DISPLACEMENT BASED APPROACH FOR SEISMIC STABILITY OF ...

DISPLACEMENT-BASED APPROACH FOR

SEISMIC STABILITY OF RETAINING

STRUCTURES

A thesis submitted to the University of Manchester for the degree

of Doctor of Philosophy in the Faculty of Science and

Engineering

2018

JUNIED AZIZ BAKR

School of Mechanical, Aerospace and Civil Engineering

PAPERS PRODUCED FROM THIS THESIS

A: Published papers

Bakr, J. and Ahmad, S. M. 2018. A finite element performance-based approach to

correlate movement of a rigid retaining wall with seismic earth pressure. Soil Dynamics

and Earthquake Engineering, 114, 460-479

Bakr, J. and Ahmad, S. M. 2018, Effect of earthquake characteristics on the permanent

displacement of a cantilever retaining wall. Proceedings of the 9th NUMGE Conference

on Numerical Methods in Geotechnical Engineering in Porto, Portugal, 25-27 June,

2018.

Bakr, J. and Ahmad, S. M. 2018. Effect of foundation soil stiffness on the seismic earth

pressure. Proceedings of the MACE PGR Conference. Rogers, B. D. (ed.). Manchester,

At the University of Manchester.

Bakr, J. and Ahmad, S. M. 2017. Risk assessment for the seismic behaviour of a

cantilever retaining wall. Proceedings of the MACE PGR Conference. Rogers, B. D.

(ed.). Manchester, p. 29-31, At the University of Manchester.

A: Submitted papers

Bakr, J. and Ahmad, S, M.. 2018, A finite element performance-based approach for

evaluating the seismic stability of a cantilever retaining wall. submitted to: International

Journal of Geomechanics (ASCE).

List of Contents

1

LIST OF CONTENTS

LIST OF CONTENTS .................................................................................................................. 1

LIST OF FIGURES .................................................................................................................... 10

LIST OF TABLES ...................................................................................................................... 18

LIST OF SYMBOLS .................................................................................................................. 19

ABSTRACT ........................................................................................................................ 26

DECLARATION ........................................................................................................................ 27

COPYRIGHT STATEMENT ..................................................................................................... 28

DEDICATION ....................................................................................................................... 29

ACKNOWLEDGEMENT .......................................................................................................... 30

CHAPTER 1 INTRODUCTION ............................................................................... 32

1.1 Background ................................................................................................................. 32

1.2 Research aim and objectives ....................................................................................... 34

1.2.1 Objectives for a rigid retaining wall.................................................................... 34

1.2.2 Objectives for a cantilever-type retaining wall ................................................... 35

1.3 Organization of the thesis ........................................................................................... 36

CHAPTER 2 LITERATURE REVIEW ................................................................... 38

2.1 Retaining wall ............................................................................................................. 38

2.2 Types of retaining wall ............................................................................................... 38

2.2.1 Gravity retaining walls ........................................................................................ 39

2.2.2 Cantilever retaining walls ................................................................................... 39

2.3 Retaining wall failure modes ...................................................................................... 40

2.3.1 Rigid-body sliding failure mode ......................................................................... 41

2.3.2 Overturning failure mode .................................................................................... 41

List of Contents

2

2.3.3 Flexural failure mode .......................................................................................... 41

2.4 Static earth pressure .................................................................................................... 42

2.4.1 Static earth pressure states .................................................................................. 42

2.4.1.1 Earth pressure at-rest ....................................................................................... 44

2.4.1.2 Active earth pressure ....................................................................................... 45

2.4.1.3 Passive earth pressure ..................................................................................... 46

2.4.2 Earth pressure theories ........................................................................................ 47

2.4.2.1 Coulomb’s (1776) earth pressure theory ......................................................... 47

2.4.2.2 Rankine’s (1857) earth pressure theory .......................................................... 48

2.4.3 Relationship between static earth pressure and wall displacement ..................... 49

2.4.3.1 Analytical methods ......................................................................................... 50

2.4.3.2 Numerical methods ......................................................................................... 53

2.4.3.3 Experimental methods ..................................................................................... 58

2.4.4 Critical discussion of the relationship between the static earth pressure and wall

displacement ....................................................................................................................... 62

2.5 Seismic design of retaining walls ................................................................................ 65

2.5.1 Force-based design methods ............................................................................... 66


2.5.1.1.1 Pseudo-static methods ............................................................................... 67

2.5.1.1.2 Critical discussion on pseudo-static methods............................................ 71

2.5.1.1.3 Pseudo-dynamic methods ......................................................................... 72

2.5.1.1.4 Critical discussion on pseudo-dynamic methods ...................................... 74



2.5.1.3.1 Shaking table tests ..................................................................................... 78

2.5.1.3.2 Centrifuge tests ......................................................................................... 79

2.5.1.4 Critical discussion of the force-based design methods ................................... 83

2.5.2 Displacement-based design method .................................................................... 87


List of Contents

3

2.5.2.1.1 One-block methods ................................................................................... 88

2.5.2.1.2 Two-block methods ................................................................................... 90



2.5.2.3.1 Shaking table tests ..................................................................................... 94

2.5.2.3.2 Centrifuge tests ......................................................................................... 95

2.5.2.4 Critical discussion on displacement-based design methods ............................ 96

2.5.3 Force-displacement hybrid design methods ........................................................ 97


2.5.3.2 Numerical methods ....................................................................................... 103

2.5.3.3 Experimental methods ................................................................................... 106

2.5.3.4 Critical discussion on force-displacement hybrid design methods ............... 108

2.5.4 Real field observations of retaining wall damage post-earthquake ................... 110

2.6 Eurocode 8: Design of structures for earthquake resistance ..................................... 114

2.6.1 General requirements ........................................................................................ 115

2.6.2 Methods of analysis .......................................................................................... 115

2.7 Summary ................................................................................................................... 116

CHAPTER 3 FINITE ELEMENT MODELLING METHODOLOGY .............. 118

3.1 Why FE modelling? .................................................................................................. 118

3.2 Overview of the PLAXIS2D software ...................................................................... 119

3.3 Domain discretisation to idealise the wall-soil system ............................................. 120

3.4 Retaining wall and soil discretisation and interface idealisation .............................. 121

3.4.1 6-noded triangular elements .............................................................................. 122

3.4.2 Plate element ..................................................................................................... 122

3.4.3 Interface element and modelling of the interface behaviour ............................. 123

3.5 Natural frequency and mode shapes of the wall-soil system .................................... 124

3.6 Initial sizing of the FE mesh considering the propagation of shear waves ............... 125

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3.7 Constitutive models .................................................................................................. 125

3.7.1 Retaining wall ................................................................................................... 125

3.7.2 Soil .................................................................................................................... 126

3.7.2.1 Hardening soil with small strain model ........................................................ 126

3.7.2.2 Reduction of soil stiffness at small strain level ............................................. 129

3.7.2.3 Damping ........................................................................................................ 131

3.7.2.4 Soil parameters required to run the FE simulation ........................................ 133

3.8 Boundary conditions for static analysis .................................................................... 134

3.9 Static analysis ............................................................................................................ 134

3.10 Boundary conditions for the seismic analysis ........................................................... 134

3.11 Seismic analyis .......................................................................................................... 135

3.12 Seismic loading ......................................................................................................... 135

3.13 Post processing approach .......................................................................................... 136

3.13.1 Acceleration and displacement ......................................................................... 136

3.13.2 Seismic wall and backfill inertia forces ............................................................ 136

3.13.3 Seismic earth pressure force ............................................................................. 137

3.14 Summary ................................................................................................................... 138

CHAPTER 4 VALIDATION OF FE MODEL ...................................................... 139

4.1 Geotechnical centrifuge modelling ........................................................................... 139

4.2 3 centrifuge tests selected from literature ................................................................. 139

4.2.1 Saito (1999) test ................................................................................................ 139

4.2.2 Nakamura (2006) test ........................................................................................ 140

4.2.3 Jo et al. (2014) test ............................................................................................ 141

4.3 FE modelling of the abovementioned 3 centrifuge tests ........................................... 142

4.3.1 Saito (1999) test ................................................................................................ 143

4.3.2 Nakamura (2006) test ........................................................................................ 144

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5

4.3.3 Jo et al. (2014) test ............................................................................................ 144

4.3.4 Material parameters........................................................................................... 145

4.4 Natural frequency and mode shapes for the 3 centrifuge tests .................................. 146

4.4.1 Saito (1999) test ................................................................................................ 146

4.4.2 Nakamura (2006) test ........................................................................................ 147

4.4.3 Critical discussion on the natural frequency of the wall-soil system ................ 148

4.5 Mesh size sensitivity analysis ................................................................................... 150

4.6 Validation of FE results ............................................................................................ 151

4.6.1 Saito (1999) test ................................................................................................ 151

4.6.2 Nakamura (2006) test ........................................................................................ 154

4.6.2.1 Horizontal displacement and rotation ........................................................... 154

4.6.2.2 Seismic earth pressure ................................................................................... 156

4.6.3 Jo (2014) test ..................................................................................................... 158

4.6.3.1 Simulation of construction process ............................................................... 158

4.6.3.2 Static earth pressure ...................................................................................... 159

4.7 Summary ................................................................................................................... 160

CHAPTER 5 FINITE ELEMENT ANALYSIS OF A RIGID RETAINING

WALL ............................................................................................................ 162

5.1 Problem description .................................................................................................. 162

5.2 FE modelling and material properties ....................................................................... 163

5.3 Seismic loading ......................................................................................................... 164

5.4 Results and discussion .............................................................................................. 166

5.4.1 Acceleration response of the soil-retaining wall system ................................... 167

5.4.2 Horizontal displacement ................................................................................... 168

5.4.2.1 Horizontal displacement of the wall-soil system .......................................... 168

5.4.2.2 Relative horizontal displacement of the retaining wall with respect to

foundation soil .............................................................................................................. 170

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6

5.4.2.3 Comparison with Newmark sliding block method (Newmark, 1965) .......... 170

5.4.2.4 Comparison with the Eurocode 8 .................................................................. 172

5.4.2.5 Rotation of the retaining wall about its toe ................................................... 172

5.4.3 Wall seismic inertia force Fw ............................................................................ 173

5.4.4 Seismic earth pressure force P .......................................................................... 174

5.4.4.1 Seismic earth pressure force time history ..................................................... 174

5.4.4.2 Comparison with M-O theory ....................................................................... 176

5.4.4.3 Comparison with Eurocode 8 ........................................................................ 177

5.4.4.4 Distribution of seismic earth pressure ........................................................... 178

5.4.5 Effect of wall seismic inertia force Fw on the earth pressure force increment ∆P

179

5.4.6 Effect of wall displacement on the wall seismic inertia force Fw ..................... 181

5.4.7 Effect of wall displacement on seismic earth pressure force P ......................... 181

5.5 Parametric study ........................................................................................................ 182

5.5.1 Effect of the earthquake acceleration level and retaining wall height .............. 183

5.5.1.1 Acceleration response ................................................................................... 183

5.5.1.2 Relative horizontal displacement .................................................................. 183

5.5.1.1 Seismic earth pressure force ......................................................................... 184

5.5.1.2 Relationship between seismic earth pressure and displacement of retaining

wall considering different retaining wall heights and acceleration levels..................... 185

5.5.2 Effect of the frequency content of the earthquake acceleration ........................ 187


5.5.2.2 Relative horizontal displacement of the retaining wall ................................. 190

5.5.2.3 Seismic earth pressure force ......................................................................... 192


wall considering different amplitudes and frequency content of earthquake acceleration

195

5.5.3 Effect of the relative density of the soil material .............................................. 198

5.5.3.1 Effect of soil material (1st combination) ....................................................... 200

5.5.3.2 Effect of the relative density backfill soil layer (2nd

combination) .............. 203

List of Contents

7

5.5.3.3 Effect of the foundation soil material (3rd

combination) .............................. 205

5.6 Summary ................................................................................................................... 208

CHAPTER 6 FINITE ELEMENT MODELLING AND ANALYSIS OF A

CANTILEVER RETAINING WALL ....................................................................... 210

6.1 Problem description ................................................................................................. 210

6.1.1 Structural integrity ............................................................................................ 211

6.1.2 Global stability .................................................................................................. 212

6.2 FE model and material properties ............................................................................. 213

6.2.1 Seismic loading ................................................................................................. 214

6.3 Seismic analysis ........................................................................................................ 215

6.3.1 Acceleration response of the retaining wall-soil system ................................... 216

6.3.2 Wall and backfill seismic inertia forces ............................................................ 217

6.3.3 Seismic earth pressure force ............................................................................. 218

6.3.3.1 Seismic earth pressure behind the stem Pstem ................................................ 218

6.3.3.2 Seismic earth pressure behind the virtual plane Pvp ...................................... 218

6.3.3.3 Comparison between Pstem and Pvp ................................................................. 220

6.3.4 Total seismic earth pressure force increments, Pstem, Pvp and wall and backfill

seismic inertia forces Fwa, Fwp, Fsa, Fsp .............................................................................. 222

6.3.5 Shear force Nw and bending moment Mw .......................................................... 224

6.3.6 Relative horizontal displacement of the wall and backfill soil with respect to the

foundation soil .................................................................................................................. 225

6.3.6.1 Total displacement response ......................................................................... 225

6.3.6.2 Relative horizontal displacement of the wall and backfill soil with respect to

the foundation soil ......................................................................................................... 227

6.3.6.3 Rotation of stem ............................................................................................ 229

6.3.6.4 Rotation of the base slab ............................................................................... 230

6.3.6.5 Deformation shape of a cantilever retaining wall ......................................... 231

6.4 Parametric study ........................................................................................................ 233

List of Contents

8

6.4.1 Effect of earthquake characteristics .................................................................. 233



6.4.1.3 Shear force and bending moment .................................................................. 236

6.4.1.4 Relative horizontal displacement .................................................................. 237

6.4.2 Effect of the natural frequency a cantilever retaining wall (height) ................. 239


6.4.2.2 Seismic earth pressure force Pstem ................................................................. 241

6.4.2.3 Seismic earth pressure force Pvp .................................................................... 243


6.4.2.5 Relative horizontal displacement of retaining wall W-F ............................... 246

6.4.3 Effect of relative density of soil ........................................................................ 247




6.4.3.4 Relative horizontal displacement of the wall ................................................ 251

6.5 Summary ................................................................................................................... 252

CHAPTER 7 ANALYTICAL METHODS ............................................................. 254

7.1 Contribution of wall seismic inertia force on the total shear force and bending moment

254

7.1.1 Problem definition............................................................................................. 255

7.1.2 Assumptions made in the simplified procedure ................................................ 255

7.1.3 Effect of wall seismic inertia force for the top ⅓H of the stem on Nw and Mw . 257

7.1.4 Effect of wall seismic inertia force for the mid-height of the stem on Nw and Mw

259

7.1.5 Effect of wall seismic inertia force for the bottom ⅓H of the stem on Nw and Mw

261

7.2 Modification of Newmark sliding block method ...................................................... 262

List of Contents

9

7.2.1 Modified Newmark sliding block method applied to rigid retaining walls ...... 264

7.2.2 Worked example and numerical validation ....................................................... 267

7.2.3 Cantilever retaining wall ................................................................................... 269

7.2.4 Worked example and numerical validation ....................................................... 272

7.3 Summary ................................................................................................................... 274

CHAPTER 8 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE

RESEARCH ............................................................................................................ 276

8.1 Conclusions of this research ..................................................................................... 276

8.1.1 FE modelling of a rigid retaining wall .............................................................. 276

8.1.2 FE modelling of a cantilever retaining wall ...................................................... 278

8.1.3 Analytical methods ........................................................................................... 280

8.2 Recommendations for future research ...................................................................... 280

REFERENCES ...................................................................................................................... 282

APPENDIX A RESULT OF THE FINITE ELEMENT ANALYSIS OF A RIGID

RETAINING WALL ................................................................................................................ 290

APPENDIX B MATLAB PROGRAMS ............................................................................... 297

List of Figures

10

LIST OF FIGURES

Figure 1.1: Failure of retaining walls during the Kumamoto earthquake in Japan occurred on the

16th of April 2016 (a) gravity retaining wall, (b) dam spillway retaining wall after Kiyota et al.

(2017) .......................................................................................................................................... 33

Figure 2.1: 3D-sketch of a rigid retaining wall ........................................................................... 39

Figure 2.2: 3D-sketch of a cantilever retaining wall ................................................................... 40

Figure 2.3: Failure modes of a retaining wall: (a) Sliding, (b) Overturning, (c) Flexure ........... 42

Figure 2.4: Direction of wall movement and soil stresses .......................................................... 43

Figure 2.5: Development of active and passive earth pressure states based on wall displacement

(after Terzaghi, 1936) ................................................................................................................. 44

Figure 2.6: Mohr circle describing active state within a soil mass ............................................. 46

Figure 2.7: Mohr circle describing passive state within a soil mass ........................................... 47

Figure 2.8: Planar failure wedge for active state (after Muller-Breslau, 1906) .......................... 48

Figure 2.9: Rankine active earth pressure behind retaining wall ................................................ 49

Figure 2.10: Relationship between active earth pressure and ..................................................... 51

Figure 2.11: Passive earth pressure versus wall displacement after Shamsabadi et al. (2005) ... 52

Figure 2.12: Modes of wall displacement generating a passive earth pressure state for a rigid

retaining wall: (a) T mode; (b) RB mode; (c) RT mode; (d) RTT mode; (e) RBT mode (after

Peng et al., 2012) ........................................................................................................................ 52

Figure 2.13: Active and passive pressure coefficients for: (a) smooth wall surface, (b) rough

wall surface (after Potts and Fourie (1986) ................................................................................. 54

Figure 2.14: Variation of earth pressure coefficient with wall displacements after Hazarika and

Matsuzawa (1996) ....................................................................................................................... 55

Figure 2.15: Comparison of passive earth pressure force from various numerical and analytical

model results with experimental measurements after Shamsabadi et al. (2009) ........................ 56

Figure 2.16: Comparison of passive earth pressure force from numerical and experimental

measurements with: (a) low interface, (b) high interface after Wilson and Elgamal (2010) ...... 57

Figure 2.17: Dimensionless earth pressure force versus wall movement - numerical and

experimental modelling results after Achmus (2013) ................................................................. 57

List of Figures

11

Figure 2.18: Relationship between active earth pressure and wall rotation after Sherif et al.

(1984) .......................................................................................................................................... 59

Figure 2.19: Variation of lateral earth pressure coefficient (Kh), relative height of resultant

pressure application (h/H) and coefficient of wall friction (tanδ) with the wall rotation about its

top After Fang and Ishibashi (1986) ........................................................................................... 60

Figure 2.20: Effect of wall movement mode on passive earth pressure after Fang et al. (1994) 61

Figure 2.21: Large-scale wall-soil model test after Wilson and Elgamal (2010) ....................... 62

Figure 2.22: Load-displacement curves for a retaining wall after Wilson and Elgamal (2010) . 62

Figure 2.23: Flow chart describing various seismic analysis methods in vogue for analysing

retaining walls ............................................................................................................................. 66

Figure 2.24: Forces acting on a soil wedge for an active case in the M-O analysis ................... 69

Figure 2.25: Forces acting on a soil wedge for a passive case in the M-O analysis ................... 70

Figure 2.26: Wall geometry considered in the Steedman and Zeng (1990) model ..................... 73

Figure 2.27: Finite difference model of a retaining wall proposed by Green et al. (2003) ......... 77

Figure 2.28: Shaking table experiment conducted by Mononobe and Matsuo (1929) ............... 78

Figure 2.29: Shaking table model used by Kloukinas et al. (2015) ............................................ 79

Figure 2.30: Cross section of the centrifuge test conducted by Nakamura (2006) ..................... 81

Figure 2.31: Cross section of centrifuge test conducted by Geraili et al. (2016): a) basement type

retaining wall and b) U-shaped retaining wall with cantilever sides .......................................... 82

Figure 2.32: Cross section of centrifuge tests conducted by Jo et al. (2017): a) Model (wall

height 5.4m), b) Model (wall height 10.8m) ............................................................................... 83

Figure 2.33: Forces acting on a wall-soil system proposed by Richards and Elms (1979)......... 89

Figure 2.34: Comparison between relative displacement predicted by FLAC, Newmark classic

and modified Newmark’s procedure After Corigliano et al. (2011) ........................................... 93

Figure 2.35: Numerical model of a retaining wall proposed by Conti et al. (2013) ................... 93

Figure 2.36: Cross section of 2 retaining walls used in shaking table tests conducted by

Sadrekarimi (2011) ..................................................................................................................... 94

Figure 2.37: Cross section of centrifuge test conducted by Zeng and Steedman (2000) ............ 95

Figure 2.38: Analytical model of a retaining wall proposed by Veletsos and Younan (1997) ... 98

Figure 2.39: Distributions of wall pressure for statically excited systems with different wall and

base flexibilities: a) dθ = 0, b) dw = 0. After Veletsos and Younan (1997) .................................. 99

List of Figures

12

Figure 2.40: Geometry of an intermediate wedge during an earthquake proposed by Zhang et al.

(1998b) ...................................................................................................................................... 100

Figure 2.41: Reduction of seismic earth pressure when the retaining wall moves away from the

backfill soil, as proposed by Zhang et al. (1998b) .................................................................... 101

Figure 2.42: Analytical model of a wall-soil system proposed by Richards et al. (1999) ........ 102

Figure 2.43: Relationship between seismic passive earth pressure and normalised wall

displacement predicted by Song and Zhang (2008) .................................................................. 102

Figure 2.44: Finite element model of xx proposed by Psarropoulos et al. (2005a) .................. 104

Figure 2.45: Distribution of earth pressure in: a) ω= ω1/6 (almost static) - dθ =0.5, b) ω= ω1/6

(almost static) - dθ =5, c) ω = ω1 (resonance) - dθ =0.5, and d) ω = ω1 (resonance) - dθ =0.5.

After Psarropoulos et al. (2005) ................................................................................................ 105

Figure 2.46: Effect of wall rotational flexibility on the amplification factor of total forces acting

on the retaining wall. After Psarropoulos et al. (2005) ............................................................. 105

Figure 2.47: Cross section of the shaking table test conducted by Ishibashi and Fang (1987) 107

Figure 2.48: Effect of wall rotation about its base on the distribution of seismic earth pressure.

After Ishibashi and Fang (1987) ............................................................................................... 107

Figure 2.49: Effect of wall rotation about the top on the distribution of seismic earth pressure.

After Ishibashi and Fang (1987) ............................................................................................... 108

Figure 2.50: Details of a typical retaining wall failure (a) actual photograph, (b) diagram

capturing the failure of the u-shaped channels After Clough and Fragaszy (1977) .................. 111

Figure 2.51: Leaning-type concrete walls a) cross section, b) sketch. After Koseki et al. (1995)

.................................................................................................................................................. 112

Figure 2.52: Gravity retaining walls a) cross section, b) sketch. After Koseki et al. (1995) .... 112

Figure 2.53: Cantilever reinforced concrete walls a) cross section, b) sketch. After Koseki et al.

(1995) ........................................................................................................................................ 113

Figure 2.54: Cantilever reinforced concrete wall supporting slope a) cross section, b) sketch.

