y ($' = 36 ) Po l y ($' = 32 ) FHWA Limitat i on $' = 40 ( FHWA ) $' = 36 (FHWA) $' = 3 2 (FHWA) Figure 5.27 E ffect of Soil Friction Angle on Total Overturnin g Moment Arm o f Total Driving Resultant 194
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900 800 700 a. OJ! 600 CJ iii < R = 0 8962 y = 6 7123 e156'" R = 0 7354 o o 0 2 0 4 0 8 1 2 Peak horizontal acceleration PHA (9) = 36 .. 32 Expon W = 40 ) Expon W = 36 ) Expon = 32 ) FHWA Limitation = 40 (FHWA) = 36 (FHWA) += 32 (FHWA) Figure 5.29 Effect of Soil Friction Angle on Maximum Reinforcement Tensile Load 195
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7 U
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5.2.4 Effects of Reinforcement Spacing Three reinforcement vertical spacing of 0.2 m 0.4 m and 0.6 m were evaluated in the parametric study. The seismic performances of maximum horizontal displacement maximum crest settlement total driving resultant, total overturning moment arm maximum bearing stress maximum reinforcement tensile load and maximum horizontal acceleration at centorid of reinforced soil mass for the three reinforcement spacings are compared in Figures 5.31 5 32,5.33 5.34 5.35 5.36 and 5.37, respectively. Results from the FEM calculation were correlated with the PHA's of the earthquake records through single predictor regression analysis where PHA's are the independent xvalue and the seismic performances are the dependent yvalue. The equations from the regression analysis along with the R 2 values are provided in these figures Exponential functions were considered for all of the seismic performances except the total overturning moment. Quadratic equation (polynomial of second degree) was considered for the location of overturning moment arm of the total driving resultant (see Figure 5.34). 1800 1600 E .s 1400 E 400 x '" ::2: 200 0 0 0 2 t 0.4 0 6 0 8 Peak horizontal accelerat i on PHA ( g) y = 23 527e39215, R 2 = 0 7186 1 2 S v = 0 2 m S v = 0.4 m S v = 0,6 m E x pon (Sv = 0 2 m ) E x pon (Sv = 0.4 m ) Expon (Sv = 0 6 m ) FHWA Limitation Figure 5.31 Effect of Reinforcement Spacing on Maximum Wall Facing Horizontal Displacement 197
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E S < R 2 = 0 7811 1 2 S = 0 2 m S = 0.4 m S = 0 6 m Expon ( S = 0 2 m) Expon ( S = 0 4 m) Expon ( S = 0 6 m) FHWA Limitation Effect o f Reinforcement Spacing on Maximum Wall Crest Settlement '*, 0 2 y = 137 35e '6527, R 2 = 0 59 7 4 0.4 0 6 0 8 Peak ho rizontal acceleration PHA (9) y = 144 .91e'5699x R 2 = 0 5231 Y = 151. 73e 512>< R 2 = 0 639 1.2 S v = 0 2 m S v = 0.4 m S = 0 6 m Expon ( S = 0 2 m) Expon ( S v = 0.4 m) Expon (S, = 0 6 m) FHWA Li mitat ion S = 0 2 m (FHWA) S v = 0.4 m (FHWA) ___ S = 0 6 m (FHWA) Figure 5.33 Effect of Reinforcement Spacing on Total Driving Resultant 198
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3 5 3 I 2 5 N E ro c 2 Q) E 0 E 0> 1 5 c E :::> 1:: Q) > 0 f0 5 0 0 0 2 y = 2.481x + 2 8784x + 2 2616 R = 0 6498 L y = 2 3548x + 2 6073x + 2 346 R = 0 5969 y = 2 154x' + 2.4006x + 2 2847 R = 0.4 731 0.4 0 6 0 8 Pea k hor izonta l acce l erati on PHA ( g ) 1 2 S = 0 2 m S = 0.4 m .. S = 0 6 m Poly (S = 0 2 m) Poly ( S = 0.4 m ) Poly ( S = 0 6 m ) FHWA Limitat ion S = 0 2 m (FHWA) 4S = 0.4 m (FHWA ) +S = 0 6 m (FHWA) Figure 5.34 Effect of Reinforcement Spacing on Total Overturning Moment Arm of Total Driving Resultant 800 Y = 21 0.42e .... 7' R = 0 7417 Y = 200 24e .... '" R = 0 6864 !l. 600 S v = 0 2 m C y = 1 93.46e '802>< '!I R = 0.7352 S v = 0 4 m 0u; .. S = 0 6 m U) U) Expon ( S v = 0 2 m ) 0> 400 Expon (S. = 0.4 m) c c Expon ( S = 0 6 m ) '" FHWA Limitation E :::> S v = 0 2 m (FHWA ) E 200 ikS v = 0.4 m (FHWA ) :E +S v = 0 6 m (FHWA ) o o 0 2 0.4 0 6 0 8 1 2 Pea k horizontal accelerat i on PHA (g) Figure 5 .35 Effect of Reinforcement Spacing on Maximum Bearing Stress 199
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I z j Ici '" iii c: C Ql E Ql e .E c: E ::J E x '" ::::E 60 40 20 y = 8 883 5 e """" R = 0 719 4 y = 5 5615e "56 R = 0 7309 o o 0 2 0.4 0 6 0 8 1 2 Pea k horizontal accelerat ion PHA (9) S = 0 2 m S = 0.4 m S = 0 6 m Expon ( S = 0 2 m) 1_Expon ( S : 0.4 m) Expon ( S 0 6 m) FHWA Limi tat i on S = 0 2 m ( FHWA ) S = 0.4 m (FHWA) __ S = 0 6 m (FHWA) Figure 5.36 Effect of Reinforcement Spacing on Maximum Reinforcement Tensile Load : 4 E <{ iii t/) '" E 0 t/) .E c: '0 "0 e c Ql o iii '" c: o N .c: o 3 2 o 0 2 y = 0 275ge ,,.,, R = 0 7272 0.4 0 6 0 8 Pea k horizontal acceleration PHA (9) I y = 0 309 3 e .3&" R = 0 7232 y = 0 290ge'''''' R = 0 7535 1 2 S = 0 2 m S = 0.4 m S = 0 6 m Expon ( S = 0 2 m) Expon ( S = 0 4 m) Ex pon ( S = 0 6 m) FHWA Limitation S = 0 2 m ( FHWA ) S = 0.4 m ( FHWA ) __ S = 0 6 m (FHWA) Figure 5.37 Effect of Reinforcement Spacing on Maximum Horizontal Acce l eration at Centroid of the Reinforced Soil Mass 200
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Seismic performances such as maximum facing horizontal displacement (see Figure 5.31) total driving resultant (see Figure 5.33) and maximum bearing stress (see Figure 5 35) are not significantly affected by the reinforcement spacing. The external stability evaluation proposed by FHW A is not to be influenced by the reinforcement spacing and this agrees with the results calculated by FEM. It should be noted that the total driving resultants and the maximum bearing stresses calculated by FEM are higher than those calculated usin g FHW A methodology. The FEM results indicate that uniform reinforcement spacing with value between 0.2 m and 0.6 m enables a coherent composite to be created Although a small reinforcement spacing is not an effective way to reduce the horizontal displacement there are added benefits to the wall system such as better quality control of backfill compaction (i.e., due to smaller lift thickness) and more tolerable against potential creep of geosynthetics Based on the FEM results effect of spacing however is more pronounced in wall crest settlement and reinforcement tensile load (see Figures 5 32 and 5 36) W a ll crest settlement and reinforcement tensile load are proportional to the reinforcement spacing Small reinforcement spacing of 0.2 m is an effective method in reducing crest settlement. The reinforcement tensile loads calculated by FEM at spacing of 0 .6 m are significantly larger than spacing of 0.4 m and 0 2 m and the difference in maximum reinforcement tensile load between spacing of 0.4 m and 0.2 m diminishe s at PHA greater than about 0 8 g (see Figure 5.36). The maximum horizontal accelerations calculated by FEM at the centroid of reinforced soil mass are higher than those predicted using FHWA methodology (see Figure 5.37) and the maximum accelerations by FEM are not significantly affected by the reinforcement spacing. 5.2.5 Effects of Reinforcement Stiffness Three reinforcement tensile loads at 5 % axial strain of 12 kN/ m 36 kN / m and 72 kN/ m were evaluated in the parametric study The three tensile loads at 5% strain also reflect the stiffnesses of the reinforcement (see Figure 4.15 for reinforcement behavior) where reinforcements with Ts% of 12 kN/ m 36 kN/ m and 72 kN/ m are considered as having low medium and high stiffnesses respectively The seismic performances of maximum horizontal displacement maximum crest settlement, total driving resultant, total overturning moment arm maximum bearing stress maximum reinforcement tensile load and maximum horizontal acceleration at centroid of reinforced soil mass for the three reinforcement stiffnesses are compared in Figures 5.38 5.39,5.40 5.41 5.42 5.43 and 5.44 respectively Results from the FEM calculation were correlated with the PHA's of the earthquake records through single predictor regression analysis where PHA's are the independent xva lu e and the seismic performances are the dependent yvalue. The equations from the regression analysis along with the R 2 va lues are provided in these figures. Exponential functions 201
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were considered for all of the seismic performances except the total overturning moment. Quadratic equation (polynomial of second degree) was considered for the location of overturning moment arm of the total driving resultant (see Figure 5.41). 2000 E .s 1600 c: ., E :'l 1200 '" a. '" 'tj s c: 2 800 .<: E :J E 400 ::!; o 0.2 0.4 0 6 0 8 Peak horizontal acceleration PHA (9) .. y = 25 375e402Cl6> R 2 = 0 714 y = 23.527e39210 R 2 = 0 7186 y = 22. 216eJ928900 R 2 = 0 7171 1 2 T o .. = 72 kN/m T o .. = 36 kN/m .. T o .. = 12 kN/m Expon (To .. = 72 kN/m) Expon (To .. = 36 kN/m) Expon (To .. = 12 kN/m) FHWA Limitation Figure 5 38 Effect of Reinforcement Stiffness on Maximum Wall Facing Horizontal Displacement 350 300 E .s 250
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800 600 z w 0 a. w C .l!l 400 :; Cl C ;; c 'C ]j 200 0 I0 0 Figure 5.40 3 5 3 I ; E 2 5 N !:f '" c 2 Q) E 0 E '" 1 5 c E ::J t:: Q) > 0 (ij (5 0 5 0 0 0 2 y = 133 .98e'699" R 2 = 0 5918 t 0.4 0.6 0 8 Peak horizontal acceleration PHA (9) y = 137 35e'6527, R 2 = 0 5974 Y = 140 7e' ""'"" R 2 = 0 579 t ... 1 2 T5 .. = 72 kN / m T ... =36kN/ m ... T ... = 12kN/ m Expon (T ... = 72 kN/m) Expon (T ... = 36 kN/m) Expon. (T ... = 12 kN/m) FHWA Limitation H = 6 m(FHWA) Effect of Reinforcement Stiffness on Total Driving Resultant 0 2 y = 2 .4726x2 + 2 .7085x + 2 .3573 R 2 = 0 5141 y = 2.3086x2 + 2 .6514x + 2 .2354 R 2 = 0 6221 0.4 0 6 0 8 Peak horizontal acceleration, PHA (9) y = 2 .3548x2 + 2 .6073x + 2 346 R 2 = 0 .5969 T ... = 72 kN/m T ... = 36 kN/m ... T ... = 12 kN/m Pol y ( T ... = 72 kN/m) Poly. (T ... = 36 kN/m) Poly. (T ... = 12 kN/m) FHWA Limitation H = 6 m (FHWA) 1 2 Figure 5.41 Effect of Reinforcement Stiffness on Total Overturning Moment Arm of Total Driving Resultant 203
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700 600 n;c.. 500 au; '" 400 '" '" c c '" 300 .2l E :::l E 200 x '" ::;; 100 0 0 Figure 5.42 70 E 60 Z lSO u '" .Q .!! iii c 40 .!l c: Q) E 30 Q) .E c 20 E :::l E x 10 '" ::;; 0 0 0 2 ' ' 0.4 0 6 y = 199 8e '7'''' R 2 = 0 7386 0 8 Peak horizontal accelerat i on PHA ( 9) .. y = 200 24e .... R 2 = 0 6864 y = 193 54e ''''''' R 2 = 0 6175 1 2 T 5 = 72 kN/ m T 5 = 36 kN/ m .. T 5 = 12 kN/ m Expon ( T 5" = 72 kN/ m ) E x pon (T5 = 3 6 kN/ m ) Expon (T." = 12 kN/m ) F H WA Limitation H =6m(FHWA) Effect of Reinforcement Stiffness on Maximum Bearing Stress 0 2 0.4 0 6 0 8 Peak horizontal accelerat i on PHA (g) y = 10.483e 6747. R 2 = 0 .711 y = 6 7123e'56<5, R 2 = 0 7354 y = 3 9874e 62891< R 2 = 0 834 1 2 T 5 = 72 kN/ m T." = 36 kN/ m .. T." = 12 kN/m E x pon ( T 5 = 72 kN/ m ) E x pon (T." = 36 kN/ m ) E x pon ( T." = 12 kN/ m ) FHWA Limitation H =6m( FHWA) Figure 5.43 Effect of Reinforcement Stiffness on Maximum Reinforcement Tensile Load 204
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3 5 < '" .. al 2 5 R = 0 762 Y = 0 .2835&0778. T s .. = 72 kN/ m e R = 0 7469 .g T s .. = 36 kN/ m c 2 '0 y = 0 290ge 9622>< .. T s .. = 12 kN/ m E x pon ( T s .. = 72 kN / m ) "0 R = 0 7535 e E x pon ( T s .. = 3 6 kN / m ) c 1 5 Q) E x pon ( T s .. = 12 k N / m ) 0 iii FHWA Lim i tation Q; 8 ctI .. IIH = 6 m ( FHWA ) c 0 0 5 N .c 0 :; 0 0 2 0.4 0 6 0 8 1 2 Peak horizontal acceleration PHA (9) Figure 5.44 Effect of Reinforcement Stiffness on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass Reinforcement stiffness is not addressed by the FHW A methodology as it is not considered in the limit equilibrium analysis The effect of reinforcement stiffness can only be accessed using finite element analysis or other numerical means. As indicated by the FEM results the magnitudes of seismic performances increase with decreasing reinforcement stiffness except the maximum reinforcement tensi l e load where the reverse is true Similar to reinforcement spacing FEM results indicate that performances related to external stability such as total driving resultant and maximum bearing stress are not significantly affected by the reinforcement stiffness (see Figures 5.40 and 5.42). The FEM results substantiate that the reinforced soil mass behaves as a single coherent composite. The effect of reinforcement stiffness is more pronounced in maximum horizontal displacement (see Figure 5 38) crest settlement (see Figure 5.39) and reinforcement tensile load (see Figure 5.43). Wall facing displacement and crest settlement can be controlled effective l y by utilizing high stiffness reinforcement. The high stiffness reinforcement however would experience higher reinforcement tensile load than the low stiffness reinforcements when experiencing similar horizontal displacement. Although the maximum reinforcement tensile load s from the high stiffness reinforcement calculated by FEM are lower than those predicted by the FHW A method they are closer to the FHW A prediction than the low stiffness reinforcements when PHA is less than 0.5 g (see Figure 5.43). The maximum reinforcement tensile load with high reinforcement stiffness surpasses the FHW A prediction at PHA greater 205
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than about 0.5 g which makes the FHWA methodology less conservative for PHA greater than 0.5 g if high stiffness reinforcements were installed The maximum horizontal accelerations calculated by FEM at the centroid of reinforced soil mass ar e higher than those predicted using FHWA methodology (see Figure 5.44) and the maximum accelerations by FEM are not significantly affected by the reinforcement stiffness 5.3 Distribution of Reinforcement Tensile Load Reinforcement tensile load is needed in the evaluation of GRS wall internal stability. In the parametric study the maximum reinforcement tensile loads calculated by FEM are in general lower than the FHW A predictions In addition to the comparison of maximum tensile load this section presents the distribution of tensile load within the reinforced soil mass. In the FHWA methodology a linear slip surface based on Coulomb's active failure wedge is assumed to pass through the reinforced soil mass FHW A methodology further assumes that the slip surface coincide the line of maximum reinforcement tensile load The anticipated tension distribution along a reinforcement layer takes the shape of bell curve with the maximum value anchored at the intersection of slip surface and the reinforcement layer ; tensile load attenuates at increasing distance away from the intersection. The maximum reinforcement tensile load decreases gradually from the lower most layer to the top most layer. With the FHWA assumptions and applying PHA of 0 568 g (No 12, Northridge earthquake P0883 ORR090) to the baseline model the maximum tensile loads under seismic condition at the lower most and upper most reinforcement layers were calculated to be 23. 2 kNl m and 3.6 kNl m respectively. The fictitious reinforcement tension distribution is shown in Figure 5.45. Also presented in Figure 5.45 are the contours of maximum reinforcement tensile loads. According to the fictitious contours a "ridge" sloping away from the toe of the wall is observed It can be inferred that the presence of a "ridge" would indicate concentration of reinforcement tensile load and the potential slip surface Effect of design parameters examined in the parametric study on reinforcement tensile load distribution was evaluated The design parameters includ e wall height wall batter angle wall friction angle reinforcement spacing and reinforcement stiffness. The models analyzed using the Northridge earthquake record (No 12, P0883 ORR090) with PHA of 0 568 g were selected and processed to generate reinforcement tensile load distribution plots. The results are shown in Figures 5.46 to 5.57. The locus of maximum tensile load and contours of tensile load are presented in these figures Comparisons were made against the baseline case which is shown in Figure 5.47. 206
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N 0 ...} 6 4 :[ J: :i 0> iii .J:: m 2 o FHWA anticipated Model : H = 6 m s., = 0.4 m (J) = 10 $' = 36 seismic 25 o 25 o 25 o 25 o __ __ 25 o 1 2 1 5 1 8 2 1 2 4 2 7 3 0 3 3 3 6 3 9 4 2 Distance from facing of block (m) (a) 6 4 :[ J: :i 0> iii .J:: m 2 o Contours of re i nforcement seismic maximum tensile load (kNlm) 0 3 0 6 0 9 1 2 1 5 1 8 2.1 2.4 2 7 3 0 3 3 3 6 3 9 4 2 Distance from facing of block (m) ( b ) Figure 5.45 Fictitious Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load
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N 0 00 6 4 I J: E 0> iii .s::. a Static Model : H = 6 m S 0.4 m ,", = 10 iii .s::. 6 4 2 a Contours of r e i nforcement static maximum tens ile l oad (kN/ m ) 0 3 0 6 0 9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 3 3 3 6 3 9 4 2 Distance from facing of b l ock ( m ) ( b ) Figure 5.46 R einforcement Tension D istribution: (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Static Condition (Model : H = 6 m S v = 0.4 m co = 10 = 36 T5% = 36 kN/ m)
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N o \0 E '" 'iii .s: 6 4 o Static Model : H = 6 m S = 0 4 m ro 10'. = 36'. T ,,. = 36 kNim Seismic Earthquake : 12. Northridge P0883 (ORR09O) Locus of seismic maxiumum tensile l oad 20 FHWA slip surface angle = 58' 20 20 ........ ......... o 1 2 1 5 1 8 2 1 2.4 2 7 3 0 3 3 3 6 3 9 4 2 Distance from facing of block (m) (a) 6 4 I :r: E '" 'iii .s: 2 o Contours of reinforcement seismic max i mum tensile load (kN/m) 0 3 0 6 0 9 1 2 1.5 1 8 2 1 2.4 2 7 3 0 3 3 3 6 3 9 4 2 Distance from facing of block (m) ( b ) Figure 5.47 Reinforc ement Tension Distribution: (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m S v = 0.4 m 0) = 10, $' = 36, Ts% = 36 kN/m)
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N ...... o +Static Model : H = 3 m S v = 0.4 m ro = 10 $' = 36 T S % = 36 kN/m +Seismic Earthquake : 12 Northridge P0883 ( ORR090 ) 3 I o Locus of seismic maximum tensile load o 20 o 20 FHWA slip line angle = 58 o 0 6 0.9 1 2 1 5 1 8 2 1 O tance from facing of block (m) (a) I 3 o Contours of reinforcement seismic maximum tensile load (k N /m) 0 3 0 6 0 9 1 2 1 5 1 8 2.1 Oistance from facing of block (m) (b ) Figure 5.48 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 3 m S y = 0.4 m ill = 10, = 36, T s % = 36 kN/ m)
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N ......I5 1: E .2' '" 4 Static Model : H = 9 m. 5 = 0 4 m. CD = 10 = 36 T l,. = 36 kNlm _ P....:O",88::3.!.COR=R:.:.09O=) ___ ..J I 1 1 1 J E J I 30 FHWA slip line angle = 5830 30 30 0 30 1 2 1 5 1 8 2 1 2 4 2 7 3 0 3 3 3 6 3 9 42 4 5 4 8 5 1 54 57 6 0 6 3 Dist a nce from faci ng o f b lock (m) C a ) I5 1: '" .2' J! 4 Contour s of reinforcement seismic ma ximum tenSile load ( kNlm ) 0 3 06 0 9 1 2 1 5 182. 1 2.4 2 7 3 0 3 3 3 6 3 9 42 45 4 8 5 1 5 4 57 6 0 6 3 D i s t a nce from facin g of block (m) C b ) Figure 5.49 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model : H = 9 m S y = 0.4 m ill = 10, = 36, Ts% = 36 kN/ m)
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tv tv 6 4 E I :i Ol iii .s= iii 2 o Static Model : H = 6 m S y = 0.4 m 0) = 5 = 36 T5 = 36 kN/m Seismic Earthquake : 12. Northr i dge P0883 (ORR090) Locus of se i smic max i umum tensile load FHWA slip line angle = 63 o E20 z 0 ..,20 0 2 0 c20 Q) E 0 Q) l:!20 0 iii 0 E ::!E 0 + 20 o 20 o O 0 9 1 2 1 5 1 8 2 1 2.4 2 7 3 0 3 3 3 6 3 9 4 2 Distance from facing of block (m) ( a ) E I :i Ol iii .s= iii 6 4 2 o Contours of reinforcement seismic max i mum tensile load (kN/m) 0 3 0 6 0 9 1 2 1 5 1 8 2 1 2.4 2 7 3 0 3 3 3 6 3 9 4 2 Distance from facing of block (m) ( b ) Figure 5.50 Reinforcement Tension Distribut i on : (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m Sy = 0.4 m (0 = 50, = 360 T5% = 36 kN/m)
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IV ...w >Stati c Model : H = 6 m S = 0.4 m ro= 15" , = 36 T ... = 36 kN/m Seismic Earthquake : 12 Northridge P0883 (ORR090) 6 4 2 o Locus of seismi c maxiumum tensile load FHWA slip line angle = 55 5 1 2 1 5 1 8 2 1 2.4 2 7 3 0 3 3 3 6 3 9 4 2 D i stance from facing of b l ock ( m ) ( a ) 6 4 2 o Contours of re i nforcement seismi c max i mum tensile l oad ( kN/m ) 0 3 0 6 0 9 1 2 1 5 1 8 2 1 2.4 2 7 3 0 3 3 3 6 3 9 4 2 D i stance from facing of b l ock (m) ( b ) Figure 5.51 Reinforcement Tension Distribution : (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m Sy = 0.4 m co = 15, $' = 36 Ts% = 36 kN / m)
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N ...... I:>. 6 4 : I Ol 'iii .J:: 2 0 Static Model : H = 6 m S = 0.4 m (j) 10', 4>' 32', T ... = 36 kNlm Seismic Earthquake : 12 Northridge P0883 (ORR090) Locus of seismic maxiumum tensile load FHWA slip surface angle = 56 20 o __ 20 20 0 E 20 Z 0 "" :;;20 co 0 20 c: 0 2! C 20 ., E 0 20 . S 0 & 20 0 i20 0 ., 20 0 20 0 20 0 1 2 1 5 1 8 2 1 2.4 2 7 3 0 3 3 3 6 3 9 4 2 Distance from facing of block (m) (a) 6 4 : I 1: Ol 'iii .J:: 2 o Contours of reinforcement seismic maximum tensile load (kN/m) 0 3 0 6 0 9 1 2 1 5 1 8 2 1 2.4 2 7 3 0 3 3 3 6 3 9 4 2 Distance from facing of block ( m ) ( b ) Figure 5,52 Reinforcement Tension Distribution : (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model : H = 6 m, Sy = 0.4 m 0) = 10, = 32 Ts % = 36 kN/ m)
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N Vl 6 4 :: I E C) iii .c 2 0 Static Model : H 6 m S = 0 4 m ro= 10, $ 40, T.% 36 kN/m Seismic Earthquake : 12 Northridge P0683 (ORR090) 20 c: 0 .!! C 20 '" E 0 20 S; 0 I!! 20 0 20 :; 0 20 0 20 0 0 Locus of seismic maxiumum tensile load 20 0 20 0 20 0 20 FHWA slip surface ang le = 60 1 2 1 5 1 8 2 1 2 .4 2 7 3 0 3 3 3 6 3 9 4 2 Distance from fa cing of block ( m ) ( a) 6 4 :: I C) 'iii .c 2 o Contours of reinforcement seismic maximum tensile load (kN/m) 0 3 0 6 0 9 1 2 1 5 1.8 2.1 2 4 2 7 3 0 3 3 3 6 3 9 4 2 Distance from facing of block (m) ( b ) Figure 5.53 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m S y = 0.4 m, (0 = 10, = 40, Ts% = 36 kN/m)
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tv ...... 0\ 6 4 : I E Cl iii .c iii 2 o Static Model: H = 6 m S = 0 2 m Ql = 10", <1>' = 36", T = 36 kN/m Seismic Earthquake : 12 Northridge P0883 ( ORR090 ) Locus of seismic max i umum tensile load FHWA slip line ang le = 58" 6 3 0 6 0 9 1 2 1 5 1 8 2 1 2.4 2.7 3 0 3.3 3 6 3 9 4 2 Distance from fac ing of block ( m ) ( a ) 4 : I E Cl iii .c iii 2 o Contours of reinf o rcement se i smic maximum tensile l oad ( kN/m ) 0 3 0 6 0 9 1 2 1 5 1 8 2 1 2.4 2 7 3 0 3 3 3 6 3 9 4 2 Distance from facing of b lock ( m ) ( b ) Figure 5,54 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m S v = 0 .2 m co = 10, = 36 Ts% = 36 kN / m)
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N ....) 6 4 g I If Cl iii .c ro 2 o S t at i c Mode l : H = 6 m S = 0 6 m co = 1 0', <1>' = 36', T." = 36 kN/ m Se i sm i c Earthquake : 12 Northridge P0883 ( ORR090 ) Locus of seism i c max i umum tens il e l oad FHWA s li p l i ne angle = 58 30 30 1 2 1 5 1 8 2 1 2.4 2 7 3 0 3 3 3 6 3 9 4 2 D i stance from f acing of block ( m ) ( a ) 6 4 g I :E Cl 'iii .c ro 2 o Conto ur s of r e i nforcemen t se i sm i c max i mum tens i le l oad ( kNlm ) 0 3 0 6 0 9 1 2 1 5 1 .8 2 1 2.4 2 7 3 0 3 3 3 6 3 9 4 2 D i stance from facing of b l ock ( m ) ( b ) Figure 5.55 Reinforcement Tension Distri b ution: (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition ( Model: H = 6 m S v = 0.6 m (j) = 100, = 36 T s % = 36 kN/m)
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N 00 6 4 :: I :c 0> iii .J::; 2 o Static Model : H = 6 m S 0.4 m CD = 10 ,41' = 36 T 72 kN/m Seismic Earthquake : 12 Northridge, P0883 (ORR090) Locus of seismic maxiumum tens i le load 20 FHWA sli p surface angle = 58 o 0 6 0 9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 3 3 3 6 3 9 4 2 D i stance from facing of block ( m ) ( a ) 6 4 :: I :c 0> 'iii .J::; 2 o Contours of reinforcement se i smic max i mum tensile load ( kNlm ) 0 3 0 6 0.9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 3 3 3.6 3 9 4 2 D i stance from facing of block ( m ) ( b ) Figure 5,56 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model : H = 6 m Sy = 0.4 m, Q) = 10, = 36, Ts% = 72 kN/m)
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tv ....... '0 6 4 :: I :E C> 'Qi .r:; 2 o S tatic Model : H = 6 m S = 0.4 m 'Qi .r:; 2 o Contours of reinforcement seismic maximum tensile load (kNlm) 0 3 0 6 0 9 1 2 1 5 1 8 2 1 2.4 2 7 3 0 3.3 3 6 3 9 4 2 Distance from facing of block (m) (b) Figure 5.57 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition (Model: H = 6 m, Sy = 0.4 m 0) = 10, = 36, Ts% = 12 kN/m)
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The effect of seismic shaking is illustrated in the maximum tension lines and the contours plots of Figure 5.46 (under static condition) and Figure 5.47 (under seismic condition). Under the static condition the tensile load is concentrated near the wall facing for the top 3/4 of the reinforcements, and the tension line bulges into the reinforced soil mass for the bottom 114 of the reinforcements ; top reinforcements experienced near zero tension at distances away from the wall facing. Under seismic condition all reinforcements experienced higher tensile load The pattern of tension distribution is similar to the static condition, where tensile load is concentrated near the wall facing for the top 3/4 ofthe reinforcements However the tensile load at bottom 114 of the reinforcements shifted further inward the reinforced soil mass as compared to the static condition. High reinforcement tensile loads in the bottom 114 of the reinforcement indicate that the reinforcements restrain the soil from lateral translation; in other words the reinforcements prevent the soil from kicking outward. An increase in tensile load in the top reinforcements toward the back of the wall was observed between the static and seismic cases indicating that these reinforcements contribute the seismic stability of the wall even though the reinforcements have near zero tension under static condition Both the maximum tension line and the contours of tensile load are different between the FHWA prediction (see Figure 5.45) and those calculated by FEM (see Figure 5.47) A distinct maximum tension line is not observed in the FEM results. In addition the tensile load contours based on FEM results are very different and are more complex than the simple "ridge" suggested by the FHW A methodology. If one were to apply the "ridge" analogy to the FEM results there could be multiple slip surfaces within the reinforced soil mass. The effect of wall height on the reinforcement tensile load distribution can be seen in Figures 5.48 5.47, and 5.49 for wall heights of 3 m 6 m and 9 m respectively. The maximum tension line for the 3 m wall is further away from the wall facing than the 6 m and 9 m walls. For the 3 m wall, the tension in the reinforcement is concentrated toward middle of the wall. Similar pattern in contours of tensile load is observed between the 6 m and the 9 m walls. The "inward bulge" of maximum tension line for the 9 m wall covers approximately bottom 1/3 of wall height as oppose to 1/4 for the 6 m wall indicating higher tendency of soil movement near the toe of the 9 m wall. The effect of wall batter angle on the reinforcement tensile load distribution can be seen in Figures 5.50 5.47 and 5 .51 for wall batter angles of 5, 10, and 15, respectively. Reinforcement tensile loads for the 5 batter wall were concentrated at some distance away from the wall facing unlike the baseline model of 10 batter where the high tensile loads were located immediately behind the wall facing. The "inward bulge" of maximum tension line for the 5 batter wall is less pronounced than the 10 and 15 batter walls. The "inward bulge" of maximum tension line for the 15 batter wall extends further upward as compared to the 10 batter wall. 220
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The effect of soil friction angle on the reinforcement tensile load distribution can be see in Figures 5.52 5.47 and 5 .53 for soil friction angles of 32, 36, and 40, respectively. According to the contours of tensile load the = 40 wall shows more vertically oriented contours and the = 32 wall shows more horizontally oriented contours. The orientation of the contours implies that the rate of increase in tensile load in each reinforcement layer toward the wall facing is higher for the vertically oriented contours than the horizontally oriented contours. The reinforcements with horizontally oriented contours would experience more uniform tensile loads within individual layer than the reinforcements with vertically orientated contours The maximum tension line for the = 40 wall is almost exclusively located immediately behind the wall facing whereas the line is further away from the wall facing for the = 32 wall. The "inward bulge" of maximum tension line for the = 40 wall is less apparent. The effect of reinforcement spacing on the reinforcement tensile load distribution can be seen in Figures 5 54 5.47, and 5 .55 for spacings of 0.2 m, 0.4 m and 0.6 m respectively. Reinforcement tensile loads for the 0.2 m spacing wall are lower than the 0.4 m and 0.6 m spacing walls. The maximum tension line for the 0.2 m spacing is less consistent then the 0.4 m and the 0 6 m spacing walls where the maximum tension line could reach as far as the end of the reinforced soil mass The distribution of tensile load in the individual layer is more uniform in the 0.2 m spacing wall than the 0.4 m and the 0.6 m spacing walls The maximum tension line for the 0.6 m spacing wall is similar to that of the 0.4 m spacing wall except that the "inward bulge" of the line for the 0.6m spacing is less apparent. The effect of reinforcement stiffness on the reinforcement tensile load distribution can be seen in Figures 5.56 5.47 and 5 57 for tensile loads at 5% strain of72 kN/m (high stiffness) 36 kN/ m (medium stiffness) and 12 kN/ m (low stiffness) respectively. The maximum tension line of the high stiffness wall is more erratic than the medium stiffness and low stiffness walls (see Figure 5.56). The locations of maximum tension lines between the medium stiffness and low stiffness walls are similar except that the "inward bulge" of the line for the low stiffness wall is less apparent (see Figure 5.57) Reinforcement tensile loads are higher in the high stiffness wall than the medium stiffness and low stiffness walls. Note that the upper reinforcements in the low stiffness wall especially toward the back of reinforced soil mass, experienced much lower tensile loads than the medium stiffness and high stiffness walls. The effect of PHA on the distribution of reinforcement tensile loads can be seen in Figures 5.58, 5.47 and 5.59 for PHA of 0.282 g (No.6, P0346 H Z14000) 0.568 g and 0 753 g (No. 16, P1020 SPV270) respectively. The location of maximum tension line for the PHA = 0.282 g wall is almost identical to the maximum tension line under static condition (see Figure 5.46) where the line is close to the wall facing The maximum tension line for the PHA = 0.753 g wall on the other hand is 221