After Koseki et al. (1995) ......................................................................................................... 113

Figure 2.55: Failure of retaining walls caused by a) Chi-Chi earthquake1999, b) Niigata-Ken

Chuetsu earthquake, 2004 ......................................................................................................... 114

Figure 3.1: Flow chart summarising the steps to model and analyse the retaining wall using

PLAXIS and AQAQUS ............................................................................................................ 119

List of Figures

13

Figure 3.2: Retaining walls analysed in the current study considering a 2D plane strain

idealization ................................................................................................................................ 120

Figure 3.3: Finite element model of the wall-soil system used for the present study for a: (a)

rigid retaining wall, (b) cantilever retaining wall ..................................................................... 121

Figure 3.4: 6-noded triangular element in local coordinates ..................................................... 122

Figure 3.5: 3-noded plate element in local coordinates ............................................................ 123

Figure 3.6: Wall-soil interface element ..................................................................................... 124

Figure 3.7: 4-noded bilinear plane strain element CPE4 .......................................................... 124

Figure 3.8: Hyperbolic stress-strain law of hardening soil model after Brinkgreve et al. (2016)

.................................................................................................................................................. 128

Figure 3.9: Shear modulus – strain behaviour of soil with typical strain ranges for laboratory

tests and geotechnical structures after Brinkgreve et al. (2016) ............................................... 130

Figure 3.10: Stiffness reduction curve Brinkgreve et al. (2016) ............................................... 131

Figure 3.11: Damping in HSsmall model Brinkgreve et al. (2016) .......................................... 132

Figure 3.12: Real earthquake time history of the 1989 Loma Prieta earthquake: a) acceleration ,

b) frequency domain ................................................................................................................. 136

Figure 4.1: Saito (1999) centrifuge test model.......................................................................... 140

Figure 4.2: Nakamura (2006) centrifuge test model ................................................................. 141

Figure 4.3: Jo et al. (2014) centrifuge test: a) model with a wall height of 10.8 cm, b) model

with a wall height of 21.6 cm .................................................................................................... 142

Figure 4.4: Finite element model of Saito (1999) centrifuge test ............................................. 144

Figure 4.5: Finite element model of Nakamura (2006) centrifuge test .................................... 144

Figure 4.6: Finite element model of Jo et al. (2014) centrifuge test ......................................... 145

Figure 4.7: Mode shapes for Saito (1999) centrifuge test model obtained from the current finite

element study: : a) 1st mode, b) 2

nd mode ................................................................................ 147

Figure 4.8: as 4.7 above ............................................................................................................ 148

Figure 4.9: Finite element mesh sensitivity analysis for modelling (a) the Saito (1999) and

Nakamura (2006) centrifuge tests, (b) the Jo et al. (2014) centrifuge test ................................ 151

Figure 4.10: a) Sinusoidal wave applied at the base of the Saito (1999) test and the FE model, b)

Horizontal displacement at the base of the wall, recorded by test and obtained from the current

FE study .................................................................................................................................... 153

List of Figures

14

Figure 4.11: Residual deformation of the wall-soil system after the end of the eartquake shaking

a) Experimental results of the Saito (1999) centrifuge, b) Current results of FE model .......... 153

Figure 4.12: a) Sinusoidal wave applied at the base of the Nakamura (2006) test and the FE

model, b) Horizontal displacement at the top of the wall, recorded by test and obtained from the

current FE study ........................................................................................................................ 155

Figure 4.13: Residual deformation of the wall-soil system after the end of the earthquake

shaking a) Experimental results of the Nakamura (2006) centrifuge, b) Current results of FE

model ........................................................................................................................................ 156


model, b) Total seismic earth pressure force increment recorded by test ,c) Total seismic earth

pressure force increment obtained from the current FE study .................................................. 157

Figure 4.15: Sinusoidal wave applied at the base of the Nakamura (2006) test and the FE model

.................................................................................................................................................. 158

Figure 4.16: Distribution of seismic earth pressure along the height of the wall recorded by test

and obtained from the current FE study: a) active state at t = 8.34 sec, b) passive state at t =

8.58 sec ..................................................................................................................................... 158

Figure 4.17: Deformation shape of a cantilever retaining wall during its construction process 159

Figure 4.18: Distribution of static earth pressures along the height of the wall for: a) H = 5.4 m,

b) H = 10 m ............................................................................................................................... 160

Figure 5.1: Sketch of a gravity retaining wall showing seismic earth pressure, wall inertia

forces, direction of wall movement and important locations of interest. .................................. 163

Figure 5.2: FE model of the gravity retaining wall ................................................................... 164


b) frequency domain ................................................................................................................. 165

Figure 5.4: Acceleration, wall seismic inertia force and wall and soil displacement directions167

Figure 5.5: Acceleration response at different locations in the wall-soil system ...................... 168

Figure 5.6: Horizontal displacement at different locations in the wall-soil system .................. 169

Figure 5.7: Comparison between relative horizontal displacement predicted by the present FE

analysis and computed by the Newmark sliding block method ................................................ 172

Figure 5.8: Rotation of the retaining wall ................................................................................. 173

Figure 5.9: Wall seismic inertia force ....................................................................................... 174

List of Figures

15

Figure 5.10: Seismic earth pressure force P: a) obtained from the FE model , b) simplified

version of (a) ............................................................................................................................. 175

Figure 5.11: Distribution of seismic earth pressure along the height of the retaining wall ...... 179

Figure 5.12: Phase difference between seismic earth pressure force increment and wall seismic

inertia force ............................................................................................................................... 180

Figure 5.13: Relative horizontal displacement between the retaining wall and backfill soil .... 182

Figure 5.14: Effect of retaining wall height on seismic response of wall-soil system considering

different amplitudes of the applied earthquake acceleration ..................................................... 185

Figure 5.15: Design chart demonstrating the relationship between seismic earth pressure and

wall displacement for different retaining wall heights .............................................................. 187

Figure 5.16: Variation of relative horizontal displacement between wall and foundation soil with

acceleration levels for different retaining wall heights ............................................................. 187

Figure 5.17: Acceleration response at the top of retaining wall and backfill soil for different

amplitudes and frequency content of the applied earthquake acceleration ............................... 190

Figure 5.18: Relative horizontal displacement between retaining wall and foundation soil for

different amplitudes and frequency content of the applied earthquake acceleration ................ 192

Figure 5.19: Seismic earth pressure force for different amplitudes and frequency content of the

applied earthquake acceleration ................................................................................................ 194

Figure 5.20: Relationship between seismic earth pressure and displacement of the retaining wall

for different amplitudes and frequency content of the applied earthquake acceleration........... 197

Figure 5.21: Relationship between relative horizontal displacement and acceleration amplitude

for different frequency content of the applied earthquake acceleration .................................... 197

Figure 5.22: Different combinations of relative densities of backfill and foundation soil (a) 1st

combination, (b) 2nd

combination and (c) 3rd

combination ....................................................... 199

Figure 5.23: Earthquake acceleration applied at the base of FE model to investigate the effect of

relative density of soil materials on the seismic response of wall-soil system ......................... 199

Figure 5.24: Effect of soil material relative density on the seismic response of wall-soil system

.................................................................................................................................................. 203

Figure 5.25: Effect of backfill soil relative density on the seismic response of wall-soil system

.................................................................................................................................................. 205

Figure 5.26: Effect of foundation relative density on the seismic response of wall-soil system

.................................................................................................................................................. 207

List of Figures

16

Figure 6.1: a) Sketch of a cantilever retaining wall showing important locations of interest, b)

Development of shear force and bending moment in the stem, c) Sliding of base slab relatively

to foundation soil and rotation of the wall about its toe ............................................................ 211

Figure 6.2: Forces acting on the cantilever retaining wall for: a) structural integrity analysis, and

b) global stability analysis ........................................................................................................ 213

Figure 6.3: FE model of the cantilever retaining wall .............................................................. 214

Figure 6.4: Acceleration response at different locations in the wall-soil system ...................... 216

Figure 6.5: Wall and backfill seismic inertia forces ................................................................. 217

Figure 6.6: Seismic earth pressure force: a) behind the stem, Pstem, b) along the xx Pstem ........ 219

Figure 6.7: Distribution of seismic earth pressures along the height of the wall-soil system: a)

Immediately before the seismic analysis at t = 0 sec, b) At t = 3.9 sec of earthquake acceleration,

c) At t = 4.5 sec of earthquake acceleration, d) At the end of seismic analysis (t = 30 sec) ..... 222

Figure 6.8: Total seismic earth pressure force increments, Pstem, Pvp and wall and backfill

seismic inertia forces Fwa, Fwp, Fsa, Fsp ...................................................................................... 223

Figure 6.9: Variation of a) Shear force, b) Bending moment at the base of the stem ............... 224

Figure 6.10: Horizontal displacement at different locations in the wall-soil system ................ 226

Figure 6.11: Relative horizontal displacement between the wall and foundation soil as well as

between the backfill soil above base slab and foundation soil .................................................. 228

Figure 6.12: Rotation of the stem.............................................................................................. 230

Figure 6.13: Rotation of base slab about the toe ....................................................................... 231

Figure 6.14: Deformation shapes of the stem and base slab at different durations during the

earthquake ................................................................................................................................. 232



Figure 6.16: Seismic earth pressure force behind the stem and along the virtual line for different


Figure 6.17: Shear force and bending moment at the base of the stem for different amplitudes

and frequency content of the applied earthquake acceleration ................................................. 237

Figure 6.18: Relative horizontal displacement of the cantilever retaining wall for different




List of Figures

17

Figure 6.20: Effect of the natural frequency of the retaining wall on the seismic earth pressure

behind the stem ......................................................................................................................... 242

Figure 6.21: Effect of the natural frequency of retaining wall on the seismic earth pressure force

along virtual plane ..................................................................................................................... 243

Figure 6.22: Effect of the natural frequency of the retaining wall on the development of shear

force predicted at the base of stem ............................................................................................ 244

Figure 6.23: Effect of the natural frequency of the retaining wall on the development of bending

moment predicted at the base of the stem ................................................................................. 245

Figure 6.24: Effect of the natural frequency of the cantilever retaining wall on the relative

horizontal displacement of retaining wall ................................................................................. 247

Figure 6.25: Effect of soil relative density of soil on the acceleration response at the top of: a)

the retaining wall and b) backfill soil........................................................................................ 248

Figure 6.26: Effect of soil relative density of soil on the seismic earth pressure forces behind the

stem and along the virtual plane ............................................................................................... 250

Figure 6.27: Effect of soil relative density of soil on the shear force and bending moment..... 251

Figure 6.28: Effect of relative density of soil on the relative horizontal displacement of the

cantilever retaining wall ............................................................................................................ 252

Figure 7.1: Free body diagram of external forces acting on the stem of the wall during the

earthquake, producing shear force and bending moment .......................................................... 256

Figure 7.2: Effect of wall seismic inertia force on Nw and Mw for the top ⅓H of the stem....... 258

Figure 7.3: Effect of wall seismic inertia force on Nw and Mw for the mid-height of the stem . 260

Figure 7.4: Effect of wall seismic inertia force on Nw and Mw for the bottom ⅓H of the stem 261

Figure 7.5: Forces acting in the wall-soil system causing sliding of the wall ........................... 265

Figure 7.6: Relative horizontal displacement comparison between the modified Newmark

procedure, current study FE results obtained from Chapter 5 and the classic Newmark sliding

block method ............................................................................................................................. 269

Figure 7.7: Forces acting on the cantilever wall-soil system causing the sliding of the retaining

wall ............................................................................................................................................ 270

Figure 7.8: Comparison between the relative horizontal displacement predicted by current

simplified procedure and FE results obtained from Chapter 6 as well as Newmark sliding block

method as above ........................................................................................................................ 274

List of Tables

18

LIST OF TABLES

Table 2.1: Major findings concerning the relationship between static earth pressure and

displacement of the retaining wall .................................................................................. 64

Table 2.2: Major findings and contradictions of the force-based design methods ......... 85

Table 2.3: Observations and contradictions in the estimation of seismic earth pressure

for a cantilever-type retaining wall ................................................................................. 87

Table 2.4: Major findings highlighting the relationship between the seismic earth

pressure and wall displacement. .................................................................................... 110

Table 3.1: Wall parameters required to run the FE model ............................................ 126

Table 3.2: Soil parameters required to run the FE model ............................................. 133

Table 4.1: Centrifuge and prototype model dimensions for Saito (1999), Nakamura

(2006) and Jo et al. (2014) test model ........................................................................... 143

Table 4.2: Parameters required to run the FE model simulations for the 3 centrifuge tests

....................................................................................................................................... 146

Table 4.3: Comparison of natural frequency of three different models predicted in

present study with results of natural frequency obtained from the previous studies .... 150

Table 5.1: Soil and retaining wall parameters chosen for the present study ................. 165

Table 5.2: Parameters required for running FE model considering different relative

densities of soil material................................................................................................ 200

Table 6.1: Parameters of soil and retaining wall used to run the FE model ................. 215

Table 7.1: Effect of wall seismic inertia force on Nw and Mw for the top ⅓H of the stem

....................................................................................................................................... 259

Table 7.2: Effect of wall seismic inertia force on Nw and Mw for the mid-height of the

stem ............................................................................................................................... 260

Table 7.3: Effect of wall seismic inertia force on Nw and Mw for the bottom ⅓H of the

stem ............................................................................................................................... 262

List of Symbols

19

LIST OF SYMBOLS

The following symbols are used in this thesis:

A = area of stem cross section;

a = acceleration;

a(g) = ratio between horizontal acceleration and gravitational acceleration;

amax = peak ground acceleration;

as(t) = acceleration response of backfill soil ;

ase(t) = elemental soil acceleration ;

aw(t) = acceleration response of retaining wall ;

awe(t) = elemental wall acceleration ;

ah = pseudo-static horizontal acceleration;

an(t) = predicted acceleration-time history for the nth element;

arel = relative acceleration of wall-soil system;

ay = yield acceleration;

av = pseudo-static vertical acceleration;

b = width of base slab;

[C] = damping FE matrix of the system;

c = cohesion of soil;

c = effective cohesion of soil;

ci = cohesion (adhesion) of the interface of soil;

Dr = relative density;

Dw = flexural rigidity of the wall;

dper = permanent block displacement;

dw = relative flexibility of the wall;

dθ = relative flexibility of rotational base constraint;

List of Symbols

20

E = modulus of elasticity;

EA = axial stiffness;

Ew = modulus of elasticity of the wall;

EI = flexural stiffness of the retaining wall;

E50 = secant modulus at 50% of maximum soil strength;

EOed = oedometer secant modulus;

Eur = unloading –reloading secant modulus;

50

refE = reference secant modulus at 50% of maximum soil strength;

ref

OedE = reference oedometer secant modulus;

ref

urE = reference unloading –reloading secant modulus;

Fdriving = total horizontal driving force;

FR = base frictional resistance force;

[F] = seismic load factor;

Fs = total horizontal seismic inertia force of backfill soil above the heel;

Fsa = seismic inertia force of backfill soil above the heel acting away from the

backfill soil;

Fsp = seismic inertia force of backfill soil above the heel acting towards the

backfill soil;

Fw = total horizontal seismic inertia force of a cantilever retaining wall;

Fwa = seismic inertia force of the wall acting away from the backfill soil;

Fwp = seismic inertia force of the wall acting towards the backfill soil;

f = frequency content of earthquake acceleration;

fa = amplification factor;

fas = amplification factor of soil;

faw = amplification factor of wall;

fmax = maximum frequency of the input acceleration;

fn1 = 1st shape mode;

fn2 = 2nd

shape mode;

List of Symbols

21

Go = initial soil shear modulus;

Gs = secant shear modulus of soil;

Gur = secant shear modulus of unloading –reloading;

50

refG = initial shear modulus at reference pressure

g = gravitational acceleration 9.81m/sec2;

H = height of a retaining wall;

Hstem = height of the stem;

h =

thickness of foundation soil;

hemax = maximum height of the element;

I = moment of inertia;

K = earth pressure coefficient;

Ka = coefficient of active earth pressure;

Kae = coefficient of seismic active earth pressure;

Kp = coefficient of passive earth pressure;

Ko = coefficient of earth pressure in at-rest;

kh = ratio between horizontal acceleration and gravity acceleration;

ky = yield acceleration coefficient;

kv = ratio between vertical acceleration and gravity acceleration;

kr = relative acceleration of the wall-soil;

[K] = stiffness FE matrix of the system;

[M] = mass FE matrix of the system;

Mw = bending moment;

m = mass;

ms = mass of the soil

mw = mass of the wall;

mn = mass of an element;

mse = mass of backfill soil element;

List of Symbols

22

mwe = mass of an wall element;

Nw = shear force;

n = number of element;

OCR = over-consolidation ratio;

P = seismic earth pressure force;

Pa = static active earth pressure force;

Pa(h) = horizontal component static earth pressure force;

Pa(v) = vertical component static earth pressure force;

Pae = total seismic active earth pressure force;

Ppe = total seismic passive earth pressure force;

Po = at-rest earth pressure force;

Pstem_n = elemental seismic earth pressure force behind the stem;

Pre = residual seismic earth pressure force;

Pstem (static) = static earth pressure behind the stem;

Pstem = seismic earth pressure force behind the stem;

Pvp(static) = static earth pressure force along the virtual plane;

Pvp = seismic earth pressure force computed along the virtual vertical plane;

Pvp(h) = horizontal component of seismic earth pressure force computed along

the virtual vertical plane;

Pvp(v) = vertical component of seismic earth pressure force computed along the

virtual vertical plane;

pref

= reference confining pressure;

pstem = earth pressure behind the stem;

pvp = earth pressure computed along the virtual vertical plane;

Qh = pseudo-dynamic backfill seismic inertia force;

qa = ultimate Soil strength;

qf = soil strength at failure;

Rf = failure ratio;

List of Symbols

23

Rinter = interface strength reduction factor;

Rθ = stiffness of the rotational base constraint;

t = time;

tw = thickness of the wall;

v = velocity;

vmax = peak ground velocity;

vs = velocity of the shear wave propagating through the soil;

u = displacement;

w = weight of the wall per unit length;

Ws = self-weight of the soil above the base slab of the retaining wall;

Ww = self-weight of the cantilever-type retaining wall;

x = elastic deflection of the stem in x-axis;

y = stress-level dependency of the stiffness of the soil;

z = depth of retaining wall;

zn = perpendicular distance between the nth element and the base of the wall;

α = mass Rayleigh parameter;

αa = Newmark integration scheme coefficient;

ae = angle of inclination of seismic active wedge with horizontal;

pe = angle of inclination of seismic passive wedge with horizontal;

β = stiffness Rayleigh parameters;

βb = Newmark integration scheme coefficient;

γ = unit weight;

γr = threshold shear strain ;

γs = unit weight of soil;

γ0.7 = reference shear strain at 70% of 50

refG

γw = unit weight of the cantilever retaining wall;

Δbase_stem = total horizontal displacement response at the top of the stem;

List of Symbols

24

Δbase_wall = total horizontal displacement response at the base of the retaining wall;

Δheel(y) = total vertical displacement response at the heel;

Δtoe(y) = total vertical displacement response at the toe;

Δtop_stem = total horizontal displacement response at the top of the stem;

Δtop_wall = total horizontal displacement response at the top of the retaining wall;

ΔP = earth pressure force increment;

ΔPae = seismic active earth pressure force increment;

ΔPpe = seismic passive earth pressure force increment;

ΔPstem = seismic earth pressure force increment behind the stem;

ΔPvp = seismic earth pressure force increment computed along the virtual

vertical plane;

ΔS-F = relative horizontal displacement between the backfill soil and the

foundation soil;

W-B = relative horizontal displacement between the retaining wall and the

backfill soil;

W-F = relative horizontal displacement between the retaining wall and the

foundation soil;

= friction angle between retaining wall and backfill soil;

b = friction angle between retaining wall and foundation soil;

θ = angle of inclination of wall-soil interface;

θslab = rotation of the base slab;

θstem = rotation of the stem;

θw = rotation of the wall;

λmin = minimum wavelength of the shear wave;

ζ = viscous damping ratio;

v = Poisson’s ratio;

vur = Poisson’s ratio for unloading-reloading;

3 = effective confining pressure;

h = horizontal effective stress at any depth behind the wall below the soil

surface;

List of Symbols

25

ha = horizontal stress at Gauss integration point in soil elements that contact

with wall

hb = horizontal stress at Gauss integration point in soil elements that contact

with wall

n = normal stress;

v = vertical effective stress at any depth behind the wall below the soil

surface

f = shear force at failure;

= effective friction angle of the soil;

cs = critical state effective friction angle of the soil;

i = friction angle of interface element;

m = mobilised effective friction angle of the soil;

ψ = dilatancy angle of the soil;

ψa = pseudo-static acceleration angle;

ψm = mobilised dilatancy angle of the soil;

𝜔 = circular frequency of earthquake acceleration;

𝜔z1 = first natural circular frequency of the FE model;

and

𝜔z2 = second natural circular frequency of the FE model.

Abstract

26

ABSTRACT

This thesis presents a unique finite element investigation of the seismic behaviour of 2

retaining wall types – a rigid retaining wall and a cantilever retaining wall. The commercial

finite element program PLAXIS2D was used to develop the numerical simulation models.

The research includes: (1) validating the finite element model with the results of 3

previously existing centrifuge tests taken from literature; (2) investigating the seismic

response of rigid and cantilever retaining walls including studying the effects of

contribution of wall displacement, wall and backfill seismic inertia and stiffness of the

foundation soil; (3) developing analytical methods to concrete the findings of the numerical

models.

Based on the results of the seismic response of a rigid retaining wall, a unique relationship

between the seismic earth pressure and wall displacement has been developed for the active

and passive modes of failure. The seismic active earth pressure has been found to be not

dependent on the wall displacement while the seismic passive earth pressure has been found

to be highly affected by the wall displacement. The maximum seismic passive earth

pressure force and relative horizontal displacement are predicted when the ground

earthquake acceleration is applied with maximum amplitude and minimum frequency

content. The seismic response of the wall was not affected by the ratio of the frequency

content of the earthquake to the natural frequency of the wall-soil system.

For the cantilever retaining wall detailed structural integrity and global analyses have been

carried out. It has been observed that the seismic earth pressure, computed at the stem and

along a vertical virtual plane are found to be out of phase with each other during the entire

duration of the earthquake, and hence, the structural integrity and global stability should be

evaluated and assessed individually. A critical case for the structural integrity is observed

when the earthquake acceleration is applied towards the backfill soil and has frequency

content close to the natural frequency of the retaining wall, while, for the global stability,

the critical case is observed when the earthquake acceleration has maximum amplitude and

is applied towards the backfill soil with minimum frequency content. The structural

integrity is also found to be highly dependent on the ratio between the frequency content of

earthquake acceleration to the natural frequency of the cantilever retaining wall.

The relative horizontal displacement of a rigid and cantilever retaining wall is found to be

highly affected by the duration of the earthquake in contrast to what has been observed for

the seismic earth pressure force. The structural integrity of a rigid and cantilever retaining

wall reduces when the backfill soil has a higher relative density, while the global stability

increases when the backfill soil has a high relative density during an earthquake.

The results obtained from the analytical methods reveal that the wall seismic inertia force

has a significant effect on the structural integrity only for the top of the stem while the base

of the stem does not get affected significantly. The modified Newmark sliding block

method provided a more reasonable estimation of the relative horizontal displacement of a

rigid retaining wall and a cantilever retaining wall compared with the classic Newmark

sliding block method.

Keywords: earthquake, retaining wall, seismic earth pressure, displacement, inertia force

Declaration

27

DECLARATION

No portion of the work referred to in the thesis has been submitted in support of an

application for another degree or qualification of this or any other university or other

institute of learning.

Copyright Statement

28

COPYRIGHT STATEMENT

The Author of this thesis (including any appendices and/or schedules to this thesis)

owns any copyright in it (the “Copyright”) and he has given The University of

Manchester the right to use such Copyright for any administrative, promotional,

educational and/or teaching purposes.

Copies of this thesis, either in full or in extracts, may be made only in accordance with

the regulations of the John Ryland’s University Library of Manchester. Details of these

regulations may be obtained from the Librarian. This page must form part of any such

copies made.

The ownership of any patents, designs, trade marks and any and all other intellectual

property rights except for the Copyright (the “Intellectual Property Rights”) and any re-

productions of copyright works, for example graphs and tables (“Reproductions”),

which may be described in this thesis, may not be owned by the author and may be

owned by third parties. Such Intellectual Property Rights and Reproductions cannot and

must not be made available for use without the prior written permission of the owner(s)

of the relevant Intellectual Property Rights and/or Reproductions.

Further information on the conditions under which disclosure, publication and

exploitation of this thesis, the Copyright and any Intellectual Property Rights and/or

Reproductions described in it may take place is available from the Head of School of

Mechanical, Aerospace and Civil Engineering.

Dedication

29

DEDICATION

To my parents for all the sacrifices they have made to ensure I obtain the

best education possible;

To my brothers and sisters for their encouragement and their support;

Acknowledgement

30

ACKNOWLEDGEMENT

First and foremost, I wish to give all the praise to Almighty God for giving me the

strength and time to complete this research.

I wish to express my deepest gratitude to my supervisor, Dr Mohd Ahmad Syed, for his

constant encouragement, wisdom guidance and helpful advices, comments and

suggestions during the undertaking of this research. He provided me with all kinds of

support during my PhD study.

I wish to express my sincere thanks for the financial support given by the Iraqi Ministry

of Higher Education and Scientific Research. The efforts made by the Iraqi

embassy/cultural attaché to assist with the financial and administration issues during my

scholarship are really appreciated.

Finally, I would like to express my deepest gratitude to my father, my mother, my

brothers, my sisters for their unflinching support, encouragement and love. Without

them, this would not have been possible.

My deepest appreciation goes to all members and friends at the School of Mechanical,

Aerospace and Civil Engineering, University of Manchester who supported me in all

respects during my PhD research. I am using this opportunity to express my deepest

gratitude to my friends Firas Maan Abdulsattar, Laith Farhan, and Bashar Ismaeel who

supported me throughout this research. I am thankful for their aspiring support.

Chapter 1: Introduction

32

CHAPTER 1

INTRODUCTION

1.1 BACKGROUND

Retaining walls are one of the most important civil engineering structures, widely used

for highways, tunnels, mines and military defences. From the geotechnical engineering

perspective, a retaining structure is constructed to provide lateral support to soil and

rock, and as such they are designed to resist the lateral earth pressure force. Hence, an

accurate estimation of the lateral earth pressure force is crucial for their safe design.