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located closer to the center of the reinforced soil mass As indicated by the contours of the tensile load the magnitude of the tensile load increases with increasing PHA and increasing wall depth The tensile load is concentrated at the toe of the wall and the rate of increasing tensile load within a reinforcement layer is much higher for PHA = 0.753 g wall than the PHA = 0 282 g wall. 5.4 Soil Thrusts and Reinforcement Resultants at Distances behind Wall Facing Variations of soil thrusts and reinforcement resultants at distances behind the wall facing are presented in this section. Soil thrusts located (1) behind the wall facing (2) at the centerline of the reinforced soil mass and (3) behind the reinforced soil mass were determined. Reinforcement resultants located (1) behind the wall facing and (2) at the centerline of the reinforced soil mass were determined. Also determined were the moment arms of the soil thrusts and reinforcement resultants measured from the base of the wall. The soil thrusts reinforcement resultants and their moment arms were determined for all 11 cases of the parametric study (see Figure 4.18) that were subjected to the 20 earthquake records (see Table 4.1) From the FEM results the soil thrust was found by summing the horizontal forces exerted at the location of interest where the individual force was found by multiplying the horizontal stress by the tributary width The location of overturning moment arm for the soil thrust from the base ofthe wall was found by dividing the sum of overturning moments due to individual horizontal forces by the sum of the horizontal forces. The reinforcement resultant at the location of interest was found by summing the tensile forces of the reinforcements The location of resisting moment arm for the reinforcement resultant was found by dividing the sum of resisting moments due to individual reinforcement tensile forces by the sum of the reinforcement tensile forces. Note that the soil thrust behind the reinforced soil mass is termed "total driving resultant" in the previous sections. Different names were devised in this section so that they are not confused with the reinforcement resultant presented in this section. As an example the static and seismic earth pressure distributions of the baseline model located behind the facing block at the centerline of the reinforced soil mass and behind the reinforced soil mass are presented in Figure 5.60 in which the model was subjected to the Northridge earthquake (No. 12, P0883 ORR090 PHA = 0.568 g). The magnitudes and locations of the soil thrusts and reinforcement resultants are compared graphically in Figure 5 61. Also included in Figures 5.60 and 5 .61 are the earth pressure distributions determined using the F HW A methodology where the maximum soil thrust acting behind the reinforced soil mass is comprised of the static thrust inertia force due to reinforced soil mass and the seismic thrust increment (i.e., see Figure 5.62) 222
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tv tv w 6 4 I J: : iii .<:: 2 0 St a tic M od el : H 6 m S. = 0 4 m ill = 10', 36, T ... 36 kNim S eismic E arthqu a k e : 6 C oal i nga P034 6 ( H Z 14000 ) 20 c: 0 C 2 0 '" E: 0 20 S S 0 e 2 0 E: 0 j 20 ;; 0 20 0 2 0 0 0 Locus of sei sm i c maxiumum t ensi l e lo ad 2 0 FHWA sli p s u rf a ce angle = 58 2 0 o 20 o 20 o 0 9 1 2 1 5 1 8 2 1 2.4 2 7 3 0 3 3 3 6 3 9 4 2 Distan ce fro m f a cing of b lock ( m ) ( a ) 6 4 I J: 1: C> iii .<:: 2 o Contours of rei n forcement sei s m ic maximum tensile loa d ( kNl m ) 0 3 0 6 0 9 1 2 1 5 1 8 2 1 2.4 2 7 3 0 3 3 3 6 3 9 4 2 Dis ta nce f ro m facin g of b lock ( m ) ( b ) Figure 5,58 Reinforcement Tension Distribution : ( a ) Distribution alon g Individual La yer ( b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition w ith Coalinga earthquake P0346 HZ14000 PHA = 0.282 g ( Model: H = 6 m S y = 0.4 m co = 10 = 36 T s % = 36 kN/ m )
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tv tv 6 4 g I :E Cl '0) .s:: (ij :s: 2 o Static Seismic Model : H = 6 m S = 0.4 m co = 10', <1>' = 36', T .. = 36 kNim Earthquake : 16 Northridge P1 020 ( SPV270) Locus of seismic maxiumum tensile load 20 FHWA slip s urface angle = 58 o 0 6 0 9 1 2 1 5 1 8 2 1 2.4 2 7 3 0 3 3 3 6 3 9 4 2 Distance from facing of block ( m ) ( a ) 6 4 g I :E Cl '0) .s:: 2 o Contours of reinforcement seismic maximum tensile load ( kNlm ) 0 3 0 6 0 9 1 2 1 5 1 8 2 1 2.4 2 7 3 0 3 3 3 6 3 9 4 2 Distance from facing of block ( m ) (b) Figure 5,59 Reinforcement Tension Distribution: (a) Distribution along Individual Layer (b) Contours of Maximum Reinforcement Tensile Load under Seismic Condition with Northridge earthquake, PI020, SPV270, PHA = 0,753 g (Model: H = 6 m Sy = 0.4 m, (j) = 10, = 36 Ts% = 36 kN/m)
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6 4 2 Model : H = 6 m CJ.) = 10, = 36, S v = 0.4 m T S % = 36 kN/m Earthquake : 12. Northridge P0883 (ORR090), PHA = 0 568 g Behind facing block (static) +Behind facing block (seismic) IIICenterline reinf soil (static) o Centerline reinf soil (seismic) IrBehind reinf soil (static) Behind reinf soil (seismic) FHWA method (static) FHWA method (s e i smic) o o 40 80 120 Lateral earth pressure (kN/m 2 ) F igure 5.60 Static and Seismic Earth Pressure Distribution s (Mo del: H = 6 m (() = = 36 S y = 0.4 m Ts% = 36 kN/m) 225
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N N 0\ Model : H = 6 m (j) = 10 $ = 36 Sv= 0.4 m TS% = 36 kN/m Earthquake : 12. Northridge P0883 (ORR090) PHA = 0 568 g Static Reinforcement Resultant Maximum Reinforcement Resultant Static Soil Thrust Behind Facing Block 6 m Crest 73. 1 kN/m (50 2%) 155 3 kN/m (47.4%) 115 4 kN/m (45 8%) 313 8 kN/m (44 1%) Om Base Maximum Soil Thrust () Location of Force Application ; % of Wall Height Static Soil Thrust Per FHWA Maximum Soil Thrust Per FHWA Centerline of Reinforced Soil 6 m Crest 128 6 kN/m (428% ) 307 9 kN/m (45 3%) 41. 3 kN/m (37 8%) 141. 8 kN/m (38.6%) Om Base Behind Reinforced Soil 6 m Crest 381. 1 kN/m (49.8%) = 323 6 kN/m (48 7%) 137.4 kN/m (39 1%) 70.6 kN/m (33 3%) Om Base Figure 5.61 Magnitudes and Locations of Soil Thrusts and Reinforcement Resultants (Model : H = 6 m ill = 10 = 36 Sv = 0.4 m T5% = 36 kN/ m)
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N N ......l Static Earth Pressure H H/3 Ka y H Equivalent Uniform Earth Pressure due to Inertial Force Inertia H/2 0 5 H y' Am Force P1R Seismic Earth Pressure Increment 0 5 (0 8 KAE Y H) Seismic soil thrust increment 1+.0 5 P AE 0 6 H 0 5 (0 2 KA E Y H) Figure 5.62 Earth Pressure Distribution s for Static Thrust Inertia Force and Seismic Thrust Increment per FHWA Methodology
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The soil thrusts determined using the FHW A methodology are considered as transmissible (or sliding) force vectors where the force vector can be applied anywhere along its line of action. Thus the locations of soil thrusts following FHW A methodology in the transverse direction are not constrained. In other words the FHW A methodology does not distinguish the transverse location of thrust application. Furthermore the maximum soil thrust of FHW A methodology is considered in the external stability evaluation (e g stability against sliding and overturning). Variation of soil thrusts and reinforcement resultants at distances behind the facing block can only be estimated using the more rigorous numerical method such as the finite element analyses adopted in this study. It should be noted that the inertia component of the maximum soil thrust proposed in the FHW A methodology cannot be compared directly with the results of finite element analyses since the inertia force is based on a rigid block assumption. The soil thrust and reinforcement resultant with the height of application as a function of PH A are presented in Figures 5 .63 through 5.73 where part (a) of these figures presents the soil thrusts and reinforcement resultants and part (b) shows the location of application. Figure 5.63 shows the results of the baseline case. Effect o f design parameters are compared in Figures 5.64 and 5.65 for the wall height Figures 5 66 and 5 67 for the wall batter angle Figures 5.68 and 5.69 for the soil friction angle Figures 5.70 and 5 .71 for the reinforcement spacing and Figures 5.72 and 5.73 for the reinforcement stiffness. General observations from these figures are summarized as follows : The magnitude of FEM calculated soil thrust increases with increasing distance away from the wall facing. The magnitude of FEM calculated reinforcement resultant decreases with increasing distance away from the wall facing At the same location (e.g. behind facing block or at the centerline of the reinforced soil mass) magnitude of soil thrust is greater than the magnitude of reinforcement resultant. The difference in magnitude also increases with increasing distance At higher PHA (e.g. > 0.5 g), the height of soil thrust moment arm calculated by FEM increases with increasing distance away from the wall facing. The height of reinforcement resultant moment arm calculated by FEM decreases with increasing distance away from the wall facing Although variable the height of moment arm of both FEM soil thrust and reinforcement resultant is l ocated within a band between 40% and 50% of the wall height and is concentrated toward 45% at higher PHA (e .g., PHA > 0 5 g). 228
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800 700 _600 E C !500 Ii i roo E 300 .. 0; =& 200 CI) 100 0 2 0 4 0 6 0 8 1 2 Peak hori zon ta l acceleration, PHA (g) (a) roy, 1; 3 5 H7L1 .. CI) 0 2 0 4 0 6 0 8 1 2 Peak horizontal accelenrtion PHA (g) (b) Soli thrust behindfadng block Sol' thrust at centerl ine lei nf loil .. Soli thrust beh i nd rein' loil x Rein' relultlnt behind f aci n g block :r Re in' felunnt al centerline lelnl sOIl FHWA ,elnf resultant Expon (Soil thru.t behind fa cing blod!.) Expon (Sad thrust al centerline reinf lOti) Expon (Soil thrust behi nd 'el nf aotl) Ellpon (Rein' ,.sultantbehindfaangblock) Expo" (Rein' resultant atcenterhne 'el n f sod) Solllhrust behind f a cing block Sod thrust at centerline reinf loll ... Sod Ih,ust beh i n d l e i nf soil )( Relnf. ultant beh l nd fadngblock X Rein' r.sultant at centert l ne ,emf sod +FHWA loil thrus t FHWA relnf resultant Poly (SoU thrust be h in d fa cing block) Poly (Soil thrust It center1 ln leln' .otl) Poly (Soil ttuust behin d r e ln f SOIl) PoIV (Reinf resultant behind faCIng block) Poly ( Remf reSUltan t 8 1 centerl i ne rein' sotl) Figure 5 .63 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Arms of Thrusts and Resu l tants (Model : H = 6 m co = 10, = 36 Sv = 0.4 m, Ts% = 36 kN/m) 229
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E j i ;; .. ., 200 100 0 2 0 4 y .,0. 615 el'Mlr R'" 0 .8<437 0 6 0 8 y. 16 35e1 R''''06758 Pe ak horizontal accelef'lltion PHA (g) (a) 1 2 ro r, :; = +4 ; i E 2 i 5 0: ;; . ., 0 2 0 4 0 6 0 8 1.2 Peak horizontal acceleration PHA (g) (b) Sotlthrust behlndfadng block Sotlthrust at canterhne re lnt SeNt a $oltthruSt behind reint soli )( Rein' lesultant beh i nd facing block x Rei n' r.sultant It cent.rllne UIIO' loti .....FHWA sOIllhrult 4FHWA rein t r.sultan t Expon. (Soi11tlruSl beh indfaang block) Expon (Soil thrust at centerline lelnf IOU) ElIpon (Sod Ihrusl behi nd lelnf soU) Expon (Rein resultant behindfadngblock) Expo" (Rein r.sultant atcenterhne reint IOU) Soli thrust behindfadngblock SOU thrust at centerline I.int lOll .. Soilduust behi ndre inf SOIl X Relnf result.nt behind f.dng blodt 1:; Re i nf resultant at centerUne relnf 8011 +FHWA loll thrust 4FHWA reinf resultant poty (Soil thrust behindf.angblock) Poly (Soil thrust at cente,lIne rein' soil) Poly (SoIl thrust behind re/nt.loll) poty (Remf resultanl beh i nd f.ang block) Poly (Reinf resultant 81 centerline rein f loll) Figure 5 64 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Arms of Thrusts and Resultants (Model: H = 3 m, co = 10, = 36, Sy = 0.4 m Ts% = 36 kN/m) 230
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1400 1200 R1,. 0 7761 I 1000 J! 3 800 i 600 ; 400 ;; .
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y II 111 37e' RJ_OS40 i .. sao i < o 300 200 1.2 12 (b) SoUthIUS' behind faCIng bloek Soilthlust at c.nterline felnl lOll ... 5011 th'Ult behind rel nr soil x Retnf r.lultant behind faang x Re in' r.lultant at cente,Une relnf lOll .....FHWA ,oIlthruI' r u l hlnl Expon (SoIl ttuust beh lndfadngblock) Expon (Soil thrust ateentelUne rel nf 1011) Expon (Soil thIUS' beh i nd relnt sotl) Expon (Ralnr r.sultant behlndf.cing block) Expo" ( R eln t r.sultant _'center line rei nf sOIl) 5011 thrust behind facing block Soil thrust al cente,l ine r elnf ,oil ... SoH Ihruat behind ,eln f soil x Rei nf I.sultan! beh lndfadngblock X Rei n' I.sultant at centen ln.,etnt soli +FHWA ,oil1ttrult FHWA ,elnf I.sultant Po ly (SoU thlust beh indfadng block.) Poly (Sod tlulIS! at centertlne leinf soU) Po ly (SoU 1t'IIUSt behind leinf sotl) Poly (Reln f ,nultant beh i nd 'ecin g block) Poly (Rein .. ullant at centerl i ne ,ein sad) Figure 5.66 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Arms of Thrusts and Resultants (Model: H = 6 m co = 5 = 36 Sy = 0.4 m T 5 % = 36 kN/ m ) 232
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800 700 _600 E i! i roo c 'ii 300 ... u 200 co 100 y 1535ge' ..... R'" 0 599 4 (b) 142 .22e'lC121 RI 0 5934 y = 137 4 e 3200 R z 0 8286 1.2 1.2 5011 thrust behmdfaang block Soil thrust at centerline re l nt. lOll .. Soil thrust beh ind leinf. loi l )( Reinf.resultant behind facing block x Rein' ,.sul tant at center U ne 'ainf. SOi l FHWA sod thrust +FHWA leint, resultant Expon (Soil thrust behi n d faang block) Expon (SotllhfUst at centerline reint SOIl) Expon (SotlltlfuSI behind leint SOIl) Expon (Rei nf 'e5ullan! behind facing block) exPO" ( Re In' resultant at centerline relnt loll) Soil th rust beh ind facing block So illhrv't at centerline re i nt loil ... SoiI1hlust beh ind r e int so i l x Reinf rnultant bel\ind facing blo ck x Reinf I.sultanlat centerline e i nt SOil +FHWA loll thrust +FHWA 1 lnf ,&su l tant Poly ( So ll lh,ust beh i nd facing blocK ) Poty ( So i l d"ust a t centerline ,e inf $Otl) poty ( So li thrust beh i nd ,ein' soli) Poly (Relnf resultant beh i nd faang block ) Poty (Reinf resultant .tcenterllnel.in' SOi l ) Figure 5 67 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Arms of Thrusts and Resultants (Model: H = 6 m co = 15, = 36 S y = 0.4 m T s % = 36 kN/ m) 233
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y 126 9 4 e ..,)
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r, =600 j i a 300 2 ii 200 0 2 0 0 6 0 8 1.2 Peak horizontal acceleration PHA (g) (a) 60 :r :: 5 5 ; '0 x i E 0 E C i! i Ii 0: 235 ii 0 ., 30 0 2 0 0 6 0 8 1 2 Peak horizontal acceleration, PHA (g) (b) Soil thrust beh ind facing block Soil thrust at centelt i ne 'el nf ,oll Soli thrus t beh i n d rel n f soi l )( R e inf ,.sultant be h i n d facin g block x Reinl ,.sultan! at centerline lelnf soli .... FHWA loll1t'!rusl 4FHWA ,e inf ,esuitant Expon ( So illtlrust behi n d faangblock) Expon ( So illtlrust a t centerline leint lOti ) Expon (Sotlltlrust beh i nd re lnt sOIl) Expon (Rein' resultant beh indfaangblock) Expon IRein' resultan l atcenlerl inerelnf sad) Soillhfust beh ind facing block Soli thrust at centerline reint lod .6. Soil thrust behind ,ein t soil x Reint resultan t beh lndtadngblock x Relnf resultant at centerline reinf SOIl +FHWA sOlllhru s t +FHWA ,eint lesultant Poly (Sod Ihrus t block ) Poly (SolllhfUlt at canterline reint soU) Poly (Soillhrus! beh i nd f eint soli) Poly (Reint resultant beh lndfaclngblodt) Poly (Rein' res u ltan t a l c e nterl i ne reln f 5011) Figure 5.69 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Arms of Thrusts and Resultants (Model: H = 6 m (0 = 10 = 40 Sv = 0.4 m Ts% = 36 kN/m) 235
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800r, y. 91e' R' 0 5231 _600 tr, i Ii roo E 6 ., "1 .. 71 1.2 1.2 (b) Soli thrust behlndtad ng block $oil thrust at center1lne,elnf soil .. Soil thrust behind leinr soi! x Rein' resultan t ben/ndtadngblod!: x Rein l r ultant at cente rllne reinf sad """+FHWA relnf r.sultant Expon (Soil thrust beh lndfadngblock) 8I:pon ( Sod thrust lit centel1l ne lei nf soli) Expon (SoIl thrust beh ind lel nf lod) Expon (Rei nf resu l tant behind fa ci ng block ) Expon (Reinf resul tant at center li ne reinf 5011) SoIl thrust benlndt,dng block Soil thrust al centerl i ne reln t lod .. Soil thrust beh i nd ralnt soil X Relnf resultant beh ind facing block x Reinf ,asultlnl at cen ten i ne rein' soli FHWAsollthIU$1: +rei n' r esu l tant Poly ( Soi l thrust behrndfadng block) Poly (Soli thrust ilt centerline rel nt soli ) Poly ( Soi l thrust beh i nd reinf soU) Poly (Rein f r esu l tant beh ind faang bfock) Poly (Rein resultant a t centerl i ne rein' so il) Figure 5.70 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Arms o f Thrusts and Resultants (Model : H = 6 m 0) = 10, = 36 S v = 0 2 m Ts% = 36 kN/ m) 236
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800 7 00+1 0 2 0 4 0 6 y: 15173e,O,llJo R'a0639 0 8 Pe .. k horizontal accel e ration PHA (g) (a) Peak horizontal acceleration, PHA (9) (b) 1.2 1.2 Sotllhrust behindf.cing block So lllhrust 8t centert l ne reinl. soli So i l thrust beh i nd reint. x Re l nl ullultanl behin d faci n g block x Rel nl resu l tant at centet1 inelei nf IOU 4FHWA ,e i nf r esultant Expon (Soillhrust behind f aCing block) Expon (Sotllhlust at cent.nine re lnt IOU) Expon ( Soli thrust beh i nd re lnt lod) Expon (Relnl resultant behlndf8C1ngbiock) Expon (Re i n l res u ltant at centerl i ne 18mf lOll) So llttlfusl beh i nd facing block Solllhrust at centerlme rel n t. loll ... 501 1 thrust beh i nd reinl. soil x Rel nl u l tant behind fadng block x Rain! ,nultant at centerline relnf loil .FHWA lOll thrust FHWA 'eln' resultant Poly ( Soi l thrust beh ind facing block ) Poly (Soil thrust a t cente r line rel nf soU) Poly ( Soi l thrust beh ind reinf aotI ) Poly (Relnf resultant behind faang block) Poly (Reint lesultant at centerl i ne rei nt loll) Figure 5 .71 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Anus of Thrusts and Resultants (Model: H = 6 m co = 10 = 36 Sy = 0 6 m T s % = 36 kN/m) 237
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800 r, y" 1<40 7e Mr R 1 0 S79 700 roo ii roo <; ., 0 2 0 4 0 6 0 8 1 2 Peak horizontal acceleration, PHA (g) (a) 60 :r = 5 S ; .. x t x ii E 0 E 1: 245 ii e 40 i! 'I '" <; 0 ., 30 0 2 0 4 0 6 0 8 1.2 Peak horizontal accelenltion PHA (g) (b) Sollthruat behind facing block $0.1 thrust at centerlIne relnf. soU Soil thrust behind reinf soil x Rein' ,esultant behindfacingblock x Rein' r.sultlnt at centerline re1 nf soil .FHWA sod thrust +FHWA rel nf resultant Expon (Soil thrust behind faCIng block) Expon (Soil thrust at centerline reinf soil) Expon (Soli thrust behind reinf sOIl) Expon (Rem' reSUltant beh i nd f aCIng block) Expon (Re l n t resultant atcenteri ine reinf soli) SoIl thrust behind fa cing block So,I thrust at c enterl i ne rei nt soil ... Soil thrust behind ,e lnf soil x Re l nf resultant behlndfaangbloc:k ::c ReJ nf resultant at centerlIne reinf soil FHWA soli thrust +FHWA r e mf resultant Poly (Soil thrust behind fadng block) Poly (Soil thrust at centerline rei nf SOil) Poly (Soli thrust behind rein' sOil) Poly (Rein' r.sultant behindfaclngblodc) Poty (Rein l resultan t at centerline reln t soU) Figure 5.72 Variation of (a) Soil Seismic Thrusts and Reinforcement Resultants and (b) Moment Arms of Thrusts and Resultants (Mode l : H = 6 m ro = 10 = 36 Sy = 0.4 m Ts% = 12 kN/m) 238
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y13398e'"'' Rl 1600 C 0 2 0 4 0 6 0 8 1 2 Peak h oriz o nta l acceleratio n PHA (0) (a) ;;; = 55 +4 ; 3 5 TIbL4 0 '" 0 2 0 4 0 6 0 8 1.2 P eak h o rizo n ta l ac.celeration PHA (g) (b) Soil thrust beh indfadngblock So,lthrust al centerline re lnf soli ... Soilthrustbehindreinf SOtI X ReIn' I.sultanl behlndfadngblock x Rein resultant a t centerl i ne rel nt. soU FHWA re lnf lesul t anl Expon (Soil thrust beh indfadngblock) Bcpon (Soil thrust a l centenine relnt soU) Expon (Soilthtust behind re lnf sotl) Expon (Reinf resultan l beh i nd facing block) Expo" (Rein resultant at center1ine reln f lod) So,lthrust behindfaangblock Soil thrust 81 center1me rein f sod ... SOil thrust behind remf soli )( Reinf resultant behindfadngblock I: Reinf resultant at centenlne reinf soli .FHWA sod thrust FHWA ,elnf resultant Poly (Solllhfust behind facing block) Poly (Solllhfust a t centenlOe relnt SOtI) Poly (Soli thrust behind refnf sod) Poly (Rein f resultant behind facing block) Poly ( Reint resultant at cen1enine f.int soil) Fi g ure 5.7 3 Variation o f ( a) Soil Sei s mic Thrust s and Reinforcement Res ultants and (b) Moment Arms of Thrusts and R e sultants (Mod e l : H = 6 m co = 10 = 36 S v = 0.4 m T s % = 72 kN / m ) 239
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Particularly the increase in soil thrust and decrease in reinforcement resultant with increasing distance away from the wall facing is more pronounced in the 9 m wall than the 3 m wall. Note also that the variation in magnitude of soil thrust and reinforcement resultant with regard to the location decreases with increasing reinforcement stiffness The magnitude of reinforcement resultant determined using the FHW A method is higher than the FEM calculated reinforcement resultant and this agrees with the findings presented in the previous sections in that the FHWA methodology provides a conservative estimate for the reinforcement tensile load. Comparable reinforcement resultants located behind the facing block were observed for cases of w = 15 (see Figure 5.67), S v = 0.2 m (see Figure 5.70) and Ts% = 72 kN / m (see Figure 5 73) between FHWA methodology and the FEM simulation. It should be noted again that the reinforcement resultant presented in this section is the sum of reinforcement tensile loads along a section within the reinforced soil mass. In the FHWA methodology the soil thrust and the reinforcement resultant are equal to each other under static condition and the soil thrust is higher than the reinforcement resultant for PHA > O. This trend was also observed in the FEM simulation. In the FHWA methodology, the moment arm of soil thrust is higher than the moment arm of the reinforcement resultant for PHA > O The FHW A moment arm of the reinforcement resultant appears to be the lowest among all of the moment arms compared. On the other hand the FHW A moment arm of soil thrust is similar to the FEM moment arm of soil thrust located behind the reinforced soil mass which has been shown in the previous sections as well. 240
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6. Multivariate Statistical Modeling Results of parametric study were analyzed by the statistical means to determine relationships between the seismic performances of freestanding simple GRS walls and the various design parameters. Correlation analysis was first performed to assess the associations between the variables where the variables can either be positively or inversely related. Regression analysis was then performed to determine equations to describe the relationship between the dependent and the independent variables. In particular prediction equations of seismic performance s of GRS wall were developed with multiple design parameters as the independent variab l es. In the preliminary design assessment the prediction equations could offer the benefit of incorporating different parameters for estimating seismic performances of GRS wall. More importantly the prediction equations allow seismic performance s to be estimated for peak horizontal acceleration beyond the limitation (i.e. 0.29 g) stipulated by the FHWA (2001) seismic design methodology. 6.1 Correlation Analysis The degree of association between two variab les can be identified by the product moment correlation coefficient or simply the correlation coefficient. The correlation coefficient r can range in value from 1.0 to + 1.0. A correlation coefficient of + 1 0 represents a perfect positive linear relationship whereas r = 1.0 indicates a perfect negative or inverse linear relationship between the two variab l es A correlation coefficient of r = 0 indicates that there is no linear relationship between the two variables Figure 6 1 shows the scatter diagrams with different degrees of correlation between two variables Note that the correlation coefficient r is appropriate for measuring the degree of relationship between variab l es that are linearly related ; a nonlinear relationship that is highly associated would result in low value of r. The correlation coefficient r between the two variab l es x and y is calculated as: r = L(x; x)(y; y ) (6.1) (n 1)sx S y where x and y are the means and S x and S y are the standard deviations of the x and y variables with n number of observations respectively. 241
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y . y .. . e. ... ... r = 0 8 ... X Y Y r = 0 . . . ... X Y Y r = 1 0 X . . . .. .. . .. .. r = 0 3 .... . .. e. .... .. r = 0 8 ... . .. ... .. .. .. r = 0 9 ... ..... ... ... ... .. X Y X X . .. . . r = 0 .. .... . .. ... . . .. .. . Figure 6 1 Scatter Diagrams with Various Degrees of Correlation between Two Variables (modified from Kachigan 1991) In this study the design parameters such as the peak horizontal acceleration PHA peak vertical acceleration PV A wall height H wall batter angle co, effective soil friction angle reinforcement spacing Sv, and reinforcement stiffness Ts% are considered as the independent variab l es (or predictor variables). The dependent variables are the FEM calculated seismic performances such as the maximum horizontal displacement !lh, maximum crest settlement !lv, total driving resultant LPOE, total overturning moment arm Zm ax, maximum bearing stress qvE, maximum reinforcement tensile load Ttotal, and the maximum horizontal acceleration at centorid of reinforced soil mass Am. With the available data a correlation coefficient r between each pair of variables can be determined and be presented in a correlation matrix. Note that the correlation matrix is square and is symmetrical about the diagonal and the diagonal coefficients have values equal to 1.0 Table 6 1 shows the correlation matrix with both the independent and dependent variables As shown in the correlation matrix linear relationship is not observed among the independent variables since values of r are zero between pairs of independent variables. Some linear relationship is observed between independent variables PHA PV A and H and the dependent variables !lh, !lv, LP O E qvE, Tto t a h and Am. Values of r in the correlation matrix suggest nonlinear relationships between the independent and dependent variables This is in agreement with the nonlinear seismic performances presented in Chapter 5. Note that some positive linear relationships are observed among the dependent variables. 242
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Table 6.1 Correlation matrix of independent and dependent variables V ariab l e Variab l e PHA PVA H ro cp' S TSY. Ilh Il" r POE Zma., qvE TtoUII Am in Ilh in llv in r POE inqvE in T,olal in Am PHA 1.00 0 ,76 0 ,00 0 ,00 0 ,00 0 ,00 0 ,00 0 ,72 0 ,74 0 ,58 0 ,09 0,60 0 ,67 0 ,75 0 ,82 0 ,79 0.57 0 ,57 0 ,72 0 ,76 P V A 0 ,76 1.00 0 ,00 0 ,00 0,00 0 ,00 0 ,00 0 ,74 0 ,73 0.54 0 ,01 0 ,54 0 ,65 0 ,80 0 .64 0 ,64 0.49 0.49 0 ,63 0 ,64 H 0 ,00 0 ,00 1.00 0 ,00 0 ,00 0 ,00 0 ,00 0 .22 0.30 0 7 1 0,94 0 ,63 0 ,35 0 ,03 0 1 9 0 ,29 0 ,73 0 6 7 0.41 0 0 1 ro 0 ,00 0 ,00 0 ,00 1.00 0 ,00 0 .00 0 ,00 0,05 0,08 0 ,01 0,03 0, 1 3 0 ,04 0 ,00 0,02 0,04 0 ,02 0,1 3 0,04 0 ,00 0 ,00 0 ,00 0 ,00 0,00 1.00 0 ,00 0 ,00 0,06 0,06 0 1 1 0, 0 1 0 2 1 0,04 0 ,08 0 ,04 0,03 0 ,10 0 1 8 0,06 0 ,02 Sv 0 ,00 0 ,00 0 ,00 0 ,00 0 ,00 1.00 0 ,00 0 0 1 0 ,07 0 ,00 0,04 0 ,02 0 ,16 0 ,00 0 .00 0 ,04 0 ,00 0.02 0 1 7 0 ,00 Ts % 0 ,00 0 ,00 0 ,00 0 ,00 000 0 ,00 1.00 0 ,03 0.17 0 ,00 0 ,03 0,01 0 .35 0,01 0,02 0 ,10 0 ,01 0 ,00 0 ,34 0 .00 Ilh 0 ,72 0 ,74 0 ,22 0 ,05 0,06 0 ,01 0 ,03 1.00 0 ,92 0 ,62 0 ,24 0 ,65 0 ,74 0 ,68 0 ,69 0 ,68 0 .57 0 ,59 0 ,69 0 ,53 Ilv 0 ,74 0 .73 0.30 0.08 0.06 0 ,07 0 .17 0 .92 1.00 0 ,72 0.33 0 ,74 0 ,70 0 .64 0 ,72 0 .78 0 ,67 0 ,68 0 ,68 0 .57 r PO E 0 ,58 0.54 0 ,71 0 ,01 0 1 1 0 ,00 0 ,00 0 .62 0 ,72 1.00 0 ,75 0 ,92 0 ,71 0 ,52 0 ,68 0 .73 0 .92 0 ,87 0 ,75 0 ,53 Zma., 0 ,09 0 ,01 0 .94 0 ,03 0, 0 1 0 ,04 0 ,03 0.24 0.33 0 ,75 1.00 0 ,67 0 ,38 0 ,02 0.39 0.46 0 ,80 0 ,74 0.48 0 .20 qvE 0 ,60 0.54 0 ,63 0, 1 3 0 2 1 0 ,02 0, 0 1 0 ,65 0 ,74 0 ,92 0 ,67 1.00 0 ,70 0 ,52 0 ,70 0 ,75 0 ,89 0 ,95 0 ,74 0.53 T,oUII 0 67 0 ,65 0.35 0,04 0,04 0 ,16 0 ,35 0 ,74 0 ,70 0 7 1 0.38 0 ,70 1.00 0 ,61 0 ,66 0 ,65 0 ,67 0 ,66 0 ,94 0.53 Am 0 ,75 0 .80 0,03 0 ,00 0 ,08 0 ,00 0 0 1 0 .68 0 ,64 0.52 0 ,02 0.52 0 ,61 1.00 0 ,65 0 .63 0.48 0.47 0 ,60 0 ,71 in Ilh 0 .82 0 ,64 0 ,19 0 ,02 0,04 0 ,00 0 ,02 0 ,69 0 ,72 0 ,68 0 ,39 0 ,70 0 ,66 0 ,65 1.00 0 ,97 0 ,75 0 ,73 0 7 6 0 9 1 in Il" 0 ,79 0 ,64 0 ,29 0,04 0,03 0 ,04 0, 1 0 0 ,68 0 ,78 0 7 3 0.46 0 ,75 0 ,65 0 ,63 0 ,97 1.00 0 ,80 0 ,79 0 ,76 0 ,88 in r POE 0.57 0.49 0 ,73 0 .02 0 ,10 0 ,00 0 ,01 0 .57 0 .67 0 ,92 0 ,80 0 ,89 0 6 7 0.48 0 7 5 0 ,80 1.00 0 ,95 0 ,78 0.58 in qvE 0.57 0.49 0 .67 0,13 0,1 8 0,02 0 ,00 0 ,59 0 ,68 0 ,87 0 ,74 0 ,95 0 ,66 0.47 0 ,73 0 ,79 0 ,95 1.00 0 ,76 0.55 in TtoW 0 ,72 0 ,63 0.41 0,04 0,06 0 1 7 0.34 0 ,69 0 ,68 0 ,75 0.48 0 ,74 0 ,94 0 ,60 0 ,76 0 ,76 0 ,78 0 ,76 1.00 0 ,62 in Am 0 ,76 0 .64 001 0 ,00 0 ,02 0 ,00 0 ,00 0 ,53 0 .57 0.53 0 ,20 0 ,53 0 ,53 0 ,71 0 ,91 0 ,88 0 ,58 0.55 0 ,62 1.00