Pioneering work on the estimation of the earth pressure for a static case was first

presented by Coulomb (1776) and Rankine (1857), who proposed the classic Coulomb’s

and Rankine’s static earth pressure theories, respectively. Both of these theories were

based on force-based methods as they disregard the effect of the wall displacement on

the development of the static earth pressure. Further research by virtue of analytical,

numerical and experimental methods, on the contrary, has revealed that the magnitude

and distribution of static earth pressure are significantly affected by the wall

displacement. Across the world, several researchers have devised different approaches

and proposed various models to study the relationship between static earth pressure and

wall displacement, but owing to the fact that such a problem is one of the most

complicated soil-structure-interaction problems, this area is still not understood very

well.

Further, as the retaining structures are also constructed in earthquake prone areas, an

accurate estimation of seismically induced lateral earth pressures on retaining structures,

called as ‘seismic earth pressure’, is crucial for a safe design of retaining walls. Okabe

(1926) and Mononobe and Matsuo (1929) did a pioneering work to propose a force-

based method by extending the Coulomb’s earth pressure theory and proposed a force-

based method (now famously called as the ‘Mononobe-Okabe method’) to estimate the

seismic earth pressure. Since then many researchers have developed this further and


33

proposed new analytical, numerical, and experimental methods and solutions to

understand the development of seismic earth pressure behind retaining structures.

Despite the fact that many theoretical, numerical and experimental studies have been

presented on the subject of seismic earth pressures and a variety of design methods have

been developed in the last several decades, there seems to be no general agreement

about the validity and applicability of design methods like M-O method and a proper

seismic design method for retaining walls (more about this in Chapter 2). Hence, it can

be noted that the seismic response of the retaining walls is quite complex compared

with their static response, and earthquakes could cause significant structural damage

with disastrous physical and economic consequences. Figure 1.1a shows a typical

damage to a retaining wall of 2-3m height as well, while Figure 1.1b shows a typical

damage to a dam spillway retaining walls immediately after the Kumamoto earthquake

in Japan occurred on the 16th

of April 2016.

Figure 1.1: Failure of retaining walls during the Kumamoto earthquake in Japan occurred on the

16th of April 2016 (a) gravity retaining wall, (b) dam spillway retaining wall (after Kiyota et al.,

2017)

On other hand, it is well-documented in available literature that during an earthquake,

retaining walls also undergo large displacements and rotations, and consequently affect

the development and distribution of seismic earth pressure. For example, Prakash and

Wu (1996) presented a real-field observation-based study and reported that the retaining

walls were damaged by excessive displacements in the form of lateral sliding and

rotation during earthquake events. In the past several decades, therefore, researchers

have developed alternative design procedures in order to account for the wall

(a) (b)


34

displacements while estimating the seismic earth pressure like Richards and Elms

(1979). However, there is no general agreement about the type of relationship between

wall displacement and seismic earth pressure and as well as the deformation mechanism

of a combined wall-soil system.

In addition, comparing the existing research methods, proposed for investigating the

seismic response of a rigid retaining wall with those proposed for a cantilever retaining

wall, it is observed that only a handful of methods are available which try to investigate

the seismic behaviour of a cantilever retaining wall. The same seismic design methods,

which have been developed for the seismic design of a rigid retaining wall, have also

been used for the seismic design of a cantilever retaining wall, despite the fact that the

rigid retaining wall behaves as a rigid structure while the cantilever retaining wall

behaves a flexible structure.

The present study, therefore, aims to improve the existing body of knowledge of the

seismic behaviour of 2 retaining wall types – a rigid-type retaining wall and a

cantilever-type retaining wall. A finite element (FE) method is proposed to be used in

the current study in order to investigate the seismic response of these 2 types of

retaining walls.

1.2 RESEARCH AIM AND OBJECTIVES

The main aim of this study is to evaluate and provide a better understanding of the

seismic behaviour of 2 types of retaining walls viz., a rigid-type retaining wall and a

cantilever-type retaining wall for a safe and economic design. Specific objectives for

each of the 2 type of walls are noted below.

1.2.1 Objectives for a rigid retaining wall

For a rigid retaining wall, the following objectives are outlined:

1. To investigate the validity of M-O and Newmark sliding block methods for the

seismic analysis of a rigid retaining wall;

2. To study the deformation mechanism of a rigid retaining wall under seismic loading;

3. To investigate, develop, and propose a relationship between the seismic earth

pressure and displacement of a rigid retaining wall;


35

4. To propose unique design charts for the above relationship;

5. To investigate the effects of: a natural frequency of a rigid retaining wall (or in other

words its height), applied earthquake acceleration amplitude and frequency content;

relative density of backfill and foundation soil on the seismic response of a rigid

retaining wall;

6. To assess and analyse the effect and contribution of seismic earth pressure to the

permanent displacement of the retaining wall;

7. To propose a simplified procedure to modify the Newmark sliding block method

(reference) for accurately predicting the seismic permanent displacement of a rigid

retaining wall.

1.2.2 Objectives for a cantilever-type retaining wall

For a cantilever retaining wall, the following objectives are outlined:

1. To investigate the effect of the development of seismic earth pressure on the

structural and global stability of the cantilever retaining wall;

2. To study the deformation mechanism of a cantilever retaining wall under seismic

loading;

3. To identify a critical loading scenario which causes the failure of a cantilever

retaining wall for the structural integrity and global stability;

4. To investigate the effects of: a natural frequency of a cantilever retaining wall

(or in other words its height), applied earthquake acceleration amplitude and

frequency content; relative density of backfill and foundation soil on the

structural integrity and global stability of a cantilever retaining wall;

5. To assess and analyse the effect and contribution of seismic earth pressure to the

permanent displacement of the cantilever retaining wall;

6. To assess and analyse the effect and contribution of wall seismic inertia force to the

shear force and bending moment developed on the stem of the wall during an

earthquake; and

7. To propose a simplified procedure to modify the Newmark sliding block method

(reference) for accurately predicting the seismic permanent displacement of a

cantilever retaining wall.


36

1.3 ORGANIZATION OF THE THESIS

The thesis titled “Performance-based approach for seismic stability analyses of retaining

walls” consists of 8 Chapters. The thesis outline is presented below:

Chapter 1 presents the problem background and motivation for the present study and

also outlines the aim and objectives of the research.

Chapter 2 presents a thorough literature review on earth pressure and various

analytical, numerical and experimental methodologies developed over the years for

the static and seismic analysis of retaining walls.

Chapter 3 details the research methodology adopted in the present study for

performing seismic analysis of retaining walls. This includes presenting an overview

of the PLAXIS2D software – a commercial specialist geotechnical software used int

eh present study. A detailed overview of the model idealisation, including details of

the FE element selection, mesh sizing, boundary conditions, seismic and static loads,

and constitutive modelling of soil and retaining wall material and as well as crucial

parameters used in the present study is also presented. This chapter ends with post

processing approach used in current research.

Chapter 4 presents the validation and comparison methodology adopted in the

present study to validate the results of the developed FE model.

Chapter 5 deals with the results and discussion obtained from the FE model of the

rigid retaining wall problem. The results include computing the accelerations,

horizontal displacement, horizontal inertia force of the retaining wall and seismic

earth pressure, while the discussion is about finding a relationship between the

seismic earth pressure and displacement of the retaining wall, and as well on

investigating the effect of the height of the retaining wall, earthquake characteristics

and relative densities of the backfill and foundation soil.

Chapter 6 details the results and discussion of the FE model for a cantilever-type

retaining wall. A clear explanation for the results is presented via figures and tables,

and the discussion is presented for analysing the development and effects of

accelerations, horizontal displacement, horizontal inertia force of the retaining wall,


37

seismic earth pressure, shear force and bending moment on the structural integrity

and global stability of the cantilever-type retaining wall. A parametric study to

analyse and investigate the effect of the height of the cantilever retaining wall,

earthquake characteristics, relative density of the backfill soil layer on the global and

structural integrity is also presented and important and unique conclusions are drawn.

Chapter 7 consists of 2 main parts: the first part includes presenting a simplified

procedure to estimate the contribution of wall seismic inertia force to the shear force

and bending moment of a cantilever-type retaining wall, while the second part

presents a simplified procedure of modifying the Newmark sliding block method to

predict accurately estimate the seismic permanent displacement for the rigid-type and

cantilever-type retaining walls.

Chapter 8 details the major findings of the present study and also outlines a brief

proposal for future work.

Chapter 2: Literature Review

38

CHAPTER 2

LITERATURE REVIEW

This chapter presents a literature survey of previous studies proposed to investigate the

static and seismic performance of retaining walls. It begins with a discussion of the

types of retaining wall and their failure modes. Then, the types of earth pressure are

presented. After that, this chapter discusses the main theories proposed to compute the

static earth pressure, and covers the previous research methods proposed to investigate

the relationship between the static earth pressure and displacement of the retaining wall.

The second part of this chapter covers the seismic design of the retaining wall. This part

presents the main design methods used in the seismic design of retaining walls. It

provides a critical discussion of previous analytical, numerical and experimental

methods proposed to investigate the seismic response of a retaining wall by using force-

based, displacement-based and force-displacement design methods. Finally, this chapter

ends with a brief discussion of real field observations of retaining wall damage reported

after some earthquakes.

2.1 RETAINING WALL

Retaining walls are one of the most important civil engineering structures. They may be

constructed from a variety of materials (concrete and steel) to provide lateral support for

any vertical or nearly vertical face of soil (both natural and made ground) or rock. They

are widely used in transportation systems, mines, underground structures and military

defences. The lateral earth pressure excreted by the retained soil material behind the

retaining wall is the most important force required to assess the stability of the retaining

wall and provide a safe design.

2.2 TYPES OF RETAINING WALL

Retaining walls are often classified according to their mass and flexibility. They can be

divided into the following two main types: gravity retaining walls and cantilever


39

retaining walls – in addition to other types of retaining wall like embedded retaining

walls, which is beyond the scope of present study.

2.2.1 Gravity retaining walls

A gravity retaining wall is one of the simplest types of retaining wall, as shown in

Figure 2.1. Its stability depends solely on its self-weight to resist the lateral earth

pressure exerted by the retained backfill soil layer. It does not bend because it is thick

and stiff, and hence it is considered as a rigid structure from an engineering perspective.

Its design procedure requires stability checks to resist’ checking for stability to resist

sliding and overturning. Sliding of the rigid retaining wall occurs because of the failure

in the friction resistance between the wall’s base and the foundation soil layer beneath

it. Overturning of the wall occurs when the retaining wall rotates about its toe due to

exceeding of moment of forces. Rigid retaining walls may be constructed from a

concrete mass, and they are usually used for low retained heights.

Figure 2.1: 3D-sketch of a rigid retaining wall

2.2.2 Cantilever retaining walls

A cantilever retaining wall is constructed as an inverted T-shape, as shown in

Figure 2.2, and it consists of the vertical part (stem) and the horizontal part (base slab).

Rigid wall

Backfill soil

Foundation soil


40

It maintains its stability from the weight of the soil above its base slab, as shown in

Figure 2.2, in addition to its self-weight. It bends because it is thin compared with a

gravity retaining wall and hence it is considered as a flexible structure from an

engineering perspective. The stability of a cantilever wall should be checked for the

same failure modes as for the gravity retaining wall (sliding and overturning). However,

the cantilever retaining wall should also be able to resist the shear force and bending

moments, which develop in the stem because of the lateral earth pressure exerted by the

backfill soil layer. The cantilever retaining wall is considered more economical

compared to the rigid retaining wall. Theoretically, in order to check the stability of the

cantilever retaining wall, the lateral earth pressure is calculated along the vertical virtual

line extending from the heel of the wall up to the backfill soil surface.

Figure 2.2: 3D-sketch of a cantilever retaining wall

2.3 RETAINING WALL FAILURE MODES

In order to design a retaining wall, the possible failure modes should be defined. The

retaining wall is acted upon by body forces related to the mass of the wall and the lateral

earth pressure exerted by the backfill soil. For a proper retaining wall design, these

forces should achieve equilibrium without reaching the shear stresses in the soil. In

certain instances, these forces may violate the equilibrium, causing the retaining wall to

fail by different modes.

Backfill soil

Foundation soil

Base slab

Stem


41

2.3.1 Rigid-body sliding failure mode

From a geotechnical engineering perspective, when the retaining wall has not been

adequately designed, it may fail because of the occurrence of a large and unacceptable

movement. For instance, when the friction resistance between the base of the retaining

wall and the foundation layer is not strong enough, the retaining wall may move as a

rigid body horizontally, as shown in Figure 2.3a, resulting in placing and compacting of

the backfill soil behind it. It may also move horizontally away from the backfill soil; as

a result, horizontal inertia forces will develop in the retaining wall and retained backfill

soil during the earthquake, and they will exceed the friction force resistance between the

retaining wall and the foundation layer.

2.3.2 Overturning failure mode

If the moment equilibrium is not satisfied in the wall-soil system, the retaining wall will

tend to completely rotate about the toe, as shown in Figure 2.3b. The bearing failure at

the foundation layer may occur because the retaining wall has been constructed on a

loose foundation layer which cannot maintain the wall’s weight. The retaining wall may

also fail by overturning mode when it has been constructed on a loose foundation layer

and it tends to liquefy during the earthquake.

2.3.3 Flexural failure mode

Most retaining walls like the cantilever retaining wall and embedded retaining wall are

considered flexible structures, so they experience both rigid body movement and

flexing. A cantilever retaining wall, as shown in Figure 2.3c, seems to slide and rotate

as a rigid body movement, while its stem seems to rotate because of elastic deflection

that occurs after lateral earth pressure has been exerted and causes increments in the

bending moment in the wall’s stem of the cantilever retaining wall. If the increment of

the bending moment exceeds the flexural strength of the wall, also seems to rotate about

a point above the bottom of the wall as a rigid body, in addition to the flexing, because

the lateral earth pressure forces behind and in front of the wall may violate equilibrium

and cause permanent deformation and flexural failure.


42

Figure 2.3: Failure modes of a retaining wall: (a) Sliding, (b) Overturning, (c) Flexure

2.4 STATIC EARTH PRESSURE

This section covers the development of static earth pressure. Three main static earth

pressure states are presented. After that, the main theories proposed to compute the

static earth pressure are discussed. Then, this section presents a critical discussion of the

research methods proposed to investigate the relationship between the seismic earth

pressure and displacement of the retaining wall.

2.4.1 Static earth pressure states

Terzaghi (1936) proposed the concept of an earth pressure coefficient (K) as below:

/h vK (2.1)


43

where, h = horizontal effective stress at any depth behind the wall below the soil

surface, and v = vertical effective stress at any depth behind the wall below the soil

surface, which for a dry sand equals the product of unit weight of the soil s and the

depth z s z . From his experiments, it was found that, when the retaining wall is

allowed to move away from the backfill soil, as shown in Figure 2.4, the horizontal

stress behind the wall (and hence the earth pressure coefficient) decreases, as shown in

Figure 2.5, after a relatively small displacement, and the minimum value of earth

pressure coefficient is reached. When the wall displacement increases no further

decrease in the pressure is observed. The minimum value of earth pressure coefficient is

called the coefficient of active earth pressure, Ka.

However, when the retaining wall is pushed towards the backfill soil from its original

position, as shown in Figure 2.5, the horizontal stress (and hence the earth pressure

coefficient) increases and it still increases for much larger displacements. However,

when the wall displacement is further increased, a constant pressure (and hence the

earth pressure coefficient) is reached, as shown in Figure 2.5. The maximum value of

lateral earth pressure coefficient is called the coefficient of passive earth pressure, Kp.

When the retaining wall is not allowed to move relative to the soil, the coefficient of

lateral earth pressure is called the coefficient of at-rest earth pressure.

Backfill soil Rigid retaining wall

Wall moves towards the

backfill soil

Wall moves away from the

z

Figure 2.4: Direction of wall movement and soil stresses


44

Figure 2.5: Development of active and passive earth pressure states based on wall displacement

(after Terzaghi, 1936)

2.4.1.1 Earth pressure at-rest

When no relative movement between the retaining wall and the soil is allowed to occur,

the soil is prevented from strain and no full shear strength is mobilised in the soil; the

at-rest earth pressure is exerted by the retained soil on the back of the retaining wall.

This condition can happen when the movement of the top and bottom of the retaining

wall is restrained, and thus the retaining wall is prevented from any movement. A

theoretical equation has been proposed by Jaky (1948) for normally consolidated soils

to compute lateral earth pressure coefficient (𝐾𝑜) in at-rest state:

1 sinoK (2.2)

where,= effective shear resistance of soil.

Mayne and Kulhawy (1982) have provided an empirical equation to compute the lateral

earth pressure in the at-rest condition taking into account the over-consolidation

condition of soil:

sin1 sin .oK OCR (2.3)

where, OCR = over-consolidation ratio.


45

2.4.1.2 Active earth pressure

As discussed above, when the retaining wall moves a relatively small displacement

away from the backfill soil, the earth pressure decreases from the at-rest value to the

minimum active earth pressure value. The earth pressure decreases from the at-rest

pressure to the active earth pressure because, when the wall moves away from the soil,

shear stresses are applied in the soil, and these shear stresses will mobilise the full shear

strength of the soil. At this state, the soil will fail.

Mohr (Clayton et al., 2014) showed that the stresses on and within a solid element in

plastic equilibrium could be represented by a circle. It can be seen from Figure 2.6 that

two points lie on the normal axis and they represent the compressive stresses of the

plane when the shear stress is equal to zero, and the normal stress is either at a

maximum or a minimum. The maximum value of compressive stresses is represented by

the vertical stress in the soil mass, while the minimum value is represented by the

horizontal stress. Coulomb was proposed at failure for non-cohesive soil that the shear

force f is related by a constant to the normal stress n as shown below:

tanf n (2.4)

It can be noted from Figure 2.6 that, within the at-rest state, the Mohr circle does not

attach to the Coulomb failure line while, as the retaining wall moves away from the soil,

the horizontal stresses seem to decrease up to the Mohr circle attaches the Coulomb

failure line, then the soil will fail and active earth pressure will develop behind the wall.


46

2.4.1.3 Passive earth pressure

On the other hand, if the retaining wall moves towards sandy soil from its at-rest

condition, the coefficient of earth pressure increases and it continues to increase for

much higher displacements. The constant value of earth pressure coefficient is reached

once again. At this condition, the full shear strength is also mobilised. Hence, a

relatively large force will be imposed on the back of the retaining wall. The maximum

earth pressure coefficient is called the passive earth pressure coefficient (Kp).

Mathematically, the development of passive earth pressure can also be represented by

using a Mohr circle, as shown in Figure 2.7. It can be noted from Figure 2.7 that, when

the retaining wall moves towards the soil, the vertical stress also remains the same and

the horizontal stress will increase up to the Mohr circle attaching the Mohr-Coulomb

failure line, then the failure occurs and the full shear strength will mobilised.

Mohr-Coulomb

failure line

At-rest state

Active earth

pressure

decreasing

Figure 2.6: Mohr circle describing active state within a soil mass


47

2.4.2 Earth pressure theories

Coulomb (1776) and Rankine (1857) made a vital contribution to the development of

earth pressure theory, and their solution is still used to determine the lateral earth

pressure behind retaining walls. Significant later efforts have also been made to improve

their solution and address its inherent limitations by considering the friction between the

wall and the backfill soil, the geometry of the wall, and the non-horizontal surface of the

backfill soil.

2.4.2.1 Coulomb’s (1776) earth pressure theory

Coulomb’s (1776) theory is based on the concept of total stress; it was later modified by

Terzaghi (1925) based on the concept of effective stress. Coulomb (1776) used a limit

equilibrium theory to determine the lateral earth pressure behind the retaining wall. It

was assumed that the soil fails along the failure plane inclined by critical with horizontal

axis , as shown in Figure 2.8, and the limiting horizontal pressure at the failure surface

is in extension and compression state, and then is used to compute the active and

passive earth pressure.

Mohr-Coulomb

failure line

At-rest state

Passive earth

pressure

increasing

Figure 2.7: Mohr circle describing passive state within a soil mass


48

Figure 2.8: Planar failure wedge for active state (after Muller-Breslau, 1906)

Mayniel (1808) modified the Coulomb solution by considering the friction between the

retaining wall and the backfill soil . Muller-Breslau (1906) extended the Coulomb

theory to take into account a non-horizontal backfill surface β, and a non-vertical wall-

soil interface θ, as shown in Figure 2.8. Hence, the active earth pressure coefficient can

be computed as below:

2

2

2

cos ( )

sin( )sin( )cos cos( ) 1

cos( )cos( )

aK

(2.5)

where, 𝜃 = inclination of the retaining wall with the horizontal, 𝛿 = friction angle

between the retaining wall and backfill soil, and 𝛽 = inclination of the backfill soil

surface with the horizontal.

For the passive state, the coefficient of passive earth pressure can be computed by:

2

2

2

cos ( )

sin( )sin( )cos cos( ) 1

cos( )cos( )

pK

(2.6)

2.4.2.2 Rankine’s (1857) earth pressure theory

Rankine (1857) derived a stress field solution for computing active and passive earth

pressure. It was assumed in his solution that back the retaining wall is smooth and non-

frictional, and vertical. The failure surface of the soil is planer. Figure 2.9 shows the


49

active earth pressure that develops behind the retaining wall based on the Rankine

solution. The active and passive earth pressure coefficients can be expressed as below:

2 2

2 2

cos cos coscos

cos cos cosaK

(2.7)

2 2

2 2

cos cos coscos

cos cos cospK

(2.8)

Bell (1915) modified Rankine’s theory to take into account the effect of the cohesion of

the soil:

2a a ap K z c K (2.9)

where, ap = active earth pressure, 𝛾 = backfill soil unit weight, z = depth of the

retaining wall, and c = cohesion of the backfill soil.

For the passive state, the passive earth pressure can be computed as below:

2p p pp K z c K (2.10)

2.4.3 Relationship between static earth pressure and wall displacement

It can be seen from Figure 2.5 that a large displacement is required to achieve the

passive state while a relatively small displacement is required to develop the active

state. The magnitude and distribution of earth pressure are significantly influenced by

Pa

β

H

Active state

β

β

H/3

Figure 2.9: Rankine active earth pressure behind retaining wall


50

the direction of the wall movement, whether the wall moves away from or towards the

soil, the amount of wall movement, and the mode of wall movement (sliding, rotation,

flexure, etc.). Previous limit equilibrium methods such as Coulomb’s and Rankine’s did

not take into account the movement of the retaining wall. More complex research

methods have been proposed in order to investigate the relationship between the static

earth pressure and wall movement. Three main research methods (numerical,

experimental and analytical methods) have been used to investigate the relationship

between the earth pressure and the movement of the retaining wall. Critical analysis of

these methods will be presented in the next section.

2.4.3.1 Analytical methods

Bang (1985) proposed an analytical method to predict the magnitude and distribution of

active earth pressure based on the various movements of the wall. The active earth

pressure was exerted by cohesionless soil behind the rigid retaining wall. The wall was

assumed to move away from the backfill soil about its base from initial active state to

fully active state. Initial active state refers to a stage of wall movement when only the

soil element at the ground surface causes wall movement to achieve an active condition

(β = 0 – see Figure 2.10). The full active condition refers to the entire soil element from

the ground surface to the base of the wall that is in an active condition (β=1 – see

Figure 2.10). A very good agreement was observed between the results obtained from

this method and those measured by shaking table test, as shown in Figure 2.10.

Bang and Hwang (1986) presented an approximate analytical solution to predict active

earth pressure exerted by horizontal cohesionless backfill soil behind a rigid vertical

retaining wall depending on various types of outward movement (rotation about the top,

rotation about the base, translation). The results of this analysis showed that the lateral

earth pressure decreased rapidly at the middle, while there was rapid reduction’ in the

lower portion of rotation about the base case. A similar reduction was observed with

rotation about the top case. The translational movement is considered the main factor

that can cause the reduction of earth pressure. The results of this solution showed a very

good agreement with the measured and calculated lateral earth pressures.


51

Figure 2.10: Relationship between active earth pressure and

wall displacement (after Bang, 1985)

Shamsabadi et al. (2005) presented a formulation of passive force-displacement

capacity for the design of an abutment-backfill system. The derived method was based

on a limit equilibrium-logarithmic spiral method, coupled with the characterisation of

the stress-strain behaviour of soil. Figure 2.11 shows the results obtained by the

proposed model. The passive earth pressure coefficient was presented as a function of

the ratio of abutment displacement to height of the abutment H. The nonlinear force-

displacement response was assessed for different types of abutment-soil combinations

(sand, clay and sandy clay soil). This method shows a very good agreement with results

obtained from experimental tests predicted by Fang (1986).

Shamsabadi et al. (2007) modified the previous model by coupling a limit equilibrium

method using a logarithmic spiral failure surface with a modified hyperbolic soil stress-

strain behaviour (LSH model). The results of the modified model were compared with

field experiments. Based on the LSH model and experimental results, a simple

hyperbolic force-displacement equation was developed.