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6.2 Regression Analysis Multiple linear regression analysis was performed to determine prediction equations of seismic performances given the multiple design parameters. The regression analysis utilized the curefitting technique where the bestfit prediction equations were developed using the least squares method based on the available F E M data The least squares solution requires that the sum of squared deviations of the predicted values from the actual observed values is at a minimum Note that the multiple linear regression analysis was performed using the statistical software Minitab As evident in Chapter 5 all seismic performances show nonlinear relationship with design parameter PHA. The observed nonlinear response can readily be described by the natural exponential function. As such it is then appropriate to take the natural logarithm of the seismic performances to be the values of dependent variables in the multiple linear regression analysis Values of coefficient of correlation between independent variables and natural logarithm of dependent variables are also included in Table 6.1. Higher values of coefficient of correlation are observed between natural logarithm of dependent variables Tto t al, and Am and independent variable PHA. In addition values of coe f ficient of correlation between natural logarithm of dependent variables and the independent variable PHA are higher than those with the independent variable PV A which suggests that PHA has a higher influence over the seismic performances than PV A. In the multiple linear regression analysis the seven independent variables or the design parameters (viz. PHA PYA H CO, Sy, and T s % ) were used to estimate one dependent variable or one seismic performance (e.g., LP D E qyE, Tto t al, and Am). The multiple linear regression equation with transformed dependent variable is defined as: f.n(Y) = bo + bl XI + X 2 + b 3 X 3 + b4 + b s X s + b 6 X 6 + b7 X7 (6 2) where f.n(Y) = transformed dependent var i able or response XI, X 2 ... X7 = independent (or predictor) variables bo = constant and bl b 2 ... b7 = regression coefficients Equation 6.2 can be expressed in terms of design parameters as: f.n(Y) = bo + bl PHA + b 2 PYA + b 3 H + b4 co + b s + b 6 S y + b7 Ts% (6 .3) The prediction equation which is a natural exponentia l function is found by taking the inverse of the natural lo garithmic function of Equation 6.3 as : Y = exp (bo + bl PHA + b 2 PYA + b 3 H + b4 co + b s + b 6 S y + b7 Ts%) (6.4) The exponent of the prediction equation consists of the linear combination of the independent variables or the design parameters resulted from the regression ana l ysis. 244
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Note that the units of predictors PHA PYA H co, Sy, and Ts% are gravitational acceleration (g), g, meter degree degree meter and kN/m, respectively Units of seismic performances i'lh, i'ly, LP O E qyE, Tto t aJ, and Am are millimeter millimeter kN/ m kPa kN/ m and gravitational acceleration (g) respectively. 6.2.1 Prediction of Maximum Horizontal Displacement The prediction equation for the maximum horizontal displacement i'lh was found as: i'lh = exp (2.900 + 4 578 PHA + 0.330' PYA + 0.251 H 0 0192 co 0 0434 + 0.0553 S y 0.00239 Ts%) ; R 2 = 0.710 (6 5) The prediction equation was applied to the conditions considered in the parametric study. Comparisons between the predicted values of i'lh and the FEM results are presented in Figures 6.2 6 3 6.4 6.5 6.6 and 6.7 to examine the influence of wall height H wall batter angle co, soi l friction angle reinforcement spacing Sy, reinforcement stiffness T s%, and peak vertical acceleration PYA respectively. In Figures 6.2 to 6.6 values of PV A were assumed to be 113 of PHA. 3000 E .s E x co o 0 2 0.4 0 6 0 8 1 2 Peak horizontal acceleration PHA ( g ) H=9m H =6m .. H=3m H = 9 m ( regression ) eH = 6 m (r egress ion) 6H = 3 m ( reg re ssion) FHW A Limita t ion Figure 6 .2 Comparison between Predicted i'lh and the FEM Results with the Effect of Wall Height H 245
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2000 .. E .s 1600 E x 400 '" o 0 2 0.4 0.6 0 8 1.2 Peak horizontal acce l erat ion, PHA (g) Figure 6.3 Comparison between Predicted and the FEM Results with the Effect of Wall Batter Angle (0 2000 .. E .s 1 600 E x 400 '" o 0 2 0.4 0 6 0 8 1 2 Pea k horizontal acceleration PHA ( g ) Figure 6.4 Comparison between Predicted and the FEM Results with the Effect of Soil Friction Angle 246
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2000 E .s 1600 c: Q) E 8 1200 '" a. VI '0 19 c: 2 800 .c E :J E 400 :;: t I o 0 2 0.4 0 6 0 .8 1 2 Peak horizontal acceleration PHA (g) S v = 0 2 m S v = 0.4 m S v = 0 .6 m S v = 0 2 m (regression) S v = 0.4 m (r egress i on ) trS v = 0 6 m ( regress ion) FHWA Limitation Figure 6.5 Comparison between Predicted and the FEM Results with the Effect of Reinforcement Spacing Sv 2000 E .s 1600
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2000 E .s 1600
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6.2 2 Pred iction of Maxim u m Crest Sett l ement The prediction equation for the maximum crest settlement I1v was found as: I1v = exp (1.397 + 3.056 PHA + 0.380 PYA + 0 272 H 0 .0222' co 0 0246 + 0.6102 Sv 0.00909 Ts%); R 2 = 0.727 (6.6) The prediction equation was applied to the conditions considered in the parametric study. Comparisons between the predicted values of I1v and the FEM results are presented in Figures 6.8 6.9 6.1 0, 6 .11, 6.12 and 6.13 to examine the influence of wall height H wall batter angle co, soil friction angle reinforcement spacing Sv, reinforcement stiffness T s%, and peak vertical acceleration PYA, respectively In Figures 6.8 to 6.12 values ofPVA were assumed to be 1 / 3 ofPHA. All of the predicted values are within the range of FEM results and the nonlinear response is consistent with those observed in Chapter 5. The prediction equation suggests that the maximum crest settlement increases with increasing wall height decreasing wall batter angle decreasing soil friction angle increasing reinforcement spacing decreasing reinforcement stiffness and increasing peak vertical acceleration. 350 300 E 250 .. H=3m E 200 CI> __ H = 9 m (r egress i on) E CI> eH = 6 m ( regress i o n)
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350 300 '" E .s 250 E 200 '" C1l = 5 a> E +C1l = 15 (regression) a> '" eC1l = 1 0 (re gress ion) ii) !!! 150 ...C1l = 5 ( regress i on ) u E FHWA Limi tat ion :::I E 100 ';( '" 50 0 o 0 2 0.4 0 6 0 8 1 2 Pea k horizontal acce leration PHA (g) Figure 6 9 Comparison between Predicted !1v and the FEM Results with the Effe ct of Wall Batter Angle co 350 300 E .s 250 E a> 200 '" cp' = 32 E a> +cp' = 40 (regression) '" ii) 150 !!! u ecp' = 36 (re gress ion) ...cp' = 32 (re gress ion) E FHWA Limitation :::I E 100 ';( '" 50 .'" 0 0 0 2 0.4 0 6 0 8 1 2 Peak horizontal acceleration PHA ( g ) Figure 6.10 Comparison between Predicted !1v and the FEM Results with the Effect of Soil Friction Angle 250
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350 300 E .s 250
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350 300 E .s 250
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350 50 /:0 .,<57. R' 0 .7958 /. / / . \ . : ,t:
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1600 f 1200 z w o a. w 800 400 ; . . o .. ,, o 0 2 0.4 0 .6 0 8 1 2 Peak horizontal acceleratio n PHA (g) H = 9 m H =6 m .. H = 3 m ___ H = 9 m ( regress ion) eH = 6 m (regress i on) 0H = 3 m ( regress i on) FHWA Limitation H=9m(FHWA) lItH = 6 m ( FHWA) H = 3 m (FHWA) Figure 6.15 Comparison between Predicted LPOE and the FEM Results with the Effect of Wall Height H 800 E 600 .. Z t ro = 15 ro = 10 w .. 0 a. w .. ro = 5 "E t ro = 15 (regression) 400 ero = 10 (regression) "S t '" trro = 5 (regression) Cl ::::::::; FHWA Limitat io n c ro = 15 (FHWA) 0 $ 200 ro = 10 (FHWA) 0 ro = 5 (FHWA) fo o 0 2 0 4 0 .6 0 .8 1 2 Peak horizontal acceleration PHA (g) Figure 6.16 Comparison between Predicted LP O E and the FEM Results with the Effect of Wall Batter Angle 0) 254
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800 E 600 Z 41' = 40 C. 41' = 36 w c Cl. '" 41' = 32 w 41' = 40 (regress i on ) .l9 400 41' = 36 (regression) :; Ul ..41' = 32 (regress i on ) Cl c FHWA Lim i tation .s; 41' = 40 (FHWA) :g r 200 41' = 36 ( FHWA) 41' = 32 (FHWA) o o 0 2 0.4 0 6 0 8 1 2 Peak horizontal acceleration PHA (g) Figure 6.17 Comparison between Predicted LPO E and the FEM Results with the Effect of Soil Friction Angle 800 700 E 600 Z C. 500 Cl. w ,.J c .l9 400 :; U) Cl c 300 .s; : 200 0 I100 0 I f o 0 2 0.4 0 6 0 8 1 2 Pea k horizontal accelerat i on PHA (g) S = 0 2 m S = 0.4 m '" S = 0 6 m S = 0 2 m ( regress i on ) S = 0.4 m ( regress i on) ..S = 0 6 m ( regress i on ) FHWA Limitat ion S = 0 2 m (FHWA ) lOIS = 0.4 m ( FHWA ) S = 0 6 m (FHWA ) Figure 6.18 Comparison between Predicted LPO E and the FEM Results with the Effect of Reinforcement Spacing Sv 255
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800 :w 600 t T s .. = 72 kN/m z t T s .. = 36 kN/m I!i 0.. .. T s .. = 12 kN/m IN c"" T s .. = 72 kN/m (regression) .'!! 400 eT s .. = 36 kN/m (re gress i on ) :;
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Figures 6 .16, 6.18 and 6.19 the total driving resultant is not influenced significantly by the wall batter angle, reinforcement spacing and reinforcement stiffness. The results suggest that so long as the reinforcements are included in the backfill hence creating a coherent soil mass, with conditions similar to those evaluated in the parametric study the total driving resultant acting behind the reinforced soil mass is irrespective of the reinforcement spacing and stiffness. In other words external stability of GRS wall is not significantly affected by the reinforcement spacing and st iffness when a coherent soil mass is achieved. From the comparison the order of significance of design parameters on the total driving resultant is (1) wall height (2) soil friction angle, (3) peak vertical acceleration (4) wall batter angle, (5) reinforcement stiffness, and (6) reinforcement spacing. As the total driving resultant is not sensitive to the wall batter angle reinforcement spacing and reinforcement stiffness the three independent variables can be removed to simplify the prediction equation. The simplified prediction equation without the contribution of wall batter angle reinforcement spacing and reinforcement stiffness was determined as: LPO E = exp (2.111 + 0.931 PHA + 0 306 PYA + 0.325 H + 0.0319 R 2 = 0.873 (6 8a) Note that the difference between Eq uations 6.8 and 6 8a is the constant bo in the exponent and the coefficients of the remaining independent variables are the same between the two equations. 6.2.4 Prediction of Total Overturning Moment Arm From the results presented in Chapter 5 the total overturning moment arm Zmax appears to be not significantly affected by the various design parameters considered in the parametric study. As Zmax is not affected by design parameters other than the peak horizontal acceleration PHA multivariate regression analysis was thus not performed. In general the total overturning moment arm can be represented closely by a quadratic equation (polynomial of secon d degree) with PHA being the independent variable The nonlinear regression equations all result in concave downward curves. Based on the available FEM results, the height of overturning moment arm normalized with the wall height plotted against PHA is shown in Figure 6.21. The prediction equation for the normalized total overturning moment arm in percentage was determined as: zmaxlH (%) = 32.49' PHA2 + 37.0' PHA + 39.86 (6 9) where PHA has units of gravitational acceleration (g). Note that values of zmaxlH predicted by Equation 6 9 range from 40% under static condition to a maximum of 50% at PHA of 0.57 g. 257
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y 32.493.' ... 37. 008x ... 39 864 R' O .SI!07 0 2 0 0 6 0 6 1 2 Peak horizontal acceleration PHA (g) Figure 6 .21 Normalized Total Overturning Moment Arm with Wall Height ve rsu s Peak Horizontal Acceleration 6 2 5 Prediction of Maximum Bearing Stress The prediction equation for the maximum bearing stress qvE was found as: qvE = exp (2.640 + 0.741 PHA + 0.207 PYA + 0.233 H 0.0266' co + 0 0465 0 .0940' S v + 0 00005 T5%); R 2 = 0 832 ( 6.10) The prediction equation was applied to the conditions considered in the parametric study. Comparisons between the predicted values of qvE and the FEM results are presented in F igures 6.22 6.23 6 .24, 6.25 6.26 and 6.27 to examine the influence of wall height H wall batter angle co, soil friction angle reinforcement spacing Sv, reinforcement stiffness T 5%, and peak vertical acceleration PV A respectively. In Figures 6.22 to 6.26 values ofPVA were assumed to be 1 / 3 of PH A. 258
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Iii' Q. 1200 J 800 iii If) If) Ol c: E E 400 x co ::2 .. . .. ../" . " o o 0 2 0.4 0 6 0 8 1 2 Pea k horizontal acce ler at ion PHA ( g ) H=9m H =6m .. H = 3 m ___ H = 9 m (re gress io n ) eH = 6 m (re gress ion ) trH = 3 m ( regress ion) FHWA Limitat i on H = 9 m ( FHWA ) H=6m(FHWA) H=3m( FHWA) Figure 6.22 Comparison between Predicted qvE and the FEM Results with the Effect of Wall Height H 900 800 / / '\ Iii' 700 / Q. / .. (i) = 15" / (i) = 10" er 600 / iii .. (i) = 5 If) 500 (i) = 15" ( regress ion ) Ol e(i) = 10" (re gre ssion) c: C 400 b(i) = 5 (re gress ion) co Q) F HWA Limitat ion .0 E (i) = 15" ( FHWA ) :::l 300 E x (i)=10 ( FHWA ) co ::2 200 +(i) = 5 ( FHWA ) 100 0 o 0 2 0.4 0 6 0 8 1 2 Peak hor i zontal acce l eration PHA (g) Figure 6.23 Comparison between Predicted q v E and the FEM Results with the Effect of Wall Batter Angle ()) 259
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900 800 m 700 a. 40 C J 600 36 iii A 32 '" 500 +40 (regress ion) U; Ol I36 (regress ion) c c 400 6,'= 32 (regression) tV Q) .0 FHWA Limitation E 40 (FHWA) :::J 300 E x ;036 (FHWA) tV 200 32 (FHWA) 100 0 o 0 2 0.4 0 6 0 8 1 2 Peak hor i zontal acceleration PHA (g) Figure 6.24 Comparison between Predicted qvE and the FEM Results with the Effect of Soil Friction Angle $' 900 800 m 700 a. S = 0 2 m C 600 I S = 0.4 m rr "," A S = 0 6 m '" 500 +S = 0 2 m (regression) U; Ol IeS = 0.4 m (regression) c 400 trS = 0 6 m (regression) Q) .0 FHWA Limitation E :::J 300 S = 0 2 m (FHWA) E S = 0.4 m (FHWA) ::;; 200 S = 0 6 m (FHWA) 100 0 o 0 2 0.4 0 6 0 8 1 2 Peak horizontal acce ler ation PHA (g) Figure 6.25 Comparison between Predicted qvE and the FEM Results with the Effect of Reinforcement Spacing Sv 260
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900 800 '" 700 0.. T s .. = 72 kN/m 6 w 600 T s .. = 36 kN/m cf .; T s .. = 12 kN/ m VI 500 T s .. = 72 kN/ m (regression) VI '" eT s .. = 36 kN/ m (regression) c c 400 ...T s .. = 12 kN/m (re g ressi on ) '" ., .a FHWA Limitation E :J 300 H =6m(FHWA) E x '" :::;; 200 100 0 o 0.2 0.4 0 6 0 8 1 2 Peak horiz ontal acceleration PHA ( g ) Figure 6,26 Comparison between Predicted qvE and the FEM Results with the Effect of Reinforcement Stiffness T 5% 900 800 '" 700 0.. H=6m 6 w 600 PYA = 1/3 PHA (r egress i on ) cf
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stress is not influenced significantly by the reinforcement spacing and reinforcement stiffness. The behaviors are consistent with those observed with the total driving resultant. A coherent soil mass was created with reinforcements placed in the backfill. The maximum bearing stress beneath the reinforced soil mass is irrespective of the reinforcement spacing and stiffness given that the reinforcements are installed with conditions similar to those prescribed in the parametric study In other words external stability of GRS wall is not significantly affected by the reinforcement spacing and stiffness From the comparison the order of significance of design parameters on the maximum bearing stress is (1) wall height (2) soil friction angle (3) peak vertical acceleration (4) wall batter angle (5) reinforcement stiffness and (6) reinforcement spacing. The order of significance is identical to the one for the total driving resultant. As the maximum bearing stress is not sensitive to the reinforcement stiffness and reinforcement spacing the two independent variables can be eliminated to simplify the prediction equation. The simplified prediction equation without the contribution of reinforcement spacing and reinforcement stiffness was determined as: qyE = exp (2.604 + 0.741 PHA + 0.207 PYA + 0.233 H 0.0266 co + 0.0465 (6. lOa) Note that the difference between Equations 6 .10 and 6.1Oa is the constant bo in the exponent and the coefficients of the remaining independent variables are the same between the two equations 6.2.6 Prediction of Maximum Reinforcement Tensile Load The prediction equation for the maximum reinforcement tensile load Ttot a l was found as: Tto ta! = exp (0.748 + 1.158 PHA + 0.437 PYA + 0.185 H 0 .0109 co + 1.194 Sy + 0.0153 Ts%) ; R2 = 0 854 (6 11) The prediction equation was applied to the conditions considered in the parametric study. Comparisons between the predicted values of Tto t a l and the FEM results are presented in Figures 6.28 6 29 6.30 6.31 6 32 and 6.33 to examine the influence of wall height H wall batter angle co, soil friction angle reinforcement spacing Sy, reinforcement stiffness T s%, and peak vertical acceleration PYA, respectively. In Figures 6.28 to 6.32 values ofPVA were assumed to be 113 ofPHA. 262
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70 60 z C j s H=9m >SO 0 H=6m ro .2 .. H=3m iii 40 +H = 9 m (re gress ion) c $ I eH = 6 m (r egress i on ) c bH = 3 m (r egress i on ) Q) E 3 0 FHWA Limitation Q) e H=9m(FHWA) .E c H=6m(FHWA ) 2 0 E +H = 3 m (FHWA ) :J E x 10 ro 0 o 0 2 0.4 0 6 0 8 1 2 Pea k horizontal accelerat i on PHA ( g ) Figure 6.28 Comparison between Predicted Ttota1 and the FEM Results with the Effe ct of Wall Height H 7 0 60 z C j ....... SO 0. ro .2 iii 40 c $ C Q) E 30 Q) e .E c 20 E :J E x 10 ro 0 .. .. _________ A .. ; ...g....::.. ....... ::: ::: 1 ..  .. o 0 2 0.4 0 6 0 8 1 2 Peak horizontal a cceleratio n PHA (g) Cll = 1So Cll = 10 +Cll = 1So (regression) eCll = 10 (regression) bCll = s o (r egression ) FHWA Limi tat i on Cll = 1So (FHWA ) Cll = 1 0 ( FHWA ) +Cll = S o ( FHWA) Figure 6.29 Comparison between Predicted Tto t a l and the FEM Results with the Effect o f Wall Batter Angle ro 263
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70 E 60 Z jj 40 I5 0 0 36 C\l .Q .. 32 iii c 40 +40 (regress ion) e36 (regress i on) C .. C1) .. 32 (regression) E 30 C1) F HWA Limitation .E 40 (FHW A ) c 2 0 36 (FHW A ) E :::> .. 32 (FHWA) E .. x 10 C\l ::2: 0 o 0 2 0.4 0 6 0 8 1 2 Peak horizontal a cceleration PHA (g) Figure 6.30 Comparison between Predicted Ttotal and the FEM Results with the Effect of Soil Friction Angle 7 0 E 6 0 Z jj B I5 0 0 C\l .Q S = 0 2 m S = 0.4 m .. S = 0 6 m iii 40 c c: C1) E 30 C1) +S = 0 2 m ( r egression) eS = 0.4 m (regression) S = 0 6 m (regression) FHWA Lim itation _____ .. 4.. .E c 2 0 E S = 0 2 m (FHWA ) lOtS = 0.4 m (FHWA ) :::> E __ S = 0 6 m (FHWA ) x 10 C\l ::2: 0 o 0 2 0.4 0 6 0 8 1 2 Peak hor iz ontal accelerat io n PHA (g) Figure 6.31 Comparison between Predicted Ttotal and the FEM Results with the Effe ct o f Reinforcement Spacing S v 264
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70 E 60 Z c jj T,,, = 72 kN/ m lSO .j T,,, = 36 kN/ m '" .Q .!! .. T,,, = 12 kN/m in 40 T,,, = 72 kN/ m ( regress i on ) c c: eT,,, = 36 kN/ m (regr ess i on ) Q) .>T,,, = 12 kN/m (regress i on ) E 30 Q) !:! FHWA Lim i tat i on .E H =6m(FHWA) c: 20 E :J E x 10 '" ::;; 0 o 0 2 0.4 0 6 0 8 1 2 Peak horizontal acceleration PHA ( g ) Figure 6.32 Comparison between Predicted Ttotal and the FEM Results with the Effect of Reinforcement Stiffness T 5% 7 0 E 60 Z C jj H =6m lSO .j PVA = 1 / 3 PHA ( regression ) '" .Q ePVA = 112 PHA ( regression ) .!! in 40 .>PV A = 2/3 PHA ( regre ssion) c: FHWA Limita t ion c: H =6m( FHWA ) Q) E 30 Q) !:! .E c: 2 0 E :J E x 10 '" ::;; 0 o 0 2 0.4 0 6 0 8 1 2 Peak horizontal acce l era tion PHA ( g ) Figure 6.33 Comparison between Predicted Tto t a l and the FEM Results with the Effect of Peak: Vertical Acceleration PV A All of the predicted values are within the range of FEM results and the nonlinear response is consistent with those observed in Chapter 5. The prediction equation suggests that the maximum reinforcement tensile load increases with increasing wall height, decreasing wall batter angle decreasing soil friction angle increasing reinforcement spacing increasing reinforcement stiffness and increasing 265
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peak vertical acceleration From the comparison the order of significance of design parameters on the maximum reinforcement tensile load is (1) wall height (2) reinforcement stiffness (3) reinforcement spacing (4) soil friction angle (5) wall batter angle and (6) peak vertical acceleration It appears that the maximum reinforcement tensile load is influenced by all of the design parameters ; hence the simplified prediction equation for the maximum reinforcement tensile load is not developed. 6.2.7 Prediction of Maximum Horizontal Acceleration at Centroid of Reinforced Soil Mass The prediction equation for the maximum horizontal acceleration at the centorid of reinforced soil mass Am was found as: Am = exp (2.437 + 2.608 PHA + 0.628' PYA 0 .0076' H + 0.00026 CO + 0.0167 $' + 0.0238 S y 0.00033 Ts%) ; R 2 = 0.590 (6.12) The prediction equation was applied to the conditions considered in the parametric study. Comparisons between the predicted values of Ttotal and the FEM results are presented in Figures 6.34 6.35 6.36 6.37 6.38 and 6 39 to examine the influence of wall height H wall batter angle CO, soil friction angle $', reinforcement spacing Sy, reinforcement stiffness T s%, and peak vertical acceleration PYA respectively In Figures 6.34 to 6.38 values ofPVA were assumed to be 113 of PH A. As indicated by the figures the maximum horizontal acceleration at the centorid of reinforced soil mass is not significantly affected by the wall batter angle reinforcement spacing and reinforcement stiffness. The remaining design parameters in the order of significance are (1) peak vertical acceleration (2) soil friction angle and (3) wall height. It was observed that the maximum horizontal acceleration at the centorid of reinforced soil mass increases with increasing peak vertical acceleration increasing soil friction angle and to a less extent with decreasing wall height. The simplified prediction equation without the contribution of reinforcement spacing and reinforcement stiffness was determined as: Am = exp ( 2.438 + 2.608 PHA + 0.628 PYA 0 .0076' H + 0 0167 $') ; R 2 = 0 590 (6.12a) Note that the difference between Equations 6 .12 and 6 12a is small change in the constant bo of the exponent and the coefficients of the remaining independent variables are the same between the two equations. 266
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6 E o 0 2 0.4 0 6 0 8 1 2 Peak horizontal acceleration PHA (g) H=9m H =6 m ... H =3m H = 9 m (regressi on ) eH = 6 m (regression) H = 3 m (regression) FHWA Limitat ion H=9m(FHWA) H = 6 m (FH WA ) +H = 3 m (FHWA) Figure 6 34 Comparison between Predicted Am and the FEM Results with the Effect of Wall Height H nI ]j C o N c o .r. 2 ... t ... ... ... t '*. t o 0 2 0.4 0 6 0 8 1 2 Peak horizontal acceleration PHA (g) w = 15 w = 10 ... w = 5 w = 15 (regress i on) ew = 10 (regression) w = 5 ( regression ) FHWA Limitat ion w = 15 (FHWA) w = 10 (FHW A ) +w = 5 (F HWA ) Figure 6 .35 Comparison between Predicted Am and the FEM Results with the Effect of Wall Batter Angle ro 267
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7 E 6 '" E 0 V> u 5 40 Q) 36 .E c .. 32 '0 4 +40 (regression) u e36 (regression) > e c 3 ...32 (regress i on) Q) u FHWA Limitat ion iii 40 (FH WA ) Qj u 2 36 (FHWA) u '" iii 32 (FHW A ) C 0 .. N c 0 .c x '" :::2; 0 o 0 2 0.4 0 6 0 8 1 2 Peak horizontal acceleration PHA ( g ) Figure 6 36 Comparison between Predicted Am and the FEM Results with the Effect of Soil Friction Angle 6 ./ 4 2 I t#. t. .. o 0 2 0.4 0 6 0 8 1 2 Peak horizontal accelerat ion PHA (g) S = 0 2 m S = 0.4 m .. S = 0 6 m +S = 0 2 m (regression) eS = 0.4 m ( regr essi on ) ...S = 0 6 m (regression) FHWA Limitation S = 0 2 m (FHWA) __ S = 0.4 m (FHWA ) _____ S = 0 6 m ( FHWA ) Figure 6.37 Comparison between Predicted Am and the FEM Results with the Effect of Reinforcement Spacing Sy 268
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6 .[ .n en co E 5 0 en ." T5 = 7 2 kN/ m cu 4 .2 T5 = 36 kN/ m c: .. T5 = 12 kN/ m '0 T5 = 72 kN/ m ( reg r ess i on ) ." 3 &T5 = 36 kN/ m ( reg r ess i on ) e c bT5 = 12 kN/ m ( regression) fl FHWA Lim i tation iii 2 H =6m( FHWA ) u co ]j c: 0 N .". 0 .c 0 o 0 2 0.4 0 6 0 .8 1 2 Peak hor i zontal acceleration PHA (g) Figure 6 38 Comparison between Predicted Am and the FEM Results with the Effect of Reinforcement Stiffness T 5% o 0 2 0.4 0 6 0 .8 1 2 Peak horizonta l acce l eration PHA (g) H =6m PVA = 1 / 3 PHA (regression ) &PVA = 1 / 2 PHA ( regress i on ) bPVA = 213 PHA ( regre s s i on ) FHWA Lim i tation H =6m( FHWA) Figure 6.39 Comparison between Predicted Am and the FEM Results with the Effect of Peak Vertical Acceleration PV A The relationships between increases in the design parameters and the resulting seismic performances of GRS walls are summarized in Table 6 2 In Table 6.2 an upward pointing arrow (t) indicates increase in value a downward pointing arrow ( .J) indicates decrease in value and a dash () indicates that the design parameter has 269
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no effect or is neutral on the seismic performance. Also included in Table 6.2 is the order of significance (or extent of influence) of the design parameter to the individual seismic performance ranging from 1 to 6 A number 1 in parenthesis indicates the most significant design parameter that could affect the seismic performance whereas a number six in parenthesis indicates the design parameter of least significance. Table 6.2 Change in seismic performance of GRS wall due to increase in design parameter Design Seismic performance parameter LP D E qvE Tt o t a l Am tH t (1) t (1) t(1) t (1) t (1) I, (3) to) I, (3) I, (4) (4) I, (4) I, (5) (6) t I, (2) I, (5) t (2) t (2) I, (4 ) t (2) t Sv (6) t (3) (6) (6) t (3) (5) tTS% I, (4) I, (2) (5) (5) t (2) (4) tPVA t (5) t (6) t (3) t (3) t (6) t (1) 6.2.8 Prediction Equations without Peak Vertical Acceleration Comparing to other independent (or predictor) variables depicted in Table 6.1 peak horizontal acceleration PHA and peak vertical acceleration PV A have a relatively high coefficient of correlation of 0.76, which indicates an interdependent relationship between the two variables. In the case of a collinearity situation one of the two interdependent variables can be removed from the prediction equation without affecting the predicted value significantly. With PHA being more readily available than PV A and due to their interdependent relationship PV A was removed from the prediction equations. The results of regression analysis to predict seismic performances of GRS walls are summarized in Tables 6.3 and 6.4 for prediction equations with all seven independent variables and for prediction equations not including PYA as one of the independent variables respectively. Note that the values ofR2 from the regression analysis are all greater than 0 .7 suggesting that the FEM results are well represented by the prediction equations; except the maximum horizontal acceleration at the centorid of reinforced soil mass Am, where R 2 are slightly less than 0.6. As indicated by Tables 6.3 and 6.4 the main differences between the two sets of equations are the values of constant bo and coefficient b l in the exponent where b l is the coefficient for PHA. The coefficients are the same for the remaining design parameters in the two sets of prediction equations. The effect of removing PV A results in negligible decrease in values of R2, which suggests that the prediction equations without independent variable PV A are 270
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close approximations to the full prediction equations The prediction equations without PYA (i.e., Table 6.4) can be used when the values ofPVA are not available during the seismic design. However it is recommended to use the full prediction equations (i .e., Table 6 3) and to assume a value ofPVA (e.g. PYA = 113 PHA to 2 / 3 PHA) even when PV A is absent in order to capture the effect of realistic seismic loading. 6 3 D es i g n C on s i dera tion s The prediction equations provided in this chapter are considered as alternatives to the FHW A methodology for assessing seismic performances of free standing simple GRS walls. The prediction equations are applicable for conditions similar to those considered in the parametric study and when PHA is greater than the limit of 0 29 g set forth by the FHW A methodology. Note that the prediction equations can be implemented in a spreadsheet or a handheld calculator. Due to the easy accessibility seismic performances can thus be computed rapidly for evaluation in both the ultimate limit state design and the serviceability limit states design For the serviceability limit states design deformation tolerance expressed in terms of horizontal displacementtowall height ratio (or verticality) from the various guidelines have bee summarized by Bathurst et al. (2010) and Huang et at. (2009). A value of 5% has been suggested by Koseki et al. (1998) and Huang et al. (2009) as the limiting verticality. The limiting facing displacement could also be governed by the required clearance to the adjacent structures found at the project site. With the advent of prediction equations it is possible to correlate maximum reinforcement tensile load with the maximum horizontal displacement. As an example Figure 6.40 shows the relationship between maximum horizontal displacement and the maximum reinforcement tensile load for the baseline model (i.e. H = 6 m 0) = 10, = 36 S y = 0.4 m Ts% = 36 kN / m) In this example when the required horizontal displacement or limiting verticality is given from the project site, a designer could use the relationship to estimate the maximum reinforcement tensile load within the GRS wall. 271