52

Figure 2.11: Passive earth pressure versus wall displacement (after Shamsabadi et al., 2005)

Peng et al. (2012) proposed an analytical study to calculate passive earth pressure on the

rigid retaining wall with different displacement modes. The backfill material behind the

retaining wall consisted of a series of springs and ideal rigid plasticity body, and the

displacement modes involved the five different modes shown in Figure 2.12:

Figure 2.12: Modes of wall displacement generating a passive earth pressure state for a rigid

retaining wall: (a) T mode; (b) RB mode; (c) RT mode; (d) RTT mode; (e) RBT mode (after

Peng et al., 2012)

Translating mode (T mode);

Rotating at the bottom of the retaining wall (RB mode);

Rotating at the top of the retaining wall (RT mode);

Rotating over the top of the retaining wall (RTT mode); and


53

Rotating over the bottom of the retaining wall (RBT mode).

This study included, firstly, providing a general function of displacement mode of the

retaining wall. Thus, the displacement mode (m) is introduced to define the ratio

between the horizontal displacement of the retaining wall and the retaining wall height

(h) Secondly, the passive earth pressure and the position of the resultant passive earth

pressure force were calculated. Finally, the analysis investigated the effect of the

displacement mode of the retaining wall on the passive earth pressure. The results of

this analysis showed a good agreement with other experimental results. The major

findings of this study were that the position and distribution of passive earth pressure

depends more distinctly on the passive displacement mode parameter than on other

factors. The distribution of passive pressure was nonlinear, and its shape was a

parabolic function with the depth of soil.

Liu (2013) proposed an analytical method to determine lateral earth pressure based on

the mode and magnitude of wall movement (translation (T), rotation about the base

(RB), and rotation about the top (RT). The backfill soil behind the wall was assumed to

be homogeneous, and the shear resistance angles of soil and soil-wall friction only

change and develop with wall movement. The result of this analytical work was

compared with investigated data and finite element results and showed that the

analytical method can predict lateral earth pressure. The limited wall movement equal to

0.3% of the height of the wall was acceptable in the calculations. The magnitude of

lateral earth pressure significantly decreased with increases in wall movement.

2.4.3.2 Numerical methods

Potts and Fourie (1986) conducted a finite element analysis to investigate the wall

movement on earth pressure. The plane strain condition was assumed in the analysis.

The soil was modelled using clay material. The analysis included investigating the

sliding and rotation of the wall about the top and bottom on the magnitude and

distribution of earth pressure. The major observations from the analysis were that the

highest value of Kp and the lowest value of Ka occur when the wall slides horizontally.

The magnitude of the displacements is required for mobilising limit conditions

depending on the mode of wall movement, and larger displacements are required for the


54

wall rotating about its base, as shown in Figure 2.13a and b. For the wall rotating about

its top or base, the distributions were far from the linear distribution.

Figure 2.13: Active and passive pressure coefficients for: (a) smooth wall surface, (b) rough wall

surface (after Potts and Fourie, 1986)

Bhatia and Bakeer (1989) also proposed finite element analysis for the earth pressure

problem. They focused on the design of the finite element mesh for modelling the earth

pressure behind a gravity retaining wall using a dry cohesionless backfill. They

investigated the effect of the type and location of boundary conditions, mesh size and

wall displacement. The results of the finite analysis were validated with a large-scale

test of a retaining wall. The typical analysis result showed that the potential failure

wedge of active condition is developed when the retaining wall translates of 0.001 mm

and rotates 0.001H.

(b)

(a)


55

Hazarika and Matsuzawa (1996) proposed a new numerical analysis by using the

coupled shear band (C.S.B) method. The analysis included studying the active earth

pressure exerted behind a rigid and rough retaining wall. The study investigated the

effect of mode of displacement of the wall on the coefficient of earth pressure (K), the

height of the point of application (h/H). Figure 2.14 shows the variation (K) with the

three modes of wall displacement: translation (T), rotation about the top (RT) and

rotation about the bottom (RB). The result from this analysis shows that the most

noticeable variation of K is in the RB mode. The study also indicates that the h/H is

influenced by the mode of retaining wall displacement. The distribution of active earth

pressure is nonlinear for the displacement modes RT and RB. These results are similar

to Potts and Fourie (1986) observations.

Figure 2.14: Variation of earth pressure coefficient with wall displacements (after Hazarika and

Matsuzawa, 1996)

Shamsabadi et al. (2009) presented a numerical study of the lateral response of an

abutment bridge. The finite element study was validated with the results recorded from

a full-scale abutment field test and the log-spiral hyperbolic analytical model (see

section 2.4.3.1). The empirical equation was developed for lateral pressure-

displacement backbone curves for varying abutment heights for two-backfill soil types.

The finite element (FE) models were developed by using the PLAXIS software package

and the stress-strain relationship of the backfill soil was simulated using the hardening

soil (HS) model.


56

Shamsabadi et al. (2009) shows the load-displacement curve predicted by 2D and 3D

FE models and compared with experimental and analytical results. It can be noted that

the lateral load is represented by the passive resistance of the backfill soil and large

displacement for the abutment bridge is necessary to reach the failure condition, and the

variation of the lateral load with the displacement of the abutment bridge is nonlinear.

The traditional methods cannot take into account this relationship.

Figure 2.15: Comparison of passive earth pressure force from various numerical and analytical

model results with experimental measurements (after Shamsabadi et al., 2009)

Wilson and Elgamal (2010) conducted a 2D FE analysis by using PLAXIS software to

predict passive load-displacement of an abutment bridge. The FE simulations

investigated the effect of the uplift component of passive force on the passive load-

displacement response. The results of the FE analysis are in good agreement with large-

scale experiment results, as shown in Figure 2.16. A nonlinear relationship between the

passive earth pressure and the displacement of the abutment bridge was observed.

Hyperbolic model approximations of passive load-displacement curves were predicted.


57

Figure 2.16: Comparison of passive earth pressure force from numerical and experimental

measurements with: (a) low interface, (b) high interface (after Wilson and Elgamal, 2010)

Achmus (2013) proposed a numerical model to estimate 3D active earth pressure forces

acting on a retaining wall. The simulations were conducted using the ABAQUS

program system version 6.7. This study included investigation of many factors on the

active earth pressure: aspect ratio n (width /height of the wall), wall deformation mode,

wall displacement, wall roughness and relative density of the soil. Different wall

movements were considered in this analysis (translation, rotation about the top and

rotation about the base of the wall. Figure 2.17 shows the relationship between the

dimensionless earth pressure force (active earth pressure force (E)) to the at-rest earth

pressure (Eo) and dimensionless wall displacement (u/h %)). The results showed a good

agreement between numerical and experimental results for the case of the translation.

Figure 2.17: Dimensionless earth pressure force versus wall movement - numerical and

experimental modelling results (after Achmus, 2013)

(a) (b)


58

For a load-displacement relationship, large displacement was necessary for mobilising

the active limit state with bottom rotation of the wall. The results of this analysis also

indicated that the largest active earth pressure occurred with top rotation and a similar

value for bottom rotation, but the smallest value occurred with parallel movement, as in

Figure 2.17.

Sadrekarimi and Monfared (2013) also conducted a series of 3D FE models to

investigate the development of static earth pressure depending on the retaining wall

displacement. The influence of many factors (like wall-backfill interaction, soil modulus

and shear resistance angle of the soil) on the relationship between the static earth

pressure and displacement of the wall was investigated. The results show that the

increase of displacement of the retaining wall causes an increase in arching, thereby

increasing the reduction of static earth pressure. The reduction of static earth pressure

with a displacement of the wall increases with increases in the subsoil-wall interface

angle and shear resistance angle of the soil. The mobilisation of active earth pressure is

independent of the backfill soil modulus.

2.4.3.3 Experimental methods

Sherif et al. (1984) carried out an experimental study by using the shaking table to

investigate the magnitude and distribution of the static lateral earth pressure behind a

rigid retaining wall that was rotated about its base. The table was designed to move in

one direction only, as shown in Figure 2.18. Dry sand was used to model the backfill

soil behind the retaining wall. The results show that the lateral earth pressure at-rest

increased because of the densification. The distribution of earth pressure was

hydrostatic. The static earth pressure decreased with increases in the wall rotation, as

shown in Figure 2.18, and the state of active earth pressure propagates towards the

bottom with increasing wall rotation.


59

Figure 2.18: Relationship between active earth pressure and wall rotation (after Sherif et al.,

1984)

Fang and Ishibashi (1986) conducted an experimental study by using the same shaking

table described above to obtain the distribution of earth pressure that was exerted by a

sand backfill behind wall that rotated about the top. Figure 2.19 shows the lateral earth

pressure coefficient (Kh), the relative height of resultant pressure application (h/H) and

coefficient of wall friction (tan (δ)) varied with the rotation of the wall about the top.


60

Figure 2.19: Variation of lateral earth pressure coefficient (Kh), relative height of resultant

pressure application (h/H) and coefficient of wall friction (tanδ) with the wall rotation about its

top (after Fang and Ishibashi, 1986)

The result of this study shows that the distribution of active stresses was nonlinear. The

stresses at the top of the rotating wall increased beyond the active stress condition due

to soil arching. The magnitude of the active lateral soil thrust exerted against the

rotating wall about the top was higher by about 17% than the values estimated by the

Coulomb equation.

Fang et al. (1994) conducted an experimental study by using the shaking table test to

investigate the effect of rotation about a point above the top (RTT) and rotation about a

point below the wall base (RBT) on the variation of passive earth pressure. The results

showed that, for a wall under translational movement, the pressure distribution was

essentially hydrostatic and it was in good agreement with Terzaghi's predictions (see

Figure 2.20a). For a wall under the rotation about a point above the top (RTT), the

passive pressure distribution was far from linear. The measured passive earth pressure

was lower than those calculated with Coulomb and Rankine methods (see

Figure 2.20b).


61

Figure 2.20: Effect of wall movement mode on passive earth pressure after Fang et al., 1994)

For a wall under rotation about a point below the base (RBT), (Figure 2.20c), high

stresses were measured near the mid-height of the wall, and the passive pressure

distribution was also nonlinear.

If the centre of rotation is still close to the top or bottom of the wall, the magnitude of

the passive thrust and its point of application are significantly influenced by the wall

movement mode. However, if the centre of rotation moves away from the top or bottom

surface of the wall (about two times the wall height), the effect of the wall movement

mode upon passive thrust becomes less important.

Bentler and Labuz (2006) presented real field observations for the performance of a

cantilever retaining wall during the construction process. The study aimed to compare

the observed wall behaviour with the assumed design. The analysis of the collected data

shows that when the wall translates about 0.1% of the backfill height, the active earth

pressure is developed. The maximum lateral earth force was close to the theoretical

active value assumed in the design. The wall rotated into the backfill soil as a rigid

body. The top of the stem deflected away from the backfill approximately equal in

magnitude and opposite in direction to the displacement of the rigid body rotation. This

study shows the development of the active earth condition and reduction of the total

lateral force because of the translation of the wall.

Wilson and Elgamal (2010) conducted two large-scale tests to predict the passive earth

pressure force behind a moveable vertical concrete wall, as shown in Figure 2.21, with

two different water contents. Based on the measured results, the load-displacement

curves were produced.

(a) (b) (c)


62

Figure 2.21: Large-scale wall-soil model test (after Wilson and Elgamal, 2010)

The results indicate that the peak passive force is measured at displacement of 3% of

wall height, as shown in Figure 2.22. The measured results are higher than log spiral

and Coulomb analysis by about 10% (for parameters derived from the triaxial test)

while, for parameters predicted by the direct shear test, they are lower than 10%.

Figure 2.22: Load-displacement curves for a retaining wall (after Wilson and Elgamal, 2010)

2.4.4 Critical discussion of the relationship between the static earth pressure and

wall displacement

The main part of the previous sections has focused on the relationship between the static

earth pressures and displacement of the retaining wall. A variety of research methods –

analytical, numerical and experimental methods – available in the literature have been

presented therein and discussed in detail. Table 2.1 shows the major findings of the


63

available research methods. It can be noted from Table 2.1that the relationship between

static earth pressure and wall displacement of the retaining wall have been well

established and there is a very good agreement among the research methods. Most of

the research methods described in Table 2.1 indicated that the horizontal movement of

the retaining wall is a major factor that affects the development of minimum active and

maximum passive earth pressure. It is also noted that the distribution of active and

passive earth pressure is highly affected by the mode of retaining wall movement and

the distribution of earth pressure being nonlinear along the height of retaining wall

when the retaining wall rotates about its top or bottom. The research methods also

indicated that the relationship between the passive earth pressure and displacement of

the retaining wall is nonlinear and larger displacement is required to reach a passive

state. Hence, it can be said that the magnitude and distribution of earth pressure is

highly related to the displacement of the retaining wall. Both the magnitude and

distribution of earth pressure are pivotal to the stability of the retaining wall. The value

of earth pressure is a main force required to check the stability of the retaining wall

against sliding; however, the distribution of earth pressure will help to determine the

location of the total earth pressure and estimate the stability of the retaining wall against

overturning.

After the critical discussion of the relationship between the static earth pressure and

displacement of the retaining wall, the next sections will discuss the earth pressure

theory for the seismic case. The seismic earth pressure theories and design techniques

will be presented in detail. A variety of analytical, numerical and experimental research

methods available in the literature will be critically discussed. More attention will be

paid to the limitations of current design techniques and research methods that have been

proposed to investigate the relationship between the seismic earth pressure and

displacement of the retaining wall. A variety of information and field observations on

the seismic behaviour and damage of retaining walls in different seismic-prone zones

are presented.


64

Table 2.1: Major findings concerning the relationship between static earth pressure and

displacement of the retaining wall

Research methods Researcher Major findings

Analytical

Bang (1985) Active earth pressure reduces with increases in the

rotation of the retaining wall

Bang and Hwang

(1986)

Translation movement is the main factor that causes the

reduction of active earth pressure

Shamsabadi et al.

(2005)

Passive earth pressure is a function of wall displacement

and nonlinear force-displacement response was observed

Shamsabadi et al.

(2007b)

Passive earth pressure is a function of wall displacement

and nonlinear force-displacement response was observed

Peng et al. (2012)

Passive earth pressure highly depends on the

displacement mode of the wall, and its distribution is

nonlinear

Liu (2013) Magnitude of active earth pressure decreases with

increasing wall movement

Numerical

Potts and Fourie

(1986)

The highest passive earth pressure and lowest active

earth pressure occur when the wall moves horizontally

Bhatia and Bakeer

(1989)

The active state reached when the wall translates 0.001

mm

Hazarika and

Matsuzawa

(1996)

The most noticeable variation of earth pressure is when

the wall rotates about the base

Shamsabadi et al.

(2009)

Large displacement is required to reach a passive state,

and the load-displacement curve is nonlinear

Wilson and

Elgamal (2010)

Nonlinear relationship between passive earth pressure

and displacement of retaining wall is observed

Achmus (2013)

Active earth pressure reduces with increases in the

displacement of the wall away from the soil. Smallest

active earth pressure occurs when the retaining wall

moves horizontally

Sadrekarimi and

Monfared (2013)

The increase of displacement of the retaining wall causes

an increase of arching, thereby increasing the reduction

of static earth pressure

Experimental Sherif et al.

(1984)

The distribution of earth pressure was hydrostatic. The

static earth pressure decreased with increasing wall

rotation


65

Fang and

Ishibashi (1986)

The distribution of active stresses was nonlinear. The

stresses at the top of the rotating wall increased beyond

the active stress condition due to soil arching

Fang et al. (1994)

For a wall under translational movement, the pressure

distribution was essentially hydrostatic. For a wall under

the rotation about a point above the top, the passive

pressure distribution was far from linear

Bentler and Labuz

(2006)

The active earth state is developed when the wall

translates away from backfill soil.

Wilson and

Elgamal (2010)

The peak passive force is measured at displacement 3%

of wall height

2.5 SEISMIC DESIGN OF RETAINING WALLS

The seismic response of the retaining walls is quite complex compared with the static

response of these structures. It has been observed that earthquakes have caused

permanent deformations of retaining structures or structural damage and they can cause

significant damage with disastrous physical and economic consequences. To accurately

evaluate the seismic stability of retaining structures against expected deformations and

additional loads, an accurate estimation of these deformations and additional loads

imposed by earthquakes on the retaining structures becomes pivotal. In the literature,

there are different analysis methods available for the seismic design of retaining walls.

Figure 2.23shows a flow chart describing the main analysis and research methods that

are used in the analysis of retaining walls under the effect of seismic loading. It can be

noted from Figure 2.23 that there are three main design methods used for seismic design

of retaining walls: force-based design method, displacement-based design method and

force-displacement design method. Under each design method, several analytical,

numerical and experimental research methods have been developed by the researchers

in order to provide a safe design for retaining walls during the seismic scenario. The

next sections will discuss the above-mentioned design methods in detail.


66

Figure 2.23: Flow chart describing various seismic analysis methods in vogue for analysing

retaining walls

2.5.1 Force-based design methods

In force-based design methods, the design of the retaining structures is entirely

dependent on the estimation of the loads imposed during the earthquake, and has to

ensure that the retaining wall can resist those loads. The additional loads computed in

this design technique are completely based upon the development of inertia forces in


67

backfill material without considering the displacement effect of the retaining wall.

Numerous researchers have attempted to modify the force-based design methods to

address their inherent limitations and provide a rational design for the retaining wall by

using analytical, numerical and experimental studies. This section summarises and

highlights the previous analysis methods and research performed using force-based

design method.


Extensive work has been achieved by using an analytical approach in order to estimate

the seismic earth pressure. The analytical approach can be divided into pseudo-static

and pseudo-dynamic methods. In pseudo-static methods, the dynamic forces can be

converted into conventional pseudo-static forces, while they are assumed to be time and

frequency dependent in pseudo-dynamic methods.

2.5.1.1.1 Pseudo-static methods

The first pioneering work which was achieved based on this design technique is found

in the work of Okabe (1926) and Mononobe and Matsuo (1929) after the Great Kanto

earthquake (1923) in Japan, and then this led to the development of the seismic earth

pressure theory. This theory is still used in practice for design and comparison because

of its simplicity.

Mononobe-Okabe (M-O) method: Mononobe and Matsuo (1929) conducted a series of

experiments by using the shaking table test. The results of these experiments and Okabe

(1926) analysis led to the development of the Mononobe-Okabe method (M-O). This

method is a direct extension of the static Coulomb wedge theory that assumed the

retaining wall moves a sufficient displacement away from or towards the dry

cohesionless backfill soil and this causes the development of a failure wedge behind the

retaining wall. In the M-O method, the total seismic active and passive earth pressure

are computed by applying pseudo-static acceleration forces on the static forces acting on

the soil wedge in both the horizontal and vertical directions. The magnitude of these

pseudo-static forces depends on the acceleration level in the horizontal and vertical

directions and the mass of the soil wedge.


68

Seismic active earth pressure: Figure 2.24 shows the forces acting on the dry

cohesionless backfill wedge. In addition to the static forces, the wedge is also under the

effect of the pseudo-static forces that are a function of the mass of the wedge and

pseudo-static accelerations (ah= kh × g and av=kv × g).

where, kh = ratio between the horizontal seismic acceleration (ah) and gravity

acceleration (g), and kv = ratio between the vertical seismic acceleration (av) and gravity

acceleration (g).

The total seismic active earth pressure force can be computed similarly to that

calculated by the Coulomb method:

21(1 )

2ae ae vP K H k (2.11)

where, 𝐾𝑎𝑒 = seismic active earth pressure coefficient and it can be computed by:

2

2

2

cos ( )

sin( )sin( )cos cos cos( ) 1

cos( )cos( )

aae

aa

K

(2.12)

where, − 𝛽 ≥ 𝜓𝑎, and 1tan / (1 )a h vk k . Zarrabi and Kashani (1979) proposed

that the critical failure surface is flatter than for the static case and it inclined at angle:

1 1

2

tan(tan E

ae a

E

C

C

(2.13)

where,

1

tan( ) tan( ) cot( ) 1 tan( )cot( )

E

a a a a a

C

(2.14)

2 1 tan( ) tan( ) cot( )E a a aC (2.15)


69

Figure 2.24: Forces acting on a soil wedge for an active case in the M-O analysis

Seismic passive earth pressure: Figure 2.25 shows the forces acting on the dry

cohesionless backfill wedge. The total seismic passive earth pressure force can be

computed by:

21(1 )

2pe pe vP K H k (2.16)

where, 𝐾𝑝𝑒 = seismic passive earth pressure coefficient and it can be calculated by:

2

2

2

cos ( )

sin( )sin( )cos cos cos( ) 1

cos( )cos( )

ape

aa a

a

K

(2.17)

The critical failure surface for the passive condition inclined at angle:

1 3

4

tan( )tan a E

pe a

E

C

C

(2.18)

where,

3

tan( ) tan( ) cot( ) 1 tan( )cot( )

E

a a a a a

C

(2.19)

4 1 tan( ) tan( ) cot( )E a a aC (2.20)


70

Figure 2.25: Forces acting on a soil wedge for a passive case in the M-O analysis

Seed and Whitman (1970) (S-W) method: Seed and Whitman (1970) conducted a

parametric study to investigate the effect of wall friction, friction angle, backfill slope

and vertical acceleration on the seismic earth pressure. They reported that the total

seismic earth pressure consisted of two parts: the static earth pressure and interment of

dynamic earth pressure. They recommended that the point of application of the dynamic

force should be about 0.6H from the base of the wall. They introduced the concept of an

inverted triangle of dynamic earth pressure distribution where the base of the triangle is

inverted to be on the top.

Following previous work, many researchers like Saran and Prakash (1968) and Madhav

and Rao (1969) computed the seismic earth pressure by using pseudo-static methods.

Choudhury and Rao (2002) presented a procedure to estimate seismic passive earth

pressure behind the retaining wall. They adopted the negative wall friction case and

assumed that the failure surface is an arc of a log spiral. They observed that the seismic

passive earth pressure decreases with an increase in the vertical seismic acceleration.

Following that, Choudhury et al. (2004) estimated the seismic passive earth pressure

and its point of application by using the horizontal slices method. Subba Rao and

Choudhury (2005) also estimated the seismic passive earth pressure by presenting a

general solution taking into account the cohesive backfill and considering composite

surface failure (planar+ log spiral). Shukla (2010) and Shukla and Habibi (2011)

presented a closed-form solution to compute the seismic earth pressure and critical

inclination of surface failure using cohesive-frictional soil backfill acting behind a


71

vertical smooth retaining wall considering the horizontal and vertical acceleration

coefficient. After that, Shukla and Habibi (2011) also presented a closed-form solution

to estimate the total seismic passive earth pressure and critical inclination of surface

failure. Shukla and Zahid (2011) presented a general solution for the total seismic

passive earth pressure considering wall geometry, soil backfill, loadings, backfill slope

angle and wall friction. Ortigosa (2005) proposed a solution to estimate the seismic

earth pressure considering the soil cohesion. Ostadan (2005) proposed an analytical

solution to compute the seismic earth pressure behind the building wall using the

concept of a single degree of freedom system. The building wall was assumed rigid

(non-yield). The frequency content of the design motion was considered in the analysis.

The comparison between the proposed method and simplified methods like the M-O

method shows a conservative result for seismic earth pressure. Further, the proposed

method was close to Wood (1973) method in terms of the magnitude and distribution of

total seismic earth pressure where the seismic earth pressure has an inverted triangular

distribution along the wall height. Mylonakis et al. (2007) presented an alternative

solution to the M-O method to compute the seismic earth pressure. The proposed

closed-form stress plasticity solution was considered symmetric because it can be

expressed by a single equation to estimate the active and passive pressure by using

appropriate signs for friction angle and wall roughness. The comparison of the proposed

method with numerical results and M-O method shows that the solution overestimated

the active pressure and underestimated the passive pressure.

2.5.1.1.2 Critical discussion on pseudo-static methods

Pseudo-static methods have been the most popular methods for designing geotechnical

retaining walls because of their simplicity. However, they have some inherent

limitations that lead to an unconservative estimation of seismic earth pressure. In these

methods, it has been assumed that the earthquake loading can be applied as a constant

load, even though the seismic loading is cyclic and it changes its magnitude and

direction with time. In addition, they did not take into account the effect of the

displacement of the retaining structure on the development of seismic earth pressure.

Choudhury et al. (2014) reported that the pseudo-static forces overestimated the risk of

earthquake failure, and the design of the retaining structure in most cases is made over-


72

safe. The National Cooperative Highway Research Program (NCHRP 611) Anderson

(2008) also reported that the M-O method is not valid for the situation of step

inclination of backfill soil because the planar failure surface will approach the backfill

slope and this will lead to the development of an infinite mass of active failure wedge.

Pseudo-static methods are still widely used in practice because of their simplicity

although they have the above limitations.