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Table 6.3 Summary of constant and regression coefficients from the multiple regression analysis including independent variable PYA [Y = exp (bo + b l PHA + b 2 PYA + b3 H + b4 0) + b s + b6 S v + b7 T s%)] Response Constant Regression coe f ficients R 2 Y bo b l b 2 b3 b4 b s b6 b7 (mm) 2 900 4.578 0 330 0.251 0.0192 0.0434 0.0553 0 00239 0.710 (mm) 1 397 3.056 0.380 0 .2 72 0.0222 0 0246 0.6102 0.00909 0.727 LPOE (kN/m) 2 033 0 931 0 306 0.325 0 0063 0 0319 0 0058 0 00035 0.874 qvE (kN/ml) 2.640 0 741 0.207 0.233 0.0266 0 0465 0.0940 0.00005 0.832 T total (kN/m) 0.748 1.158 0.437 0 185 0.0109 0.0209 1 194 0.0153 0.854 Am (g) 2.437 2.608 0.628 0 0076 0.00026 0.0167 0.0238 0.00033 0.590 Table 6.4 Summary of constant and regression coe f ficients from the multiple regression analysis without independent tv variable PYA [Y = exp (bo + b l PHA + b 2 H + b3 0) + b4 + b s S v + b6 T s%)] ....) tv Response Constant Regression coefficients R 2 Y bo b l b 2 h 3 b4 b s b6 (mm) 2.892 4.811 0.251 0.0192 0.0434 0 0553 0 00239 0.709 (mm) 1.388 3 324 0.272 0 0222 0 0246 0.6102 0 00909 0.724 LPOE (kN/m) 2.025 1.147 0.325 0.0063 0 0319 0 0058 0.00035 0 865 qvE (kN/ml) 2 635 0 887 0.233 0 0266 0.0465 0 0940 0.00005 0.825 Ttotal (kN/m) 0 738 1.466 0.185 0.0109 0 0209 1 194 0 0153 0.838 Am (g) 2.453 3.051 0 0076 0.00026 0.0167 0 0238 0 00033 0.582
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30 E z i! 25 0 <0 .Q 20 (j) c .& C Q) E 15 .E c E 10 ::::l E x <0 Baseline model : I I H = 6 m (J) = 10 = 36 S v = 0.4 m T S % = 36 kN/m .../ / I ::2 5 I I I I o 400 800 1200 1600 Maximum horizontal displacement (mm) Figure 6.40 Variation of Maximum Reinforcement Tensile Load with Maximum Horizontal Displacement for the Baseline Model 273
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7. Conclusions and Recommendations for Future Studies This study was performed to evaluate the effectiveness of FHW A methodology in assessing the seismic stability of freestanding simple GRS walls. The first part oftms undertaking involved literature review on: (1) the insitu seismic performance of GRS walls and structures (2) physical and numerical modeling of GRS walls under dynamic shaking (3) methods of seismic analysis of simple GRS wall and (4) geosynthetic material behavior under dynamic loading The remainder involved assessment of FHW A methodology using the finite element method (FEM) computer program LSDYNA. Its predictive capability was examined using the computer program validation process where the calculated values were compared against the measured results from the fullscale shaking table tests of GRS walls found in the literature. The backfill and geosynthetic reinforcement material characterizations were performed in the validation process Material model parameters were determined form the available laboratory data. A cap model with stressdependent parameters was adopted for the backfill. Model calibration was also performed to fine tune the input parameters such as the viscousdamping constant mass damping coefficient and the soilgeosynthetic interface friction coefficient. The calibrated values were adopted for the subsequent parametric study. The calibrated values along with the material characterization approaches were adopted for the subsequent parametric study. Qualities of validation tests were evaluated and discussed. Factors responsible for comparison discrepancy were identified. Variability within the measured data is thought to have contributed to some of the comparison discrepancies. Prior to the parametric study the extent of finite element model boundary was determined with the intent to minimize the boundary effect. A parametric study program was carried out to assess the effectiveness of different design parameters such as peak horizontal acceleration peak vertical acceleration wall height wall batter angle soil friction angle reinforcement spacing and reinforcement stiffness. Different design parameter values were evaluated and the FEM results were compared against the baseline model with a wall height o f 6 m Each case in the parametric study was subjected to 20 freefield earthquake records with PHA ranging from 0.114 g to 0.990 g and PV A ranging from 0 093 g to 1 048 g. Results of parametric study were then compared against the corresponding values determined using the FHWA methodology. In this comparison all of the parametric study results were presented as functions of PHA and the correlation of performances with PHA were determined through single predictor variable regression analysis The seismic performances evaluated include maximum horizontal displacement maximum crest settlement total driving resultant total overturning moment arm 274
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maximum bearing stress maximum reinforcement tensile load, and maximum horizontal acceleration at centroid of reinforced soi l mass. Multivariate statistical modeling was performed using the results of the parametric study. Correlation analysis was first performed to assess the associations between the variables through the construction of correlation matrix. Multivariate regression analysis was then performed to determine prediction equations of the seismic performances based on the design parameters. The prediction equations provide the firstorder estimates of the seismic performances given the various design parameters. In addition the prediction equations allow seismic performances to be estimated for PHA beyond the limitation (i.e 0.29 g) stipulated by the FHWA methodology. Simplified prediction equations were also developed to omit variables not contributing to the seismic performance in question 7.1 Conclusions The findings and conclusions drawn from this study are separated into two categories. The first category pertains to the general observations and seismic performance of freestanding simple GRS wall. The second category identifies the discrepancy between the FHW A methodology and the FEM calculated results. General Observations and Seismic Behavior of FreeStanding Simple GRS Wall 1. Lessons learned from the field case histories to ensure seismic stability of GRS walls include : (1) specify a stringent backfill compaction requirement (2) perform ground improvement on weak foundation (3) avoid large reinforcement spacing (e.g. spacing > 800 mrn), and (4) evaluating the impact of peripheral structures added to the GRS walls. 2 Through the model calibration study it was found that (1) wall face response is not sensitive the range of viscousdamping constants examined (e g., 20 40 and 60) (2) mass damping coefficient equal to O.IScon yielded good agreement between the measured and calculated wall face responses and (3) good agreement is also observed with soilgeogrid interface friction coefficient equal to 2 / 3 3 The discrepancy noted in the computer program validation process between the calculated results and measured wall response is thought to have been attributed to the deficiencies of the constitutive models and idealization of the numerical model. However these primitive constitutive models conversely offer the benefit of limited material parameters and still achieve reasonable agreement with the measured response. 4. Natural period of the wall model increases with increasing wall height and decreasing friction angle. The natural period is not affected as much by the wall batter angle the reinforcement spacing, and the reinforcement stiffness. 275
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Variation of natural period with wall height and soil friction angle is depicted in Appendix L. 5. For the cases analyzed the amplification of seismic performances or resonance was not observed when the predominant period of the input motion is close to the natural period of the GRS finite element model. It appears that PHA and PV A have greater influence on the performances than the predominant period of the input motion. The nonoccurrence of resonance could be due to nonlinear effects such as inelastic materials and contact interfaces specified in the FEM model. 6. The average percent increase from horizontal shaking only to combined shaking for horizontal wall displacement crest settlement, bearing stress and reinforcement tensile load are 12.6% 26.0% 3.2% and 5 7% respectively. The overall average percent increase for all performances was found to be 11.9%. Since the increases are not negligible vertical component cannot be ignored in numerical modeling of GRS walls especially with PHA greater than 0.3 g. 7. When subjected to seismic loading GRS wall facing displacement involves the forward tilting where top of wall displaced more than bottom of wall. 8. Failure of GRS walls due to toppling of top facing blocks is more likely than other failure modes. Seismic stability of GRS walls can be improved by joining the top two or three courses of facing block with grout or using other mechanical stabilization mechanism. 9. Failure wedges were noted in the horizontal displacement contours in the retained earth behind the reinforced soil mass and the pattern is consistent with the twopart wedge failure observed in the physical model tests. 10. High concentration of bearing stress is observed in the foundation soil directly beneath the facing block and the stress distribution is fairly uniform beneath the reinforced soil mass. Seismic stability of GRS wall can thus be improved by incorporating sound foundation material at the toe of the wall. 11. It is observed that the reinforcement tensile load is inversely related to the wall displacement at the location of the reinforcement layer. The high tensile load near the toe of the wall is a result of both high overburden stress and restraining effect of the geosynthetic reinforcement. 12. Both FEM results and FHW A analysis values show that all seismic performances increase with decreasing wall batter angle. GRS walls are less stable at a small batter angle. 13. Magnitudes of seismic performances increase with decreasing friction angle as the stiffness of the soil is positively related to friction angle Exceptions were noted in total driving resultant and bearing stress where these responses are positively related to the soil friction angle or the soil stiffness. The calculated displacements and settlements concur with the notion that a GRS wall built 276
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using higher friction angle backfill would be more stable than that of a lower friction angle backfill. 14. Seismic performances such as maximum facing horizontal displacement total driving resultant and maximum bearing stress are not significantly affected by the reinforcement spacing. The results indicate that uniform reinforcement spacing between 0.2 m and 0.6 m allows a coherent composite to be created. Although a small reinforcement spacing is not an effective way to reduce the horizontal displacement it does provide additional benefits to the wall system such as added quality control of backfill compaction (i.e., due to smaller lift thickness) and more tolerable against potential creep of geosynthetics 15. Wall crest settlement and reinforcement tensile load are positively related to the reinforcement spacing. Small reinforcement spacing of 0 2 m is an effective method in reducing crest sett lement. 16. Similar to reinforcement spacing FEM results indicate that the performances related to external stability such as total driving resultant and maximum bearing stress are not significantly affected by the reinforcement stiffness 17. The effect of reinforcement stiffness is more pronounced in maximum horizontal displacement crest settlement and reinforcement tensile load Wall facing displacement and crest settlement can be controlled effectively by utilizing high stiffness reinforcement. The high stiffness reinforcement however would experience higher reinforcement tensile load than the low stiffness reinforcements when experiencing similar horizontal displacement. 18. An increase in tensile load in the top reinforcements toward the back of the wall was observed between the static and seismic cases indicating that these reinforcements contribute the seismic stability of the wall. 19. Although reinforcement maximum tension line under static condition is not exactly identical to the maximum tension line under seismic condition both are close to the wall facing 20. The maximum tension line in genera l is located close to the facing blocks for the top 3/4 of the reinforcements. The maximum tension line shifts toward the midpoint of the reinforcement for the bottom 114 of the reinforcements, which results in a high reinforcement tensile load mound at the toe of the wall. Discrepancies between FHW A Methodology and FEM Results 1. FEM results indicated that multidirectional loading yielded greater seismic wall responses than the loading with only horizontal component. Wall facing displacements begins to deviate from one another at PHA of approximately 0.3 g and the threshold value coincides with the maximum acceleration coefficient of 0.29 specified in the FHWA design methodology 277
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2. The FEM calculated total driving resultants are greater than the FHWA analysis values and the deviation is more pronounced for wall heights greater than 3 m. 3. The FEM calculated locations of total overturning moment arm are in general in agreement with the FHW A analysis values. The correlation agrees with the FHW A methodology in that the location of resultant application decreases with increasing PHA for PHA greater than approximately 0.6 g. The location of total driving resultant application varied from 1 / 3 (static condition) to 112 (PHA 0.6 g) of the wall height. 4. For wall height greater than 6 m the bearing stresses calculated by FEM are greater than the FHW A analysis values under both the static condition and the seismic condition 5 The reinforcement tensile loads calculated by FEM were lower than the FHWA analysis values The prediction equation developed in this study can be used to estimate the maximum reinforcement tensile load based on design parameters including the reinforcement stiffness which is not considered in the FHWA design methodology 6. Outofphase oscillations between the reinforced soil mass and the retained earth is only observed in the initial portion (i .e., first 5 seconds) of the acceleration time histories. The inphase oscillations persisted in the remainder of time histories. This observation contradicts the FHW A assumption of phase lag between the reinforced soil mass and the retained earth 7. The linear slip surface that passes through the reinforced soil mass assumed by the FHW A methodology could not be substantiated with results calculated by FEM. The assumption that the maximum tension line coincides with the linear slip surface is also not supported by the FEM results. In other words a distinct slip surface could not be identified from the FEM results As suggested by the FEM results multiple slip surfaces may exist within the reinforced soil mass. 8. In light of maximum tension line being close to the wall facing connection strength is the prevalent failure mode in the internal stability evaluation Prediction equation for estimating the maximum reinforcement tensile load developed in this study can be used in the evaluation of connection strength 9. Since the maximum horizontal displacement and the maximum crest settlement could not be determined using the current FHW A design methodology prediction equations for estimating and developed in this study are recommended for potential adoption in the future design guidelines. 278
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7.2 Recommendations for Future Studies The recommendations for future studies are presented as follows : 1. Perform additional validation and calibration activities with LSDYNA on welldocumented fullscale model tests as they become available to further refine the prediction capability of LSDYNA. 2. Increase the sample size of earthquake records to further refine the correlations of seismic performances with the design parameters. 3. For material characterization, augment the relationship of shape factor R with initial relative density for different soil types as the laboratory test results become available 4 Use the current modeling approach to simulate complex GRS structures subjected to seismic loading. Examples of complex GRS structure include tiered walls (or superimposed walls) backtoback walls and bridge abutments. 5 Perform additional numerical simulation to evaluate the effect of reinforcement length wall embedment depth and other design details on the seismic performances of GRS wall. 6. For future numerical investigations the effects of sequential loading versus gravity tumon method in simulating gravity load should be evaluated as it may have a potential influence on the seismic performances of GRS walls. 7 For the future fullscale model tests numerical simulations ofthe model tests should be performed prior to the actual experiment. Results of numerical simulations could serve as guidelines for the instrumentation planning. For example due to the high tensile loads near the wall facing observed in this study strain gages on the reinforcements should be placed more densely close to the wall facing than toward the back of the wall in order to monitor and capture the anticipated high tensile loads. 8. Earthquake records with higher PHA and PV A in the range of 0 5 g to 1.0 g should be emphasized in the future numerical studies since the GRS walls eva lu ated in this study appeared to be stable with PHA and PYA less than about 0.5 g. 9. The soilgeogrid interface specified in this study followed the idealized contact interface mechanism based solely on coefficient of friction However real interaction between soil and geogrid involves both passive and interface shear resistances. Factors influencing the interaction include geogrid characteristics soil density and confining pressure (e.g., see Farrag et al. 1993 and Teixeira et al. 2007). Numerical investigation on soilgeogrid interaction could be performed and the results could potentially enhance the fullscale simulations. 279
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APPENDIX A RambergOsgood Material Model The RambergOsgood model is often used to represent the nonlinear behavior with hysteretic energy dissipation of the granular material when subjected to cyclic shear deformation. The RambergOsgood model is a quasi linear (piecewise linear ) model in which the empirical backbone stressstrain relationship subjected to monotonic loading can be expressed by: Y 't 't = +a y y 'ty 'ty Y 't 't (for y < 0) (A.I) =a y y 'ty 't y where y = shear strain 't = shear stress y y = reference shear strain 't y = reference shear stress a = stress coefficient 0 and r = stress exponent. Note that 'tyiy y = Gmax, where Gmax is the initial shear modulus. The stress coefficient (a) can be varied to adjust the position of the curve along the strain axis and the stress exponent (r) can control the curvature of the curve. When r = I a linear relationship between the shear stress and the shear strain is observed. To account for the unloading and reloading the stressstrain relationship of Equation A.I needs to be augmented according to the extended Masing rules (e.g. Figure A.I). The first stress reversal is detected by 1'(8y /8t) < O. After the first stress reversal the stressstrain relationship is augmented by a factor of 2 to: y y 't't ('t't Jr __ 0 = __ 0 +a __ 0 2y y 2't y 2't y (for y 0) yYo = 't'to _a('t'toJr 2y y 2't y 2't y (for y < 0) (A.2) where Y o and 't o represent the values of shear strain and stress at the point of stress reversal. The subsequent stress reversals are detected by (y yo)(8y/ 8t) < O 280
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1: F Backbone curve y (8) (b) Figure Al Extended Masing Rules: (a) Variation of Shear Stress with Time ; (b) Resulting StressStrain Behavior with Backbone Curve indicated by Dashed Line (after Kramer 1996) The RambergOsgood (RO) constitutive relations are considered one dimensional and are assumed to apply to the shear components. To generalize this theory to the multidimensional case it is assumed that each component of the deviatoric stress tensor and the deviaotric strain tensor are independently related to the onedimensional stressstrain equations Furthermore the NewtonRaphson iterative method is used in the RO model for finding the solution of the nonlinear stressstrain equation. For the initial monotonic loading and with deviatoric strain being positive the RO stressstrain equation is expressed as: e(s) = y + a r l (A3) 'ty 'ty where e(s) = deviatoric strain as a function of deviatoric stress (s) y y = reference shear strain 'ty = reference shear stress and r = stress exponent Note that y y, 'ty and r are material parameters to be deduced from the experimental results. Newton Raphson iterative method utilizes the linear portion of the Taylor series and takes the form of: e(s I) = e(s o ) + de (A4) ds where e(sl) = value of the function at a specific value (Sl) of the independent variable s e(so) = value of the function at so, = the difference (Sl s o ) and de / ds = first derivative of the function With e(sl) prescribed to be E Equation AA can be expressed as: E = e(s o)+ de (Sl so ) (AS) ds 281
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Solving Equation AS for the unknown value of Sl yields: E e{so) s, = S o + / de ds (A6) The first derivative de/ds is to be eval uated at so, hence it is denoted as e'(so); then Equation A6 becomes: E e{so ) s, = S o + () e's o Equation A 7 can be iterative l y evaluated as: E e{sJ Sj+ = Sj + '() e S j (A.7) (A8) From the RO stressstrain relationship of Equation A3, the first derivative of the function at Si is: r+' ,( ) y y Y y Sj e Sj = + ur(A9) 'ty 'ty 'ty As implemented in the finite element computer code NIKE3D, the default convergence criterion for each load increment is: IE e{sJ 0.001 (AI0) with the maximum number of iteration defaulted to 100. RO model inherently uses the isotropic linear elastic stressstrain relations in combination with variable shear modulus G to account for the nonlinear behavior. From the isotropic linear elastic stressstrain relations (i.e. generalized Hooke's law) the total incremental strain tensor dEij in indicial notation can be expressed as : 1 1 III dE = dEkk 8 +de. =dp8. +ds = d{I,)8 + ds (All) I) 3 I ) I ) 3K I) 2G I) 9K I) 2G I ) where yjEkk = mean strain (or hydrostatic strain), 8ij = Kronecker delta, eij = deviatoric strain tensor, p = mean stress Sij = deviatoric stress tensor (Sij = 2Geij) and II = first invariant of the stress tensor. Equation All indicates that the total response considers both the volumetric and the shear deformations It is further assumed that the volumetric behavior is elastic ; hence the mean pressure p is determined using the elastic relation as : p=KEkk (AI2) Alternatively, the total incremental stress tensor dcrij in indicial notation can be expressed as : dcrj j = KdEkk 8jj +2Gdej j (AI3) As a variable moduli model the value of tangent shear modulus Gt in RO model varies as the state of stress changes during the incremental loading. Based on Equation A.3, Gt is found as: 282
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Gt =(de)' r 'J' (A. 14) ds 't y 't y 't y As implemented in NIKE3D the tangent shear modulus Gt in RO model is a variable while the bulk modulus K remains constant during each load increment. Furthermore an uncoupled behavior is observed for the RO model between the volumetric deformation and shear deformation. Note that NIKE3D program and the RO model has been used to study problems of soilpile interaction (Jiang 1996) retaining walls (Lee 2000) and composite dams (Oncul 2001) Table A.l summarizes the material parameters of the R 0 model. It is to be noted that the failure state of a soil can not be simulated by the RO model as it is an equivalent linear model. Table A.l Material parameters for the RambergOsgood model Parameter Description y y Reference shear strain 'ty Reference shear stress a Stress coefficient r Stress exponent K Elastic bulk modulus The RO material model parameters yy, 'ty a and r are to be determined from the laboratory experiment results. Ueng and Chen (1992) have developed an iterative procedure to obtain the RO material model parameters for soils using modulus reduction (G / Gmax) and damping ratio versus shear strain curves in which the RO relationship is rearranged so that the results can best fit the data of both modulus reduction and damping ratio curves. Note that the secant shear modulus G and the damping ratio are referred to as the equivalent liner material parameters Gmax is the maximum shear modulus and can be computed as Gmax = pv/, where p = density of soil and V s = shear wave velocity. Figure A.2 shows the definitions of the secant shear modulus and the associated backbone curve Figure A.3 shows the modulus reduction and damping ratio curves of average sand determined by Seed et al. (1986). By rearranging Equation A.I the secant shear modulus for the backbone curve can be expressed as : 283
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r I For very small strain, i.e. Y + and 't + 0 since r > 1 {G )y=o = G m ax = Y y Then the backbone relation can be rewritten as: Y 't [1 't r I J y; = G ma x Y y + a G max Y y (A.1S) (A. 16) (A. 17) Hence besides Gmax, there are three other parameters (YY' a and r) left to be determined for the RO model. By substituting 't = G'Y and rearranging Equation A.17 one gets the following equation: r I Gmax 1 = a G y G G ma., Y y (A. 18) By applying logarithm to both side of E quation A.18, one obtains : 10g(Gmax l)=IOga+{rl)IOg( Gy J G Gmax y y (A.19) From Equation A.19 with an assumed value for the reference shear strain yy and the experimental modulus reduction curve (e.g. Figure A.3a), the values of a and r can be determined from the intercept and th e slope respectively of the b es t fit straight line shown in Figure A.4. The values of a and r are further refined by using the damping ratio curve (e.g., Figure A.3b). The damping ratio for a hysteresis loop with the tip at (Yc,'tc ) (see Figure A.2a) can be expressed as: L'lE 2a{rl)( G Jr(YcJr1 21t1:c y c = n{r + 1) G rna., y; where L'lE = energy dissipation in one loading cycle. Equation A.20 the damping ratio is: 2{r n{r + 1) G max or 284 (A.2 0) Substituting E quation A.18 in (A.21)
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Gmax 2(r 1) (A. 22) Substituting Equation A.22 in Equation A.19 then yields: 109[ 1 = IOgu+(r1)IOg([12(r1) Y y (A.23) Similar to Equation A.19 a best fit straight line can be found from the damping ratio curve data applied to Equation A.23 (similar to Figure A.4). The final u and r are said to have included the effect of both the modulus reduction and damping ratio curves. The value of the reference shear strain Y y can affect the shape of the backbone curve and the hysteresis loop. Y y can be refined through an iteration procedure listed in the following: (1) Assume a value for yy and obtain the values of u and r by plotting the data according to Equations A.19 and A.23. (2) Compute y y according to Equation A.20 from the given modulus reduction and damping ratio data and obtain an average value of y y (3) Compare the new value ofyy with the previous value. Repeat steps 1 and 2 if the difference is too large Finally, the reference shear stress 'ty can be calculated using Equation A.16. y (a) (b) Backbon e cu r ve or Figure A.2 Definitions of (a) Secant Shear Modulus and (b) Backbone Curve (after Kramer 1996) 285
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(a) Modulu s rtducti o n c urv es 00001 0001 001 0 1 Shr.ar s irain y (e/_) ( b ) Dampin g ralio curvtS 30.r,,.." 00001 0001 001 01 Figure A.3 Modulus Reduction and Damping Ratio Curves of Average Sand (after Seed at al. 1986) 286
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( Gmax ) lo g I G Best fit str a i g ht line 10 Ex p erime nt d a t a I lo g ( G Y ) G max Yy Figure A.4 Example of a Best Fit Straight Line for Determining Parameters a and r (after Ueng and Chen 1992) 287
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10 :[ C 2 o ., 0 Q) iIi .. g 2 0 ill (c) APPENDIXB Global Stability of Models considered in Parametric Study (a) ... 8 ( b ) Bas_lne m:>del '10 6 ... o tl H = 3 m Sv = 0.4 m wall batter = 10, soil f riction = 36 1 _'__ '_'_ 1 0 12 Dstcrce(rr) H 6 m Sv 0 4 m ,.,.1 batter10". soil frictkln,. L1 L 10 12 ,. 1lsta1ce(rrj .l.;l H 9 m Sv '" 0 m well baiter" 10 soU 'rictlon '" 36 Clstrce(n) Figure B.l Effect of Wall Height on Global Factor of Safety 288 ,.
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(a) .U1 H = 6 m Sv = 0.4 m v.9U batter = 5, 501 friction = 362 c 0 "" '" > 0 ., ill .. 0 ., ill .. 0 ., ill ..
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:s 2 c: o ., .. > Q) [j] .. ( a ) H ,. 6 m Sv :II 0 4 m wal batter" soil friction "" 32L__ _L ____ L_ __ ____ L_ __ ____ __ ____ L_ __ _L ____ L_ __ _L ____ L_ __ ____ L_ __ _L G .. w a M W re :s 2 c: o ., .. a; [j] .. '2 :s 2 c: o ., 0 [j] (b ) Baseine rmdel '0 .. ( c ) H .6 m $" .0.4 m waU batter = 10", soi l friction" 36 '0 '2 ,. ,. ,. Dstn::e ( m H = 6 m Sv = 0 4 m wa l batter = 10", soil friction = 40 L__ ____ ____ L__ ____ ____ L__ ____ ____ L__ ____ L ____ L__ ____ L ____ L__ '2 '0 .. '0 '2 ,. ,. ,. Figure B.3 Effect of Soil Friction Angle on Global Factor of Safety 290
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( a ) S 2 C 0 ,., ra > Q) 0 ill .. 12 10 l ( b ) Baseine rrodel 1 E i 2 ,., 0 ill t J S 2 c o "" ra a; 0 ill .. 1 2 10 .( c ) ,,'''1 2 10 " H 6 m Sv .. 0 2 m wall batt e r 10 soil fricllon a 36.1 ..LJ.. .1 .. 2 10 12 I C1sta"ce (rri .1.Z1 H e m Sv .. 0.4 m wa l baUer 10 soi l fridlon '" 36 I .. 10 12 ,. C1sta"ce (rri .11l1 H = 6 m Sv = 0 6 m wall batte r = 10 soil f riction = 36 [l &ana! (m) Figure BA Effect of Reinforcement Spacing on Global Factor of Safety 291 .J 16 I '" .J 16 I. '" '"
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APP E NDIX C Analysis o f GRS W all f ollowin g F HWA Methodolo g y FHWA D e s ign at Geosynthetl c Re inforce d Soil W all under Static a n d Seis m i c Loads 1 Willi geometry an d soli prope rties W an height and extemallo a d s : Wall height H ::: Wall batter (j) ::: Surch arge / back slope angle p :;: Height with backs tope h Max. ground accelerat io n coefficien t A :;: Wall embedm en t depth 0, Fou nd a t i on s o il : Moi st unit weight 'Yf :;: Dra i ned frict ion ang le .', :;: Dra i ned cohesion c ,:;: 6 m 10 ;O=90 +m= 0 6 .000 m 0 ,29 100 o m (needed for find i ng bearing capa city of foundat ion soil) 20, 8 kNlm' 36 o k Pa Re tatned earth Rel nfon :e d rill : Yrf n c'r( R einforced fill: Moi s t uni t we ight, YI1 :; Drained friction angle .'rf :;: Dra ined coh es ion C 'lf:;: Interf ace friction an g l e 6..t :;: 213 '11 :;: Reta i n e d earth : Moi st uni t wei ght "f, :;: Drai ned frictio n ang le .'re :;: Dra i ned cohes i on e 're :;: I nterface frictio n ang l e s... :;: 213 II .',. ::;: Rei nforcement : R e inforcement type :;: Re inforcement strength T ul :;: 20, 8 kNlm 36 o kPa 2 4 .000 20, 8 kNlm' 36 o kPa 24. 000 User defined geogrid 35 kNim II Founda u o n"ll Y r r c r Re inforcement length L :;: Length wit h facing dimens i on B = Re inforcement spacing Sv = 3 9 m ( ten t a tive un i f o rm reinforcement length ) 4 2 m 0.4 m ( tenta tiv e un iform rein forcemen t spacing ) 2 P erform. ne e criterl. Externa l stability: Slid i ng : F S 1 5 Eccentricity at base : e s 816 i n so i l and 814 i n rock Seism i c eccentricity at base : e s Bl3 Bearing capacity : F S 2 5 Se i sm i c sta b i lity : F S.;;:: 75 % of static F S for all fa il ure modes Internal stability: Pu llou t res i stance : F S .2: 1 5 Allowable tens ile strength for geosynthetic re i nforcemen t : T:s T. = T .. / 1 5 Sei smic pullout resistance : Friction coefficient F reduce d by SO% General requi rements: Length o f r einforcement: Minimum length i s 0 7H and i s not less than 2 5 m Min imum t raffic loads: Un i form loa d equiva len t to 0 6 m o f so i l over trafftc la nes Wall embedment: HJ20 'Nith horizontal slope i n front o f wall and m inimum of 0 5 m Design life : M i n imum service l i fe of permanent reta i ning wall i s 75 years A wall 'Nith face batter of l ess than S i s conskie r ed as a v ert ical wall ( Note : M SEW uses 10 as the crite rion ) M aximum re inforcement spacing : Sv:s 800 mm S e ismi c design i s need ed when peak a cceleration coefficient (A) i s grater than 0 .05. The FHWA sei sm i c desi gn procedure is appl i cable for A S 0 .29 (i e ., 0 .05 < A S 0 29) 292 h[ L Sin!! +' ']COIIb 11n(90!DP ) COltll
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3 S t.Uc An.lys/s of GRS W.1f Exte rnal stability e v al uat i on Active earth pressure for external stability: For fI) < 8 with horizontal backslope: K,= tan' (45' 1<,,= N / A For (j) < 8 'Nith backslope angle p : [COS c os cos' ] K COS + cos' f R 1<,,= N / A For (i) 8 with backs l ope angle P: I n Ihe calculation 6,. = 6,. = = 0 000 K = Sin' ( 0 + .'R ) [ Sin' O s in(O 0.) J + / ,'':"::"','':"::''s in(O 0.) sin(O + I<" = 0 198 F T == Active thrust F T = 1 ,. h 2 K. F = 74 18 1 kNim Govem i ng I<" = 0 198 FTH == horizontal component of active thrust F", = F,' "') = 71. 978 kN/ m F", = F, cos ( P ) = 74 .181 kN/ m ( based on FHWA assumption ) The greater of the two values controls. Thi s i s t o maxi m ize overturning m oment F", = 74 .181 k Ni m FTV = vertical component of active thrust FTV = FT' sin( 8,. CiI) = 17 940 kN/ m FTV = F s i n(p) = 0 000 kN / m ( based on FHWA assumption) The lesser of the two values controls. Thi s i s to m i n imize resisting moment. FTV:; 0 .000 kNim V = weight of the rei nforced soi l mass V = 111' H B (Note: MSEW assumes a vertical back edge i n finding area. ) V = 524 160 kNim V 2 = weight of backfill on top of reinforced soi l mass V = (112 ( L s i np ) ( L cos p + L s i np Ilan(90' ",. P V = 0 000 kNim t V:: summation of forces i n vertical direction t V:: V + V2 + FTV = 524 160 kNim R :: vertical r e action at the base of r einforced soi l mass from e q u i l ibrium R = R = 524 160 kNim IMoc = sum of overturning moment about point C = F", (hI3) = 148 363 kNlm m t MRc = sum o f resi sting moments about point C = V,' xVle + V 2 XV2c + FTV XFTVc + R e XVlc = 0 529 m XV2e = 1 .858 m Xf:'TVe = 2 453 m I MRC = AI M R c + R e I MR C = 277 271 kNlm m + R e = AIMR C + R e For moment eqUilibrium condition: t Moc = I MRC e = eccentricity at the base o f the reinforced vol ume 0 = IF", (hl3) V Xv,. V x",. FTV "nvJ I R = R 0 = 0 246 m BI6 = 0 700 m 1 0 1 < BI6 ( OK) 293 hIJ
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I P o:;: summation of d rivi ng forces i n horizontal direction E P o = F", E Po = 74 .181 kNim I PR :;: summati on of res i s ti ng forces i n horizontal d i rect ion I PR :;: R :;: tan.', R ( Note : I gnores the contribution of foundation coehs ion) EP. = 380 825 kN/ m :;: facto r of safety aga i nst base slid i ng for the re i nforced volume FS ...... = EP.I E Po FS ...... = 5 13 ( OK) 8 :;: effective foundation width B = B 2 Iel B = 3 708 m q... :;: equ i valent un iform v ert i cal pressure at the base of the re i nforced volume q,=E V / B q, = 141. 354 kPa cu :;: u l timate bearing capacity of the founda tion soil q ul :; c', N c + 1 D, Nq + 0 5 ." 8 N N = 37 75 ( N = tan (45 + ,', / 2) e '""') N = 50 5 9 = ( N 1 ) cot+',) N = 56 3 1 (N, = 2 ( N, + 1 ) tan+',) q ... = 2 171. 60 kPa FSt..e = fador of safety aga i nst bearing capa ci ty 15 38 ( > 2 5 ; OK) Intema' s t a bility evaluation Active ea rt h pressure for i nternal stability : For (,,) < 8 with ho ri zon t al back s lope: K = tan ( 4 5 l K, = N / A For (I) 8 with ba ckslope ang l e p : __ f+7""/ No n rf,//{ No n I \I I n the calculation 6rf:; = m i n ( p %+'.) = 0 000 K = sin (9 + +'dl [ S in(+ 'd+OdlSl n ( +',,Pl]' Sin 9 s in(9 0" l 1 + I:''::':""''."c:'' sin( 9 0" lSln(9 + P l K,= 0 198 Governi ng K.:; 0 198 0'2 :;: uniform vert ica l o verbu rden stress from back s l ope 0'2:;: V2/ L 0'2 :;: 0 000 kPa 0' ... :; vertical overburden stress at depth Z ( Z s H ) o ... :;:y,., Z +02 0'" :; horizontal stress at each re inforcement layer at depth Z 0',,:; K.. 0' ... :: vertica l reinforcement spa ci ng T mu :; maximum force i n the re inforcement l ayer at depth Z T mu :: 0'" S ... ( Note : th i s equat io n only approximat es tensile force ; actual calculated value i s based on pressure distribut ion) T. :: allowable ten sil e force per uni t width of re inforcement T = T ... I ( RFCR RF o RF o FS ) T ul :; ultimate geosynlhet i c tensile strength RF CR:: creep reduction factor Polymer Type Polyester Polypropylene H igh Dens i1y Polyethylene RFCR = 1 Creep Reduction F adors 2 .5to 1 6 5 t04 5 102. 6 2 94 Cnlu:a l hp .urflte
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RFo = durability reduction factor 1 1 (minimum of 1 1 ) RFo = 1 RF10 = installation damage reduction factor 1 0 5 RFlo 3 (minimum o f 1 1 ) RF,o= 1 FS = overall factor of safety Minimum FS = 1 5 FS = T.= 35 kNim FS",,*" = factor of safety against reinforcement rupture FS_=T.lTmu Requires: FSrupeur. O!: 1 a. = inclination of ctitical sli p surface from the horizontal For!l):s;8 : a. = 45 + .'1f/ 2 For > 8': a,. = ", + tan Ian( ,', + C ,) I c,] C, = (Ian(,', P) (Ian( +', P) + COI(,', + .,) (1 + lan(6" .,) coil+', + mo, C, = 1 010 C, = 1 + (lan(6" ,,) (Ian(, p> + c oi l+', + m c, = 0 702 Governing 0.. = 58. 00 (This i s the same i n both static and seismic anatyses.) P = pullout resistance P = F' a (0 4) C R. F = pullout resistance factor F' = 2/3 Ian ,', F = 0 484 a. = scale correction factor (0. 6 S a. S 1 0 for geosynthetic rei n forcements) For geogrid: a. = 0 8 For geotextile: (1 = 0 6 (1= 1 L. = length of embedment i n the resistant zone behind failure surface at depth Z 4 = L 4 (Nole : MSEW uses B Insle ad of l.) L. = length of embedment i n tha active zone at depth Z 4 = (H Z) (1 I lana,. Ian,,) C = reinforcement effective unit perimeter C = 2 for strips, grids, and sheets C= 2 R e = coverag e ratio = 1 for full coverage reinforcement R.= FSp&6MA = factor of safety against reinforcement pullout failure Requires: O!: 1 5 Summary of f a ctors of saf ety a ain st rei nforcemen t rupture and pullout under static l oading No. Depth Spacing 0 "" Tmu T. L 4 Z S, (m) (m) (kPa) (kPa) (kN /m) (kNlm) (m) (m) 1 5 6 0 6 116 460 23 079 14. 094 35 000 3 900 0 179 2 5 2 0 4 108 .160 21.430 8 572 35 000 3 9 00 0 359 3 4 8 0.4 99 840 19 782 7 913 35. 000 3 900 0 538 4 4 4 0.4 91. 520 18 133 7 253 35. 000 3 900 0.718 5 4 0 .4 83. 200 16 485 6 594 35 000 3 900 0 897 6 3 6 0.4 7 4 860 1 4 836 5 935 35 000 3 900 1 077 7 3 2 0.4 66 560 13 188 5 275 35 000 3 900 1 256 8 2 8 0 .4 58 240 11. 539 4 616 35 000 3 900 1.435 9 2 4 0 4 49 920 9 891 3 956 35 000 3 900 1 615 10 2 0.4 41. 600 8 2 4 2 3 297 35. 000 3 900 1 794 1 1 1 6 0 4 33 280 6 594 2 6 38 35 000 3 900 1 974 12 1 2 0 4 2 4 960 4 945 1 978 35. 000 3 900 2 153 13 0 8 0 4 16 640 3 297 1 319 35. 000 3 900 2 332 14 0 4 0 3 8 320 1 .848 0 556 35 000 3 900 2 512 15 0 2 0 3 4 160 0 824 0 18 5 35 000 3 900 2 602 295 4 P FS FS,....... (m) (kNlm) 3 721 4 19 819 2 .48 OK 29 79 0 3 541 371. 033 4 .08 OK 43 28 0 3 362 325 139 4 42 OK) 41. 09 0 3 182 282 138 4 83 OK) 38 90 0 3 003 242 028 5 3 1 36 70 0 2 823 204 811 5 .90 OK) 34 .51 0 2 844 170 466 6 .63 OK) 32. 32 OK) 2 465 139 053 7 58 OK) 30 13 OK 2 285 110 512 8 85 OK) 27 93 OK) 2 106 84 863 10 62 OK) 25 74 0 1 926 62 106 13 27 OK 23 55 0 1 7 4 7 42.241 17 69 OK 21. 35 0 1 568 25.269 26. 54 OK) 19 16 0 1 3 88 11. 188 62 .91 OK) 20 .11 0 1 298 5 .233 188 73 OK 28 22 0
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4. Seis m ic An.'ys/s of GRS W." External stability evaluation PIA :z: horizo ntal i nert i a force of the reinforced fill mass PIA:z: p. + p .. = (M.+ M .. ) Am Am = maximum horizontal wall acceleration coeffICient at the centroid of the reinforced fill mass; design acceleration coeffICient "'"" (1. 45 A) A (Note : MSEW sets"," to Awhen A > 0 45 ) "'" = 0 3364 p. = inertia force due to reinforce d fill i n constderation P = M "'" = ( y A,) "'" A.. = area of re i nforced fill affeded by I nert i a force A, ( 0 5 H) H A,:z: 18 m2 (Note : MSEW d i scounts the batter area ) J i 0 8 .! 0 6 8 0 4 I , , , , , I \ M = weight of reinforced fin affected by Inertia force ;. 0 2 / I ","=(1.45/>!j4M .'"'TrfA.. M P = 374 40 kNim 125 95 kNim 0 2 0 4 0 6 0 8 Peak honzont alacceleratl o n c oefficient A p :z: ine rt i a f orce due to I nclin ed backslope i n cons i derat ion p M ","= ( y . A.J. "'" A.. = area of i ncl i ned backs lope on top of A.. affeded by ine rtia force A..=%(12+IJ ) I I 1 H (9 L)] sln p = = 1 H (9 L)]cosp = b = I I tan(90 "'P ) = A. = 0 000 m 0 000 m 2 700 m 0 .000 m M .. = weight o f incli ned backs lope on top of A. affected by i nert i a force M = T,. As M = 0 00 kNl m p. "" 0 00 kN/m P,AE = dynamic thrust exerted on re i nforced fill by retained earth PAE = Ft + i1PA,E _7 I I I 1\ : _____________ .J/ 0' 1lIBl) KAE "" total se i sm ic earth pressure coeffilcient based on the MononobeOkabe general express ion KAE "" K. + Conventional expression for determ ini ng KAE where a,. p .. = 85 7 04 kNIm 296
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= horizontal component of seismic active thrust increment A P .... = APAE 00.(6,. "') = 83. 158 kNim 85. 704 kNim (based on FHWA assumption) FHWA value i s used. To be conservative, the greater of the two should thi s is to maximi ze overtuming moment APAEH = 85. 704 kN / m 50 % o.PAf.H = 42. 852 kNim (based on FHWA assumption) = vertical component of seismic active thrust increment AP"", = A PAE "') = 20 734 kNim AP"", = A PAE = 0 000 kNim (based on FHWA assumption) FHWA value i s used. To be conservative, the l esser of the two shouk1 control; thi s is to mini mize resisting moment. AP"", = 0 000 kNim 50% = 0 000 kNim IVe = summation of forces i n vertical direction I VE = V + V, + FTV + ( 50 % A P .... ) I V E = 524 160 kNim Re = vertical reaction at the base of reinforced soil mass from equilibrium R e = IVe R E = 524 160 kNim IMoec = sum of overturning moment about point C IMOEc = F". (h/3) + (50 % A P....,.) (0. 6 H,) + p (y...J + p,. (y..J YPK = IMoec = 6 000 m 3 000 m 680 473 kNlm m I MREC = sum of resisting moments about point C MRfc = VI xV,c + V2 XV2c; + FTV XFlV c + (50 % + RE e E = 2 735 m t MREC = + RE ee t MREc = 277 .271 kN/ m m + RE eE = o.I MREC + RE e E For moment equilibrium condition: IMoec = IMREC eE = eccentricity at the base of the reinforced volume e E = ( I MOEc AIM.Eell RE e E = 0 769 m BI3 = 1 400 m le i < BI3 (OK) E POE = summation of driving forces in horizontal d irection IPOE = F'JloI + p + Pi' + (50% I POE = 242 981 kNim I PRE = summation of resisting forces in horizontal direction r PRf = Il R E = tan+ R E IPRE = 380 825 kN/ m FS.ading = factor of safety against base sliding for the reinforced volume FS ..... = I P E I I POE FS ..... = 1 57 (> 75 % of 1 5 ; OK) Be' = effective foundation width BE=B2 I"EI B E = 2 682 m = equivalent uniform vertical pressure at the base of the reinforced volume q", = IVEI BE q", = 196 940 kPa qui = ultimate bearing capacity of the foundation soil Qua = c Nc + 1 0, Nq + 0 5 8' E N., N, = 37 75 N = 50 59 N, = 56 .31 q.. = 1558 67 kPa = fador of safety against bearing capacity Fs,.. = q..1 q", 7 .91 ( > 75 % of 2 5 ; OK) 297
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I nternal stability evaluat i on WI. = weight of active zone WI. = + 111 + An' 'Ire (Note : In MSEW, slip plane ema nates from toe o f wall.) At.... = 1 800 m' (area of the f a cing b l ocks ) = 8 074 m2 ( area of t h e reinforced volume) A.. = 0 000 m ( are a of the ba ci
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5 Design Summary Stat i c Analys is External Stabi ljty Eccentricity: l e i = Factor of safety against sliding : FS ...... = Factor of safety against bearing capacity: Intemal Stability Minimum factor of s afety against pullout: FS,.....= Minimum factor of safety against rupture: FSrupu.. ;: Sei smic Analys is External Stabjlffv Eccentricity: 10,1 = Factor of safety against sliding: FS ...... = Factor of safety against bearing capacity: Internal Stability M i n imum facto r o f safety against pud out: FS,.....= Minimum factor of s afety against rupture: FS_= Additional geosynthetic rupture condition 1 : {T ma:Jmin == Addi tional geosynthetic rupture condi tion 2 : (T mcJm .. == 0 246 m < 816 5 13> 1 5 1 5 36 > 2 5 19. 1 6 > 1 5 2 48 > 1 0 0 7 69 m < 813 1 57 > 1 125 7 .91 > 1 875 1 .61 > 1 1 2 5 2 .22 > 1 0 0 .19 2 4 2 299 ( OK) ( OK) ( OK) ( OK) ( OK) ( OK) ( OK) ( OK) ( OK) ( OK) 1(75 % RFCft' RFo RF,o' FS) S.I ( 75 % RFo RF" FS) (OK) ( OK)
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APPENDIXD Maximum Wall Facing Horizontal Displacement Profiles 300
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:[ I 1:Cl iii .s:: :[ I 1: Cl iii .s:: c;; 6 01Duzce P1557 020rovill e P0115 4 03 Kobe P1054 &06Coa li ngaP0346 07NorthridgeP0887 08 LomaPrietaP0736 09 LomaPrieta P0745 2 10Northridge P0833 o o 6 4 2 100 200 300 Max i mum wa ll facing horizontal d i splacement ( mm ) 400 11CapeMendoc i noP0810 12Northridge P0883 13 LomaPrietaP0745 14Kobe P 1 056 15 ChiChi P 1461 16Northridge P1020 17Northr i dge P1005 18Northridge P1023 19ChiChi P 1 532 20 Northridge P0935 o o 400 800 1 200 1600 Ma ximum wall facing horizontal displacement ( mm ) Figure D.1 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m OJ = 10 = 36 Sv = 0.4 m Ts% = 36 kN /m) 301
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I I .r: '" iii .r: I I 1: '" iii .r: iii 3 2 03 KobeP1054 D04CoyoteLake P0149 05 Kobe P1054 06Coalinga P0346 07 Northridge P0887 08LomaPrieta P0736 09LomaPrieta P0745 10NorthridgeP0833 o o 3 2 40 80 120 Maximum wall facing horizontal displacement (m m ) 11CapeMendocinoP0810 12Northridge P0883 13LomaPrietaP0745 e14KobeP1056 15ChiCh iP146 1 +16Northridge P1020 17NorthridgeP1005 18North ridgeP1023 19ChiChiP1532 20NorthridgeP0935 o o 200 400 600 800 Ma ximum wall facing horizontal displacement ( mm) Figure D.2 Maximum Wall Facing Horizontal Displacement Profiles (H = 3 m, 0) = = 360 Sy = 004 m Ts% = 36 kN/ m) 302
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10 I 6 I :i '" ";:; L:: 4 2 o 10 8 I 6 I :i '" ";:; L:: 4 2 o 02 0rovillePOl15 03 K obeP1054 e04 C oyoteLakeP0 1 49 05 K obeP1054 06CoalingaP0346 07 N orthridgeP0887 08L omaPrietaP 0 736 09 L omaPrieta P0745 10Northridge P0833 200 400 600 800 Maximum wall fac ing horizontal d i spla cement (m m ) 11CapeMendocino P0810 12 Northr i dge P0883 13LomaPrietaP0745 e1 4 KobeP1056 15ChiChiP146 1 16Northridg e P1020 17NorthridgeP1005 18NorthridgeP1023 19Ch iChiP1532 20NorthridgeP0935 400 800 1200 1600 2000 2400 Maximu m wa ll fac ing horizontal displacemen t ( mm ) Figure D.3 Maximum Wall Fac ing Horizontal Displacement Profiles (H = 9 m 0) = = 36, S v = 0.4 m Ts % = 36 kN/m) 303
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6 4 I I 2 0 0 6 4 I I :c C> iii .r: iii 2 0 0 100 200 3 00 400 01DuzceP1557 020rovilleP0115 03KobeP1054 <>04CoyoteLakeP0149 05KobeP1054 006CoalingaP0346 07NorthridgeP0887 08LomaPrietaP0736 09LomaPrietaP0745 10NorthridgeP0833 500 Max i mum wall facing horizontal displacement ( mm) l 11CapeMendocinoP0810 12NorthridgeP0883 13LomaPrietaP0745 014 K obe P1056 15Ch iChiP1461 16NorthridgeP1020 17 NorthridgeP1 005 18NorthridgeP1023 19Ch iChiP1532 20NorthridgeP0935 400 800 1200 1600 2 000 Ma xi mum wall facing horizontal displacement (mm) Figure DA Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m ()) = = 36, Sv = OA m Ts% = 36 kN/ m) 304
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g I :E '" iii .c iii g I :E '" iii .c iii 6 4 01 Duzce PI557 02roville POI15 03KobePl054 04CoyoteLakePOI49 05 Kobe Pl054 06 Coa li nga P0346 2 07Northridge P0887 08 LomaPrieta P0736 09 LomaPr i eta P0745 1 0 Northridge P0833 o o 6 4 2 100 200 300 400 Max i mum wall facing horizo n tal displacement (mm) IICapeMendocinoP0810 12 NorthridgeP0883 13 LomaPrie t a P0745 14 Kobe Pl056 15ChiCh iPI461 16 NorthridgePl020 17Northridge Pl005 18 Northridge Pl023 19 Chi Ch iPI532 20 NorthridgeP0935 o +,,,,,,,, o 400 800 1 2 00 1600 Maximum wall facing hor i zontal d i splacement ( mm ) Figure D.5 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m 0) = 15 = 36 Sy = 0.4 m T5% = 36 kN / m) 305
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I I I I :E Ol "iii .c 0; 6 4 2 02 0rovilleP0115 03KobeP1054 04CoyoteLakeP0149 05 Kobe P1054 06 Coa li ngaP0346 07Northr i dgeP088 7 08LomaPrieta P0736 09LomaPrietaP0745 10 NorthridgeP0833 o 1,.,.,. o 6 4 2 o 200 400 600 Max i mum wall fac ing hor i zontal displacement ( mm ) 400 800 1200 11CapeMendocinoP0810 12NorthridgeP0883 13LomaPrietaP0745 14KobeP1056 15 ChiChiP1461 16Northridge P1020 17NorthridgeP1005 1 8 NorthridgeP1 023 19Ch i Chi P1532 20NorthridgeP0935 1600 2000 Ma xi mum wall fac ing horizontal displacement (mm) Figure D.6 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m co = 10, = 32 S y = 0.4 m T s % = 36 kN/ m) 306
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I I I I :i 0> 0 0; .J:: Iii :!: 6 4 2 o 6 4 2 100 200 300 01Duzce P1557 +020rovi ll e P0115 03Kobe P1054 <>04CoyoteLakeP0149 05Kobe P1054 906Coal i ngaP0346 07Northr i dgeP0887 08 LomaPrieta P0736 09LomaPrietaP0745 10NorthridgeP0833 400 Max i mum wall facing horizontal displacement ( mm ) 11CapeMendocino P 0810 12NorthridgeP0883 13LomaPrietaP0745 &14KobeP1056 15ChiChiP1461 16 NorthridgeP1 020 17 NorthridgeP1 005 18NorthridgeP1023 19ChiChiP1532 +20NorthridgeP0935 o 4.,.,..., o 400 800 1200 1600 Max imum wall facing horizontal displacement (mm) Figure D.7 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m co = 10, = 40 Sy = 0.4 m, Ts% = 36 kN/ m) 307
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I I .E 0> "iii .<::: I I .EO> "iii .<::: (ij 6 4 2 004 CoyoteLakeP0149 05 KobeP1054 006 Coalinga.P0346 07 Northridge P0887 08 LomaPrieta P0736 09 LomaPr i etaP0745 10NorthridgeP0833 o o 6 4 2 100 200 300 400 Max i mum wall facing hor i zontal displacement ( mm ) 11CapeMendocinoP0810 12Northridge P0883 13 LomaPrietaP0745 14KobeP1056 15 Ch i Chi P1461 016Northridge P1020 17 Northridge P1005 18NorthridgeP 1 023 19 Ch i Ch iP1532 20Northridge P0935 o o 400 800 1200 1600 Max i mum wall facing horizontal displacement ( mm ) Figure D.S Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m, 0) = 10, = 36, Sv = 0.2 m, Ts% = 36 kN/m) 30S
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I I I I '" "0; .<: 6 4 2 020roville P0115 03Kobe P1054 04 CoyoteLake P0149 05KobeP1054 006 CoalingaP0346 07 Northridge P0887 08 LomaPr i etaP0736 09LomaPrietaP0745 10 Northridge P0833 o o 6 4 2 100 200 300 400 Maximum wall fac ing horizonta l d i splacement ( mm ) 11CapeMendocinoP0810 12 Northridge P0883 13 LomaPrieta P0745 <>14 Kobe P1056 15 ChiCh iP1461 016 Northridge P1020 17 Northridge P1005 18NorthridgeP1023 19 Ch iChiP1532 20NorthridgeP0935 o o 400 800 1200 1600 Max i mum wall facing horizontal d i splacement (mm) Figure D 9 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m, co = 10, = 36 Sv = 0.6 m Ts% = 36 kN/m) 309
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I I I 0> iii .s:; ni 6 4 2 01DuzcePI557 020rovillePOI15 03KobePl054 e04CoyoteLakePOI49 05Kobe Pl054 006 Coali nga P0346 07NorthridgeP0887 08LomaPrietaP0736 09LomaPrietaP0745 10Northri dge P 0833 o o 6 4 2 100 200 300 400 500 Max i mum wall facing horizontal d i splacement (mm) llCapeMendocinoP0810 12NorthridgeP0883 13LomaPrieta P0745 e14KobePl056 15Ch iChiPI46 1 016Northridge Pl020 17 NorthridgePl005 18NorthridgePl023 19Ch iChiPI532 20North ridgeP0935 o o 400 800 1200 1600 2000 Maximum wall facing horizontal displacement ( mm ) Figure D .10 Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m, (0 = 10 = 36, Sv = 0.4 m Ts % = 12 kN/m) 310
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I I I I '" iii .s:: 6 4 2 01DuzceP1557 020rovilleP0115 03Kobe P1054 <>04CoyoteLakeP0149 05Kobe P1054 06Coal i nga P0346 07NorthridgeP0887 08LomaPrietaP0736 09LomaPrietaP0745 10NorthridgeP0833 o o 6 4 2 100 200 300 400 Max i mum wall facing horizontal d i splacement ( mm ) 11CapeMendocinoP0810 12NorthridgeP0883 13LomaPrietaP0745 14KobeP1056 15ChiChiP1461 16 Northridge P1020 17NorthridgeP1005 18NorthridgeP1023 19ChiChiP1532 20NorthridgeP0935 o o 400 800 1200 1600 Maximum wall facing horizontal displacement (mm) Figure D_ll Maximum Wall Facing Horizontal Displacement Profiles (H = 6 m, ill = = 36, Sy = 0.4 m, Ts% = 72 kN/m) 311
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E g c CI> E CI> E CI> E :2 E g C CI> E CI> E CI> E :2 0 20 40 APPENDIXE Maximum Wall Crest Settlement Profiles +01Duzce P 1557 020rovilleP0115 03Kobe P1054 04CoyoteLake P0149 05Kobe P1054 <>06Coal ingaP0346 07 Northridge P0887 08LomaP rietaP0 736 09LomaP rietaP0 745 10 Northr i dge P0833 60 o 2 D ista nce from facing block (m) 0 40 80 120 160 200 0 2 D i stance from facing b l ock (m) 3 4 3 4 11CapeMendocinoP0810 12Northr i dgeP0883 13LomaPrietaP0745 14KobeP1056 15Ch i Ch iP 1461 16NorthridgeP1020 17 Northr i dge P1 005 18NorthridgeP1023 19Ch i Ch iP1532 20NorthridgeP0935 Figure E.1 Maximum Wall Crest Settlement Profiles (H = 6 m ill = = 36 Sv = 0.4 m T 5% = 36 kN / m) 312
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0 E .s c 4 C1l E C1l E 8 C1l If) iii 12 (ij E :::J 16 E x '" :2 20 0.4 12 E .s c 16 C1l E C1l E 20 C1l If) iii (ij 24 E :::J 28 E x '" :2 32 0.4 0 8 1 2 1.6 Distance from facing block (m) 0 8 1 2 1 6 D istance from facing block (m) 2 2 01DuzceP1557 +020roville P0115 03Kobe P 1054 04CoyoteLakeP0149 05KobeP 1 054 <>06Coalinga P0346 +07N orthridgeP0887 08LomaP rietaP0736 09LomaPr i eta P0745 10 N orthr id geP0833 11CapeMendocino P 0810 12N orthridgeP0883 13LomaPr i etaP0745 14Kobe P1056 15ChiChiP1461 16 N orthridge P1020 17NorthridgeP1005 18 Northridge P1023 19ChiChiP1532 20NorthridgeP0935 Figure E.2 Maximum Wall Crest Settlement Profiles (H = 3 m, co = 100 = 360 Sv = 0.4 m, Ts% = 36 kN/m) 313
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0 E :: .s :: c: OJ E OJ 40 E OJ E 120 E .s o c: 5 0 OJ E OJ 100 iii 150 E E 200 o 2 4 6 D i stance from facing block ( m ) 25 0 o 2 4 6 Distance from facing b l ock (m) 01Duz ceP1557 020rovill eP0115 03 Kobe P1054 e04 Coy oteLakeP0149 05 Kobe P 1054 06Coa lingaP0 346 07 Nort hr i dge P0887 08LomaPrieta P0736 09LomaPrieta P0745 10Nort hr i dgeP0833 11CapeMendocino P0810 12 Northridge P0883 13 LomaP rieta P0745 e14Kobe P1056 15ChiChiP1461 16 Northrid ge P1020 17 No rthridge P 1 005 18NorthridgeP1023 19 Ch i Ch iP 1532 20 North ridge P0935 F igure E.3 Maximum Wall Crest Settlement Profiles (H = 9 m {J) = 10 = 36, Sv = 0.4 m Ts % = 36 kN/ m) 314
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o c Q) 20 E Q) E Q) '" iii 40 u 60 E x ., 80 o 2 3 4 D i stance from facing block (m) 0 E .s c Q) E Q) 100 E Q) '" iii (ij 200 E :::J E x ., 300 0 2 3 4 Distance from facing b lock (m) 01DuzceP1557 020rovill e P0 115 03Kobe P1054 04 CoyoteLake P0149 05Kobe P1054 &06 CoalingaP0346 07 N orthridgeP0887 08LomaP riet a P0736 09LomaP rie ta P0745 10North ridgeP0833 11CapeMendoc in o P0810 12Northridge P0883 13 LomaPr i etaP0745 14Kobe P1056 15ChiChiP1461 16 NorthridgeP 1 020 17 NorthridgeP 1 005 18NorthridgeP1023 19 ChiChi P1532 20NorthridgeP0935 Figure E.4 Maximum Wall Crest Settlement Profiles (H = 6 m 0) = 5, = 36, Sv = 0.4 m Ts% = 36 kN/m) 315
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0 E .s 10 c Q) E Q) 20 E Q) In iii 30 0 (ij 40 E ::J E 50 x '" 60 20 E .s 40 c Q) E Q) 60 E Q) In iii 80 (ij 100 E ::J E 120 x '" 140 0 0 01Duzce P1557 +02 0rovi llePOl15 tI 2 3 4 Distance from facing block (m) 2 3 4 Distance from facing block (m) 03Kobe P1054 e04CoyoteLake P0149 05 KobeP1054 06Coalinga P0346 07 N orthridge P0887 08LomaPrieta P0736 09LomaPrieta P0745 10Northr i dgeP0833 11CapeMendocinoP 0810 12N orthridgeP0883 13 LomaPr i etaP0745 e14Kobe P1056 15ChiCh i P146 1 16N orthridge P1020 17 N orthridge P1005 +18N orthridgeP1023 +19ChiCh i P1532 20N orthr i dgeP0935 Figure K5 Maximum Wall Crest Settlement Profiles (H = 6 m, co = 15, = 36 Sv = 0.4 m Ts% = 36 kN/m) 316
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0 E .s : C Cl) 20 E Cl) E Cl) 06Coal inga P0346 07 North ridgeP0887 08LomaP riet a P0736 09LomaPrieta P0745 1 0 N orthr i dgeP0833 11CapeMendocino P0810 12NorthridgeP0883 13LomaPr ietaP0745 14Kobe P 10 56 15Ch i C hiP 1461 16NorthridgeP 1 0 2 0 17Nort hri d ge P 1 005 18 Northridge P1023 19ChiChiP1532 20Northridge P093 5 Fi gure E .6 Maximum Wall Crest Settlement Profiles (H = 6 m co = 10, $' = 320 Sv = 0.4 m Ts % = 36 kN/ m) 317
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o c: 4> 20 E 4> E 4> '" U; 40 60 E :; 80 o 2 3 4 D i stance from facing block ( m ) 0 E .s c: 40 4> E 4> E 4> '" U; 80 iii E 120 ::l E x "' :; 16 0 0 2 3 4 D i stance from facing bloc k (m) .01Duzce P1557 020rovill e P0115 03 Kobe P1054 04CoyoteLake P0149 05Kobe P1054 006Coa linga P0346 07 Northr i dgeP0887 08LomaP riet aP0736 09LomaPrieta P0745 10Northr i dgeP0833 11CapeMendocino P0810 12Northridge P0883 13LomaPr iet a P0745 14Kobe P 1 056 15Ch i Ch i P 1 46 1 16Northridge P1020 17Northridge P1005 18No rthridgeP1023 19Ch i Ch iP1532 2 0 Nort hridgeP0935 Figure E.? Maximum Wall Crest Settlement Profil es (H = 6 m co = 100 $' = 400 Sv = 0.4 m Ts% = 36 kN/m) 318
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0 E .s c 10 Q) E Q) E 2 0 Q) In in () 30 (ij E ::> 40 E x .. 50 0 E .s c 40 Q) E Q) E Q) In in 80 (ij E 120 ::> E 160 0 2 3 4 D i stance from facing block (m) 0 2 3 4 D i stance from facing block (m) 01DuzceP1557 +020rovill e P0115 03 KobeP1054 &04 CoyoteLake P0149 0 5 Kob e P1054 06Coalinga P0346 07North r i dge P0887 +08LomaPrietaP0736 09 LomaPrieta P0745 10Northr i dgeP0833 11CapeMendoc i no P0810 12 North ridge P088 3 13LomaPrieta P0745 14Kob e P1056 15ChiCh i P1461 16Nort hridge P1020 17 Nort hridgeP1 005 +18No rthridge P1023 19Ch iChiP1532 2 0 No rthri dgeP0935 Figure K 8 Maximum Wall Crest Settlement Profil es ( H = 6 m 0) = 10 = 360 Sv = 0.2 m Ts% = 36 kN/m) 319
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0 E .s c Q) 20 E Q) :a Q)
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0 E ====: g 20 C Q) E Q) 40 E Q) II) Cii 60 u (ij 80 E :::> E 100 x '" 120 0 2 Distance from fac i ng block (m) 50 E g C 100 Q) E Q) E 150 Q) II) Cii 200 (ij E :::> 250 E 300 0 2 Dis tanc e from fac i ng block (m) 3 4 3 4 01Duzce P 1557 +020rovill e P0115 +03 KobeP1054 04CoyoteLake P0149 05KobeP1054 06Coalinga P0346 .07N orthridge P0887 08LomaP rietaP0736 09 LomaP rietaP0745 10NorthridgeP0833 11CapeMendoc i no P0810 12 N orthridge P0883 13 LomaPr iet a P07 45 14 Kobe P1056 15 Ch i Ch iP 1461 16 Northridge P1020 17 NorthridgeP1005 18 Northridge P1023 19 ChiChi P 1 532 20NorthridgeP0935 Figure E l 0 Maximum Wall Crest Settlement Profiles (H = 6 m co = 100, = 360 S v = 0.4 m Ts% = 12 kN / m) 321
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0 E .s c 10 Ql E Ql E Ql 20 40 E x '" :::; 50 0 E .s c 40 Ql E Ql E Ql E x '" :::; 160 0 2 3 4 D i stance from facing block (m) 0 2 3 4 D i stance from fac i ng block (m) 01Duzce P 1 557 020rovill e P0115 03KobeP1054 it04CoyoteLakeP0149 05Kobe P1054 <>06Coalinga P0346 07NorthridgeP0887 08LomaPrieta P0736 09LomaPrietaP0745 10Northr i dgeP0833 11CapeMendocinoP0810 12N orthridgeP0883 13L omaPri etaP0745 14Kobe P1056 15ChiCh i P146 1 16 N orthridge P1020 1 7Northridge P1005 18NorthridgeP1023 19ChiCh i P1532 20Northr i dge P0935 Figure E.!! Maximum Wall Crest Settlement Profiles (H = 6 m 0) = 10 = 36 S v = 0.4 m, Ts% = 72 kN/m) 322
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APPENDIX F Maximum Lateral Eart h Stress Distribution s 323