2.5.1.1.3 Pseudo-dynamic methods

To cope with the inherent limitations of pseudo-static methods like ignoring the

dynamic nature of earthquake loading, dynamic response of backfill layer, phase

difference and amplification effects within backfill soil, Steedman and Zeng (1990a)

attempted to propose a new method to estimate the seismic earth pressure considering

the phase difference because of finite shear wave propagation in the backfill soil behind

a retaining structure.

Steedman-Zeng (1990) method: Steedman and Zeng (1990a) proposed an analytical

solution to estimate the seismic active earth pressure considering finite shear wave

propagation within backfill soil. A fix-base vertical cantilever wall of height H is

assumed to support a cohesionless backfill material with definite soil friction, as shown

in Figure 2.26. The backfill soil is considered horizontal in the analysis. The base of the

backfill soil is assumed to be subject to harmonic horizontal acceleration of amplitude

ah. The horizontal seismic acceleration acting in the backfill soil is not constant, but it is

dependent on the time, frequency and phase difference in a shear wave (vs) propagating

in the vertical direction within the backfill soil. The horizontal seismic acceleration at

any depth z below soil surface and time can be expressed as:

( , ) sinh h

s

H za z t a t

v

(2.21)

where, t = time and ω = frequency of sinusoidal earthquake acceleration.

The seismic active earth pressure is assumed to develop from the backfill soil with a

triangle wedge inclined at with the horizontal, as shown in Figure 2.26. The mass of

the thin element of the soil wedge at depth z can be computed:


73

( )tan

H zm z dz

g

(2.22)

where, γ: is the unit weight of soil.

In the pseudo-dynamic method, as a particular case is assumed that the soil wedge

behaves as a rigid body having an infinite shear wave, then the pseudo-dynamic method

can be reduced to a pseudo-static method of analysis, as shown below:

limvs→∞

(Qh)

max=

γH2ah

2gtan α=

ah

gW=khW (2.23)

Figure 2.26: Wall geometry considered in the Steedman and Zeng (1990) model

The total (static + seismic) active soil thrust can be obtained by resolving the forces on

the wedge and can be expressed as follows:

( )cos( ) sin( )

cos( )

hae

Q t WP

(2.24)

The total active soil thrust can be maximised with respect to the trial inclination angle of

the failure surface and then the seismic earth pressure distribution 𝑝𝑎𝑒 can be computed

by differentiating Pae with respect to depth z:

sin( ) cos( )sin ( )

tan cos( ) tan cos( )

ae hae

s

P k zz zp t

z v

(2.25)


74

The first term in Equation (2-24) represents the static earth pressure and an increase

linearly with depth as well as it does not vary with time. However, the second term in

Equation (2-24) represents the seismic earth pressure, and it increases in a nonlinear

fashion with the depth.

Choudhury and Nimbalkar (2005) modified the pseudo-dynamic method proposed by

Steedman and Zeng (1990a). Furthermore, Choudhury and Nimbalkar (2006) extended

their previous work for estimation of the seismic active earth pressure. Ghosh (2008)

proposed a solution by using the pseudo-dynamic method to estimate the seismic active

earth pressure acting behind a non-vertical retaining wall. Kolathayar and Ghosh (2009)

proposed a solution by using the pseudo-dynamic method to compute the seismic earth

pressure behind a bilinear rigid retaining wall considering the effect of uniform shear

modulus with depth. Nimbalkar and Choudhury (2008b) modified their previous work

by considering the soil amplification to compute the seismic passive earth pressure and

the horizontal and vertical acceleration. They observed that the soil parameters have

more effect on the seismic passive earth pressure than the seismic active earth pressure.

Basha and Babu (2008) modified the work of Choudhury and Nimbalkar (2005) by

proposing an approach to compute the seismic passive earth pressure by using the

composite curved rupture surface (arch of log spiral + linear). Ghosh and Sharma

(2010) modified the pseudo-dynamic approach to estimate the seismic earth pressure

behind a non-vertical retaining wall supporting c- backfill soil considering the planner

rupture surface.

2.5.1.1.4 Critical discussion on pseudo-dynamic methods

The pseudo-dynamic methods have been modified to consider the characteristics of

seismic acceleration force (effect of the time and frequency); the geometry of the

retaining walls; effect of the phase difference of shear and primary waves of the backfill

soil; and the effect of amplification in both shear and primary waves through the vertical

direction. These methods have been used by some researchers like Nimbalkar and

Choudhury (2008a) to propose design factors for the weight of the retaining wall for

seismic conditions under seismic active earth pressure. However, other researchers like

Ahmad and Choudhury (2008a), Ahmad and Choudhury (2008b), Ahmad and

Choudhury (2009) and Ahmad and Choudhury (2010) analysed the seismic stability of


75

vertical and non-vertical water-front retaining walls by using the pseudo-dynamic

method. In spite of the fact that the pseudo-dynamic methods do consider the seismic

forces and backfill soil characteristics and have been widely used in the analysis of the

geotechnical retaining wall, they still have some limitations that could affect the

magnitude and distribution of seismic earth pressure. For instance, they ignore the

seismic response of the retaining wall and the phase difference between the seismic

response of the retaining wall and backfill soil by assuming that the backfill soil is

supported by a rigid retaining wall. Another important limitation is that they ignore the

displacement of the retaining wall by assuming that the retaining wall has a fixed

connection with the foundation layer, and ignoring the sliding of the retaining wall. The

seismic earthquake acceleration is assumed in these methods to be a uniform sinusoidal

wave, while the real earthquake acceleration is more complicated and may have multi-

amplitude and a wide range of the frequency contents. These methods also ignore the

effect the soil foundation layer below the wall-soil system on the development of

seismic earth pressure. Pain et al. (2017) reported that the pseudo-dynamic methods do

not satisfy the boundary conditions. Furthermore, the pseudo-dynamic methods do not

take into account the damping properties of backfill soil.


The seismic response of the geotechnical retaining walls can also be estimated by using

different numerical methods like finite element and finite difference methods. These

methods have shown their robustness and ability to model the actual and complex

behaviour of materials and capture the failure modes of retaining walls under both static

and seismic loading. A variety of computer programs have been used to analyse the

retaining structures problem numerically like PLAXIS, ABAQUS, FLAC, ANSYS, etc.

The use of numerical methods is required to overcome the challenges related to the

modelling boundary conditions, mesh design, seismic loading and the actual stress-

strain behaviour of the materials (soil and retaining wall). This section discusses the

numerical methods that have been adopted to investigate the seismic earth pressure

problem by using a force-based design method.

Wood (1973) presented a finite element study to investigate the static and dynamic

response of non-yielding walls and effect of bounded walls and uniform soil stiffness.


76

The soil behaviour was assumed to be homogeneous linear elastic during the analysis.

The study revealed that there was no significant influence of the smooth wall and

bounded wall contact on the frequency response or earth pressure distribution.

Siddharthan and Maragakis (1989) proposed a finite element model to investigate the

seismic response of a flexible cantilever retaining wall. This study investigated the

effect of wall flexibility and relative density of soil on the dynamic response of the wall.

The bending moment estimated by the finite element model was compared to those

computed by Seed and Whitman’s (1970) procedure and the comparison shows that

Seed and Whitman’s (1970) procedure gave conservative results. In this study, the base

of the cantilever wall was assumed to be in rigid connection with the foundation layer

and this does not reflect the real behaviour of the wall, because the wall is rarely rigidly

connected to the foundation layer in real situations.

Green et al. (2003) conducted a series of finite difference dynamic analysis of a

cantilever retaining wall to assess the M-O method for estimating the seismic earth

pressure induced on the stem of the wall. The finite difference models were built by

using the FLAC computer program, as shown in Figure 2.27. Both backfill (zone 2) and

foundation soil (zone 1), as shown in Figure 2.27, were simulating as elastoplastic. The

results of the analysis show that the computed seismic earth pressures were in general

agreement with those predicted by the M-O method at low acceleration levels.

However, when the acceleration level increased, the computed seismic earth pressures

were larger than those predicted by the M-O method.

After that, Green et al. (2008) used the same finite difference dynamic analysis to

investigate the structural and global stability of the cantilever retaining wall under

seismic condition. The study concluded that the critical load case for structural design is

when the seismic acceleration is applied away from the backfill soil, and it differed

from that for the global stability.


77

Figure 2.27: Finite difference model of a retaining wall proposed by Green et al. (2003)

Pathmanathan (2007) conducted a series of finite element models to determine the

seismic earth pressure on a flexible diaphragm, a flexible cantilever wall and gravity

wall. The results obtained from the finite element model for a cantilever retaining wall

did not correspond with those estimated by the M-O method. However, for the rigid

retaining wall, the predicted seismic earth pressure corresponded with those calculated

by the M-O method when the shaking level was small, while they did not correspond

with those obtained by M-O when the shaking level was large.

Al Atik and Sitar (2008) developed a 2D nonlinear finite element model by using the

OpenSees program to evaluate the ability of a numerical model to simulate the seismic

response of retaining structures observed in centrifuge experiments. The finite element

model was developed to estimate seismic earth pressure behind two U-shaped cantilever

retaining walls, one flexible and one stiff. The result shows that the seismic earth

pressure depends on the magnitude and intensity of the shaking and flexibility of the

retaining wall. The distribution of dynamic earth pressure can be approximated to a

triangular shape. The dynamic earth pressure and inertial forces did not act in the same

phase. The seismic earth pressure can be neglected at acceleration levels below 0.4g.

The finite element analysis for denser soil backfill soil shows that the seismic earth

pressure reduced by about 23-30%.

Geraili et al. (2016) presented a finite difference analysis by using FLAC2D

to simulate

two centrifuge experiments. The retaining walls were modelled to simulate the

basement wall type and cantilever retaining wall type to support dry medium-dense

sand backfill. The results of the analysis show the same observations as those recorded

by Al Atik and Sitar (2008).


78


Physical model tests are widely used to study the performance of the retaining wall for

static and seismic conditions. They are generally considered a very useful method to

identify important phenomena and verify numerical and analytical models. The physical

model tests can be classified into two main types: the first type is performed under the

gravitational field of the earth (1g model test) and it is commonly performed by using

the shaking table test. However, the second type is performed under increased

gravitational fields to overcome the sensitivity of soil behaviour to the stress level and it

can be performed by using the geotechnical centrifuge test. The next section presents a

discussion of the previous experimental studies conducted using a force-based design

method.

2.5.1.3.1 Shaking table tests

Experimental shaking table studies of seismic earth pressures acting on retaining walls

were begun by Mononobe and Matsuo (1929) after the Great Kanto earthquake of 1923

in Japan. Mononobe and Matsuo (1929) carried out experiments on the dry loose sand

in a rigid 1-g shaking table in order to record the dynamic earth pressures on retaining

walls and verify the analytical method proposed by Okabe (1926), as shown in

Figure 2.28.

Figure 2.28: Shaking table experiment conducted by Mononobe and Matsuo (1929)

The accuracy of 1-g shaking table experiments is limited because they are inherently

unable to replicate the real soil stress conditions. The results from the 1-g shaking table

experiments were reported by Matsuo (1941), Matsuo and Ohara (1960), Sherif et al.


79

(1982), Bolton M. D. and Steedman (1982), Sherif and Fang (1984b), Steedman (1984

), Bolton and Steedman (1985), and Ishibashi and Fang (1987) concluded that the M-O

method is able to predict the total seismic earth pressure force.

Kloukinas et al. (2015) carried out a shaking table experiment to investigate the seismic

response of a cantilever retaining wall. The shaking table experiment was conducted by

scaling the retaining wall model and assuming the retaining wall has a compliant base

under different geometries of the wall and input shaking, as shown in Figure 2.29 The

backfill and foundation soil was considered to be dry silica sand. The shaking table

results show that the rotation of the retaining wall is more sensitive to the strong seismic

shaking than the sliding mechanism.

Figure 2.29: Shaking table model used by Kloukinas et al. (2015)

The plastic deformation of the foundation layer under the wall toe dominates the

amount of rotation in the majority of the tests. It is also observed that the maximum

bending moment on the stem develops when the retaining wall is moving towards the

backfill soil (the system in the passive state). However, the maximum displacement and

rotation happen when the retaining wall is moving away from the backfill soil (the

system in the active state).

2.5.1.3.2 Centrifuge tests

Centrifuge experiments have been widely used in recent years to study the seismic

performance of geotechnical retaining walls. The scaled model in a centrifuge test (1/N)

is usually rotated to raise the acceleration in the model to N times the gravity


80

acceleration. Then, the stress conditions at any point in the model should be similar to

those corresponding points in the full-scale prototype. This section discusses the

centrifuge experiments used to investigate the seismic performance of retaining

structures by using a force-based design method.

Ortiz et al. (1983) conducted a centrifuge test to investigate the seismic response of a

cantilever retaining wall supporting medium dense sand. The parameters measured from

the experiment include bending moment, shear force, pressure and displacement over

the height of the retaining wall, and they are predicted as a function of time. The test

results show that the total seismic earth pressure is in reasonable agreement with those

computed by the M-O method but the bending moment can be different. The seismic

earth pressure distribution along the height of the wall is nonlinear. The static and

dynamic reaction parameters (bending moment, pressure) appear to be independent of

wall stiffness.

Dynamic centrifuge tests of retaining walls with dry and saturated cohesionless soil

were also conducted by Bolton and Steedman (1985), Zeng (1990;Steedman and Zeng

(1991), Stadler (1996) and Dewoolkar et al. (2001). Sinusoidal earthquake accelerations

were used in the majority of these dynamic centrifuge experiments. Bolton and

Steedman (1982) conducted dynamic centrifuge experiments on concrete cantilever

retaining walls, and Bolton and Steedman (1985) conducted a centrifuge experiment on

aluminium cantilever retaining walls to support dry cohesionless backfill. Their results

as measured from the centrifuge experiment support the M-O method. Steedman (1984 )

conducted centrifuge experiments on cantilever retaining walls to support dry dense

sand backfill. The measured seismic earth pressures were in good agreement with those

computed by the M-O method.

Stadler (1996) performed 14 dynamic centrifuge experiments on cantilever retaining

walls. Dry medium-dense sand backfill is used in the experiments. The experiment

observations show that the total seismic lateral earth pressure is linear with depth while

the incremental seismic lateral earth pressure distribution is between triangular and

rectangular.


81

Nakamura (2006) conducted an experimental study using a centrifuge model test in

order to assess the M-O method. Figure 2.30 shows a cross section of the model. Dry

Toyoura sand with relative density of 88% was used as a backfill in the model test.

After applying a centrifuge acceleration of 30g, horizontal shaking was conducted using

different types of base earthquake acceleration. The results of this study show that the

inherent assumptions of the M-O method do not appropriately express the real

behaviour of the backfill and gravity retaining walls during earthquakes. The

experimental data of this study show that a part of the backfill follows the displacement

of the retaining wall and plastically deforms while in the M-O conditions a rigid wedge

is formed in the backfill. The distribution of earth pressure in the back face of the

retaining wall is not triangular while the M-O conditions assume the distribution of

earth pressure to be triangular.

Figure 2.30: Cross section of the centrifuge test conducted by Nakamura (2006)

Al Atik and Sitar (2008) conducted two sets of dynamic centrifuge tests to estimate the

magnitude and distribution of seismic earth pressure induced behind two U-shaped

cantilever-retaining structures, one flexible and one stiff, which were constructed to

support dry sand backfill material. The result shows that the seismic earth pressure

depends on the magnitude and intensity of the shaking and flexibility of the retaining

wall. The distribution of dynamic earth pressure can be approximated to a triangular

shape. The dynamic earth pressure and inertial forces did not act in the same phase. The

seismic earth pressure can be neglected at acceleration levels below 0.4g.

Geraili et al. (2016) conducted two sets of dynamic centrifuge tests. The first

experiment includes modelling of a retaining wall basement type (see Figure 2.31a) and


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the second experiment includes modelling a U-shaped wall with cantilever sides (see

Figure 2.31b). Dry sand backfill material was used in both experiments. The results of

the analysis show the same observations as those recorded by Al Atik and Sitar (2008).

Figure 2.31: Cross section of centrifuge test conducted by Geraili et al. (2016): a) basement type

retaining wall and b) U-shaped retaining wall with cantilever sides

Jo et al. (2017) carried out two centrifuge experiments to investigate the seismic

response of inverted T-shape cantilever retaining walls in dry sand by using a real

earthquake and sinusoidal earthquake acceleration. The height of the retaining wall in

the first and second centrifuge tests was 5.4 m (see Figure 2.32a)) and 10.8m (see

Figure 2.32b)) at the prototype scale respectively.

(a)

(b)


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Figure 2.32: Cross section of centrifuge tests conducted by Jo et al. (2017): a) wall height 5.4m,

b) wall height 10.8m

The centrifuge tests revealed that the seismic earth pressure changed with time and its

distribution is close to a triangular shape, as well as the point of total seismic earth

pressure force is located at 0.33H above the wall base. The M-O and S-W methods

underestimate the seismic earth pressure for the wall model with height 5.4m, while

they overestimate the seismic earth pressure for the wall model with height 10.8m. The

phase difference between the wall and soil has an important effect on the dynamic earth

pressure distribution and force. The critical load case for the structural design of a

cantilever retaining wall is when the seismic acceleration is applied towards the backfill

soil.

2.5.1.4 Critical discussion of the force-based design methods

The previous sections have discussed a variety of numerical and experimental methods

proposed in the literature to estimate the seismic earth pressure as well as to verify the

analytical solutions like pseudo-static and pseudo-dynamic methods. In all the previous

numerical and experimental methods, the concept of force-based design has been used.

The effect of the displacement of the retaining wall on the development of seismic earth

pressure was not considered. The results predicted by numerical and experimental

methods related to the estimation of seismic earth pressure show that no clear and

unified trend can be drawn to compute the seismic earth pressure and its distribution.

The contradictions are found in the literature and for clarity are summarised in

(a) (b)


84

Table 2.2. The results predicted by some researchers like Matsuo (1941), Matsuo and

Ohara (1960), Sherif et al. (1982), Bolton M. D. and Steedman (1982), Sherif and Fang

(1984b), Steedman (1984 ), Bolton and Steedman (1985), Ishibashi and Fang (1987),

Green et al. (2003), Ortiz et al. (1983), Bolton and Steedman (1985), Zeng

(1990;Steedman and Zeng (1991), Stadler (1996) and Dewoolkar et al. (2001) support

the M-O method. However, other researchers like Siddharthan and Maragakis (1989),

Nakamura (2006), Al Atik and Sitar (2008), Geraili et al. (2016), Candia et al. (2016)

and Jo et al. (2017) revealed that the M-O method does not appropriately express the

real behaviour of the backfill and gravity retaining walls during earthquakes.

Some researchers like Nakamura (2006), Al Atik and Sitar (2008), Geraili et al. (2016),

Candia et al. (2016) and Jo et al. (2017) show that the seismic earth pressure is highly

influenced by the phase difference between the retaining wall and retained backfill soil.

Some researchers like Matsuo (1941), Matsuo and Ohara (1960), Sherif et al. (1982),

Bolton M. D. and Steedman (1982), Sherif and Fang (1984b), Steedman (1984 ), Bolton

and Steedman (1985), Ishibashi and Fang (1987), Green et al. (2003), Ortiz et al.

(1983), Bolton and Steedman (1985), Zeng (1990;Steedman and Zeng (1991), Stadler

(1996), and Dewoolkar et al. (2001) Siddharthan and Maragakis (1989), Geraili et al.

(2016), Candia et al. (2016) and Jo et al. (2017) consider that the wall-soil system is

only in the active state, while other researchers like Nakamura (2006), Pathmanathan

(2007), Green et al. (2008), Al Atik and Sitar (2008) and Kloukinas et al. (2015) prove

that the wall-soil system is subjected to both active and passive states during an

earthquake. Relating to the distribution of seismic earth pressure, some researcher like

Stadler (1996), Al Atik and Sitar (2008), Geraili et al. (2016), Candia et al. (2016) and

Jo et al. (2017) show that the distribution can be approximated to a linear shape, while

others like Ortiz et al. (1983) and Nakamura (2006) demonstrate that the distribution of

seismic earth pressure is nonlinear. Candia et al. (2016) reveal based on their centrifuge

results that the seismic earth pressure is independent of cohesion and this conclusion is

in contrast to hypotheses of the previous pseudo-static method like Subba Rao and

Choudhury (2005), Ortigosa (2005), Shukla (2010), Shukla and Habibi (2011) and

Shukla and Zahid (2011). The discrepancy between the researchers’ results can reflect

the complexity of the seismic earth pressure problem, which can be considered as one of

the most complicated cases in soil-structure interaction.


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Table 2.2: Major findings and contradictions of the force-based design methods

Case study Researchers Major findings

Seismic earth

pressure

Matsuo (1941), Matsuo and Ohara (1960), Sherif

et al. (1982), Bolton M. D. and Steedman

(1982), Sherif and Fang (1984b), Steedman

(1984 ), Bolton and Steedman (1985), Ishibashi

and Fang (1987), Green et al. (2003), Ortiz et al.

(1983), Bolton and Steedman (1985), Zeng

(1990; Steedman and Zeng (1991), Stadler

(1996) and Dewoolkar et al. (2001)

Results obtained for

seismic earth pressure are

similar to the results

obtained by M-O method

Siddharthan and Maragakis (1989), Nakamura

(2006), Al Atik and Sitar (2008), Geraili et al.

(2016), Candia et al. (2016) and Jo et al. (2017)

M-O method does not

appropriately express the

real behaviour of the

backfill and gravity

retaining walls during

earthquakes

State of

seismic earth

pressure

Matsuo (1941), Matsuo and Ohara (1960), Sherif

et al. (1982), Bolton M. D. and Steedman

(1982), Sherif and Fang (1984b), Steedman

(1984 ), Bolton and Steedman (1985), Ishibashi

and Fang (1987), Green et al. (2003), Ortiz et al.

(1983), Bolton and Steedman (1985), Zeng

(1990; Steedman and Zeng (1991), Stadler

(1996), and Dewoolkar et al. (2001) Siddharthan

and Maragakis (1989), Geraili et al. (2016),

Candia et al. (2016) and Jo et al. (2017)

Wall-soil system is only in

the active state

Nakamura (2006), Pathmanathan (2007), Green

et al. (2008), Al Atik and Sitar (2008) and

Kloukinas et al. (2015)

Wall-soil system is

subjected to both active

and passive state during an

earthquake

Distribution of

seismic earth

pressure

Stadler (1996), Al Atik and Sitar (2008), Geraili

et al. (2016), Candia et al. (2016) and Jo et al.

(2017)

Distribution can be

approximated to linear

shape

Ortiz et al. (1983) and Nakamura (2006) Distribution of seismic

earth pressure is nonlinear

For a cantilever retaining wall, it is observed that few efforts have been made to

investigate the development of seismic earth pressure compared with research methods

proposed for the rigid retaining wall. However, in the available literature a great deal of

attention has been paid to the estimation of the seismic earth pressure for the gravity-

type retaining walls, while little emphasis has been given to the estimation of the

seismic earth pressure for the cantilever-type retaining walls. This is despite the basic


86

difference that the cantilever-type retaining walls behave as flexible members, while the

rigid retaining walls behave as rigid members.

It can also be noted that the results obtained from numerical modelling, as reported by

Green et al. (2008), as well as those obtained by Kloukinas et al. (2015) when using the

shaking table test, show that the critical loading case for seismic design for a cantilever-

type retaining wall is when the seismic acceleration is applied away from the backfill

soil, thereby rendering the retaining wall-soil system in a passive earth pressure

condition. This, however, is contradicted by the centrifuge test results reported by Jo et

al. (2017), who observed that the critical loading case for the retaining wall will be the

one in which the seismic acceleration is applied towards the backfill soil, thereby

rendering the retaining wall-soil system in an active earth pressure condition. However,

Candia et al. (2016) and Geraili et al. (2016) did not discuss the critical load case when

they investigated the seismic behaviour of a cantilever retaining wall by using a series

of centrifuge tests.

Similar contradictions are found in the literature and for clarity are summarised in

Table 2.3. It is important to highlight that the identification of a critical loading case for

the proper design of a retaining wall is extremely important; however, as evidenced

from Table 2.3, the results available in the literature are contradictory.


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Table 2.3: Observations and contradictions in the estimation of seismic earth pressure for a

cantilever-type retaining wall

Research method Researcher Observations

Numerical Green et al. (2008)

The critical load case causing maximum bending moment

in the stem of the wall occurs when the earthquake

acceleration is applied away from backfill soil, thereby

rending the wall-soil system in a passive state.

Shaking table test Kloukinas et

al.(2015) The same observations reported by Green et al. (2008).

Centrifuge test &

Numerical Geraili, et al. (2016)

The static earth pressure is in an active state along the

height of the wall. The seismic earth pressure is lower than

M-O value. The seismic earth pressure distribution is

linear. There is no clear discussion about the critical load

case.