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4 I I :if en iii .c: 2 o 40 SO Max imum latera l eart h s tress (kN/ m 2 ) 6 4 I I .E en iii .c: 2 o 40 SO 120 Max i mum lateral earth stress (kN/m 2 ) Stat i c 01DuzceP1557 02 0rovill e P0115 03KobeP1054 04CoyoteLake P0149 05 Kobe P1054 <>06 Coa li ngaP0346 07 Northridge POSS7 OSLomaPr i etaP0736 09 LomaPrietaP0745 10 North r i dge POS33 120 Stat i c 11CapeMe n docinoPOS10 12NorthridgePOSS3 13 LomaPrietaP0745 14KobeP1056 15Ch iC h i P 1461 16NorthridgeP1020 17 Northr i dge P1 005 1 S Northridge P1 023 19ChiCh i P1532 20 Northridge P0935 160 Figure FJ Maximum Lateral Earth Stress Distributions (H = 6 m 0) = 10 = 36 S y = 04 m Ts% = 36 kN/ m) 324
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I I :i Cl "Qi .c I I :i Cl "Qi .c 3 2 o 3 2 10 20 30 40 50 Max imum latera l earth stress (kN / m 2 ) Stat i c 01D u zceP1557 +02 0rovillePO 115 03KobeP1054 04CoyoteLakeP0149 05Kobe P1054 06 Coa li nga P0346 +0 7NorthridgeP0887 08 LomaPrietaP07 3 6 09 LomaPr i etaP0745 10 No rt hr i dgeP0833 60 Stat i c 11CapeMendodnoP0810 12NorthridgeP0883 13 LomaPr i eta P0745 14 Kobe P1056 15 ChiChiP1461 16Northridge P1020 17 Northridge P1 005 18 NorthridgeP1023 19ChiChiP 1 532 20Northridge P0935 o o 20 40 60 80 100 Max i mum latera l earth stress (kN / m 2 ) Figure F.2 Maximum Lateral Earth Stress Distributions (H = 3 m ()) = 100 = 36 S y = 0.4 m Ts% = 36 leN/m) 325
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10 8 I 6 I :i Cl "iii .. 4 2 o 10 8 I 6 I ;: Cl "iii 4 2 o 40 80 120 Max i mum lateral earth stress (kN/m 2 ) Static 01DuzceP1557 020rovilleP0115 03Kobe P1054 04CoyoteLakeP0149 05KobeP1054 <>06Coa li nga P0346 07NorthridgeP0887 08LomaPrietaP0736 09 LomaPrietaP0745 10 NorthridgeP0833 160 Static 11CapeMendoc i noP0810 12NorthridgeP0883 I 13 LomaPrieta P0745 14KobeP1056 15 ChiChiP1461 16 Northridge P1020 17Northridge P1005 18NorthridgeP1023 19 ChiChiP1532 20NorthridgeP0935 40 80 120 160 200 Maximum lateral earth st ress (kN/m 2 ) Figure F.3 Maximum Lateral Earth Stress Distributions (H = 9 m co = 10, = 36 Sy = 0.4 m Ts% = 36 kN/ m) 326
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I I E '" "iii .s::: I I E '" "iii .s::: 6 Stat i c 01DuzceP1557 4 020rovilieP0115 03 KobeP1054 04CoyoteLakeP0149 05Kobe P1054 <>06Coa li nga P0346 .07NorthridgeP0887 08 LomaPrietaP0736 2 09 LomaPrieta P0745 10NorthridgeP0833 o o 40 80 Max i mum latera l ea rth s t ress (kN / m ) 6 4 2 120 Static 11CapeMendocinoP0810 12 Northridge P0883 13 LomaPrieta P0745 14 KobeP1056 15 Ch i Ch iP 1461 016 NorthridgeP1020 17NorthridgeP1005 18 NorthridgeP1023 19Ch iChiP 1 532 20NorthridgeP0935 o o 40 80 120 160 Maximum lateral earth stress (kN / m ) Figure F.4 Maximum Lateral Earth Stress Distributions (H = 6 m (i) = 5 $' = 36 S v = 0.4 m Ts% = 36 kN /m) 327
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6 Static 01DuzceP1557 4 02 0rovill eP0115 I 03KobeP1054 I e04CoyoteLakeP0149 E 05KobeP1054 '" 06Coa li ngaP0346 "0; .c 07NorthridgeP0887 08LomaPrieta P0736 2 09LomaPrietaP0745 10 NorthridgeP0833 o o 40 80 120 Max i mum lateral earth stress (kN/m 2 ) Static 11CapeMendodnoP0810 12NorthridgeP0883 13LomaPrietaP0745 4 14KobeP1056 15Chi Ch i P1461 I 16NorthridgeP1020 I 17 NorthridgeP1 005 E 18 Northridge P1023 '" "0; .c 19Ch iChi P 1532 20NorthridgeP0935 2 o 40 80 120 160 Maximum lateral earth stress (kN/m 2 ) Figure F.5 Maximum Lateral Earth Stress Distributions (H = 6 m, 0) = 15, = 36 Sy = 0.4 m Ts% = 36 kN/ m) 328
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4 I I :E 0> iii .s::: (ij 2 o 20 40 60 SO Max i mum lateral earth stress (kN/m ) 6 4 I I :E 0> iii .s::: 2 o 40 SO Max i mum lateral earth stress (kN/m ) Static 01DuzceP1557 020rovill eP0115 03KobeP1054 04CoyoteLakeP0149 05 KobeP1054 006 Coalinga P0346 07NorthridgePOSS7 OSLomaPrietaP0736 09 LomaPrietaP0745 10NorthridgePOS33 100 11CapeMendocinoPOS10 +12 Northridge POSS3 13LomaPrietaP0745 e14 KobeP 1 056 15ChiCh iP1461 16NorthridgeP1020 +17NorthridgeP1005 1SNorthridgeP1023 19Ch i Ch iP1532 20NorthridgeP0935 120 Figure F 6 Maximum Lateral Earth Stress Distributions (H = 6 m 0) = 10, = 32 Sv = 0.4 m Ts% = 36 kN/ m) 329
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I I .E OJ '0; .<:: I I .E OJ '0; .<:: Iii !i: 6 Stat i c 01Duzce P1557 4 020rovilleP0115 03 KobeP1054 04 CoyoteLakeP0149 05 KobeP1054 06 Coalinga P0346 07 Northridge P0887 08LomaPrieta P0736 2 09 LomaPr i eta P0745 10Northr i dge P0833 o +,.,.,, o 40 80 Maximum lateral eart h stress {kN/ m 2 } 6 4 2 120 Stat ic 11CapeMendocinoP0810 +12Northri dge P0883 13LomaPrietaP0745 14KobeP1 056 15ChiChiP1461 16NorthridgeP1020 17 NorthridgeP 1 005 18NorthridgeP1023 +19ChiChiP1532 20 NorthridgeP0935 o +.,,,,r,, o 40 80 120 160 Maxi mum latera l earth stress {k N/m 2 } Figure F 7 Maximum Lateral Earth Stress Distributions (H = 6 m 0) = 10, = 40, S y = 0.4 m Ts% = 36 kN/ m) 330
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4 I I E 0> '0; 2 o 40 80 Max i mum lateral earth st r ess (kN / m 2 ) 6 4 I I E 0> '0; 2 o 40 80 120 Maximum lateral earth stress (kN / m 2 ) 01D u zce P1557 02 0roviil eP0115 03 KobeP1054 04CoyoteLake P0149 05KobeP1054 06Coa lin ga P0346 07 NorthridgeP0887 08LomaPrietaP0736 09 LomaPrieta P0745 10Northr i dgeP0833 120 Stat i c 11CapeMendocinoP0810 12NorthridgeP0883 13 LomaPrietaP0745 14 KobeP1056 15Ch iChiP 1461 16 Northridge P 1 020 17 NorthridgeP1 005 18Northridge P1023 19 ChiChiP 1 5 3 2 20 NorthridgeP0935 160 Figure F.8 Maximum Lateral Earth Stress Distributions (H = 6 m ill = 10, $' = 36 S y = 0.2 m T 5% = 36 kN / m) 331
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I I :E C> iii .t:: I I :E C> iii .t:: 6 Static 01Duzce P1557 4 02 0roville P0115 03KobeP1054 004 CoyoteL akeP0149 05KobeP1054 +06 CoalingaP0346 07 NorthridgeP08 87 08Lom aP rietaP0736 2 09LomaPrietaP0745 10NorthridgeP0833 o ;,,,.,,,.,,,. o 6 4 2 20 40 60 80 100 Maximum latera l earth stress (kN/m 2 ) 120 Stat ic 11CapeMendocinoP0810 12NorthridgeP0 883 13LomaP rietaP 0745 14KobeP1056 15Ch iChiP 1461 16NorthridgeP1020 17NorthridgeP10 05 18Northr i dgeP1023 19Ch iChiP1532 20NorthridgeP0935 o o 40 80 120 160 Max i mum lateral earth stress (kN/m 2 ) Figure F.9 Maximum Lateral Earth Stress Distributions (H = 6 m co = 10, = 36, Sv = 0.6 m, Ts% = 36 kN/ m) 332
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4 I I Cl iii .r::. 2 o 6 4 I I Cl iii .r::. 2 o Static 01DuzceP1557 020roviliePOl15 03 KobePl054 04CoyoteLake P0149 005 KobePl054 006Coal i ngaP0346 07 NorthridgePOSS7 OSLomaPrietaP0736 09LomaPrietaP0745 10Northr i dgePOS33 20 40 60 SO 100 120 Max i mum late ral earth stress (kN/m 2 ) 40 SO 120 Maximum lateral earth stress (kN/m 2 ) Static llCapeMendodnoPOS10 12NorthridgePOSS3 13LomaPrieta P0745 14KobePl056 15ChiChi P1461 16NorthridgePl020 17 NorthridgePl 005 lSNorthridgePl023 19Ch i Ch iP1532 20NorthridgeP0935 160 Figure F.10 Maximum Lateral Earth Stress Distributions (H = 6 m, ill = = 36 Sv = 0.4 m Ts% = 12 kN/m) 333
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g I :E 0> "iii .<::: g I :E 0> "iii .<::: 6 Stat i c 01DuzceP1557 4 02 0rovilleP0115 03KobeP1054 e04CoyoteLakeP0149 05KobeP1054 e06Coa li ngaP0346 07 NorthridgeP0887 08 LomaPrietaP0736 2 09 L omaPr i etaP0745 10Northr i dgeP0833 o o 40 80 Maximum lateral earth stress (kN/m2) 6 4 2 120 Static 11CapeMendocinoP0810 12NorthridgeP0883 13 LomaPrietaP 0745 14KobeP1056 15ChiCh iP1461 e16 NorthridgeP1 020 17NorthridgeP1005 18NorthridgeP1023 19Ch iChi P1532 20NorthridgeP0935 o o 40 80 120 160 Maximum l ateral earth stress (kN/m 2 ) Figure F .11 Maximum Latera l Earth Stress Distributions (H = 6 m, co = 10 = 36 Sy = 0.4 m Ts% = 72 kN/m) 334
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100 z 200 VI VI iil C) 300 c Q) .c E :::I 400 E ::; 500 0 0 f z 200 VI VI VI C) 400 c c ro Q) .c E :::I 600 E x ro ::; 800 0 APPENDIXG Maximum Bearing Stress Distributions 2 3 4 5 D i stance from facing block (m) 2 3 4 5 D istance from facing block (m) Static 01DuzceP155 7 02 0 rovilleP0115 03 Kob e P1054 &04 CoyoteLa keP0149 05 Kob eP1054 06 Coalinga P0346 07 N orthridgeP0887 08LomaPrieta P0 73 6 09LomaP rieta P0745 10 N orthr i dgeP0833 Stati c 11CapeMendocinoP0810 +12 Nort hridgeP0883 13LomaPr i etaP0745 &14Kobe P1056 15ChiCh i P1461 16No rthridgeP1020 17 N orthridgeP1 005 18 N orthridgeP1023 19ChiCh i P1532 20 N orthridgeP0935 Figure G.l Maximum B earing Stress Distribution s (H = 6 m co = 10, = 36, Sy = 0.4 m Ts % = 36 kN/m) 335
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40 f z 80 (J) (J) e! 1ii Cl 120 c C III Q) .0 E ::l 160 E x III 200 0 40 80 z (J) (J) 120 (J) Cl C 160 Q) .0 E ::l E 200 x '" 240 0 0 5 1 1 5 2 D ist ance from facing block ( m) 0 5 1 1 5 2 D i stance from fac i ng block ( m) 2 5 2 5 Static 01Duzce P155 7 020rovilleP011 5 03 KobeP1054 04CoyoteLakeP0149 05Kobe P1054 06Coalinga P0346 07 N orthr i dge P0887 08LomaPrieta P0736 09LomaPrietaP0745 10Northr i dgeP0833 Static 11CapeMendocino P0810 12N orthridge P0883 13LomaPrietaP0745 14Kobe P1056 15ChiCh iP1461 16Northr i dge P 102 0 17 Northr i dge P1 005 18N orthr i dge P1023 19ChiChiP1532 20N orthr i dge P0935 Figure G.2 Maximum Bearing Stress Distributions (H = 3 m, (0 = 10, = 36, Sv = 0.4 m Ts% = 36 kN/m) 336
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0 f 200 z rn rn l!? 400 u; 0> c 600 E :::> E x 800 '" :::;; 1000 0 0 t::' 200 z rn rn l!? 400 u; 0> C 600 Q) .0 E :::> E 800 :::;; 1000 0 2 4 6 8 Distance from facing block (m) 2 4 6 8 Distance from fac ing b lock (m) Static 01Duzce P1557 +020rovill ePOl15 03 KobePl054 <>04CoyoteLakeP0149 05KobePl054 06CoalingaP0346 07Northr i dge P0887 08LomaPrietaP0736 09LomaPr ietaP 0745 10NorthridgeP0833 Static llCapeMendoc i noP0810 12Northr i dgeP0883 13 LomaPr i etaP0745 14Kobe Pl056 15ChiChiP1461 16Northr i dgeP1020 17Northr idgePl005 18Northr i dgeP1023 19 ChiChiP1532 20Northr i dgeP0935 Figure G.3 Maximum Bearing Stress Distributions (H = 9 m co = 10, = 36 Sv = 0.4 m Ts% = 36 kN/m) 337
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100 f z 200 UJ UJ i!! U; CJ) 300 C C '" Q) .0 E :::l 400 E x '" 500 0 ;;E Z 200 UJ UJ UJ CJ) 400 c C '" Q) .0 E :::l 600 E x '" 800 0 2 3 4 Distance from facing block (m) 0 2 3 4 D istance from facing block ( m) 5 5 Static +01DuzceP1557 020rovi lleP0115 03KobeP1054 e04CoyotelakeP0149 05KobeP1054 <>06CoalingaP0346 07 No rthr i dgeP0887 08lomaPrietaP0736 09 lomaPrietaP0745 10Northr idge P0833 Static 11CapeMendoc i noP0810 12Northr idgeP0883 13lomaPr i etaP0745 14Kobe P1056 15ChiCh iP1461 16Nort hr i dge P1020 17 Nort hr i dgeP1 005 18Northr idgeP1023 19Ch iCh iP1532 20 North ridge P0935 Figure G.4 Maximum Bearing Stress Distributions (H = 6 m co = 5, = 36, Sv = 0.4 m, Ts% = 36 kN/m) 338
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0 z 100 III III ii) Ol 200 c c .0 E :::l 300 E x .0 E 400 :::l E 500 :!E 600 0 2 3 4 Distance from fa cing block (m) 0 2 3 4 D istance from fac ing block (m) 5 5 Static 01Duzce P1557 02 0rovill e P0115 03 Kobe P105 4 <>04CoyoteLake P0149 05Kobe P 1 054 <>06Coalinga P0346 07 NorthridgePOBB7 OBLomaPrietaP0736 09LomaPrietaP0745 10 Nort hridgePOB33 Static 11CapeMe ndocinoPOB10 12Northr i dgePOBB3 13 LomaP rietaP0745 14KobeP 1056 15 ChiCh iP1461 16North ridgeP1020 17 Northr i dgeP1 005 1BNorthridgeP1023 19ChiCh iP1532 20Northr i dgeP0935 Figure G.5 Maximum Bearing Stress Distributions (H = 6 m ill = = 36, Sv = 0.4 m Ts% = 36 kN/m) 339
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0 z c100 VI VI !!:' 0 0> 200 c: Q) .Q E :::l 300 E x '" 400 0 0 100 z cVI VI !!:' 200 0 0> c: 'C: '" 300 .2l E :::l E 400 500 0 2 3 4 Distance from fac ing block (m) 2 3 4 D i stance from facing b l ock ( m) 5 5 Static +01DuzceP 1557 02 0rovi lleP0115 0 3 Kob eP1054 004CoyoteLa keP0149 05K obe P1054 06Coa lingaP0346 07 N orthr i dgeP0887 08 LomaP rieta P0736 09L omaPrieta P0745 10Northr i dgeP0833 Static +11CapeMendoc i no P0810 1 2 N orthr i dge P0883 13LomaPr i eta P0745 14 Kobe P1056 15 ChiChiP 1461 <>16N orthr i dge P1020 17 N orthr i dge P1 005 18 Nort hridge P1023 19ChiCh iP15 32 20 No rthridge P0935 Figure 0 6 Maximum B earing Stress Distributions (H = 6 m CD = 100 = 32 Sv = 0.4 m Ts% = 36 kN/ m) 34 0
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100 Z 200 '" '" 300 iii Cl c: 400 Q) D E ::> E 500 x '" 600 0 0 N' E Z 200 '" '" iii Cl 400 c: Q) D E ::> 600 E x '" 800 0 2 3 4 5 D istance from fac ing block (m) 2 3 4 5 D istance from fac ing block ( m ) Stat ic 01DuzceP 1557 02 0rovillePOl15 03Kobe Pl054 <>04CoyoteLa keP0149 05Kobe Pl054 <>06Coa lingaP0346 0 7 N orthridge P088 7 08LomaP rietaP073 6 09LomaP riet a P07 45 10Northr i dge P0833 Static llCapeMendocino P0810 12North ridgeP0883 13LomaPr i eta P0745 14KobePl056 15ChiChiP1461 16No rthr idgePl020 17 No rthridgePl 005 18North ridgePl023 19ChiCh iP 1532 20Nort hridge P0935 Figure 0.7 Maximum Bearing Stress Distributions (H = 6 m (j) = 100 = 400 S y = 0.4 m Ts% = 36 kN /m) 341
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0 100 z D E ::> E x 400 D E ::> 600 E ::i! 800 0 2 3 4 5 D is tance from facing block (m) 2 3 4 5 Distance from fac ing b lock (m) Static 01Du zceP1557 +020rovilleP0115 03 Kobe P1054 &04 CoyoteLake P0149 05Kobe P1054 006 Coa linga.P0346 07 No rthridgeP0887 08 LomaPrietaP0736 +09LomaPrieta P0745 10Northr idge P0833 11CapeMendocino P0810 1 2N orthr i dge P0883 13 LomaPr i eta P0745 14 Kobe P 1056 15 ChiChi P1461 16 North ridgeP1020 17 North ridgeP1 005 18 N orthr i dgeP1023 19ChiCh i P1532 20 Northr idge P0935 Figure G 8 Maximum Bearing Stress Distributions (H = 6 m OJ = 10 = 360 Sv = 0.2 m Ts% = 36 kN/m) 342
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100 'f z C 200 U) U) "' 0> 300 c C '" C1l Il E ::> 400 E x '" 500 0 0 'f z C 200 U) U) U) 0> 400 c C1l Il E ::> 600 E x '" 800 0 2 3 4 5 Distance from facing block (m) 2 3 4 5 D i stance from facing block (m) 01Duzce P1557 020rovilie P0115 03Kobe P1054 e04CoyotelakeP0149 05KobeP1054 <>06Coalinga P0346 07 Northr i dgeP0887 08lomaPrietaP073 6 09loma PrietaP07 45 10Northr i dgeP0833 Static 11CapeMendoc i noP0810 12Northr i dge P0883 13lomaPrie taP0745 14Kobe P1056 15ChiCh i P1461 16Northr i dgeP1020 17 Northr idgeP10 05 18 Nort hridge P1023 19 ChiCh iP1532 20 Northr idgeP0935 Figure G.9 Maximum Bearing Stress Distributions (H = 6 m (0 = 10, = 36 Sv = 0.6 m Ts% = 36 kN/m) 343
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100 '[ z 200 If) If) U; 0) 300 c c '" Ql .0 E ::J 400 E x '" 500 0 0 '[ Z 200 If) If) U; 0) 400 c c '" Ql .0 E ::J 600 E x '" 800 0 2 3 4 5 Distance from facing block (m) 2 3 4 5 D i stance from fac i ng b lock (m) Sta t ic 01Duzce P1557 020rovill eP0115 03K obeP1054 04CoyoteLake P0149 05KobeP1054 <>06Coalinga P0346 07 N orthr i dgeP0887 0 8 LomaPrieta P0736 09 L omaPrietaP 0 745 10N orthr i dge P0833 Static 11CapeMendocinoP0810 12N orthridge P0883 13LomaPrietaP0745 14K obeP1056 15ChiChiP1461 16N orthr i dge P1020 17 N orthr i dgeP1 005 18N orthr i dgeP1023 19ChiChiP1532 20N orthr i dge P0935 Figure G 10 Maximum Bearing Stress Distr ibu tions (H = 6 m co = 10, = 36 Sv = 0.4 m Ts% = 12 kN/m) 344
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100 z ::s 200 (/) (/) U; '" 300 c c co Q) .0 E ::J 400 E x co 500 0 100 200 z ::s (/) (/) 300 (/) '" c c III 400 Q) .0 E ::J E 500 600 0 2 3 4 5 D istance from fac ing block (m) 2 3 4 5 D i stance from fac i ng bloc k (m) Static 01Duzce P1557 +020rovilleP0115 +03Kobe P1054 04CoyotelakeP0149 05 KobeP1054 06Coalinga P0346 07 Northr i dgeP0887 ...08lomaPrietaP0736 09lomaPrietaP0745 10N orthr idgeP0833 Static 11CapeMendoc i no P0810 12Northridge P0883 13lomaPrietaP0745 14KobeP1056 15ChiChi P1461 16Northr i dge P1020 17 Northr i dgeP1 005 18 Northr i dge P1023 19ChiCh iP1532 20Northr i dge P0935 Figure GJ 1 Maximum Bearing Stress Distributions (H = 6 m 0) = 10 = 36 Sv = 0.4 m, Ts% = 72 kN/m) 345
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APPENDIXH Maximum Reinforcement Tensile Load Profiles 346
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I I E Cl iii .r. I I E Cl iii .r. 6 4 2 o 6 4 2 Stat i c 01DuzceP1557 +02 0roville P0115 +03 KobeP1054 G04CoyoteLake P0149 05KobeP1054 06Coa li ngaP0346 0 7Northridge P088 7 08 LomaPrietaP0736 09 LomaPrietaP0745 10 Northr i dge P0833 4 8 12 16 20 24 Max imum re i nforcement tens ile load (kN / m ) Static 11CapeMendocino P0810 12NorthridgeP0883 13 LomaPrietaP0745 14KobeP1056 15Ch i Ch iP1461 16 NorthridgeP1020 17 NorthridgeP1 005 18Northridge P1023 19Ch i Ch i P1532 20NorthridgeP0935 o +,,,,,,,,,,, o 10 20 30 40 50 Maximum reinforcement tens ile load (kN/m ) Figure H.i Maximum Reinforcement Tensile Load Profiles (H = 6 m (0 = 10 = 36 Sv = 0.4 m Ts% = 36 kN/m) 347
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I I :E C) '0; ..c I I :E C) '0; ..c 3 2 5 Static 01DuzceP1557 2 020rovilleP0115 03 Kobe P1054 04CoyoteLakeP0149 05 Kobe P1054 1 5 06 Coa li ngaP0346 0 7NorthridgeP0887 08 LomaPrietaP0736 09 LomaPrieta P0745 10Northridge P0833 0 5 o 2 3 2 5 2 1 5 0 5 o 468 Max imum reinforcement tens ile load (kN / m ) 4 8 1 2 16 20 Max i mum rei nforcement tens il e load (kN / m ) 10 Static 11CapeMendocinoP0810 12 Northridge P0883 13LomaPrietaP0745 014KobeP1056 15ChiCh i P1461 16 Northridge P1020 17 Northridge P1005 18Northridge P1023 19Ch i Chi P1532 20NorthridgeP0935 24 Figure H 2 Maximum Reinforcement Tensile Load Profiles (H = 3 m, co = 10 = 36 Sy = 0.4 m Ts% = 36 kN/m) 348
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I I 1: '" iii .r:; I I Ii '" iii .r:; 10 8 01Duz c e P1557 +02 0rovillePO 115 6 03 K obeP1054 &04CoyoteLakePO 149 05 K obe P1054 06 C oa li ngaP0346 07 NorthridgeP0887 4 08LomaPrietaP0736 +09 LomaPrietaP0745 10 N orthridge P0833 2 o +,,,,,,,, o 10 20 30 Maximum reinforcemen t tensile load (kN/m) 10 8 6 4 2 40 Stat ic 11Ca peMendocino P0810 12North rid geP0883 13LomaPrietaP0745 14 Kobe P1056 15ChiChiP1461 16NorthridgeP1020 17NorthridgeP1005 18NorthridgeP1023 19ChiChiP1 532 20NorthridgeP0935 o +,.,,,,,,,, o 10 20 30 40 50 Ma xim um reinforcement tens il e load (kN/m ) Figure H.3 Maximum Reinforcement Tensile Load Profi l es (H = 9 m co = 10 = 36, Sv = 0.4 m Ts% = 36 kN/m) 349
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I I ...c 0> iii .c I I :i 0> iii .c n; 6 Sta tic 01DuzceP1557 +020rovill eP0115 4 03 K obe P1054 04CoyoteLake P0149 05 K obe P1054 006CoalingaP0346 07 N orthridge P0887 +08 LomaPri eta P0736 09 L omaPrieta P0745 10Northridge P0833 2 o +..,,,,,.., o 4 8 12 16 Max imu m rei nforcement tens il e load (kN/m) 6 4 2 2 0 Stat ic 11CapeMendocino P0810 12NorthridgeP0883 13LomaPr i etaP0745 14KobeP1056 15Ch i ChiP1461 16NorthridgeP1020 17NorthridgeP1005 18NorthridgeP1023 19ChiChiP1532 +20NorthridgeP0935 o +,,,,,,,,,, o 10 20 30 40 50 Max imum reinforcement tensile l oad (kN/m) Figure H.4 Maximum Reinforcement Tensile Load Profiles (H = 6 m 0) = 5 = 36, S v = 0.4 m Ts% = 36 kN/ m) 350
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I I '" iii .r: I I :c '" iii .r: iii 6 Static 01DuzceP1557 4 020rovilleP0115 03KobeP1054 e04CoyoteLakeP0149 05 KobeP 1 054 06CoalingaP0346 +07NorthridgeP0887 08LomaPrietaP0736 09LomaPrietaP0745 2 10NorthridgeP0833 o 4.,,,,,.... o 6 4 2 4 8 12 16 Max i mum reinforcement tensile loa d (kN/m) 20 11CapeMendocinoP0810 12NorthridgeP0883 13LomaPrietaP0745 14Kobe P1056 15ChiChiP1461 16Northridge P1020 17NorthridgeP1005 18Northridge P1023 19ChiChiP 1532 20NorthridgeP0935 o 4,,,,,.,..,, o 10 20 30 40 50 Maximum reinforcement tensile load (kN/m) Figure H.5 Maximum Reinforcement Tensile Load Profi l es (H = 6 m co = 150, $' = 36, Sv = 0.4 m, T5% = 36 kN/m) 351
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I I C> "., .r:. I I C> "., .r:. n; 6 01Duzce PI557 4 020rovillePOI15 03KobePl054 04CoyoteLakePOI49 05KobePl054 +06CoalingaP0346 07NorthridgeP0887 08 LomaPrietaP0736 2 09LomaPrietaP0745 10NorthridgeP0833 o t,,,,.,,,,,, o 5 10 15 20 Max imum reinforcement tens il e loa d (kN/m) 6 4 2 25 Static llCapeMendocinoP0810 12NorthridgeP0883 13LomaPrietaP0745 14KobePl056 15Ch i ChiPI461 e16NorthridgePl020 e17NorthridgePl005 18NorthridgePl023 19 Chi Ch iPI532 20NorthridgeP0935 o t,,,,,,,,,, o 10 20 30 40 50 Max i mum reinforcement tensile load (kN/m) Figure H.6 Maximum Reinforcement Tensile Load Profiles (H = 6 m, co = 10 = 32 Sv = 0.4 m Ts% = 36 kN/m) 352
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I I ... .c '" iii .c 6 4 2 01DuzcePI557 020roville POI15 03Kobe Pl054 <>04 CoyotelakePO 149 05KobePl054 &06Coa l ingaP0346 07 NorthridgeP0887 08lomaPrietaP0736 09 lomaPrieta P0745 1 0NorthridgeP0833 o 5 10 15 2 0 25 Maximum rei nforcement tens ile load ( kNlm ) 6 4 2 o 4.,,.,.,,.. o 10 20 30 40 50 Maximum reinforcement tensile load (kN/ m ) Stat i c llCapeMendoc i noP08 1 0 12 Northridge P0883 13 lomaPri etaP0745 14 Kobe Pl056 15 Chi Ch iPI461 16Northridge Pl020 17NorthridgePl005 18 Northridge Pl023 19Ch iChiPI532 20Northridge P0935 Figure H.7 Maximum Reinforcement Tensi l e Load Profiles (H = 6 m, co = 10 0 = 400 Sv = 0.4 m Ts% = 36 kN/m) 353
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I I 1f Cl "iii .c I I 1f Cl "iii .c iii :!: 6 0 1 Duzce P1557 4 +0 2roville P0115 03Kobe P1054 e04.CoyoteLake .P0149 05 Kobe P1054 +06CoalingaP0346 07Northridge P0887 +08LomaPrietaP0736 09 Loma Prieta P0745 2 10Northr i dge P0833 o +.,...,,.,,,. o 5 10 15 20 25 Maximum rei nforcemen t tensile loa d (kN / m ) 6 4 2 30 Stat ic 11 CapeMendocinoP0810 +12 Northr i dgeP0883 +13 LomaPri eta P0745 e14.Kobe.P1 056 15ChiChiP1461 16Nort hridg eP1020 17 NorthridgeP1005 18NorthridgeP1023 19Ch iChi P1532 20NorthridgeP0935 o 4,,,,...... o 10 20 30 40 50 Maximum reinfor cement tensile load (kN/m) Figure H.8 Maximum Reinforcement Tensi le Load Profiles (H = 6 m (J) = 10, = 36, Sy = 0.2 m Ts% = 36 kN / m) 354
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:[ I C> iii .c :[ I 1:: C> iii .c 6 4 2 o 5 1 0 1 5 20 2 5 Maxim u m reinforcement t ens il e load (kN/ m ) 6 4 2 01DuzceP1557 +020r o vill ePOl15 03 Kobe P1054 04 CoyoteLake P0149 05 Kobe P1054 +06Coa li ngaP0346 +07 Northridge P0887 +08L omaPrieta P0736 09 LomaPrieta P0745 10 Northridge P0833 30 Stat i c 11CapeMendoc inoP0810 12 Northr i dge P0883 13 LomaPrieta P0745 14Kobe P1056 15Ch i Chi P1461 16 NorthridgeP1020 17NorthridgeP1005 18 NorthridgeP1023 19ChiC hiP1532 +20Nort hri dge P0935 o o 10 20 3 0 40 50 Maxi mum reinfor cement tensile load (kN/ m ) Figure H.9 Maximum Reinforcement Tensi l e Load Profiles (H = 6 m 0) = 10, = 36 Sy = 0.6 m Ts% = 36 kN /m) 355
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I I :E Ol 0., .s::; I I :E Ol 0., .s::; (ij :J: 6 Stat ic 01Duzce P1557 4 02 0rovilleP0115 03KobeP1054 &04 CoyoteLakeP0149 06 C oa li nga P0346 07NorthndgeP0887 08LomaPnetaP0736 09 LomaPnetaP0745 2 10NorthridgeP0833 o t,,,,,,,,.. 2 4 6 8 10 Maximum r einforcement tens il e load (kN/ m ) 6 4 2 12 11C apeMendocinoP0810 12 N orth n dgeP0883 13LomaPneta P0745 14KobeP1056 15Ch iChiP1461 16 North n dge P1020 17 Northridge P1 005 18 North n dge P1023 19 ChiChiP15 32 20 North n dge P0935 o t,rr,rr,rr,,. o 4 8 12 16 20 24 Maximum reinforcement tens ile l oad (kN / m ) Figure RIO Maximwn Reinforcement Tensile Load Profiles (H = 6 m co = 10 = 36 Sv = 0.4 m Ts% = 12 kN/m) 356
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:r: '" 'Qi :r: '" 'Qi 6 Stat ic 01DuzceP1557 020rovilleP0115 4 03 KobeP1054 <>04CoyoteLake P0149 05KobeP1054 06 Coal ingaP0346 07 Northri dge P0887 08 L omaPrieta P0736 09LomaPrieta P0745 10NorthridgeP0833 2 o o 10 2 0 Max imum reinforcement tens il e load (kN/ m ) 6 4 2 30 Stat i c 11CapeMendoc inoP0810 12 North ridgeP0883 13LomaP riet a P0745 14Kobe P1056 1 5ChiChiP1 46 1 16Northri dge P 1 0 2 0 17 Northri dgeP 1 005 18Northri dge P 1 023 19Ch iChiP1532 20NorthridgeP0935 o +...,.,,, o 2 0 40 60 80 Maximum reinforcement ten sile load (k N / m ) Figure H.11 Maximum Reinforcement Tensile Load Profiles (H = 6 m ()) = 10, = 36, Sy = 0.4 m Ts% = 72 kN /m) 357
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APPENDIX I Correlations with Peak Vertical Acceleration In addition to correlations with the peak horizontal acceleration (PHA) provided in Chapter 5 seismic performances (e .g., maximum facing horizontal displacement maximum wall crest settlement total driving resultant total overturning moment arm oftotal driving resultant maximum bearing stress maximum reinforcement tensile load and maximum horizontal acceleration at centroid of the reinforced soil mass) were correlated with the peak vertical acceleration (PVA) Similar to the arrangement in Chapter 5 results are organized according to the design parameters (e.g. wall height wall batter angle soil friction angle, reinforcement spacing and reinforcement stiffness) A total of 35 charts are included in this appendix. The same single predictor variable regression models (or equations) from Chapter 5 were adopted in this appendix. Discussion is provided below based on each design parameter. Effects of Wall Height Effect of wall height on the seismic performances is illustrated in Figures I.l to I.7 The results were compared to Figures 5 7 to 5 .13 that were correlated with PHA. Due to the lower R 2 values correlations with PVA are considered weaker than those with PHA when the same regression model was used. In general the R 2 value indicates the goodness of fit and a higher value indicates that the regression model fits better with the data In some cases negative R 2 values were calculated from the nonlinear regression analyses (e.g., see Figures 1.4 and I.5) Note that a negative R 2 value is legitimate since the second power is not involved in the computation ofR2 value ; rather the sum of square of the vertical distances of the points from the curve is used. In this study negative R 2 value occurred when the nonlinear regression curve was anchored with the static value (i.e., at zero acceleration). The negative R 2 value suggests that the imposed regression model may not be appropriate 358
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y,620nell1 R' 0 .4416 I / I j I I I E .......... y 18. 9620'0 2 0 0 6 R 0 5633 .... .......... : .. .. .. 0 8 P eak vertica l acce leration PYA (g) 1.2 H =9m H =6m .&. H =3m Expon ( H = 9m) Expon (H=6m) Expon ( H = 3 m) Figure 1.1 Effe ct of Wall Height on Maximum Wall Facing Horizontal Displacement (C orrelated with PV A) 350 300 t200 j 150 E E 100 50 Y "24el2Oeb R' 0 4337 / / H = 9m / / / / / / H =6m Expon ( H = 9 m) Expon ( H = 6 m) ;/ / y" 19 08Sel"''". R' 0517 / .,Expon ( H = 3 m) // . ,/ .... y S .695e2 cs.. . ..,................ R" 0 .<4638 :.. ...... .. ... .. . .' . ... .. .. . . 0 2 0 0 6 0 8 1 2 P eak vertical acceleration PYA (g) Figure 1.2 Effe ct of Wall Height on Maximum Wall Crest Settlement (Correlated with PYA) 359
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'f .310 58e11e01i11o R' 0.071 .. .... y" 137 3Se'''''' R' 0 288 / e / .. / / / // .... / .... : ..... .J ..... ..... .... ... Jo. .....  . 0 2 0 0 6 0 8 1 2 Peak vertical acceleration, PVA ( g) H =9m H=6m ... H=3m E x p o n ( H = 9 m) Expon ( H=8m ) Expo n ( H =. 3 m) Figure 1.3 Effect of Wall Height on Total Driving Resultant (Correlated with PYA) __ / j R'O. m E :. . . i _E '. E 2+ ________________________________________ f ,...., .. 1 5 .  .. _... . .. .. _... _... .. y '" SS4Sx2 + 1 7932x ... 1 0 7 05 R'0313 0 5 ti 0 2 0 0 6 0 8 1 2 Peak vfitical PVA ( g) H= 9 m H =6m ... H= 3 m P o ly ( H = 9 m ) P oly ( H = 8m) Poly ( H = 3 m ) Figure 1A Effect of Wall Height on Total Overturning Moment Arm of Total Driving Resultant (Correlated with PYA) 360
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1.00 1200 1000 ... 0f 800 co < .: j 600 E E = ,. .00 200 y 318 068 133b R'O. 1.21 / / : / ,fa 200 24e '.tM51o R '00903 .. .... 0 ... ,/' ,/' I .... .... / ,/ / ......... ;. .2038 .. ,e,. R' 00283 ..... . .. ... : .. 0 2 0 0 6 0 8 1 2 Peak .... rtical acceleration, PYA (g) H=9m H=6m .. H=3m Expon (H=9m) Expon (H=6m) Expon (H=3m) Figure 1.5 Effe ct of Wall Height on Maximum Bearing Stress (Correlated with PV A ) 90 y .11. 504e'''''' R' O .len / 80 / / y671238' .... R 2 0 4 892 / / / / " . / ,,/ / / /. ,/ ./ Y 3 30408e' 1)79. ... .. ",,R' 0 2526 . :,., .. .. .. ......... _ .. . .. ... .. L:1 _... ... __ .... a10 0 2 0 0 6 0 8 P eak vertical accel era ti on PYA (g) 1 2 H=9m H=6m .. H= 3m Expon (H = 9 m) Expon (H=6m) Expon (H = 3 m) Figure 1.6 Effec t of Wall Height on Maximum Reinforcement Tensile Load (Correlated with PV A) 361
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'. 0 2 0 4 0 6 0 8 Peak vertic al acc e leration PYA ( g) y 0 4262e2'oq" R' 0 5988 0 4 27 8 '1lII:0I ,/ 0 6343 / H= gm H=6m ... H=3m Expon (H=gm ) Expon. ( H=6m) ( H = 3 m) 1 2 Figure 1.7 E ffect of Wall Height on Maximum Horizontal Acceleration at C e ntroid o f the Reinforced Soil Mas s (Correlated with PYA ) Effects of Wall Batter Angle The effect of wall batter angle on the seismic performances is depicted in Figures 1.8 to 1.14. The results were compared to Figures 5.17 to 5.23 that were correlated with PHA. The R 2 values of correlations with PV A are lower than those with PHA which suggests a poor fit of data with PV A. Negative R 2 values were calculated for correlations of total overturning moment arm and maximum bearin g stress (see Figures 1.11 and 1.12). The same arguments discussed above (i e effect o f wall height) also apply here Note that the total overturning moment arm from base of wall tends to decrease with increasing PYA (see Figure 1.11) 362
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y ,. 69. 738ellO&h Rl0 4 38 , ""'/ I 0 2 0 4 0 6 0 8 Peak vertical ac c eleration PVA (g) 1 2 Batter = 15 I Batter = 10 Batter = 5 Expon (Batter = 1 5 ) Expon (Batter = 10 ) exPO" ( Batter = S) Figure 1.8 E ffect of Wall Batter Angle on Maximum Wall Facing Hori z ontal Displacement (Correlated with PYA) 350 300 250 E .s .; t200 I 150 E E = :IE 1 00 50 y 18 98Je2 R' = 0 5397 .. . : . . . . . . "T". 0 2 0.4 0 6 0 8 P eak verti c a l accel eratio n PVA (g ) /y:: 19 08Se12 "1Ix / Rl",0517 Batter = 15 I Baller = 10 .. Balter = 5 Expon ( Batter = 15" ) Expon (Bat1er = 10) . Expo" ( Batter = 5") 1 2 Figure 1.9 E ffect of Wall Batter Angle on Max imum Wall Crest Settlement ( Correlated with PYA ) 363
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1 600 1400 1200 E ;j! 1000 Z 0W or 800 .. 5 600 I400 200 y.'1'. 37e241101b R" .0. 365 I t / , . I 'I. .... ..... 0 2 0 4 0 6 0 8 Peak vertic.l acceleration PYA (g) 1 .,37 J5e1fR11h / W="()268 :' I /tj ,: I i / 1 2 Batter = 15 Satter = 10 ... Batter = 5 Expon (Batter = 15) Expon (Batter = 10) . Expo" (Bauer = 5 ) Figure 1.10 Effect of Wall Batter Angle on Total Driving Resultant (Correlated with PYA) 3 5 I 2 5 j 1 E .. = 1 5 E i o 1 I0 5 y 3 .3214.2 + 374211t + 2 145 ;, .. .. R" .0.71 'rt . .......... ... / : ,,. ..... ... 2 _'+ W y26634x2 + 28213x + 2346 R".o 455 0 2 0 4 0 6 0 8 Peak vemall acceleration PYA ( g) 7911lx + 2.261 8 559 1 2 Satter = 1S" Satter = 10 ... aattlr = So Poly ( Battel = 15) POly (Batter = 10) . Poly ( Bauer = 5) Figure I.11 Effect of Wall Batter Angle on Total Overturning Moment Arm of Total Drive Resultant (Correlated with PYA) 364
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1 200 1 000 .. 800 ..