Centrifuge test &

Numerical Candia, et al. (2016)

The static earth pressure is in an active condition along the

height of the wall. The seismic earth pressure is lower than

M-O value. The seismic earth pressure distribution is

linear. There is no clear discussion about the critical load

case.

Centrifuge test &

Finite difference

Joe et al. (2017) The static earth pressure at the bottom of the cantilever

retaining wall is between the active and at-rest values while

it is active at the top of the wall. The seismic earth pressure

changed with time and its distribution is close to a

triangular shape. The M-O method overestimates the

seismic earth pressure. Dynamic wall moment was induced

by seismic earth pressure and inertia force of the wall. The

inertial moment of the wall cannot be ignored. The critical

load case causing maximum bending moment in the stem

of the wall occurs when the earthquake acceleration is

applied towards the backfill soil, thereby rendering the

wall-soil system in an active state.

2.5.2 Displacement-based design method

Although the analysis methods discussed previously have provided valuable

information about the development of seismic earth pressure that is induced on the back

of retaining walls during an earthquake, the real field observations from post-

earthquakes (see section 2.5.4) indicated that many retaining walls had failed because of


88

the excessive displacement. Hence, an effort has been made to propose design methods

to predict the permanent retaining wall displacement and design a retaining wall based

on the allowable displacement. This design technique is called the ‘displacement-based

design method’. Several analytical, numerical and experimental methods have been

proposed in the literature to estimate the permanent displacement of the retaining wall,

and they will be discussed in this section.


Several analytical methods have proposed to estimate the permanent displacement of

retaining walls based on different concepts like one-block analysis concept and two-

block analysis concept.

2.5.2.1.1 One-block methods

Richards-Elms method (1979): Richards and Elms (1979) developed a method for

seismic design of gravity retaining walls, depending on allowable permanent wall

displacements. The method predicts the permanent displacement of the retaining wall in

a procedure similar to the Newmark sliding block procedure ((Newmark, 1965)

proposed for estimation of seismic slope stability.

The Richard-Elms procedure requires an evaluation of the yield acceleration for the

retaining wall-soil system. When the active wedge is subjected to acceleration acting

toward the backfill, this will cause inertial force acting away from the backfill, as shown

in Figure 2.33. Richards and Elms assumed that a soil wedge in the retained backfill

reaches an active stress state and they used the M-O method to estimate the total seismic

active earth pressure. The level of acceleration that is required to cause the wall to slide

on its base is the yield acceleration, and it can be expressed as:

cos( ) sin( ) tantan ae ae b

y b

P Pa g

W

(2.26)

where, ya = yield acceleration, W = weight of the wall, and Pae = seismic active earth

pressure and is calculated using the M-O method as recommended by Richards and

Elms. They also proposed the following formula to calculate permanent block

displacement:


89

2 3

max max

40.087perm

y

v ad

a

max 0.3

ya

a (2.27)

where vmax = peak ground velocity, amax = peak ground acceleration, and ay the yield

acceleration for the wall-backfill system. The estimation of the yield acceleration can be

computed by using an iteration manner. Firstly, the total seismic earth pressure can be

calculated by using Equation 2.10 by assuming a trial yield acceleration equal to the

pseudo-static acceleration. Then, a new yield acceleration can be calculated from

Equation 2-29. If the computed yield acceleration is inconsistent with the assumed

pseudo-static acceleration, a new iteration will be required until the computed yield

acceleration becomes close to the assumed pseudo-static acceleration.

Figure 2.33: Forces acting on a wall-soil system proposed by Richards and Elms (1979)

The Richard-Elms procedure includes estimating the weight of the retaining wall

required to ensure that the expected permanent displacement is equal to or less than the

allowable value. The design procedure can be described as follows:

Select an allowable displacement of the retaining wall

Compute the yield acceleration using Equation 2.29

Calculate the total seismic earth pressure using Equation 2.10

Compute the weight of the retaining wall using Equation 2.29 to limit the

permanent displacement to the allowable displacement


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Apply the factor of safety to the weight of the wall. Richard and Elms

recommended a factor of safety of 1.5.

Similarly, Nadim and Whitman (1983), Steedman (1984 ;Whitman RV and S (1984)

proposed a procedure using a one-block method to compute the displacement of the

retaining wall. To account for the permanent seismic displacement as well as the

permanent rotation of the retaining wall. Zeng and Steedman (2000) proposed a pseudo-

static rotating block method to estimate the permanent displacement of a gravity

retaining wall. The method can be extended to estimate the displacement where sliding

and rotation of a retaining wall is coupled. The method takes into account the influence

of the ground motion characteristics. This method assumed that the backfill soil behaves

as a rigid plastic material and seismic earth pressure can be computed using the M-O

method. The retaining wall cannot rotate towards the backfill material. The foundation

layer below the retaining wall is assumed to be a rigid layer. Wu and Prakash (2001)

proposed one-block sliding analysis methods to estimate the seismic displacement of the

retaining wall taking into account the effect of the stiffness of the foundation layer. The

one-block method is considered the iterative approach, and there is no closed-form

solution to compute the horizontal critical acceleration coefficient. The effect of backfill

material on the retaining wall was only expressed by using a pseudo-static approach like

the M-O method, and this approach has some practical limitations, as discussed in

section (2.5.1.1.2). Corigliano et al. (2011) have proposed a novel procedure to improve

the applicability of the Newmark method in computing the permanent displacement of

gravity earth-retaining structures induced by earthquake loading introducing the effects

of the double-support seismic excitation in the foundation layer and backfill retained

soil. The results predicted from the modified Newmark procedure show that the

standard Newmark method underestimates residual displacement.

2.5.2.1.2 Two-block methods

The effect of the backfill wedge sliding is considered in this analysis procedure to

evaluate the seismic displacement of the retaining wall. When the retaining wall slides

on its base, the backfill wedge will slide downwards with the inclination of the least soil

resistance. Hence, the wall-soil system consists of two blocks. Zarrabi and Kashani

(1979) computed the angle of the critical wedge during the retaining wall displacement.


91

Stamatopoulos and Velgaki (2001) proposed a two-block procedure to compute the

displacement of the retaining wall considering only horizontal seismic acceleration.

They found that the earth thrust acting on the wall does not coincide with the active

earth thrust predicted by the M-O method and the angle of the critical wedge does not

coincide with that predicted by Zarrabi and Kashani (1979). Stamatopoulos et al. (2006)

extended the previous solution proposed by Stamatopoulos and Velgaki (2001) to take

into account the case of cohesive and frictional backfill and foundation soils. Caltabiano

et al. (2012) also used the two-block procedure and proposed closed-form solutions for

computing the critical horizontal acceleration and critical backfill wedge, considering

different surcharge and boundary conditions. Conti et al. (2013) proposed a new two-

rigid block model for sliding gravity retaining walls. The study shows that the proposed

method is capable of fully describing the kinematics of the whole system’s wall-soil

under dynamic loading. Biondi et al. (2014) proposed a method to derive an equivalent

seismic earth pressure coefficient related to the sliding of the retaining wall based on the

two-block approach. The induced earthquake displacement was introduced by an

alternative definition of the wall safety factor. The new displacement model has been

used to predict reliable values of an equivalent seismic earth pressure coefficient and

can be used to check the performance of a retaining wall.


Many numerical methods have been proposed to evaluate the seismic response of

retaining structures using a displacement-based design philosophy. The numerical

analysis involves using either finite element or finite difference methods. A plane-strain

condition was considered in most numerical methods. Thus, this condition is valid for

the retaining structures problem and provides an economical solution to the numerical

problem. The wall-soil domain in all numerical models is discretised to a large number

of elements. The domain is bounded by vertical and horizontal boundary conditions. A

variety of interface elements have been used to model the soil-structure interaction. The

stress-strain behaviour of the backfill soil is modelled by a variety of constitutive

models available in the literature and commercial programmes like the Mohr-Coulomb

model, hardening soil model with small-strain, etc. The behaviour of the retaining wall

is modelled as rigid elastic. During the seismic analysis, in most cases, the absorbing


92

boundaries were used in the lateral boundaries to reduce the effect of wave reflection.

Either a real earthquake or sinusoidal earthquake acceleration is used to simulate the

seismic loading. The major findings from the numerical models based on displacement-

based design philosophy will be discussed as follows.

Nadim and Whitman (1983) proposed a finite element model to compute the permanent

displacement of a rigid retaining wall taking into account the amplification of

earthquake acceleration. The finite element result shows that the amplification of

earthquake acceleration in the backfill has an important effect on the permanent

displacement of the wall when the ratio of dominated frequency of the earthquake

acceleration to the fundamental frequency of backfill soil is greater than 0.3.

Madabhushi and Zeng (1998) conducted finite element analysis to investigate the

seismic response of a rigid retaining wall. The displacement predicted by numerical

modelling agrees reasonably well with experimental observations.

Bhattacharjee and Krishna (2009) conducted numerical analysis to investigate the

dynamically induced displacement of a retaining wall. The numerical analysis was

carried out using a computer package, FLAC 3D, investigating the effects of ground

acceleration, frequency and properties of backfill soil.

Corigliano et al. (2011) proposed finite difference analysis to compute the relative

displacement between the retaining wall and the soil foundation layer to verify the

modified Newmark’s procedure. The predicted results of the numerical model support

the proposed modified Newmark’s procedure, as shown in Figure 2.34.


93

Figure 2.34: Comparison between relative displacement predicted by FLAC, Newmark classic

and modified Newmark’s procedure (after Corigliano et al., 2011)

Tiznado and Rodríguez-Roa (2011) carried out a series of two-dimension finite element

analysis by using PLAXIS software to investigate the seismic behaviour of a gravity

retaining wall. The results showed that seismic amplification effects in the soil

foundation and backfill have a significant role in determining the permanent

displacements of these walls.

Conti et al. (2013) conducted numerical analysis, as shown in Figure 2.35, for assessing

the capability of the analytical model of the dynamic behaviour of the gravity retaining

wall. The results showed that there was a good agreement between the numerical

method and the two-rigid block model.

Figure 2.35: Numerical model of a retaining wall proposed by Conti et al. (2013)

Ibrahim (2014) conducted FE analysis by using PLAXIS2D software. The results of this

study found that numerical seismic displacements are either equal to or greater than

corresponding pseudo-static values. It was also found that seismic wall displacement is

directly proportional to the positive angle of inclination of the back surface of the wall,


94

soil flexibility and with the earthquake maximum ground acceleration. Seismic wall

sliding is dominant, and rotation is negligible for rigid walls when the ratio between the

wall height and the foundation width is less than 1.4, while for greater ratios the wall

becomes more flexible and rotation (rocking) increases till the ratio reaches 1.8, where

it is susceptible to overturning.


2.5.2.3.1 Shaking table tests

Sadrekarimi (2011) conducted shaking table tests of two wall model types, as shown in

Figure 2.36, to study the displacement of broken-back quay walls under a seismic event.

The study observed that the loose foundation layer significantly contributed to the quay

walls’ horizontal displacement and rotation. The sliding displacement was a function of

the walls’ acceleration. The backfill ground settlement was observed at a long distance

behind the wall.

Figure 2.36: Cross section of 2 retaining walls used in shaking table tests conducted by

Sadrekarimi (2011)

Kloukinas et al. (2015) conducted a shaking table test, as shown in Figure 2.29, to study

the seismic displacement of a cantilever retaining wall. The configurations of the

shaking table test were described in section 2.5.1.3.1. The experimental study found that

there is no rigid block response in the backfill soil observed during shaking. The critical

acceleration (yield acceleration) required for the wall to slide increases as the wall toe

penetrates the foundation layer, resulting in more sliding resistance. The study also


95

observed that the sliding was more sensitive to shaking time while the rotation was

more sensitive to the acceleration level.

2.5.2.3.2 Centrifuge tests

Saito et al. (1999) conducted a series of centrifuge tests to study the seismic behaviour

of a rigid retaining wall. Both backfill and foundation layers were modelled by using

dry Toyoura sand with a relative density of 82%. A sinusoidal wave was applied with

an amplitude of 0.4g and frequency 1.5 Hz for 25 s and used to predict the horizontal

displacement at the base of the retaining wall and rotation about the base. The

permanent displacement at the base of the wall was 1.4m while the rotation of the wall

was 4° away from the backfill soil.

Zeng and Steedman (2000) carried out a centrifuge test to verify their analytical solution

to compute the permanent rotational displacement. Figure 2.37 shows the cross section

of the wall-soil system in the centrifuge test. The gravity retaining wall was constructed

of three concrete blocks and constrained against base rotation. A series of earthquakes

were applied at the base of the centrifuge model.

Figure 2.37: Cross section of centrifuge test conducted by Zeng and Steedman (2000)

The major finding of the experimental study is that the amplification of acceleration was

a significant factor and it should be accounted when computing rotational displacement.

The amplitude of the displacement of the wall increased from the wall’s base to its top.

One of the controversial observations in the test is that there was no wall sliding

displacement after the test but the permanent displacement was caused by rotation of the

wall.


96

Nakamura (2006) carried out a series of centrifuge tests (see Figure 2.30) to study the

influence of input seismic motion in the behaviour of a retaining wall. All

configurations of the centrifuge test were presented in section 2.5.3.1.2. Many

sinusoidal earthquake accelerations were applied to study the deformation of the

retaining and backfill soil during the shaking event. The main observation of the test is

that the retaining wall and backfill soil move in the active and passive directions and

they oscillate as a result of a change in the acceleration direction. A rigid block is not

formed in the backfill soil behind the retaining wall as assumed in force-based design

methods.

2.5.2.4 Critical discussion on displacement-based design methods

The displacement-based design methods are considered a more reliable design

technique to investigate the seismic performance of retaining walls. Analytical methods

derived to compute the permanent seismic displacement have shown their efficiency in

assessing the seismic performance of retaining structures when they are compared with

numerical and experimental results. For example, numerical methods proposed by

Green et al. (2008) and Corigliano et al. (2011) as well as experimental methods

conducted by Conti et al. (2012) indicated that the Newmark sliding method provides a

reasonable estimation of seismic wall displacement.

However, traditional methods like one-block and two-block methods account for the

effect of retained backfill soil by considering the seismic active earth pressure induced

behind the retaining wall. The seismic active earth pressure used for evaluating the

permanent seismic displacement is usually estimated by using the M-O method.

However, as discussed in section 2.5.1.1.2, the M-O method has inherent limitations

which might lead to an inaccurate estimation of seismic earth pressure, and this could

cause an inaccurate prediction of the permanent seismic displacement of a retaining

wall. Hence, efforts will be required to produce an accurate estimation of the

contribution of the seismic earth pressure force to the permanent seismic displacement

of the rigid-type retaining wall and cantilever-type retaining wall. The effect of the

foundation layer below the retaining wall on the retaining wall’s displacement was

noted but not clearly discussed. Further investigations are also required to estimate the


97

effect of backfill material and earthquake characteristics on the permanent seismic

displacement of the retaining wall.

2.5.3 Force-displacement hybrid design methods

As discussed previously, efforts have been made to draw the relationship between the

static earth pressure and the displacement of the retaining wall. Most studies have

proven that the minimum active earth pressure develops behind the retaining wall when

the retaining wall is moving relatively a small displacement away from the retained

backfill soil, while the maximum passive earth pressure develops when the retaining

wall is moving a large displacement towards the retained backfill soil. The distribution

of either active or passive earth pressure is noted as being highly related to the mode of

retaining wall displacement (sliding and/or rotation about the top or base of the wall).

For the seismic condition, the relationship between the seismic earth pressure and

displacement is still very difficult to understand because the behaviour of a retaining

wall under seismic loading is much more complicated than under static loading.

As discussed in the static earth pressure section, the relationship between seismic earth

pressure and displacement of the retaining wall is pivotal to the design of retaining

walls. It was noted that this relationship is a main component in designing the bridge

abutment. This relationship can provide an accurate estimation of earth pressure at any

stage of retaining wall movement. In contrast to the static state, a few efforts have been

made to understand the relationship between the seismic earth pressure and the

displacement of the retaining wall, and evaluate the contribution of seismic earth

pressure to the permanent displacement of the retaining wall. The next subsections will

present an in-depth discussion of the analytical, numerical and experimental methods

that have been proposed in the literature to investigate the relationship between seismic

earth pressure and the displacement of the retaining wall.


Veletsos and Younan (1997) proposed an analytical elastic solution for estimating the

magnitude and distribution of dynamic earth pressure, displacement and forces induced

by horizontal excitation in walls assumed to be flexible and elastically constrained

against rotation at their base. The soil was assumed as a uniform visco-elastic medium


98

of height H, as shown in Figure 2.38, and it was defined by the density, shear modulus,

Poisson’s ratio and damping ratio. The retaining wall was assumed to be vertical,

flexible and elastically constrained against rotation at its base.

A harmonic excitation was used as an earthquake acceleration, and it was controlled by

the ratio between the dominant cyclic frequency of earthquake acceleration and the

fundamental frequency of the soil layer. In this analysis, there is no de-bonding or

relative displacement allowed to occur at the wall-soil interface. The retaining wall was

considered massless, and on vertical stresses developed. The two main parameters that

were considered to affect the response of the wall-soil system are:

Figure 2.38: Analytical model of a retaining wall proposed by Veletsos and Younan (1997)

(1) Relative flexibility of the wall

3

ow

w

G Hd

D (2.28)

where, Go = initial soil shear modulus, H = height of the wall and Dw = flexural rigidity

per unit length of the wall:

2

212 1

w ww

E tD

v

(2.29)

where, 𝐸𝑤 = modulus elasticity of the wall, tw = thickness of the wall and v = Poisson’s

ratio.

(2) The relative flexibility of the rotational base constraint


99

2

oG Hd

R

(2.30)

where, 𝑅𝜃 = stiffness of the rotational base constraint.

The analytical solution presented that the dynamic earth pressure strongly depends on

the wall flexibility and foundation rotational compliance. The results obtained from this

analysis show that the dynamic earth pressure was lower than for pressure for a rigid

and fixed-base wall, as shown in Figure 2.39. It was found that the dynamic earth

pressure may reduce to the level of the M-O solution (rigid plastic method) if either the

wall or base flexibility increases.

Figure 2.39: Distributions of wall pressure for statically excited systems with different wall and

base flexibilities: a) dθ = 0, b) dw = 0. (after Veletsos and Younan, 1997)

The above solution was limited by the assumption of complete contact between the wall

and soil, which leads to the development of tensile stress on the wall. The analysis also

ignored the effect of the horizontal translational displacement. The behaviour of the soil

layer was assumed to be linear elastic, and the retaining wall was considered massless.

All these limitations were considered to oversimplify the response of the wall-soil

system, and therefore this analysis did not reflect the realistic response of a wall-soil

system.

Zhang et al. (1998a) proposed the intermediate wedge concept. This concept is

developed depending on the soil frictional resistance during the retaining movement

between the active and passive direction for the static condition. Zhang et al. (1998b)


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used the same concept to estimate the seismic earth pressure as a function of retaining

wall displacement. Figure 2.40 shows the formation of an intermediate wedge between

the active and passive states during an earthquake. A new equation for seismic earth

pressure has been proposed and separated into four components according to their

formation: the static earth pressure force, seismic inertial force, surcharge load and

residual earth pressure force. The first three components of seismic earth pressure are

physically related to mode and level of wall movement.

Figure 2.40: Geometry of an intermediate wedge during an earthquake proposed by Zhang et al.

(1998b)

The proposed method assumed the seismic earth pressure coefficients for active and

passive conditions derived by the M-O method can be varied with horizontal wall

displacement. The new equations for seismic active and passive earth pressure were also

extended to consider different levels and modes of wall movement (translation –

rotation about base – rotation about top). The derived equations can be reduced to the

M-O equations.

The same assumptions adopted for the M-O method were used in the above current

method. The seismic loading was applied as a constant load even though seismic

loading is cyclic and it changes its magnitude and direction with time. Further, this

method was established based on the assumption that the displacement of the retaining

wall is not predicted as a response of the fluctuation of earthquake acceleration. The

seismic earth pressure amplitude is only related to the fluctuation of the earthquake

acceleration, and the fluctuated seismic earth pressure tends to reduce when the

retaining wall moves away from the backfill soil, as shown in Figure 2.41. This


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assumption did not reflect the real behaviour of the wall-soil system under seismic

loading. However, the observations from the centrifuge test conducted by Nakamura

(2006) revealed that both seismic earth pressure and displacement of retaining wall are

dependent on earthquake acceleration.

Figure 2.41: Reduction of seismic earth pressure when the retaining wall moves away from the

backfill soil, as proposed by Zhang et al. (1998b)

Richards et al. (1999) proposed a kinematic method to calculate the seismic earth

pressure against retaining structures. The dry backfill soil was represented by elastic,

perfectly plastic material with a Mohr-Coulomb yield criterion and it was retained on a

vertical rigid wall. The backfill soil was modelled by a series of spring or subgrade

moduli, as shown in Figure 2.42. The retaining wall was assumed to be either fixed or

moveable.

The subgrade modulus was related to the value of the elastic or secant shear modulus of

soil. The seismic earth pressure was determined based on the free-field stress and

deformation compared to the movement of the retaining wall. The horizontal

acceleration was assumed to be uniform within the soil layer. The point of application

of seismic earth pressure was found to vary with different wall movements, rotation

about the base, rotation about the top and horizontal translation. The seismic active

condition was only considered in the analysis. It can also be noted that the analysis

assuming the displacement of the retaining wall is not related to the seismic acceleration

response. The seismic earth pressure is only acceleration-dependent; also, this

assumption did not reflect the real behaviour of the wall-soil system.


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Figure 2.42: Analytical model of a wall-soil system proposed by Richards et al. (1999)

Song and Zhang (2008) also used the concept of the intermediate wedge with a curved

sliding surface to propose a new methodology to compute the seismic passive earth

pressure under any level of horizontal displacement of a rigid retaining wall. The

approach can compute the seismic passive earth pressure of normally consolidated

cohesionless soil under any lateral deformation between the isotropic compression and

passive state. Figure 2.43 shows the relationship between the seismic passive earth

pressure coefficient and normalised wall displacement in the passive side under

different horizontal seismic coefficients for wall friction angle 2 / 3.

Figure 2.43: Relationship between seismic passive earth pressure and normalised wall

displacement predicted by Song and Zhang (2008)


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However, the analysis also assumed that the displacement of the retaining wall is not

seismic acceleration-dependent and the seismic acceleration is simulated as a pseudo-

static force.


Psarropoulos et al. (2005a) presented an FE analysis to predict the distribution of

dynamic earth pressures behind a rigid and flexible non-sliding retaining wall. The

numerical result was compared with available analytical results proposed by Veletsos

and Younan (1997) (section 2.5.3.1). The wall-soil system in this analysis consisted of a

rectangular gravity retaining wall constructed on a horizontally visco-elastic soil

foundation layer and retained a semi-infinite layer of visco-elastic soil backfill layer.

The whole wall–soil system was simulated by two-dimensional, plane-strain,

quadrilateral 4-noded finite elements and the finite-element mesh was truncated by the

use of viscous dashpots, as shown in Figure 2.44. The critical damping ratio was

assumed to be 5% for both soil layers. The wall was modelled by a rigid elastic

behaviour. No de-bonding and relative slip were assumed to occur in the interface

between the wall and retained soil as well as foundation soil. The harmonic excitation

was introduced by a prescribed acceleration time history on the nodes of the base of the

foundation layer.

Veletsos and Younan (1997) assumed that the effect of the foundation layer is simulated

by a rotational elastic constraint at the base of the wall, and this assumption is expected

to affect the retaining wall response. To assess this effect, in current finite element

analysis, the retaining wall and backfill soil were constructed on the linear visco-elastic

soil foundation layer. The following parameters were investigated in the finite element

analysis:


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Figure 2.44: Finite element model of xx proposed by Psarropoulos et al. (2005a)

The base width B to the wall height H ratio (B/H), the relative flexibility factor (it was

defined by changing the shear velocity of the foundation layer), and two values of the

excitation cyclic frequency ω were investigated: ω= ω1/6 (almost static) and ω = ω1

(resonance), where ω1 is the fundamental cyclic frequency of the two layers’ profiles.

For the static response (ω= ω1/6), the earth pressure distribution was compared for those

predicted by Veletsos and Younan’s spring model considering the different values of

rotational flexibility and B/H ratio for the two-layer finite element model. It is evident

from Figure 2.45a and b that the increase in the rotational flexibility of the system leads

to a reduction of wall pressure. Replacing the Veletsos and Younan spring at the base of

the wall by an actual foundation layer leads to a further reduction of wall pressure. For

resonance excitation (ω = ω1), as shown in Figure 2.45c and d, the earth pressure

increases when the wall flexibility (simulating the foundation layer with low shear

velocity) and B/H ratio increase.


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Figure 2.45: Distribution of earth pressure in: a) ω= ω1/6 (almost static) - dθ =0.5, b) ω= ω1/6

(almost static) - dθ =5, c) ω = ω1 (resonance) - dθ =0.5, and d) ω = ω1 (resonance) - dθ =0.5.

(after Psarropoulos et al., 2005)

Figure 2.46: Effect of wall rotational flexibility on the amplification factor of total forces acting

on the retaining wall. After Psarropoulos et al. (2005)

It was also noted that the amplification factor of shear force and moment in the

rotational spring model increases when the wall flexibility increases, while these

amplification factors reduce when the wall flexibility of the real two-layer model

increases, as shown in Figure 2.46a and b.

(a)

(b)

(c)

(d)


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It can be observed that the current finite element analysis also has some limitations and

assumptions which lead to an unrealistic response of the wall-soil system. For example,

the soil behaviour of the backfill and foundation layer was assumed to be elastic while,

in reality, the soil materials show highly nonlinear behaviour. The complete contact

assumed between the wall and backfill soil layer leads to the development of an

unrealistic tensile stress on the wall near the ground surface (see Figure 2.45). The

effect of the sliding of the retaining wall also was not taken into account in the finite

element analysis because complete contact was assumed between the wall and

foundation layer. It was revealed that the wave propagation in the foundation layer

might have an effect that cannot be simply accounted for with a rocking spring at the

base of the wall. Finally, for the analytical and numerical models, it is noted that they

did not predict the exact value of the earth pressure at the heel of the wall. However,

this value has a significant effect on the magnitude of total dynamic earth pressure and

overturning moment.


Early investigations were conducted by Ichihara and Matsuzawa (1973), Ichihara et al.

(1977) and Sherif et al. (1982) by using shaking table tests to study the effect of wall

displacement on the development of seismic earth pressure. They found that the

seismic earth pressure and its distribution along retaining wall height varied with the

amplitude of seismic acceleration and magnitude and mode of wall movement.

Following that, Sherif and Fang (1984a), Sherif and Fang (1984b) and Ishibashi and

Fang (1987) conducted experiments using the shaking table test. A moveable retaining

wall, as shown in Figure 2.47, was installed at one side of the vibrating soil box.

Dry sand was used in the experiments with relative density of 53%. During the shaking

test, a moveable retaining wall was moved away from the backfill with different modes

of rotation about the base mode, rotation about the top mode and translation mode,

while the soil box was vibrated with a constant horizontal acceleration at a frequency of

3.5 Hz. The result obtained from the shaking table experiments showed that:


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Figure 2.47: Cross section of the shaking table test conducted by Ishibashi and Fang (1987)

When the retaining wall rotated about the base, the distribution of dynamic earth

pressure was nonlinear and a high residual stress zone was observed near the base

of the wall, as shown in Figure 2.48.

Figure 2.48: Effect of wall rotation about its base on the distribution of seismic earth pressure.

After Ishibashi and Fang (1987)

When the retaining wall rotated about the top, the distribution of dynamic earth

pressure was also nonlinear and there was a residual stress zone observed near the

top, as shown in Figure 2.49.


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Figure 2.49: Effect of wall rotation about the top on the distribution of seismic earth pressure.

(after Ishibashi and Fang, 1987)

The study recorded that, for the lower horizontal acceleration, the dynamic earth

pressure was controlled by the displacement geometry of the retaining wall while, for

high acceleration, the effect of dynamic inertia forces was dominant and the effect of

wall displacement became negligible.

It can be noted that, in the experiments, the displacement of the retaining wall was

performed by the controlled displacement during the shaking process and the study

ignored the wall displacement predicted from the dynamic response of the retaining wall

itself. The inherent limitation of the shaking table tests may lead to unconservative

estimates of the seismic earth pressures. Another limitation observed in this study is that

the effect of foundation layer deformability on the wall displacement and retaining wall

displacement was ignored. This experimental study also just discussed the development

of the seismic active condition while later experimental investigations conducted by

Nakamura (2006) using centrifuge tests pointed out that both the seismic active and

passive conditions developed during the shaking event.

2.5.3.4 Critical discussion on force-displacement hybrid design methods

Previous discussion on the use of force-displacement design methods has shown that the

relationship between seismic earth pressure and displacement of retaining wall is not

very well understood like in static case. Analytical method proposed by Veletsos and

Younan (1997) revealed that the seismic earth pressure is highly affected by the


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flexibility of retaining wall and base rotational constraint. Numerical study proposed by

Psarropoulos et al. (2005) in order to verify the previous analytical work (Veletsos and

Younan (1997)) indicated that the seismic earth pressure is related to the flexibility of

the retaining wall and base deformation. It can also be noted that the results obtained

from analytical modelling, as reported by Veletsos and Younan (1997) as well as those

obtained by Psarropoulos et al. (2005) show that the seismic earth pressure reduces

when the flexibility of retaining wall and base rotational constraint (shear velocity of

foundation soil) increases. Veletsos and Younan (1997) reported that the amplification

factor of structural forces increases when the flexibility of retaining wall and base

rotational constraint (shear velocity of foundation soil) increases. This, however, is

contradicted by the numerical modelling results reported by Psarropoulos et al. (2005),

who observed that the amplification factor of structural forces reduces when the

flexibility of retaining wall and base rotational constraint (shear velocity of foundation

soil) increases. Other researchers like Zhang et al. (1998b) and Song and Zhang (2008)

modified the pseudo-static methods by proposing the intermediate wedge concept to

investigate the relationship between the seismic earth pressure and displacement of

retaining wall. However, the pseudo-static methods have already criticised and they are

found not represent the seismic behaviour of retaining wall. Richards et al. (1999)

proposed a kinematic method to calculate the seismic earth pressure against retaining

structures considering the effect of retaining wall displacement and rotation. Sherif and

Fang (1984a), Sherif and Fang (1984b) and Ishibashi and Fang (1987) conducted

experiments using the shaking table test to investigate the effect of wall rotation on the

seismic earth pressure. However, the previous methods assumed that the retaining wall

is rigid, and they ignored the seismic response of the retaining wall. Hence, the

displacement of the retaining wall is assumed to be not seismic acceleration-dependent.

Table 2.4 summarises the major finding in the literature concerning the relationship

between the seismic earth pressure and displacement of retaining wall.


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Table 2.4: Major findings highlighting the relationship between the seismic earth pressure and

wall displacement.

Research method Researcher Major findings

Analytical

Veletsos and Younan

(1997)

The seismic earth pressure reduces while amplification

factor of structural forces increases when the wall

flexibility and base rotational constraint increases.

Zhang et al. (1998b), The seismic active earth pressure reduces when the

displacement of the wall increases

Song and Zhang

(2008)

The seismic passive earth pressure increases when the

displacement of the wall increases

Richards et al.

(1999)

The seismic active earth pressure reduces when the

displacement and rotation of the wall increases.

Numerical

Psarropoulos et al.

(2005)

The seismic earth pressure and amplification factor of

structural forces reduce when the wall flexibility and base

rotational constraint increases.

Experimental

Sherif and Fang

(1984a), Sherif and

Fang (1984b) and

Ishibashi and Fang

(1987)

When the retaining wall rotated about the base, the

distribution of dynamic earth pressure was nonlinear and a

high residual stress zone was observed near the base of the

wall. However, when the retaining wall rotated about the

top, the distribution of dynamic earth pressure was also

nonlinear and there was a residual stress zone observed

near the top.

2.5.4 Real field observations of retaining wall damage post-earthquake

This section presents a variety of information and field observations of the seismic

behaviour and damage of retaining walls in different seismic-prone zones. For non-

liquefied soil, there has been limited information related to the field performance of the

retaining wall during recent major earthquakes. However, collected information has

shown that some retaining walls experience damage or large deformation during an

earthquake although they have been constructed to support dry cohesionless backfill

materials. In this section, selected case histories are presented to describe the behaviour

of the retaining wall during seismic scenarios.

Clough and Fragaszy (1977) reported that open U-shaped channels were constructed to

support a dry medium-dense sand soil. The structures were designed for static earth


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pressure by using the Rankine method, and there were no seismic provisions included in

the design. They reported that the cantilever walls collapsed during the San Fernando

earthquake (1971), as shown in Figure 2.50.

Chang et al. (1990) reported that the field measurements of seismic earth pressure

behind the embedded retaining walls were similar to or less than those computed by the

M-O method.

Figure 2.50: Details of a typical retaining wall failure (a) actual photograph, (b) diagram

capturing the failure of the u-shaped channels (after Clough and Fragaszy, 1977)

Prakash and Wu (1996) demonstrated that some retaining walls were rotated about 1 or

2 degrees during the Hokkaido-Nansi-Oki earthquake (1993). During the Northridge

earthquake (1994), the retaining wall, which had a height of 1.5 m and was constructed

on Holocene sediments, was moved 5-6 m outward. Other concrete crib walls with

height of 9m and a conventional concrete retaining wall with height of 5m were

observed to experience complete failure. The masonry retaining wall, which was 4m in

height and constructed on a gravel sand layer, completely failed during the Kobe

earthquake (1995). Two other lean-type unreinforced concrete retaining walls 2.6m and

5m in height, constructed on sandy gravel and reclaimed land, respectively, were

completely overturned.

Koseki et al. (1995) attempted to explore the behaviour of retaining walls during the

Hyogoken-Nambu earthquake of January 17, 1995. They noted that many retaining

walls with dry backfill were damaged and this can be summarised as follows:


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Leaning type concrete walls: the unreinforced retaining wall extends for a length of 500

m. Some parts of the wall were broken and upper part of the wall was completely

overturned while other parts that have a small embedment depth they were completely

overturned about the bottom as shown in Figure 2.51.

Figure 2.51: Leaning-type concrete walls a) cross section, b) sketch. (after Koseki et al., 1995)

Gravity retaining walls: several parts of the gravity-unreinforced wall, which extends

for a length of 400m, were largely tilted while other parts, which extend for a length of

200 m, were broken at the mid-height and overturned, as shown in Figure 2.52.

Figure 2.52: Gravity retaining walls a) cross section, b) sketch. (after Koseki et al., 1995)

Cantilever reinforced concrete walls: the sections of the cantilever retaining wall with

a length of 30 m were observed to suffer cracking at the mid-high level, significantly

tilting away from the backfill soil, as shown in Figure 2.53. Other cantilever-reinforced

walls, which were constructed to support a sloped embankment for a length of 200m,

were observed to suffer extensive cracking at the mid-height and significant sliding and

tilting outward, as shown in Figure 2.54.

(a) (b)

(a) (b)


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Figure 2.53: Cantilever reinforced concrete walls a) cross section, b) sketch. (after Koseki et al.,

1995)

Figure 2.54: Cantilever reinforced concrete wall supporting slope a) cross section, b) sketch.

(after Koseki et al., 1995)

Lew et al. (1995) demonstrated that the temporary pre-stressed-anchored walls were

deflected by about 1 cm without significant damage observed when they were subjected

to an acceleration level of 0.2g and in some cases close to 0.6g during the Northridge

earthquake in 1994.

Pathmanathan (2006) reported that many of the fill slopes failed because of excessive

deformation of the gravity retaining wall during the Niigata-Ken Chuetsu earthquake in

2004. Figure 2.55a shows the failure of a retaining wall because of the seismic

excessive displacement during this earthquake. The author also reported the failure of

retaining walls during the Chi-Chi earthquake in 1999. Figure 2.55b shows that

retaining walls that were structurally damaged during the two earthquakes.

(a) (b)

(a) (b)


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Figure 2.55: Failure of retaining walls caused by a) Chi-Chi earthquake1999, b) Niigata-Ken

Chuetsu earthquake, 2004

Gazetas et al. (2004) demonstrated that the retaining structures constructed in the

Kerameikos metro station were able to resist a 0.5g acceleration level without damage,

even though they were not designed for seismic conditions. They reported that the

maximum wall displacements did not exceed a few centimetres.

Kiyota et al. (2017) conducted a preliminary damage survey immediately after the 2016

Kumamoto earthquake in Japan. According to their observations, the gravity-type and

cantilever-type of retaining wall of 2-3m height as well as dam spillway retaining walls

were structurally damaged, as shown in Figure 1.1.

According to the previous field observations of the seismic behaviour of retaining

structures, it can be remarkably noted that many of these structures failed because they

experienced excessive displacements. However, many reinforced cantilever retaining

walls were observed to fail due to structural damage. Hence, an accurate estimation of

the seismic performance of the retaining structures could provide a safer and economic

design and reduce the disastrous physical consequences.

2.6 EUROCODE 8: DESIGN OF STRUCTURES FOR EARTHQUAKE

RESISTANCE

This code covers the design of different foundation systems, the design of earth

retaining structures, and soil structure interaction under seismic actions. The design of

retaining walls is dealt with in Chapter 7 of Eurocode8-Part 5.

(a) (b)


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2.6.1 General requirements

Eurocode 8 stipulates that “1) Earth retaining structures shall be so conceived and

designed as to fulfil their function during and after the design earthquake, without

suffering significant structural damage; 2) Permanent displacements, in the form of

combined sliding and tilting, the latter due to irreversible deformation of the foundation

soil, may be acceptable if it is shown that they are compatible with functional and/or

aesthetic requirements”.

2.6.2 Methods of analysis

For methods of the analysis, it is stated for general methods that: “

1) Any established method based on the procedures of structural and soil dynamics, and

supported by experience and observations, is in principle acceptable for assessing the

safety of an earth-retaining structure.

2) The following aspects should properly be accounted for:

a) The generally non-linear behaviour of the soil in the course of its dynamic interaction

with the retaining structure;

b) The inertial effects associated with the masses of the soil, of the structure, and of all

other gravity loads, which may participate to the interaction process;

c) The hydrodynamic effects generated by the presence of water in the soil behind the

wall and/or by the water on the outer face of the wall;

d) The compatibility between the deformations of the soil, the wall, and the tiebacks,

when present.”

For Simplified methods: pseudo-static analysis, it is stated that “

l) The basic model for the pseudo-static analysis shall consist of the retaining structure

and its foundation, of a soil wedge behind the structure supposed to be in a state of

active limit equilibrium (if the structure is flexible enough), of any surcharge loading

acting on the soil wedge, and, possibly, of a soil mass at the foot of the wall, supposed

to be in a state of passive equilibrium.


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2) To produce the active soil state, a sufficient amount of wall movement is necessary to

occur during the design earthquake; this can be made possible for a flexible structure by

bending, and for gravity structures by sliding or rotation. For the wall movement needed

for development of an active limit state

3) For rigid structures, such as basement walls or gravity walls founded on rock or piles,

greater than active pressures develop, and it is more appropriate to assume at rest soil

state. The same applies for anchored retaining walls if no movement is permitted.

The seismic design of retaining structures was based on pseudo-static analysis. The

pseudo-static method is based on the well-known M-O theory (see Equations 2.11 and

2.12). Pseudo-static seismic actions both in the horizontal and vertical directions should

be taken into account. The vertical seismic coefficient kv is a function of the horizontal

kh:

kv = ± 0.33 × kh or kv = ± 0.5 × kh (2.31)

The horizontal seismic coefficient kh is:

kh = a × R × γI × S / (g × r) (2.32)

where, γI = importance factor of the structure; r = factor that depends on the allowable

wall displacements (in the Final Draft of EC8-5 the formula is kh = α×S / r, where α =

(ag ×R/g) ×γI ). The seismic coefficient shall be taken a constant along the height for

walls are not higher than 10 m. The value of the factor r should be taken equal to 1 for

structures cannot accept any displacement, while it assumes 1.5 and 2 values as the

acceptable displacement increases. The threshold values of the displacement dr are

proportional to the peak ground acceleration (α×S) expected at the site.

2.7 SUMMARY

This chapter firstly discussed the retaining wall type and static earth pressure theories.

Then, the previous analytical, numerical and experimental studies proposed to

investigate the relationship between the static earth pressure and displacement of the

retaining wall were critically discussed. The second part of this chapter covered the


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seismic design of retaining walls. The main design techniques used to investigate the

seismic stability of retaining structures were briefly discussed. This chapter discussed

the previous analytical, numerical and experimental studies proposed using the force-

based design method. A critical discussion of the force-based design techniques was

presented. Then, this chapter discussed the displacement-based design technique and the

analytical, numerical and experimental studies proposed using this technique. Critical

discussion of the displacement-based design technique was also presented. After that,

this chapter critically discussed the relationship between the seismic earth pressure and

retaining wall displacement using the analytical, numerical and experimental methods.

The next chapter will discuss the finite element method that will be adopted for this

research. All steps required for building up the numerical model will be presented in

detail in order to use the finite element model to bridge the knowledge gaps discussed

previously.

Chapter 3: Finite Element Modelling Methodology

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CHAPTER 3

FINITE ELEMENT MODELLING METHODOLOGY

This chapter discusses the detailed methodology used to develop the FE model for the

rigid- and cantilever-type retaining walls by using the commercial specialist

geotechnical software – PLAXIS2D Brinkgreve et al. (2015) . The first section of this

chapter presents an overview of the PLAXIS2D software, followed by a discussion of

the geometric idealisation of the wall-soil problem, mesh sizing and element selection

for the FE model. A detailed discussion on the constitutive model chosen to simulate

the behaviour of retaining wall and soil materials, boundary conditions, both for the

static and seismic analyses performed using the FE model is also presented, along with

the methodology to interpret and extract the output from the developed fe models.

3.1 WHY FE MODELLING?

The FE method, due to its versatility, is used to analyse a wide range of geotechnical

engineering problems. For example, it is used to solve simple problems involving stress

and strain computations, slightly complex problems of assessing flow characteristics

and pore pressure distribution in soils, more complex soil-structure interaction problems

involving the assessment of interaction between a retaining structure and its

foundations, and extremely complex problems of tunnelling methods and tunnelling is

soft soils. The FE method provides an approximate solution for the above geotechnical

problems, especially when finding their analytical solutions is either not possible or

extremely tedious. The FE method can consider the effect of a variety of loading

patterns including static, seismic, blast, impact, and vibratory loading and thus, it is

often considered to be an efficient and cost-effective alternative to experiments. Once a

FE model has been calibrated and validated, it can be used for investigating the effect of

a wide range of parameters obtained from real-case history and/or experimental/

analytical modelling results. In regards to the retaining walls, previous studies available

in literature by Al Atik and Sitar (2008), Geraili et al. (2016), and Candia et al. (2016)


119

suggest that the FE modelling method is capable of capturing essential features of both

static and dynamic response of retaining wall-soil systems. Because of these reasons,

the FE modelling method has been chosen as the main research method for the present

study to simulate the seismic behaviour of the retaining wall-soil problem. For the

present study, the FE modelling is carried out using commercial specialist geotechnical

software PLAXIS2D Brinkgreve et al. (2015).

3.2 OVERVIEW OF THE PLAXIS2D SOFTWARE

PLAXIS2D Brinkgreve et al. (2015), a 2D FE method based geotechnical software,

developed by the PLAXIS Company, is used to analyse a variety of geotechnical

engineering research problems, including deformation and stability analyses,

groundwater flow problems, static and dynamic problems, and thermal problems. It can

conveniently model the non-linear and time-dependent behaviour of soil and rock

materials, and is equipped to handle problems involving retaining walls, piles, anchors,

and tunnels. Researchers in the past have successfully used PLAXIS2D to model,

assess and analyse the seismic behaviour of retaining wall problems for example,

Tiznado and Rodríguez-Roa (2011) and Ibrahim (2014). A summary of the steps

involved in developing a FE model by using PLAXIS2D is presented in Figure 3.1.

Figure 3.1: Flow chart summarising the steps to model and analyse the retaining wall using

PLAXIS and AQAQUS


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3.3 DOMAIN DISCRETISATION TO IDEALISE THE WALL-SOIL SYSTEM

As shown in Figure 3.2, a retaining wall-soil system consists of 3 main parts: (1) the

retaining wall, (2) backfill soil, and (3) foundation soil. In order to simulate the

retaining wall problem using PLAXIS2D, it is necessary to simplify the geometry of the

problem by considering it to be a plane-strain problem. For such a case, a wall-soil

system is considered to have one dimension very large in comparison to the other two

dimensions (see Figure 3.2); thereby inherently assuming that the displacements are

occurring only in the x-y plane, which are not affected by the displacements in the z-

direction (i.e., the larger dimension). Such an assumption facilitates modelling a

retaining-wall soil problem, which, in reality is 3D problem, to be modelled as a 2D

problem.

Figure 3.2: Retaining walls analysed in the current study considering a 2D plane strain

idealization

For studying the seismic behaviour of a retaining wall soil system such a simplification

has also been made in the past by several researchers like Green et al. (2003); Green et

al. (2008); Al Atik and Sitar (2008); Geraili et al. (2016); Candia et al. (2016);

Corigliano et al. (2011);Tiznado and Rodríguez-Roa (2011); Conti et al. (2013); Bao et

al. (2014); Conti et al. (2014). Similarly, past researchers who modelled the seismic

behaviour of retainign wall using shaking table and centrifuge tests like Saito et al.

(1999); Nakamura (2006); Al Atik and Sitar (2008); Kloukinas et al. (2015); Geraili et

al. (2016); Candia et al. (2016)) also compared their results using numerical FE plane-

strain models. In view of the above, a 2D plane strain has also been made for the present

z x 𝜀𝑧 = 0

y

z x 𝜀𝑧 = 0

y

Rigid wall Cantilever wall

Backfill soil


121

study as well. Figure 3.3a and b show plane strain FE model of the rigid and cantilever

retaing wall respectively, and it will be used as a reference for demonstrating all steps of

the numerical model of the wall-soil system.

Figure 3.3: Finite element model of the wall-soil system used for the present study for a: (a)

rigid retaining wall, (b) cantilever retaining wall

3.4 RETAINING WALL AND SOIL DISCRETISATION AND INTERFACE

IDEALISATION

For a rigid-type retaining wall, both the wall and soil have been modelled by the 6-

noded triangular elements, while for the cantilever-type retaining wall, the wall has been

modelled by plate elements, and soil by 6-noded triangular elements. The interface

behaviour has been modelled using the interface elements. All these elements all

1.5m

17m 3m

10m

4m

25m

Foundation soil

Backfill soil

Retaining Wall6-noded triangular element Interface element

25m

Earthquake acceleration

9 m

2.6 m 40.25m

42.85 m

5.4 m

1.15 m16m

Backfill soil

Foundation soil


Stem

Base slab

6-noded triangularelement

Interfaceelement

(a)

(b)


122

available in the PLAXIS2D library and a brief description about them is presented

below.

3.4.1 6-noded triangular elements

A 6-noded triangular element as shown in Figure 3.4, has 6-nodes, marked as n1, n2, n3,

n4, n5 and n6, with 2 degrees of freedom at each node, and 3 Gauss integration points,

marked as x1, x2, x3. The shape functions for each of the 6-nodes, in their

corresponding local coordinates of ξ and η, are shown in Equation 3.1.

N1= ζ(2ζ ˗ 1) , N2= ξ (2ξ ˗ 1) , N3= η (2η ˗ 1), N4=4ζ ξ , N5=4ξ η , N6=4η ζ (3.1)

Figure 3.4: 6-noded triangular element in local coordinates

3.4.2 Plate element

A plate element, as shown in Figure 3.5, has 3 nodes, marked as n1, n2, and n3, with 2

degrees of freedom at each node (displacements in the x (ux)- and y-direction (uy) and

rotation in the xy-plane (z)), and 2 Gauss integration points, marked as x1, and x2.


123

Figure 3.5: 3-noded plate element in local coordinates

3.4.3 Interface element and modelling of the interface behaviour

The interaction between the soil and the retaining wall is modelled by using interface

elements. Each element has three pairs of nodes and 3 integration points as shown in

Figure 3.6. Figure 3.3 shows the location of the interface elements in the FE model

where they used to connect the elements of retaining wall with soil elements.

The interaction between the wall and soil was modelled by using the 6-noded interface

elements of the PLAXIS2D library (Brinkgreve et al. 2016). For the chosen interface

element, the interface roughness is controlled by using an interface strength reduction

factor Rinter; Rinter = 0 for a perfectly smooth interface, Rinter = 1 for a perfectly rough

interface; and 0 < Rinter > 1 for a partially rough interface.

For the present study, the interface behaviour has been modelled by using elastic-plastic

model. For elastic behaviour, the shear stress is given by PLAXIS2D Brinkgreve et al.

(2016):

tann i ic (3.2)

where, = shear stress, n = effective normal stress, i = friction angle of the interface,

and ci = cohesion (adhesion) of the interface, while for the plastic behaviour, the shear

stress is given by:

tann i ic (3.3)

The strength properties of the interface described above are associated with the strength

of a surrounding soil layer by applying the following equations:

inti er soilc R c (3.4)


124

inttan tan tani er soil soilR (3.5)

Figure 3.6: Wall-soil interface element

3.5 NATURAL FREQUENCY AND MODE SHAPES OF THE WALL-SOIL

SYSTEM

To run the seismic analysis for the retaining walls, it is essential to first obtain the

natural frequencies and mode shapes of the wall-soil system. For the present study,

these have been obtained by using the commercial software ABAQUS Simulia (2013)

because such an analysis cannot be performed in PLAXIS2D. For this, a quadratic

plane strain 2D CPE4 element, as shown in Figure 3.7 , has been chosen to simulate the

retaining wall and soil. A CPE4 is defined by 4 nodes, marked as n1, n2, n3 and n4,

each having 2 degrees of freedom with 4 Gaussian integration points, marked as x1, x2,

x3 and x4 ( Figure 3.7).