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3 5 E 3 <: :: E 2.5 ] f 2 .. ... g 8 1 5 .. < .! 1 jj < 0 5 0 2 0 4 0 6 0 8 Peak vertical acceleration, PVA (g) y 0 .4852.' 722eIc R" 0 5343 Batt.r = 15 Batter = 10' .. Batter = S o upon (Batter = 15) Expon ( Batter = 10') . Expon (Batter = 5) 1 2 Figure 1.14 Effect of Wall Batter Angle on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass (Correlated with PYA) Effects of Soil Friction Angle The effect of soil friction angle on the seismic performances is illustrated in Figures 1.15 to 1.21. The results were compared to Figures 5.24 to 5.30 that were correlated with PHA The R 2 values of correlations with PV A are lower than those with PHA which suggests a poor fit of data with PV A. Negative R 2 values were calculated for correlations of total overturning moment arm and maximum bearing stress (see Figures 1.18 and 1.19). The same arguments discussed above (i.e., effect of wall height) also apply here 366
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y 78.7&&tJ "'2'" R' 0 4355 'It' 62 one" // / 0 4416 I / / I 53. 58gel 112l1o R' 0 4433 // / e a o 0 2 0.4 0 6 0 6 1 2 Peak vertical ac ce leration PYA (g) Fnc angle = 40" Fric angle = 36" .. F ric ilngle = 32" Expon (Fric. angle = 4 0) Expon (Frtc angle = 36") Expon (Flic angle= 32") Figure 1.15 Effect of Soil Friction Angle on Maximum Wall Facing Horizontal Displacement (Correlated with PYA ) 300 250 I 200 .; oJ Ii i 150 e e 100 a 50 yo 21.1)950'''''''' R' 04796 'I It: 19 08581 2nQk ,/ R'0517 < A ,/ / :' I R'0 507 :/ .. / / / .. / "./ & '/ /  t ... 1 I _"Y./"" .. ...... .::::.; . 0 2 0.4 0 6 0 8 1 2 Peak vertical acceleration PYA (g) FtM:. angle = 40" Fric angle = 36" .. Fric angle = 32" Expon (Fnc. angle = 40 ) Expon (Fric. angle = 36) Expon (Fric. angle = 32") Figure 1.16 Effect of Soil Friction Angle on Maximum Wall Crest Settlement (Correlated with PV A) 367
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1600 1400 1200 e ;j! 1000 ... .. 1 800 .. .E 600 >400 I I 200 0 2 I 0 4 0 6 0 8 Peak wrneal acceleration PYA (g) y 156 .5Se1011 h / ,,'y. 126 8M7. .' 7 1 2 Fric angle = 40 Fric angle = 36" F ric angle = 32" Expon ( Fric. angle = 40") Expon (Fnc. .ngle = 36 ) Expo" ( Foc. .ngl. = 32 ) Figure 1.17 Effect of Soil Friction Angle on Total Driving Resultant (Correlated with PYA) 3 5 .. ,: V I 2 5 j i e 0 e .. < 1 5 e i! >0 5 0 2 .. .. ... 0 4 0 6 y. 2 .4524.' ... 2 .7164)( + 2 3532 R'. ...... ........::: y 2 .3756)(2... ',. 6498. + 2 3358 432 "y. 2 .66304)(2 ... 28213. + 2346 455 0 8 1.2 Ffle ang le = 4 0 Fric. angl e = 36" A Fr;c angl e = 3r Poly (Frie angle = 40') Poly ( Frie. angle= 36) Poly (Fnc angle = 32 ) Peak vertical acceleration PYA (g) Figure 1.18 Effect of Soil Friction Angle on Total Overturning Moment Arm of Total Drive Resultant (Correlated with PYA) 368
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1200 1000 .. 800 .. m& ... I R' 0,1 I .' 1 ,/ / .' / Fr\c angle = 40 Fric angle = 36" Frjc angle = 32" ;f/ .. ",I'," // ... / Expon (Fric. angle =40" ) Expon (Foc. angle = 36" ) Expon (Fric. angle= 32 ) .. / y .. t., / .. .. ... .. .. ... ... : .. .. .... : 10 0 2 0 4 0 6 0 8 1 2 Peak vertica l accelerati on PYA (g) Figure 1.20 Effect of Soil Friction Angle on Maximum Reinforcement T ensile Load (C orrelated with PYA ) 369
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0 y O 4107e2lGb / R" 0 .6n8 Fric. angle:; 40 / '0_''R" 0 5&42 / // 0 ,/ 0./' Frlc angle = 36.. Fric. angle = 3r Expon (Fric. angle:; 40 ) Expon (Fric. angle:; 36 ) . Expo" (Fric angle:; 32) ...... / ..... : 0 .437.'m". _____ ........ .,....., R20.5411 ...  0 I 0 2 0 4 0 6 0 8 1 2 Peak vertical acceleration, PVA ( g) Figure I.21 E ffect of Soil Friction Angle on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass (Correlated with PYA) Effects of Reinforcement Spacing The effect of reinforcement spacing on the seismic performances is illustrated in Figures I.22 to I.28. The results were compared to Figures 5.31 to 5.37 that were correlated with PHA. The R 2 values of correlations with PV A are lower than those with PHA which suggests a poor fit of data with PV A. Negative R 2 values were calculated for correlations of total overturning moment arm (see Figure I.25). The same arguments discussed above ( i .e effect of wall height ) also apply here 370
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2000 1800 1600 / / . t / .1 .9' 200 0 2 0 0 6 0 6 Peak vertical acceleration PVA (g) y % 62 Onell1ll!l1 R'04416 I / I I 1 2 SV=02m sv=o .. m .. Sv=06 m Expon (Sv=02m) Expon (Sv=04 m) Expon (Sv=O 6 m ) F igur e I.2 2 Effe ct of Reinforcement Spacing on Maximum Wall Facing Horizontal Displacement (Correlated with PYA) y 20 652&2 : neJa R'0 4945 , : / / E /'f. 19 085e H"g, 5. I R" 0 517 f y. 17.588e2 Olth R',.O.531 5 .. / / E .. /' a ... ,.,r/ ;'/o o 0 2 0 0 6 0 8 1 2 Pe ak vertical acceleration. PVA (g) $\'=02 m SV = 0 m .. SV=08m Expon (Sv=02m) Expon (Sv=04 m) Expon (Sv=06m ) F igure I.23 Effect of Reinforcement Spacing on Maximum Wall Crest Settlement (C orrelat e d with PYA ) 371
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Y' 137 3Se' ..... R'. 0 288 y 151.73e' 1m7,. R" 0 203 .. 0 2 0 4 0 6 0 8 1 2 P u k vertica l ac celeration PYA ( g) SV=02 m 5 ... =04 m .. 5\1=06 m Expon (S v=02m) Expon (Sv= 0 4 m) Expon (SV=06m) Figure I.24 Effect of Reinforcement Spacing on Total Driving Resultant (Correlated with PYA) 3 5 ,., y 2.883x:l + 3 .2(93)1 .. 2 .2616 R' 0 468 . . ::::._... ,'"1 _:::" 1 i,;;i _ ...... ... .6 .. .. ....... :::::.. .. ",1/' .. "+" 282 3 ... 2346 I 2 5 J i E "' V y 2 3435x' + 2 .S459x + 2 28047 R" 0 693 1 5 +j E 1+1 0 5 tj 0 2 0 4 0 6 0 8 1 2 Peak vertical acceleration, PYA (g) 5\1=02 m SV=04 m .. $..,=06 m Poty{SV=02m) Poty (5'1=04 m) Pot)' (Sv = 0 6 m) Figure 1.25 Effect of Reinforcement Spacing on Total Overturning Moment Arm of Total Drive Resultant (Correlated with PYA) 372
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1000 900 800 .. 7 00 .. /!. :t 600 l co 500 0 0 .i 400 E E = 300 :E 200 1 00 1 y=2002 ...... / RJ00903 / y 210 42e .11. R" 0 2303 / / I . / .. # ,' I : :::i;:" .... ... .::::. ... 0 2 0 4 0 6 0 8 1 2 Peak vertical ac celeration, PYA (g) SV=02m SV=04m & Sv=06m Expon (Sv=02m) Expon (SV=04 m ) Expon (8v= 06 m) Fig ure 1.26 Effec t of R e inforcement Spacing on Maximum Bearing Stress (Correlated with PYA ) 90 y 8 88J5e2 R'.0104 .' .' / ,/ y S .561Se2l8h 0 6895 / / / I / RJ = a 4892 / / A A / / / .. A t A A / / A 6 ........ .... ' .. A A A . .... .... .. ., .. !: . 10 0 2 0 0 6 0 8 1 2 P eak vertica l PYA (g) Sv=02 m 5v=0. 4 m .. Sv=06m upon (Sv=02m) Expon (SV=04 m) Expon (Sv=OBm) Figure 1.27 Effe ct of Reinforcement Spacing on Maximum Reinforcement Tensile Load (Correlated with PV A) 373
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yo 0 422""' ..... W 0 .6122 3 5 +,.___j J 5 y:: 0 458 ,1!Jc W OS 1 'E 2 5 +r+.,..,..___j f .. ... 0 2 b 1i 5 1 5 ., = I .c 0 5 +""""=..___j :E 0 2 0 4 0 6 0 8 1 2 Peak vertical acceleration PYA (g) SV=02m Sv=04m ... SV=06m Expon (Sv=02m) Expon (SV=04 m ) Expon (Sv: 06 m) Figure 1.28 Effect of Reinforcement Spacing on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass (Correlated with PV A) Effects of Reinforcement Stiffness The effect of reinforcement stiffness on the seismic performances is illustrated in Figures 1.29 to 1.35. The results were compared to Figures 5 38 to 5.44 that were correlated with PHA. The R 2 values of correlations with PV A are lower than those with PHA, which suggests a poor fit of data with PV A. Negative R 2 values were calculated for correlations of total overturning moment arm and maximum bearing stress (see Figures 1.32 and 1.33). The same arguments discussed above (i.e., effect o f wall height) also apply here. Note that the total overturning moment arm from base of wall tends to decrease with increasing PYA (see Figure 1.32) 374
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2000 1 800 1 600 E !. 1 400 400 200 rr y R" 0 33S6 , yf : / 1 ./ / ,/ /. / y66. 28e'''''' .. ,' /.. R''' 0 332 4 .' / j.? .' ..' .. ............ _ ...... 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 Peak vertical a cceleration PYA (g) T5%=72kN1m .. TS%= 12 kNlm Expon (T5% = 72 kNlm) Expon (T5% = 36 kNlm) Expon. (15% = 12 kNIm) Figure 1.29 Effect of Reinforcement Stiffness on Maximum Wall Facin g Hori z ontal Displacement (Correlated with PYA) 400., y 2 787.., R' 0 .4732 , / y 19 Q85e1111h R' 0 517 Qj ,t' 2u / ./ // E 1 ........ / / L a .1S4e'''''' ,. ..,;" / .... ... !t ......... .. ;" . . ................ ... " . ''':'' . _... ;::::::...;. ... 0 2 0 4 0 6 0 8 1 2 Peak .... rtical a c c elerat ion, PYA (g ) T5% = 72 kNlm T5%=38kNhn ... T5%= 12 kNlm Expon (T5% = 72 kNlm) Expon (T5" = 36 kNlm) Expon (T5% = 12 kNlm) Figure 1.30 Effect of Reinforcement Stiffness on Maximum Wall Crest Settlement (Correlated with PYA ) 375
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y 137.35.""'" y. 133.98e2 Rl :II: CI 288 R' ..(1. 295 / / r : yo 140.7." "'''' W 0281 1" t TS" = 72 kNIm TS" = 36 kNIm TS" = 12 kNlm Expon = 72 kNlm) Expon (T5% = 36 kNlm ) . Expon (T5" = 12 kNIm) o 0 2 0 4 0 6 0 8 1 2 Peak vertiCilI acceleratio n PYA ( g) Figure 1.31 E ffect of Reinforcement Stiffness on Total Driving Resultant (Correlated with PYA ) 3 5 12 5 j E .. .5 1.5 E i 1 0 5 I : . ... ... f 'P " .,. . ... / ... v.; .....0 2 : I R' 0 583 AA .... 2354 2 8213x + 2 346 0455 R y 2 .8OO9x' + 2.9205x + 2 3573 R' O.4n T5% = 72 kNhn TS% = 36 kNAn T5% = 12 kNJm Poly (T5" = 72 kNhn) P oly (T5'" = 36 kNhn) Pol y (T5"= 12kNhn) 0 4 0 6 0 8 1 2 P ea k vertical accelerati on PYA (g) Figure 1.32 Effect of Reinforcement Stiffness on Total Overturning Moment Arm o f Total Drive Resultant (C orrelated with PYA) 376
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1200 y 2002. 8 R' 0 0903 1000 .. 800 :B. Ii 600 g0 j E E 400 ,. y 199.8e'!'JOIiIa ,', R' 0 .,9, 6.;/ /i' P y 193 548 "11 y V R' ., 239 .;:f r ,./; ,., I ..;::: a I . "" T5" = 72 kNIm T5% = 36 kN/m T5" = 12 kNlm Expon ( T5" = 72 k N Im) Expon (T5% = 36 kNlm) exPO" (T5% = 1 2 kN/m) 200 0 2 0 4 0 6 0 8 1 2 Peak vertical acceleration PYA (g) Figure 1.33 Effect of Reinforcement Stiffness on Maximum Bearing Stres s (Correlated with PYA) 100 y 10 4838'0921. 90 80 E 70 1 60 j 50 40 .I! E 30 E ij ,. 20 10 R' 0 1499 I / / / yo 6 7123e'''''' R' = 0 4692 / / / / / / / // .// / L ,1''' y 3 .987482 08121 "" '''/' R' 0 1&44 k/ ""'/ ... ..  &. .. i.. .. ........... .. ' .. ...... _._ ..  T5%=72kNJm TS ... =38kN1m .. T5%= 12 kNlm Expon (T5% = 72 kN/m) Expon. (T5" = 36 kNlm) Expon (T5% = 12 kNhn) 0 2 0 4 0 6 0 8 1 2 Peak wrtical accel.,..tio n PYA (a) Figure 1.34 Effect of Reinforcement Stiffness on Maximum Reinforcement Tensile Load (Correlated with PYA) 377
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'f. 0 4 258' 0675J R' 0 6296 3 5 ++_____1 I / // /1 ../ 'f. 0448ge11J18x I R' 0 5542 1 5 +"'7..,..;."....:_____1 0 5 +=+''_____1 0 2 0.4 0 6 0 8 1 2 Peak vertica l acceleratio n PYA (g) T5'" = 72 kNhn T5'" = 36 kNIm 4 T5% = 12 kNJm Expon (T5,," = 72 kNIm) Expon. (T5% = 36 kNlm) . Expon (T5% = 12 kNIm) Figure I.35 Effect of Reinforcement Stiffness on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass (Correlated with PV A) 378
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APPENDIXJ Regression Analysis for Low Peak Horizontal Acceleration Range The FHW A design methodology is applicable for peak horizontal acceleration (PHA) up to 0.29 g Therefore a separate regression analysis was performed using FEM results with PHA less than 0.29 g in order to attain a direct comparison with the FHW A analysis values The results of multivariate regression analysis are summarized in Table J l where the prediction equations have the same form as the exponential functions presented in Chapter 6. It should be noted that the linear multivariate regression equations were also evaluated but they typically result in negative performance values when both the design parameter and PHA have low values. Since negative performance values are not realistic linear multivariate regression equations are not presented to avoid erroneous interpretation. Comparisons of seismic performances from the FEM results the predictions and the FHW A analysis values with PHA less than 0.29 g are presented in this appendix and the comparisons are grouped based on design parameters. The seismic performances evaluated include maximum horizontal displacement maximum crest settlement total driving resultant LP D E maximum bearing stress qyE, maximum reinforcement tensile load Tto t al, and maximum horizontal acceleration at centroid of reinforced soil mass Am. The effects of wall height H wall batter angle 0), soil friction angle reinforcement spacing Sy, and reinforcement stiffness T s % on the wall performances are shown in Figures 1.1 to J.6 Figures J.7 to 1.12, Figures 1.13 to J .18, Figures J.19 to 1.24 and Figures J 25 to 1.30 respectively The seismic performances are consistent with those presented in Chapter 6 where F E M results show higher total driving resultant maximum higher bearing stress and lower maximum reinforcement tensile load than the FHW A analysis values. The regression analysis indicated that the R 2 values are generally higher than those presented in Chapter 6 (see Table 6.3), which suggests that the prediction equations have a better fit with the FEM results. However the higher R 2 values could also be due to smaller sample size (i .e., six earthquake records). With the limited sample size the prediction equations presented in Table J.l should be used with caution from a practical standpoint. 379
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w 00 o Table 1.1 Summary of constant and regression coefficients from the multiple regression analysis using earthquake records with PHA less than 0.29 g [Y = exp (b o + bl PHA + b 2 PYA + b3 H + b4 co + b s + b 6 S v + b7 T s%)] Response Constant Regression coefficients R 2 Y b o bl b2 b3 b4 b s b 6 b7 flh (mm) 0 523 17.526 1.938 0.2496 0.0067 0 0475 0.0110 0 00158 0 866 fl v (mm) 0.182 10. 702 0.895 0.2626 0.0135 0.0321 0.0965 0.00531 0.766 LP D E (kN/m) 1.662 3.063 0.414 0.3162 0.0183 0.0260 0 0117 0.00015 0.909 qvE (kN/ mL ) 2.801 2.070 0 690 0.2373 0.0336 0 0346 0.1642 0 00080 0 870 Ttotal (kN/m) 1.220 2 307 1.031 0.2143 0.0122 0.0429 1.4681 0.01339 0.803 Am (g) 4.683 13.123 4.497 0.0051 0 0018 0.0109 0 2694 0 00011 0.904
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E .s 400 300 c Q) E '" a. '" 15 200 c o N c o .s:: E E 100 o 0 1 Q2 Peak horizontal acceleration PHA (g) 0 3 H=9m H=6m ... H=3m H = 9 m (regression) eH = 6 m (regression) .!rH = 3 m (regression) Figure 1.1 Effect of Wall Height on Maximum Wall Facing Horizontal Displacement (for PHA s 0 .29 g) 100 E .s E x '" o 0 1 0 2 0 3 Peak horizontal acce leration PHA (g) Figure 1.2 Effect of Wall Height on Maximum Wall Crest Settlement (for PHA s 0.29 g) 381
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E Z "' o 01000 800 W 600 200 o Q1 0 2 Peak horizontal acce l eration PHA (g) 0 3 H=9m H =6 m .. H=3m +H = 9 m (regression) H = 6 m (regression) ...H = 3 m (regression) H=9m(FHWA ) H = 6 m ( FHWA) H=3m(FHWA) Figure 1.3 Effect of Wall Height on Total Driving Resultant (for PHA 0 29 g) 600 J 400 iii '" '" 0> C c '" Q) Il E E 200 x '" :2 .. .. .. .. .. o 1,,,,,, o Q1 Q2 0 3 Peak horizontal acceleration PHA (g) H=9m H = 6 m .. H=3m __ H = 9 m (regression) H = 6 m (reg ress ion) ...H = 3 m (regression) H = 9 m (FHWA) H = 6 m (FHWA) H = 3 m (FHWA ) Figure 1.4 Effect of Wall Height on Maximum Bearing Stress (for PHA 0.29 g) 382
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40 Z jj 30 H = 9 m ,,; H = 6 m '" .Q .. H=3m H = 9 m (regression) iii c 20 eH = 6 m (re gression ) C bH = 3 m (regr ess ion) Q) E H=9m(FHWA) Q) e H = 6 m (FH WA ) .E c H=3m(FHWA) E 10 ::l E x .. '" .. .. .. .. .. o o 0 1 0 2 0 3 Peak horizontal acceleration PHA (g) Figure 1.5 Effect of Wall Height on Maximum Reinforcement Tensile Load (for PHA 0.29 g) .9 1 2 < o 0 1 0 2 Pea k horizontal acce leration PHA (g) 0 3 H=9m H = 6 m .. H=3m H = 9 m (reg ression ) eH = 6 m ( regression ) bH = 3 m (reg ress ion) H=9m(FHWA) H=6m(FHWA) H=3m(FHWA) Figure 1.6 Effect of Wall Height on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass (for PHA 0.29 g) 383
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E ,S E 2l '" Ci '" '6 m C 0 N .c: 0 E :J E x '" 200 160 120 80 40 o 0 1 0 3 Peak horizontal accelerat i on PHA (g) 00 = 15 00=10 00 = 15 (regression) e00 = 10 (regression ) b00 = 5 ( regress i on ) Figure 1,7 Effect of Wall Batter Angle on Maximum Wall Facing Horizontal Displacement (for PHA 0.29 g) 40 E 30 ,S E .. 00 = 5 a> E 20 00 = 15 (regression ) a> '" e00 = 10 ( regress i on ) U; b00 = 5 (regress i on ) u E :J E x 10 '" o o 0 1 0 2 0 3 Peak horizontal acce leration PHA (g) Figure 1.8 Effect of Wall Batter Angle on Maximum Wall Crest Settlement (for PHA 0.29 g) 384
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400 E 300 Z OJ = 15 OJ = 10 Cl. w .. OJ = 5 c' OJ = 15 (regression) 200 eOJ = 10 (regression) "5 '" .!rOJ = 5 (regression) Cl OJ = 15 (FHWA) c OJ = 10 (FHWA) 0 f100 OJ = 5 (FHW A ) o 4,,,,,, o 0 1 0.2 0 3 Peak horizontal acceleration PHA (g) Figure 1.9 Effect of Wall Batter Angle on Total Driving Resultant (for PHA 0.29 g) 400 .. Ii Cl. OJ = 15 300 OJ = 10 CT
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30 I z li C1l = 15 f0 20 C1l = 10 til .Q .. C1l = 5 .91 iii C1l = 15 (re gression ) c $ eC1l = 10 (regression) c Q) bC1l = 5 (regression) E Q) OfC1l = 15 (F HWA) S 10 C1l = 10 (F HWA ) c .. +C1l = 5 (F HWA) E E x til o o Ql 0 2 0 3 Peak horizontal acceleration PHA (g) Figure J.11 Effect of Wall Batter Angle on Maximum Reinforcement Tensile Load (for PHA 0.29 g) 1 2 .f
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200 E S 160 '" 0. VJ '6 IV C 0 80 N c 0 .r: E E x 40 '" o 0 2 0 3 Pea k horizontal acceleration PHA ( g) ,'= 40 ,'= 36 ... = 32 ,'= 40 (regress ion) e,'= 36 (regression) ...,'= 32 (regression) Figure J.13 Effect of Soil Friction Angle on Maximum Wall Facing Horizontal Displacement (for PHA 0 29 g) 40 ... E 30 S E E x '" 10 o o 0 1 0 3 Pea k horizontal accelerat i on PHA (g) Figure J .14 Effect of Soil Friction Angle on Maximum Wall Crest Settlement (for PHA 0 29 g) 387
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400 E Z 300 cjl' = 40 cjl' = 36 c.. w '"' cjl' = 32 C cjl' = 40 (regression) .l!l 200 ecjl' = 36 (r egress io n ) :; If) 6cjl' = 32 (regression) 0> cjl' = 40 (FHWA) c .;; : cjl' = 36 (FHWA) 100 cjl' = 32 ( FHWA) 0 o o 0 1 0 2 0 3 Peak horizontal acceleration PHA ( g ) Figure 1.15 Effect of Soil Friction Angle on Total Driving Resultant (for PHA 0 .29 g) Ol c.. YJ 0u; If) u; 0> C Q) Il E E x '" ::2! 400 300 200 100 o 0 1 0 2 0 3 Pea k horizon tal acce l erat i on PHA ( g ) 9' = 40 cjl' = 36 '"' cjl' = 32 cjl' = 40 (regression) ecjl' = 36 ( r e gress io n ) 6cjl' = 32 ( regress i on ) cjl' = 40 (FHWA) cjl' = 36 (FHWA) cjl' = 32 (FHWA) Figure 1.16 Effect of Soil Friction Angle on Maximum Bearing Stress (for PHA 0.29 g) 388
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30 Z li g I 40 .,j 20 '" .Q 3 6 .. 32 .u; c 40 (r egress ion) .J!! C I36 ( regression ) Ql E 632 (regression ) Ql e 40 ( FHWA) S 10 c +36 (FHWA) E 3 2 ( FHWA ) :::> E x '" ::E o o Q1 Q2 0 3 Pea k hor izonta l acceleration PHA (g) Figure 1.17 Effect of Soil Friction Angle on Maximum Reinforcement Tensile Load (for PHA 0.29 g) u '" OJ C o N .s::: 0 o 0 1 0 2 0 3 Pea k h orizontal acce l erat i on PHA (g) 40 = 36 .. 3 2 41' = 40 ( regress i on ) I41' = 3 6 (regression ) 63 2 ( regress i on ) 41' = 40 ( FHWA ) += 36 ( FHWA ) 41' = 3 2 ( FHWA ) Figure 1.18 Effect of Soil Friction Angle on Maximum Horizontal Acceleration at Centro id of the Reinforced Soil Mass (for PHA 0 29 g) 389
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200 E .s E x IV ::!E 0 0 t 0 1 0 2 0 3 Peak horizontal acceleration PHA ( g ) S = 0 2 m S = 0.4 m .. S = 0 6 m __ S = 0 2 m (regres s ion) eS = 0.4 m (regression) ___ S = 0 6 m (re gression ) Figure 1 .19 Effect of Reinforcement Spacing on Maximum Wall Facing Horizontal Displacement (for PHA ::; 0.29 g) 40 E 30 .s .{ S = 0 2 m c' S = 0.4 m Q) S = 0 6 m E .. Q) E 20 __ S = 0 2 m (re gress ion) Q) E x 10 IV ::!E o o 0 1 0 2 0 3 Peak horizontal acceleration PHA (g) Figure 1.20 Effect of Reinforcement Spacing on Maximum Wall Crest Settlement (for PHA ::; 0.29 g) 390
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400 I 300 S v = 0 2 m z w S v = 0.4 m 0 a. & S v = 0 .6 m w ... S v = 0 2 m (regression) c 200 BS v = 0.4 m (regression) ::l '" ...S v = 0 .6 m (regression) '" S v = 0 2 m (FHWA) c ;; c S v = 0.4 m (FHWA) 0 100 S v = 0 .6 m (F HWA) 0 Io 4...... o Q1 Q2 0 3 Peak horizontal accelerat i on PHA (g) F igure 1.21 Effect of Reinforcement Spacin g on Total Drivin g Resultant (for PHA 0.29 g) 400 .. a. S v = 0 2 m 30 0 S v = 0.4 m C"
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30 E z c J S = 0 2 m 0 20 '" .Q S = 0.4 m .. S = 0 6 m in c C +S = 0 2 m (regression) eS = 0.4 m (regression) Q) E cS = 0 6 m (regression) Q) .E 10 c E S = 0 2 m (FHWA ) iItS = 0 4 m (FHWA ) +S = 0 6 m (FHW A ) ::J E x '" o 1,,,,,. o Q1 0 2 0 3 Peak horizontal accelerat ion PHA (g) Figure J .23 Effect of Reinforcement Spacing on Maximum Reinforcement Tensile Load (for PHA 0.29 g) 1 2 '" c 2 c o .r:: 0 .8 0.4 0 o 0 1 0 2 0 3 Peak horizontal acceleration PHA (9) S = 0 2 m S = 0.4 m .. S = 0 6 m +S = 0 2 m (regression) eS = 0.4 m (regression) cS = 0 6 m (regression) S = 0 2 m (FHWA) S = 0.4 m (FHWA) +S = 0 6 m (FHWA) Figure J .24 Effect of Reinforcement Spacing on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass (for PHA 0 .2 9 g) 392
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200 E .. 160
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400 : 300 T ,,, = 72 kN/m z T ,,, = 36 kN/ m n. .. T ,,, = 12 kN/m w .; T ,,, = 72 kN/m (regression) c S 200 eT ,,, = 36 kN/m (regression) "5 '" II! bT ,,, = 12 kN/m (regression) Cl H=6m(FHWA) c "0 19 100 0 fo o Q1 Q2 0 3 Peak horizontal accelerat i on PHA (g) Figure 1.27 Effect of Reinforcement Stiffness on Total Driving Resultant (for PHA 0.29 g) 400 cu 300 n. Ts = 72 kN/m w Ts = 36 kN/m cJ
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30 E E .. .. .. ::;; o o 0 1 0 2 0 3 Peak horizontal acceleration PHA (g) T ,, = 72 kN/m T,,, = 36 kN/m .. T ,,, = 12 kN/m T,,, = 72 kN/m (regression) eT '" = 36 kN/m (r egression ) &T,,, = 12 kN/m (re gress ion) H = 6 m ( FHWA ) Figure 1.29 Effect of Reinforcement Stiffness on Maximum Reinforcement Tensile Load (for PHA 0.29 g) 2 o co lii c: o N o .t:: ... 0 o 0 1 0 3 Pea k horizontal accelera tion PHA ( g ) T,,, = 72 kN/m T,,, = 36 kN/m .. T ... =12kN/m T,,, = 72 kN/m ( regress i on ) eT,,, = 36 kN/m (r egress i on ) &T ... = 12 kN/m ( regress i on ) H =6m(FHWA) Figure 1 30 Effect of Reinforcement Stiffness on Maximum Horizontal Acceleration at Centroid of the Reinforced Soil Mass (for PHA 0.29 g) 395
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APPENDIXK Data for Statistical Modeling 396
PAGE 440
Table K.l Data for statistical modeling PHA PYA H ro $' S" Tl% 8" Zmax q"E T,olal Am (g) (g) ( m ) ( d eg r ee) (degree) (m) (kN/m) ( mm ) (mm) (kN/m) (m) (kN/m2) (kN / m) (g) 0 0 9 1 0 36 0.4 36 0 0 0 0 3 1 0 6 3.429 318. 1 11.5 0 000 0 114 0 093 9 10 36 0.4 36 28.5 18. 6 416 9 3 836 358 5 1 2 0 0.224 0 1 68 0 073 9 10 36 0.4 36 10. 1 4.4 37 1 6 3.756 340.7 1 2 1 0 .271 0 212 0 059 9 10 36 0.4 36 1 35 9 49. 8 522.4 4.351 5 7 6 7 18. 1 0.450 0 229 0 1 6 9 1 0 36 0.4 36 90 2 22 2 55 7.5 4.485 463 7 14. 0 0.412 0.243 0 059 9 10 36 0.4 36 202 9 46. 3 554.1 4 337 544 8 21.5 0 622 0 282 0 .097 9 1 0 36 0.4 36 297. 0 61.9 70 6 1 4 503 565 3 31.3 0.694 0.344 0 552 9 10 36 0.4 36 24 1 1 60.4 701.1 4 .116 583 9 24 2 0 615 0.367 0 338 9 1 0 36 0.4 36 1 63.4 34.9 626 9 4 .141 451.0 21.9 0 768 0.479 0.455 9 1 0 36 0.4 36 677.9 106.2 829.4 4 334 819 1 31.6 0 894 0 5 1 4 0.2 1 7 9 1 0 36 0.4 36 718 7 103.4 74 5 0 4.443 660. 6 33 8 0 900 0 549 0 1 95 9 10 36 0.4 36 449. 6 83.3 653 9 4.454 532 2 22 6 0 765 0.568 0.217 9 1 0 36 0.4 36 465.3 85 3 743 7 4 5 4 5 543.0 30 1 0 894 0 644 0.455 9 1 0 36 0.4 36 659. 8 103. 2 878 1 4.464 5 7 4 6 36 1 0 986 0 694 0.433 9 1 0 36 0.4 36 530 7 89.5 893 1 4 365 722 9 25.2 1.7 64 0 7 1 2 0.255 9 1 0 36 0.4 36 354.3 68 3 628 2 4 328 495.4 20 8 0 964 0 753 0.467 9 1 0 36 0.4 36 7 6 1 .4 156 6 96 7 0 4 393 917, 6 42 1 1 203 0 838 0 852 9 10 36 0.4 36 431.1 142 5 1 095 7 4 122 7 95.4 42 7 3.425 0 897 0 586 9 1 0 36 0.4 36 932 8 1 44 9 104 2.3 4 300 955 1 35 2 1 .559 0 958 0.311 9 10 36 0.4 36 522 5 97 3 781.7 4 275 497. 1 42 1 1 .066 0 99 1 048 9 1 0 36 0.4 36 2068 2 233 6 1 078.4 4 ,077 647.5 48 1 2 1 92 0 0 6 1 0 36 0.4 36 0 0 0 0 137.4 2.346 200. 2 6 7 0 .000 0 114 0 093 6 1 0 36 0.4 36 1 3.4 1 2 5 1 89 7 2.590 237. 6 7.3 0 186 0 168 0 0 73 6 10 36 0,4 36 7.3 5 0 1 65.3 2.523 222. 3 7 5 0 .216 0 2 1 2 0 .059 6 1 0 36 0,4 36 99 8 28 7 276.4 3 .044 346 6 10.3 0 .601 0 .229 0 .16 6 1 0 36 0.4 36 47. 1 14. 6 270.4 3 .034 292. 8 18. 9 0.493 0.243 0 059 6 10 36 0.4 36 106. 8 21.5 291.9 3 .004 320 2 10. 1 0 640 0.282 0.097 6 10 36 0,4 36 86.4 24,4 289 2 2 994 316,4 10.4 0 564 0.34 4 0.552 6 10 36 0.4 36 1 65 2 32 9 324 7 2 893 380 2 1 3 0 0 .721 0 36 7 0.338 6 10 36 0.4 36 119 2 24 9 301.9 3 .056 305 8 11.5 0 616 0.479 0.455 6 1 0 36 0.4 36 360 6 52 0 394 1 3 059 445. 7 15,4 1 085 0.514 0 217 6 1 0 36 0.4 36 351. 0 59 1 391.3 3 022 417. 7 21.0 1.015 0.549 0 .195 6 1 0 36 0,4 36 322 0 62.7 405 8 3 052 464,4 1 5 6 0 809 0.568 0 217 6 1 0 36 0,4 36 3 1 8.3 58 8 381. 1 2 985 327, 9 15. 8 0 978 0 .644 0.455 6 10 36 0,4 36 451.8 62 5 435 7 3 025 43 3.4 1 8 0 0 955 0 .694 0,433 6 10 36 0.4 36 326 1 58.4 500 2 2 857 450,4 18. 8 1 320 0 712 0 255 6 1 0 36 0,4 36 278 8 54 2 380 3 3 0 1 9 355 5 17. 1 1 .257 0 753 0,467 6 1 0 36 0,4 36 49 1 0 81.3 5 1 5 2 2 895 567 9 21.3 1.146 0 838 0 852 6 1 0 36 0,4 36 299. 3 84 0 576 7 2 66 7 490 1 29 7 1 868 0 89 7 0 586 6 1 0 36 0,4 36 544 8 82.4 477.8 2 833 526.3 2 1.7 1 .333 0 958 0.311 6 10 36 0.4 36 467. 9 69 5 381. 0 2 9 1 9 444. 1 17. 8 1 065 0 99 1 048 6 1 0 36 0,4 36 1 52 7 9 184 1 554.2 2 .650 619.5 47, 2 2.338 397
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Table K.l (Cont.) PHA PYA H OJ S,. TS% 6.h 6..,. L POE Zma..1( q vE T, otal A m (g) (g) ( m ) (degree) (degree) (m) (kN/m) (mm) (mm) (kN /m) (m) (kN / m 2 ) (kN /m) (g) 0 0 3 1 0 36 0.4 36 0 0 0 0 36.4 1.071 76.2 3 3 0 000 0 114 0 093 3 1 0 36 0.4 36 7 7 4 8 50.4 1.236 86.4 6 1 0 214 0 1 68 0 073 3 to 36 0.4 36 4 8 1.6 48 0 1.199 83 5 4.3 0 232 0 212 0 059 3 1 0 36 0.4 36 1 9 0 4 6 68 9 1.407 1 04.7 4 5 0 482 0 229 0 1 6 3 10 36 0.4 36 21.4 5 6 72.9 1.443 1 07.3 4 3 0.484 0 243 0 059 3 10 36 0.4 36 1 8 8 3 2 71.1 1 .435 96 5 4 1 0.479 0 282 0 097 3 1 0 36 0.4 36 21.0 6 5 84.3 1 .49 1 1 08 8 5 0 0 7 04 0 344 0 552 3 10 36 0.4 36 47 1 9 7 93.5 1 382 1 38 8 6.3 0 7 0 1 0.367 0.338 3 to 36 0.4 36 48.3 11.0 91.9 1 551 111.1 6 7 0 753 0.479 0.455 3 1 0 36 0.4 36 1 04 1 17. 3 124 2 1.552 14 4 0 9 0 2.397 0 5 1 4 0 217 3 10 36 0.4 36 80 0 16. 9 1 07 7 1.494 163.4 8 5 0 957 0.549 0 195 3 1 0 36 0.4 36 1 3 7 2 17. 8 119 0 1.516 1 89 8 12. 9 1.190 0.568 0 217 3 1 0 36 0.4 36 1 32 0 17.4 107 9 1 526 1 96 7 13. 1 1.215 0 64 4 0.455 3 1 0 36 0.4 36 1 47 9 22.2 134.4 1.470 1 68 3 1 5 7 0 885 0 694 0.433 3 1 0 36 0.4 36 63 1 21.5 117 8 1.496 1 62 6 1 0 2 0 997 0 712 0 255 3 to 36 0.4 36 129. 1 17. 6 125. 5 1 537 202 6 12. 5 1 075 0 753 0.467 3 to 36 0.4 36 90 7 18. 8 119 8 1.461 1 64 5 8 8 1 092 0 838 0 852 3 to 36 0.4 36 2 1 7 8 22 6 175 7 1 392 1 83 2 17.4 1.589 0 897 0 586 3 to 36 0.4 36 1 02 8 24 5 151. 6 1.464 1 55 7 10.4 1 268 0 958 0.31 1 3 10 36 0.4 36 176.5 23 1 117 2 1.472 201.4 15. 7 1 086 0 99 1 048 3 10 36 0.4 36 707 1 31.6 1 77.8 1.407 221.5 20 3 5.321 0 0 6 15 36 0.4 36 0 0 00 1536 2 262 1 45 3 6 9 0000 0 .114 0 093 6 15 36 0.4 36 1 4 0 12.3 206 0 2.49 1 1 85 6 11.4 0 1 93 0 168 0 073 6 15 36 0 4 36 7 6 4 9 181.4 2.433 167. 3 7 6 0.22 6 0 212 0 059 6 15 36 0.4 36 98 8 24 0 286.4 2 95 1 294 5 9 8 0 584 0 229 0 .16 6 15 36 0.4 36 46 6 11.7 283 9 2 953 256 6 8 0 0.487 0 243 0 059 6 15 36 0.4 36 1 04 8 17. 9 30 7 6 2 938 257 5 1 0 1 0 655 0 282 0 097 6 15 36 0.4 36 87 2 19. 9 296 8 2 895 246 0 18. 6 0 604 0 344 0 552 6 15 36 0.4 3 6 1 60 0 36 1 337 0 2 771 325 0 13. 9 0 827 0 36 7 0.338 6 15 36 0.4 36 114.3 19. 1 30 7.3 2 954 249 1 10. 0 0 740 0.479 0.455 6 15 36 0.4 36 335 9 50 3 400 3 2 944 395 9 17. 9 1.124 0 5 1 4 0 217 6 15 36 0.4 36 335 0 48 5 39 7.5 2 934 383 7 19. 7 1.2 1 9 0 549 0 .195 6 15 36 0.4 36 299.4 55 0 407 6 2 98 1 428 7 18. 8 1 0 7 0 0 568 0 217 6 15 36 0.4 36 286 8 49 1 379 2 2 90 7 296 0 18.4 0 .881 0 644 0.455 6 15 36 0.4 36 436 0 50 2 450 5 2 975 370 1 18. 5 1 217 0 694 0.433 6 15 36 0.4 36 299 1 6 4.4 490.3 2 8 1 5 383 7 18. 3 1 .253 0 712 0 255 6 15 36 0.4 36 2 7 6.3 50 5 377.4 2 959 297 7 16. 9 1 277 0 753 0.467 6 1 5 36 0.4 36 471.6 73. 8 5 1 9 8 2 863 508 1 20 6 1.523 0 838 0.852 6 15 36 0.4 36 247 9 88.4 584 9 2.718 436 7 30 1 1.73 9 0 89 7 0 586 6 15 36 0.4 36 505 8 68.4 471.8 2 835 445 1 22 2 1 593 0 958 0.311 6 15 36 0.4 36 483.5 56 7 385 9 2 828 412 5 23.4 1 098 0 99 1 048 6 15 36 0.4 36 1 407.9 132.4 541.0 2 69 1 491.3 40.4 2 256 398
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Table K.l (Co nt. ) PHA P Y A H w S v T s % t.h t.". L PO E Zma.v qvE T,otal A m (g) (g) ( m ) ( d e g ree) (degree) (m) (kN /m) ( m m ) ( mm) (kN /m) (m) (kN / m2) ( kN /m) (g) 0 0 6 5 36 0.4 36 0 0 0 0 111.4 2 1 45 2 1 8 2 9.4 0 000 0 .114 0 093 6 5 36 0.4 36 1 3.3 11.3 163. 5 2 529 267 0 9 7 0 1 9 1 0 1 68 0 0 7 3 6 5 36 0.4 36 7.4 4 8 1 36 9 2.4 1 2 242 5 9.4 0 24 9 0.2 1 2 0 059 6 5 36 0.4 36 1 07 8 29 9 259 0 3 08 1 381.5 1 3 0 0 54 9 0.229 0 1 6 6 5 36 0.4 36 51.4 1 4 2 238.4 3 0 47 324 9 1 0 6 0.5 1 4 0 243 0 059 6 5 36 0.4 36 1 27 9 23.5 28 1 2 3 0 7 6 346 7 1 3 8 0.572 0 282 0 097 6 5 36 0.4 36 1 03.4 28 0 2 7 3 2 3 .027 372.4 1 2 7 0 564 0 344 0.552 6 5 36 0.4 36 201.6 34 5 306 5 2 942 390 2 14. 5 0 806 0.36 7 0.338 6 5 36 0.4 36 1 41.6 24 1 298 7 3 1 38 352.3 1 2 9 0 592 0.479 0.455 6 5 36 0.4 3 6 442 5 57 8 402 0 3 0 70 474 8 18. 6 0 979 0 5 1 4 0.217 6 5 36 0.4 36 44 7 7 63 2 402.4 3 09 1 450 5 1 9.3 0 .981 0 549 0 .195 6 5 36 0.4 36 355.4 6 7 2 422 9 3 077 4 9 7 5 1 9.3 1.561 0 56 8 0 217 6 5 36 0.4 36 386 I 68.3 3 7 6 0 2 946 39 1 I 1 8 7 0 946 0 644 0.455 6 5 36 0.4 36 551.4 78 3 437 2 3 025 468.4 24. 6 1 2 92 0 69 4 0.433 6 5 36 0.4 36 4 01.9 65 I 533 8 2 .951 5 1 2.4 22 0 2 006 0 7 1 2 0 255 6 5 36 0.4 36 326 1 63 0 359 7 2 994 412. I 1 6 9 1 .555 0 753 0.467 6 5 36 0.4 36 598.5 8 7 3 535 0 2 9 1 2 598 7 30. I 1.27 6 0 838 0 852 6 5 36 0.4 36 363 8 1 0 0.3 6 1 5 5 2 764 531.3 27 7 1.903 0 897 0 586 6 5 36 0.4 36 686.5 1 0 7 1 460 2 2.90 1 6 1 2 3 30. 7 1 504 0 958 0 3 1 1 6 5 36 0.4 36 584 2 83 9 370 9 2.900 463 2 21.2 0 987 0 99 1 048 6 5 36 0.4 36 1 829 6 28 7 7 553 3 2 755 657 9 44 3 1.757 0 0 6 1 0 40 0.4 36 0 0 0 0 1 56 6 2 336 226 8 6 2 0 00 0 0 .114 0 093 6 1 0 40 0.4 36 11.6 9 3 205 7 2 559 2 7 4 9 6 5 0 194 0 1 68 0 073 6 10 40 0.4 36 7 0 4 8 1 89 2 2.48 1 263 I 7 1 0 223 0 2 1 2 0 059 6 1 0 40 0.4 36 7 5 8 21.7 3 1 5 I 2 968 395 0 1 0 0 0 639 0 229 0 1 6 6 1 0 40 0.4 36 40 8 1 3 7 293 0 2 899 359 8 8 2 0 502 0 243 0 059 6 1 0 40 0.4 36 7 9 7 19. I 330.4 2 994 363 3 1 0 2 0 559 0 282 0097 6 1 0 40 0.4 36 79 7 22 2 326 3 3 035 354.4 1 0 1 0 64 0 0.344 0 552 6 1 0 40 0.4 36 135. 2 31.1 359 9 3 025 417.5 1 2 5 0 868 0 36 7 0.338 6 1 0 40 0.4 36 119 0 25. I 336 6 2 869 352 0 11.0 0 578 0.4 7 9 0.455 6 1 0 40 0.4 36 280 0 59.2 4 1 0 8 2 98 1 505 0 17. 8 1.021 0 514 0 217 6 1 0 40 0.4 36 329 6 60 6 449 9 3 009 473 5 24 6 1 .08 0 0 549 0 .195 6 1 0 40 0.4 36 279 I 63 2 465 7 2 985 622 0 1 7 2 1 057 0 568 0 2 1 7 6 1 0 4 0 0.4 36 2 71.7 50 7 430 9 2 850 419. I 1 7 8 0 934 0 644 0.455 6 1 0 40 0.4 36 384 9 53. 6 456 6 3 068 525 7 21. 1 1.160 0 69 4 0.433 6 1 0 40 0.4 36 257 5 631 596 1 2 852 530 2 1 9 5 2.201 0 7 1 2 0 255 6 10 40 0.4 36 234.5 5 7 1 408 0 2 950 3 94 5 1 5 7 1 059 0 753 0.46 7 6 1 0 40 0.4 36 4 1 6 8 69.4 538 8 2 889 634 8 24 9 1.5 1 5 0 838 0 852 6 1 0 40 0.4 36 256 0 85 1 693 8 2 689 634 6 25.4 2 .173 0 89 7 0.586 6 1 0 40 0.4 36 415 6 7 5 5 5 1 9.3 2 854 630 7 24 7 1 640 0 958 0 3 1 1 6 1 0 40 0.4 36 505 9 65 0 434.2 2 945 525 I 30.4 1.012 0 99 1.048 6 1 0 40 0.4 36 1 4 0 7 2 143. 9 6 71.7 2 7 5 1 666 0 48.3 5 .891 399
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Table K.l (C ont. ) PHA PYA H (() S T sl'. 6h 6y LPOE Zona< q,' E T,otal Am (g) (g) (m) (degree) (degree) (m) (kN /m) (mm) (mm) (kN /m) (m) (kN / m 2 ) (kN /m) (g) 6 10 32 0.4 36 0 0 0 0 1 26 9 2.353 184. 9 7 8 0 000 0 .114 0 093 6 10 32 0.4 36 14. 2 12. 3 169. 5 2 525 214 3 10. 7 0 1 90 0 .168 0 073 6 10 32 0.4 36 10. 2 4 2 150 7 2 .561 206 5 8 6 0 176 0.212 0 059 6 10 32 0.4 36 126 1 34.3 254 2 3 025 288 2 14.4 0.542 0.229 0 .16 6 10 32 0.4 36 62 7 15. I 245.3 3 065 244 5 9 9 0.436 0.243 0 059 6 10 32 0.4 36 159 9 34 6 259 I 2 994 290 9 16. 5 0 513 0 282 0 097 6 10 32 0.4 36 123.4 37 I 270.5 3 060 257 2 15. 7 0 665 0.344 0 552 6 10 32 0.4 36 185. 0 36 8 269 3 2 885 277.9 16. 1 1 028 0 367 0.338 6 10 32 0.4 36 147 6 24. 9 254 8 2 979 267 6 14.4 0 578 0.479 0.455 6 10 32 0.4 36 545 3 73. 6 362 5 2 926 392 6 22 9 1.403 0 514 0 217 6 10 32 0.4 36 456 1 67 3 344 6 3 073 309.4 20.3 0 772 0 549 0 .195 6 10 32 0.4 36 358.3 61.1 322 5 3 082 326.4 18.4 1.013 0 568 0 217 6 10 32 0.4 36 383.3 71.6 334 9 3 086 278 9 20 0 1.051 0 644 0 455 6 10 32 0.4 36 562 I 80.4 380 0 2 986 337 I 22 6 1 205 0 694 0.433 6 10 32 0.4 36 371.8 58 7 391. I 2 800 378 I 18. 8 1 342 0 712 0 255 6 10 32 0.4 36 299 6 53.4 293.5 3 .165 297 1 16.3 0 988 0 753 0.467 6 10 32 0.4 36 586.3 97 9 434 1 2.920 462 5 25 2 1.103 0 838 0 852 6 10 32 0.4 36 387 7 91.1 486 8 2 770 364 0 34 2 1.861 0 897 0.586 6 10 32 0.4 36 723 I IOU 424 8 2 738 421.9 32 9 1 .359 0 958 0.311 6 10 32 0.4 36 497 6 67.4 328 4 3 063 315 I 23.0 1 003 0 99 1 048 6 10 32 0.4 36 1 773.2 243 2 475 8 2 815 360 9 46 9 1.719 6 10 36 0 2 36 0 0 0 0 1 44 9 2.262 210.4 5 6 0 000 0 .114 0 093 6 10 36 0 2 36 1 4 0 13.3 1 95 6 2 542 257 I 6.3 0 204 0 .168 0 073 6 10 36 0 2 36 7.5 5 5 175. 2 2.475 234 6 6.2 0 .178 0 212 0 059 6 10 36 0 2 36 1 02 6 27 1 302 8 3 055 330 7 8 5 0 .551 0 229 0 .16 6 10 36 0 2 36 48 6 13. 8 273 9 2 992 306 5 9 1 0 505 0 243 0 059 6 10 36 0 2 36 113. 7 20.3 3 1 6 0 2 976 297 5 8 8 0 590 0 282 0 097 6 10 36 0 2 36 89 9 22 8 295 2 3.013 328 I 10. 1 0 600 0.344 0.552 6 10 36 0 2 36 174. 7 24 8 338.4 2 929 376.4 25.4 0 774 0.367 0.338 6 10 36 0 2 36 125. I 20 6 3 1 4.3 3 065 308 2 8 6 0 615 0.479 0.455 6 10 36 0 2 36 382 2 41.7 404.2 2 996 445 2 15.3 1.018 0.514 0 217 6 10 36 0 2 36 368 8 46 2 398.4 3 035 4 1 4 5 14. 7 0 998 0 549 0 .195 6 10 36 0 2 36 333.1 52. 6 396.4 3 058 461.4 13. 7 0 889 0.568 0 217 6 10 36 0 2 36 345 I 49 0 395.3 2 912 345 8 15.4 1 074 0 644 0.455 6 10 36 0 2 36 481.8 47 7 423 2 3 086 408 7 17. 9 1.463 0 694 0.433 6 10 36 0 2 36 339.4 56.5 496 8 3 006 444.3 15.4 1 52 0 0 712 0 255 6 10 36 0 2 36 301.7 47 7 390.4 3 019 376 6 12. 6 1 .175 0 753 0.467 6 10 36 0 2 36 528 6 62 7 507 0 2 989 571. I 25 2 1 .119 0 838 0 852 6 10 36 0 2 36 282.3 82.4 5 1 7 0 2 775 485.3 27.0 1.739 0 897 0 586 6 10 36 0 2 36 588 6 65 0 482 7 2 833 528 0 22 6 1 .403 0 958 0 .311 6 10 36 0 2 36 506 0 51.7 382 5 2 943 437 8 19. 0 0 933 0.99 1 048 6 10 36 0 2 36 1 550 6 1 49 9 532 6 2 737 679 9 48 3 3.601 400
PAGE 444
Table K.1 (Cont.) PHA PYA H OJ S v T s % Il", I PO E zma., qvE TIOIaI A m (g) (g) (m) (deg r ee) (degree) (m) (kN /m) (mm) (mm) (kN /m) (m) (kN / m 2 ) (kN/m) (g ) 0 0 6 1 0 36 0 6 36 0 0 0 0 1 5 1.7 2 285 193. 5 8 9 0 000 0. 114 0 093 6 10 36 0 6 36 1 4 7 11.4 202 1 2.5 1 2 227. 1 18. 0 0 .241 0 168 0 073 6 10 36 0 6 3 6 7 1 4 8 180 2 2.469 2 1 6 6 11.4 0 244 0 212 0 059 6 1 0 3 6 0 6 36 101.6 3 0 7 283 7 2 964 332 1 18. 9 0 551 0 229 0 1 6 6 1 0 36 0 6 36 48 5 15. 0 2 71.7 2 958 285.4 12. 8 0 578 0 243 0 059 6 10 36 0 6 36 112 2 24.4 298.4 2 9 36 294 0 14. 1 0 592 0 282 0 097 6 1 0 36 0 6 36 89 9 27 7 292.4 2 920 299 0 1 4 7 0 687 0.3 44 0 552 6 10 3 6 0. 6 36 168 6 38 9 329 8 2 752 361.8 1 9 6 0 693 0.367 0 338 6 1 0 36 0 6 36 119 5 29 2 306 8 2 9 98 292 7 16. 0 0 583 0.479 0.455 6 1 0 3 6 0 6 36 35 9 8 59 5 403 6 2 886 434 1 2 6 9 1.041 0 514 0 217 6 1 0 36 0 6 36 361.9 7 0 8 396 0 2 880 413 2 26.3 0 905 0 549 0 .195 6 10 36 0 6 36 326 8 73.0 409 8 2 959 450 7 28 8 1 .30 0 0 568 0 2 1 7 6 1 0 36 0 6 36 322 2 68 7 380.3 2 810 331.9 27 6 0 856 0. 644 0.455 6 1 0 36 0 .6 3 6 460.4 76 2 438 1 2 943 435 1 3 0 1 0 765 0 694 0.433 6 10 36 0 6 36 34 1 0 62 3 482.4 2 814 44 1 2 26. 0 1 265 0 7 1 2 0 255 6 10 36 0 6 36 283 5 61.5 384 9 2 813 347 0 2 5 7 1.14 6 0 753 0.467 6 10 36 0 6 36 500 1 89 9 507.2 2 84 1 568 5 34 3 1 .371 0 838 0 852 6 1 0 36 0 6 36 3 1 8 6 92 5 565 0 2.564 487 0 35 1 301 9 0 897 0 586 6 10 36 0 6 3 6 549 8 99 1 495.4 2 699 5 1 3 6 33.4 1 .308 0 958 0.311 6 1 0 36 0 6 3 6 464 5 88 3 383.4 2 8 0 3 447 8 27.4 1 004 0 99 1 048 6 10 36 0 6 36 1 558 7 212.5 552 0 2 670 607.4 48 2 2.48 6 0 0 6 10 36 0.4 72 0 0 0 0 134 0 2 357 1 93.5 1 0 5 0 00 0 0 114 0 093 6 10 36 0.4 72 1 2 5 11.7 184. 8 2 616 253 8 12.3 0 .185 0 168 0 073 6 1 0 36 0.4 72 6 8 5 1 161.5 2 555 209 1 11.6 0 219 0 212 0 059 6 10 36 0.4 72 97 6 25 1 280 9 3 089 340 5 17. 0 0 54 5 0 229 0 .16 6 10 36 0.4 72 46 5 11.6 266 0 3 104 28 1 .4 12.3 0 .541 0 243 0 059 6 10 36 0 .4 72 1 02.5 1 7 3 293 9 3 045 332 9 17. 8 0 615 0 282 0 097 6 10 36 0.4 72 81.4 19. 3 286 6 3 024 327.4 17. 0 0 567 0 344 0.552 6 1 0 36 0.4 72 1 51.5 23 8 324 8 2 963 351.1 22 5 0 8 7 9 0.36 7 0.338 6 1 0 36 0.4 72 110 7 18.3 303.5 3 1 0 9 318 0 15. 8 0 56 2 0.479 0.455 6 1 0 36 0 .4 72 34 1 6 40 1 400 9 3 064 445 8 28 7 1.012 0 514 0 217 6 10 36 0.4 72 33 1 1 46 2 396 6 3 059 405.4 27.4 0 888 0 .549 0 195 6 10 36 0.4 72 304 6 51.6 407 8 3 06 7 45 3.7 65 5 1 132 0 568 0 217 6 1 0 36 0.4 72 299.3 46 6 379 2 3 000 338 3 28 6 0 927 0 644 0.455 6 1 0 36 0.4 72 428.3 49 5 437 6 3 027 4 04 7 4 7 7 1.110 0 694 0.433 6 10 36 0 .4 72 303.3 48.4 502.4 2 872 424 6 27.3 1.237 0 712 0 255 6 10 36 0.4 72 270 8 45 8 386 3 3 086 361.2 2 5 9 1 225 0 753 0.467 6 10 36 0 4 72 456.3 65 2 5 1 2 6 2 880 556 1 43 7 1 .450 0 838 0 852 6 10 36 0.4 72 283 7 6 6.3 586 7 2 63 1 461.8 32.0 1.859 0 89 7 0 586 6 10 36 0.4 72 532.0 65 0 482 3 2 785 5 1 5 0 38 0 1.401 0 958 0 .311 6 10 36 0.4 72 430 6 52 9 382.5 3 018 447 0 36 7 0 954 0 99 1 048 6 1 0 36 0.4 72 1 506.3 143. 9 562.4 2 653 542 7 58 7 3 202 401
PAGE 445
Table K.l (Cont.) PHA P Y A H ro $' S T s % 6h L POE Z ma q,E T otal A m (g) (g) ( m ) ( d egree) (deg r ee) ( m ) (kN / m ) ( m m) (mm) (kN / m ) (m) (kN / m 2 ) (kN /m) (g) 0 0 6 1 0 36 0.4 1 2 0 0 0 0 1 40 7 2 235 1 99 8 4 0 0 000 0 114 0 093 6 1 0 36 0.4 1 2 1 3.4 11. 9 1 91.6 2.495 233 6 4 9 0 .251 0 168 0 073 6 10 36 0.4 1 2 7.4 5 1 1 6 7 2 2.42 1 2 1 5 2 5 1 0 1 89 0 212 0 059 6 1 0 36 0.4 1 2 105. 0 38. 7 284 9 2 946 333.4 7.4 0 546 0 229 0 1 6 6 1 0 36 0.4 1 2 50 6 21.5 271.6 2 945 289 1 5 8 0.480 0 2 4 3 0 059 6 1 0 36 0.4 1 2 123. 7 38 5 301.7 2 905 309 7 7 5 0 528 0 282 0 097 6 1 0 36 0.4 1 2 1 05.4 43. 1 292.4 2 922 292 0 7 6 0 585 0 344 0 552 6 10 36 0.4 1 2 1 99 1 61.8 325 5 2 778 3 7 8.4 8 6 0 713 0 36 7 0 338 6 1 0 36 0.4 1 2 1 41.7 4 6 1 29 1 0 2 960 308 3 7 9 0 634 0.479 0.455 6 1 0 36 0.4 1 2 401.6 9 0 7 396 1 2 961 44 0 9 11. 5 0 952 0 514 0 217 6 1 0 36 0.4 1 2 4 0 7 3 1 02 6 3 92 5 3.003 4 1 6.4 11.4 1.140 0 549 0 .195 6 1 0 36 0 4 1 2 358 2 95. 9 4 07 0 2 967 45 1 2 1 0 1 1 .182 0 568 0 2 1 7 6 1 0 36 0.4 1 2 371. 5 9 4 5 369 5 2 908 367 9 1 0 7 0 8 7 4 0 644 0 .455 6 1 0 3 6 0.4 12 5 1 8 9 112 1 443.4 2 987 45 0 1 12. 0 0 .961 0 694 0.433 6 10 36 0.4 12 389.4 86 5 493 5 2 882 463.3 1 3.4 1 534 0 7 1 2 0 255 6 1 0 36 0.4 12 322 5 87 8 363 2 2 .821 3 7 8 6 1 0 7 0 8 6 1 0 753 0.467 6 1 0 36 0.4 12 56 1 6 123. 7 514 9 2 830 5 733 1 4 0 1 622 0 838 0 852 6 1 0 36 0.4 12 368 6 126 8 593 1 2 68 7 5003 1 4 8 3 09 6 0 89 7 0 586 6 1 0 36 0.4 1 2 621.5 138. 7 4 7 3 0 2 77 3 560 8 1 4 5 1 65 5 0 958 0 .311 6 10 36 0.4 12 536 5 128. 0 380 5 2 845 4 1 0 5 12. 6 1 06 6 0 99 1 048 6 1 0 36 0.4 1 2 1745 3 287 9 535.2 2 6 7 9 6 1 9 9 20. 8 2 588 402
PAGE 446
0 8 0 7 0 6 U OJ 0 5 .!!!.c: Ic 0 0.4 c OJ a. ro .... :::J 0 3 ro Z 0 2 0 1 0 3 APPENDIX L Natural Period of GRS Walls GRS wall W = 32 ) f++4Ij GRS wall W = 36 ) GRS wall W = 40) FEM data points 2 Tn = a1x H + a2x H + a3 ( 0 ) a1 a2 a3 32 0 0015 0 0776 0 0458 36 0 0013 0 0678 0 0400 40 0 0012 0 0617 0 0364 4 5 6 7 8 9 10 Wall height H (m) Figure L.l Variation of Natural Period with Wall Height and Soil Friction Angle 403
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