Figure 3.7: 4-noded bilinear plane strain element CPE4


125

The interaction between the wall and soil has been analysed in ABAQUS by using

surface-to-surface contact. This type of interaction involves defining the surfaces of soil

and retaining wall that can be contacted together. After that, the contact interaction

property is introduced by mechanical contact type that involves using two options; the

tangential behaviour and normal behaviour. The tangential behaviour is represented by

penalty friction formulation that is defined by the friction coefficient parameter (angle

of friction between the soil and retaining wall, and sometimes between two layers of

soil).

3.6 INITIAL SIZING OF THE FE MESH CONSIDERING THE

PROPAGATION OF SHEAR WAVES

A seismic analysis involves generation and propagation of shear waves through the wall

and backfill and foundation soils, and thus, the FE model should cater to this effect.

Kuhlemeyer and Lysmer (1973) suggest that to properly model this via a FE model, the

maximum size of any FE element should not be greater than 20% of the wavelength of a

shear wave propagating through the medium. This means, that if 𝜆𝑚𝑖𝑛 and 𝑣𝑠 is the

wavelength and velocity of the shear wave, respectively, and 𝑓𝑚𝑎𝑥 is the maximum

frequency of the earthquake acceleration, then the maximum height of an element of the

FE mesh ℎ𝑒𝑚𝑎𝑥 should be:

minmax

max5 5

svhe

f

(3.6)

It is to be noted that Equation 3.6 is inherently satisfied by the mesh generation

procedure adopted by PLAXIS2D Brinkgreve et al. (2016).

3.7 CONSTITUTIVE MODELS

Appropriate constitutive models must be chosen for the wall and backfill and

foundation soils to model their behaviour appropriately.

3.7.1 Retaining wall

The retaining wall is modelled using a linear viscoelastic constitutive model in the

present study. A viscoelastic model has an elastic component and a viscous component,


126

in which for the elastic component the stress changes linearly with strain while the

viscous component handles the energy dissipation during the application of the seismic

loading. Past researchers like Gazetas et al. (2004) Geraili Mikola (2012); Agusti and

Sitar (2013); Conti et al. (2013); Bao et al. (2014); Conti et al. (2014) have also

modelled a retaining wall a linear viscoelastic constitutive model. A viscous damping

has been used for the retaining wall modelled using the Rayleigh formulation. Table 3.1

shows all the rigid and cantilever wall parameters that are required to run the FE model.

Table 3.1: Wall parameters required to run the FE model

Rigid retaining wall Cantilever retaining wall

Parameter Symbol Unit Parameter Symbol Unit

Elastic modulus E kN/m2 Bending stiffness EI kN.m

2

Poisson’s ratio v - Axial stiffness EA kN

Unit weight γ kN/m3 Poisson’s ratio v -

Damping ratio ξ % Weight w kN/m

Thickness tw m

Damping ratio ξ %

3.7.2 Soil

For modelling soils, there is a plethora of constitutive models available in literature.

For example, there is a Mohr-Coulomb model, Hoek-Brown model, hardening soil

model, hardening soil with small strain model, soft soil model, modified Cam-clay

model, etc. In this study, a hardening soil with small strain model has been selected to

model the soil used for the backfill and as well as the foundation underneath the

retaining wall.

3.7.2.1 Hardening soil with small strain model

The hardening soil with small strain model Benz (2007) to in this thesis from this point

onwards as HSsmall) has been selected because it can efficiently simulate variety of soil

and types and can also simulate stress-dependency of soil stiffness, loading and

unloading circles, nonlinear shear modulus reduction with shear strain, and generation


127

of hysteretic damping during cyclic loading. The salient features of the HSsmall model

are:

It is an elasto-plastic constitutive model;

The stiffness of the soil is simulated by considering the secant modulus E50, which,

is defined as the modulus of soil corresponding to 50% of the soil’s strength at

failure qf , as depicted in Figure 3.8. Mathematically, E50 is given by:

350 50

cos sin

cos sin

y

ref

ref

cE E

c p

(3.7)

where 50

refE = secant modulus at 50% of the soil’s strength corresponding to a

reference confining pressurerefp , c = effective soil cohesion, = effective soil

friction, 3 = effective confining pressure, y = a constant, which takes into account

the stress-level dependency of the stiffness of the soil. For PLAXIS2D Brinkgreve

et al. (2016), pref

is taken as 100 kN/m2, and the constant y is considered to be

between 0.5 – 1. From Figure 3.8, it is observed that the soil’s strength at failure qf

is marginally smaller than the asymptotic value of the ultimate strength of soil qa.

The ratio qf/qa is called as the failure ratio Rf, and in PLAXIS2D Brinkgreve et al.

(2016) this is usually considered as 0.9, thereby suggesting that the failure criterion

is not reached and perfectly plastic yielding does not occurs as described by the

Mohr-Coulomb failure criterion.


128

Figure 3.8: Hyperbolic stress-strain law of hardening soil model after Brinkgreve et al. (2016)

To simulate the stress history of the soil, a tangent stiffness modulus of primary

oedometer loading 𝐸𝑜𝑒𝑑, is considered in the HSsmall, which is computed by:

3cos sin

cos sin

y

ref

Oed Oed ref

cE E

c p

(3.8)

where ref

OedE = secant oedometer modulus corresponding to a reference confining

pressure pref

. In PLAXIS2D Brinkgreve et al. (2016) , ref

OedE = 50

refE

For the unloading and reloading, the stiffness of the soil is simulated by using an

unloading-reloading modulus Eur, determined by:

3cos sin

cos sin

y

ref

ur ur ref

cE E

c p

(3.9)

where, ref

urE = secant modulus for the unloading-reloading cycles corresponding to a

reference confining pressure refp .

The soil dilatancy, is also considered in the HSsmall model and its mobilized value

is computed by Brinkgreve et al. (2016)

3: sin sin : 0

4

sin sin3: sin sin 0 : sin

4 1 sin sin

3: sin sin 0 :

4

0 : 0

m m

m csm

m cs

m m

m

for

for and

for and

if

(3.10)

asymptote

failure line

axial strain -ε1

deviatoric stress

|σ1 − σ2|

qa

qf

EiE50

Eur

11

1

‗

‗


129

where , m = mobilised dilatancy angle, = dilatancy angle, = effective shear

resistance angle, and m = mobilised effective shear resistance angle, computed by:

1 3

1 3

sincot

mc

(3.11)

and cs = critical state effective shear resistance angle of the soil, computed by:

sin sin

sin1 sin sin

cs

(3.12)

3.7.2.2 Reduction of soil stiffness at small strain level

As mentioned previously, the soil behaviour during unloading-reloading is assumed to

be purely elastic, however, increasing strain for larger unloading-reloading cycles, the

soil stiffness will experience a nonlinear decay. This decay of soil stiffness at small

strain can be associated with the loss of intermolecular and surface forces within the soil

skeleton. Figure 3.9 shows the characteristic of stiffness reduction against strain of soil

with typical strain ranges for laboratory test and structures. The HSsmall model takes

into account this nonlinear reduction of soil stiffness with strain amplitude, by virtue of

2 additional parameters:

Go - initial or very small-strain shear modulus; and

γ0.7 - shear strain at which the secant shear modulus Gs is reduced to about 70%

of Go.


130

Figure 3.9: Shear modulus – strain behaviour of soil with typical strain ranges for laboratory

tests and geotechnical structures after Brinkgreve et al. (2016)

The shear modulus Go is calculated from:

350

cos sin

cos sin

y

ref

o ref

cG G

c p

(3.13)

where, 50

refG = initial shear modulus corresponding to a reference confining pressure

𝑝𝑟𝑒𝑓. The relationship between Gs and Go is described by using the Hardin and

Drnevich (1972) hyperbolic law as:

1

1

s

o

r

G

G

(3.14)

where, Gs = secant shear modulus, Go = initial shear modulus, γ = shear strain, γr =

reference shear strain. More straightforward and less prone to error is the use of a

smaller threshold shears strain. Santos and Correia (2001) suggested that the secant

shear modulus is reduced to about 70% of its initial value (𝛾𝑟=γ0.7) and can be expressed

as:

0.7

1

1 0.385

s

o

G

G

(3.15)


131

Figure 3.10 shows the above reduction of shear modulus relationship according to

Equation 3.15.

Figure 3.10: Stiffness reduction curve Brinkgreve et al. (2016)

A lower cut-off in the small-strain stiffness reduction curves as shown in Figure 3.10 is

introduced at shear strain c where the tangent shear stiffness is reduced to level of

unloading –reloading stiffness Gur, which is related to HS model parameter Eur as

follow:

2 1

urur

ur

EG

v

(3.16)

3.7.2.3 Damping

On cyclically loading the soil, as will be done while carrying out the seismic analysis,

due to the internal friction of the soil particles, a lot of energy will be dissipated, often

referred to as hysteretic damping. Figure 3.10 shows an example of predicted damping

in HSsmall model along with the reduction of shear modulus reduction, as shown in

Figure 3.11.


132

Figure 3.11: Damping in HSsmall model Brinkgreve et al. (2016)

However, as the chosen HSsmall model is (almost) linear at very small strain with no

hysteretic damping (see Figure 3.10), a viscous damping, which is frequency dependent,

is used in the present study. A similar approach was adopted by Tiznado and

Rodríguez-Roa (2011), who modelled wall-soil problem. The viscous damping is

considered by using the Rayleigh damping formulation (Rajasekaran, 2009), given by:

C M K (3.17)

where, [𝐂], [𝐌] and [𝐊] = damping, mass and stiffness matrices of the system,

respectively, 𝛼 and 𝛽 = Rayleigh parameters, computed by using Equation 3.18,

(Rajasekaran, 2009).

1 2

1 2

2

1

z z

z z

(3.18)

where, = viscous damping ratio, 𝜔z1 and 𝜔z2 = first 2 natural circular frequencies of

the wall-soil system.

It is important to note that the Rayleigh damping formulation is based on the first 2

modes of natural frequency of the retaining-soil wall system because the response of the

retaining walls is usually governed by their first two modes of vibrations (Candia et al.,


133

2016). Candia et al. (2016) reported that over 97% of the total displacement of a

retaining wall gets captured in the first and second modes of vibration, while only about

3% of the total displacement comes from its third mode of vibration. As outlined in

Section 3.5, ABAQUS software (Simulia, 2013) has been used to determine the natural

frequencies of the first two modes of vibration for the retaining wall-soil system, which

will then be used to compute the Rayleigh parameters given by Equation 3.18. The

Rayleigh parameters will be then be used in the FE model in PLAXIS2D (Brinkgreve et

al., 2016) to run the simulation.

3.7.2.4 Soil parameters required to run the FE simulation

As per the above discussion, several soil parameters are required to run the FE model,

and they are listed in Table 3.2.

Table 3.2: Soil parameters required to run the FE model

Parameter

Variable

Unit

Unit weight γ kN/m3

Effective friction angle of the soil ' o

Reference secant modulus at 50% of the soil’s

strength 50

refE MPa

Reference tangent stiffness modulus of primary

oedometer loading

ref

OedE MPa

Reference stiffness modulus of unloading reloading ref

urE MPa

Dilatancy angle of the soil o

Poisson’s ratio for unloading-reloading ur -

Stress-level dependency of the stiffness of the soil y -

Initial shear modulus 50

refG MPa

Reference shear strain at 70% of 𝐺𝑜𝑟𝑒𝑓

γ0.7 -

Reference confining pressure pref kN/m

2

Damping ratio %

Failure ratio Rf -


134

3.8 BOUNDARY CONDITIONS FOR STATIC ANALYSIS

For the problem under consideration (see Figure 3.3), the lateral boundaries are

restrained against horizontal movement while the base boundaries are restrained against

horizontal and as well as vertical movement. The lateral and base boundaries are placed

at a distances equal to 6H and 2H respectively based on the recommendations suggested

by Bhatia and Bakeer (1989) and Green et al. (2008). After the boundary conditions

have been fixed, a static analysis is first carried out by applying the gravitational loads

of soil and retaining wall so as to define the initial stress state for the FE model as

discussed in next section

3.9 STATIC ANALYSIS

After applying the boundary conditions, a static analysis is first performed to compute

the initial stresses in the FE model. The initial stress in the FE model is affected by the

weight of the material (i.e., by gravity). These stresses are defined by an initial vertical

effective stresses ( v ) and the initial horizontal effective stresses ( h ). These are

correlated with each other by the coefficient of lateral earth pressure (Ko). The value of

Ko (= h / v ) is defined in the PLAXIS2D (Brinkgreve et al., 2016) by using Equation

2.2. After that, the plastic calculation is used by PLAXIS2D in order to carry out an

elasto-plastic deformation analysis for FE model and produce the deformation of

retaining walls and the lateral earth pressures.

3.10 BOUNDARY CONDITIONS FOR THE SEISMIC ANALYSIS

The demarcation of the boundaries for carrying out the seismic analysis is very different

from the static analysis. This is because, in a seismic analysis, a reflection of the

outward propagating shear waves back into the model takes place, which does not allow

necessary energy radiation. There are different techniques that can be adopted to cope

with this problem. One of the techniques is to use a significantly larger domain so that

the boundary effects are reduced, but this would lead to a huge increase in the

computational time for completing the seismic analysis. Another technique is to use

absorbing boundaries, which are defined by using horizontal and vertical dashpot on the


135

vertical boundaries and they are used by Psarropoulos et al. (2005b); Tiznado and

Rodríguez-Roa (2011). For the present study, absorbing boundaries have been used.

3.11 SEISMIC ANALYIS

After the static analysis, the FE model will be subjected to a seismic analysis using the

PLAXIS2D software. Behind the scenes, the solution for the seismic analysis is

obtained by solving a dynamic time-dependent equation for the seismic loading,

expressed as PLAXIS2D (Brinkgreve et al., 2016):

a v u M C K F (3.19)

where, a , v , and u = acceleration, velocity, and displacement time-varying

vectors respectively, M = mass matrix and, C = damping matrix, and K = stiffness

matrix F = seismic load vector also varying with time,. To solve Equation 3.19,

PLAXIS2D (Brinkgreve et al., 2016), uses the Newmark numerical integration, which

when used to compute displacement and velocity at time t+t gives:

21

2

t t t t t t t

a au u v t a a t

(3.20)

1t t t t t t

a av v a a t (3.21)

where, a , beta = coefficients, equal to 0.25 and 0.5, respectively, according to the

default setting of PLAXIS2D (Brinkgreve et al., 2016).

3.12 SEISMIC LOADING

The earthquake effect on the retaining wall-soil system has been simulated by applying

horizontal earthquake acceleration at the base of the retaining wall-soil FE model. In

the present study, 2 type of horizontal earthquake accelerations are applied at the base

of the FE model – (1) a real earthquake acceleration of the Loma prieta 1989 earthquake

(Database, 2015), and (2) an equivalent sinusoidal acceleration of varying amplitudes


136

and frequency contents. For a real earthquake acceleration, the Loma Prieta (1989)

earthquake, having a peak ground acceleration of 0.264g (Figure 3.12a) and dominant

frequencies of 0.7 Hz and 2.5 Hz (Figure 3.12b).

0 5 10 15 20 25 30

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0 2 4 6 8 10 12 14 16 18 20

0.00

0.05

0.10

0.15

0.20

(b)

Acce

lera

tio

n,

a(g

) (m

/se

c2)

Time, t (sec)

a(max)

= 0.256g

(a)

f = 2.5Hz

f = 0.7Hz

Fo

urie

r a

cce

lera

tio

n

am

plit

ud

e,

a(g

) (m

/se

c2)

Frequency, f (Hz)


b) frequency domain

3.13 POST PROCESSING APPROACH

The FE output, obtained after completing the seismic analysis is used to estimate

accelerations, seismic wall and backfill inertia forces, displacements and seismic earth

pressure forces. The next section describes the procedure adopted in order to produce

abovementioned quantities.

3.13.1 Acceleration and displacement

The acceleration and displacement is readily extracted from the FE model output.

3.13.2 Seismic wall and backfill inertia forces

The wall and backfill inertia forces are deduced from the FE model output by following

the procedure as outlined below:

The wall seismic wall and backfill inertia forces are estimated by using the following

procedure:

1- Elemental acceleration ae is obtained by the FE model:


137

a) For the rigid retaining wall awe, and it is obtained only for those elements which

lie in the middle of retaining wall

b) For cantilever retaining wall, it obtained for elements of the stem and base slab,

c) For the backfill soil, ase is obtained only for those elements which lie in the

middle of backfill soil above the base slab.

2- The corresponding masses of the elements (for the case of the cantilever retaining

wall), and masses of the horizontal strips (for the case of the rigid retaining wall and

backfill soil above base slab) are multiplied with the elemental accelerations, awe and

ase to get the elemental seismic wall and backfill inertia forces.

3- These elemental seismic wall and backfill inertia forces are summed together, for the

rigid retaining wall, cantilever retaining wall and backfill soil, to get the seismic wall

inertia forces – Fw and seismic backfill inertia force Fs.

3.13.3 Seismic earth pressure force

The seismic earth presure forces behind the rigid retaining wall P and for the cantilever

retaining wall, behind the stem, Pstem, and total seismic earth pressure force at the virtual

plane extending from the heel to ground surface, Pvp, have been estimated by adopting

the following procedure:

1- The elemental lateral stresses are obtained from the FE model in the Gauss

integration points for all those elements of the backfill soil which are in contact with

the wall (rigid retaining wall) and the stem (cantilever retaining wall) and as well

those which are along the virtual vertical plane (cantilever retaining wall).

2- These elemental lateral stresses are multiplied with the corresponding element

heights, to get the elemental total seismic earth pressure forces.

3- These elemental seismic earth pressure forces are summed together to get P, Pstem

and Pvp.


138

3.14 SUMMARY

In the present study, the finite element (FE) method is found to be a very efficient and

cost-effective alternative to experiments to bridge the gaps and solve contradictions

found in literature in the preceding chapter. It is selected to investigate the seismic

response of the rigid and cantilever retaining wall to achieve the objectives of the

present study. In the present chapter, the geometric idealisation of the wall-soil system

is simplified based on the special geometric characteristics to the well-known plane

strain condition. The mesh and element types that will be used in the finite element

analysis are discussed in detail, in which the retaining wall and soil are modelled by

using 6-node triangular elements while the retaining wall is connected to the soil by

using interface elements. The behaviour of the soil material is simulated by using

HSsmall constitutive model. However, the behaviour of the retaining wall is assumed to

be linear viscoelastic. The natural frequency is predicted by using ABAQUS software to

define the coefficients of Rayleigh damping formulation in which they will be used to

define the damping term in seismic analysis at very low strain levels. The vertical

boundary conditions of the FE model are assumed to be free for vertical movement and

fixed against horizontal movement, while the base boundary condition is assumed to be

fixed against both vertical and horizontal movement. Then, the static analysis is

performed to define the initial stress state of the finite model. For seismic analysis, the

seismic load is defined by applying earthquake acceleration at the base boundary of the

finite model. The absorbing boundaries are also applied at the vertical boundaries in the

seismic analysis to reduce the reflection of the seismic wave in the domain.

After preparing the FE models, it is important to validate them with experimental results

available in the literature so that its reliability can be ascertained before detailed results

are obtained from them. The next chapter covers the validation of the finite element

model with 3 centrifuge experiments available in the literature.

Chapter 4: Validation of FE Model

139

CHAPTER 4

VALIDATION OF FE MODEL

To validate the proposed FE model, 3 centrifuge tests have been taken from literature.

Details about these 3 tests and the methodology adopted to simulate them with the FE

model, and the comparison of result are presented in the next sections.

4.1 GEOTECHNICAL CENTRIFUGE MODELLING

Geotechnical centrifuge modelling is a technique in which scaled-down yet prototype-

representative models of geotechnical engineering systems like earth retaining walls are

tested by spinning the models by high accelerations. The spinning effect increases the

g-force on the models such that the stresses in the models are equal to the stresses in the

prototype. Owing to this convenience, centrifuge modelling has frequently been

adopted to model a huge variety of geotechnical engineering systems including bridge

foundation interaction with the surrounding soils, slope stability analysis, retaining

structures, and pile foundations, and contaminant transport and more complex soil-

structure interaction problems involving earthquake loading In the present study, 3 such

centrifuge tests have been selected and they have been remodelled using the FE

modelling approach as outlined in Chapter 3. Results from the remodelling tests have

been used to validate the FE model.

4.2 3 CENTRIFUGE TESTS SELECTED FROM LITERATURE

4.2.1 Saito (1999) test

Saito (1999) (Okamura and Matsuo, 2002) conducted a series of centrifuge tests to

study the seismic behaviour of a rigid retaining wall. The centrifuge model, as shown in

Figure 4.1 had a height of 30 cm, base width of 15 cm, top width of 5 cm to support a

horizontal backfill layer that poured with the same level of retaining wall and extends

80 cm. The retaining wall was seated on a 10 cm thick foundation layer that extends 80

cm behind the wall and 55 cm in front the wall. The backfill and foundation soils were


140

prepared by using dry Toyoura sand with a relative density of 82% (Okamura and

Matsuo, 2002). The centrifuge was rotated with a centrifugal acceleration 30 g. The

results predicted by this centrifuge test are presented in Chapter 2, section 2.5.2.3.2.

Figure 4.1: Saito (1999) centrifuge test model

4.2.2 Nakamura (2006) test

Nakamura (2006) carried out a series of centrifuge tests in Public Works Research

Institute Japan to study the influence of input seismic motion in the behaviour of the

retaining wall. The centrifuge test was operated by applying of 30 g centrifugal

acceleration. The tests were developed to examine the assumptions used in the M-O

theory. 26 cases of earthquake acceleration input motion were used in these tests. Earth

pressures were measured with load cell and the displacements were measured by using

displacement transducers. The results of the tests are presented in Chapter 2, section

2.5.1.3.2. Figure 4.2 shows the cross-section of the centrifuge model test. Dry, dense

Toyoura sand with a relative density of 88% was placed into the model to depth 10 cm.

After that, a rigid retaining wall 30 cm in height, with a 12.5 cm base width, and 5 cm

embedded depth was constructed to support the horizontal backfill layer that extends

83cm behind the retaining wall (Nakamura, 2006).

Dry Toyoura sand (Dr =82%)

10 cm

55 cm 15 cm 80 cm

10 cm

30 cm

80 cm5 cm

Retaining wall

shaking direction


141

Figure 4.2: Nakamura (2006) centrifuge test model

4.2.3 Jo et al. (2014) test

Jo et al. (2014) conducted two dynamic centrifuge tests to simulate the dynamic

behaviour of inverted T-shape cantilever retaining wall models. Both centrifuge tests

were rotated by 50 g of centrifugal acceleration. The first centrifuge test was performed

to simulate a cantilever retaining wall 10.8 cm in height, with a base slab width of 7.34

cm, and stem and footing slab thickness 0.44 cm as shown in Figure 4.3a. However, the

second centrifuge test was conducted to simulate a cantilever retaining wall 21.6 cm in

height, with a footing slab width of 0.7 cm, and stem and footing slab thickness of 0.7

cm, as shown in Figure 4.3b. Dry silica sand with a relative density of 60% and dry unit

weight of about 14.23 kN/m3 was placed behind and below both retaining walls. Both

retaining walls were made of aluminium alloy with a modulus of elasticity 68.9 GPa.

The results predicted by these tests are presented in Chapter 2, section 2.5.1.3.2.

Dry Toyoura sand (Dr =88%)

5 cm

5 cm 83 cm

30 cm

10 cm

12.5 cm 83 cm54.5 cm

shaking direction

Retaining wall


142

Figure 4.3: Jo et al. (2014) centrifuge test: a) model with a wall height of 10.8 cm, b) model

with a wall height of 21.6 cm

4.3 FE MODELLING OF THE ABOVEMENTIONED 3 CENTRIFUGE TESTS

This section describes the FE modelling procedure adopted to simulate the Saito (1999)

Nakamura (2006) and Jo et al. (2014) centrifuge tests. It is to be noted that the

centrifuge model tests were conducted on scaled-down models, while to remodel them

using the FE software PLAXIS2D, actual dimensions of the prototype have to be used.

The actual dimensions of the prototype have been deduced by using scaling laws,

representing the ratio between the prototype and centrifuge dimensions; for Saito

(1999) and Nakamura (2006) tests, they have the same the scaling law (N = 30) while

for the Jo et al. (2014) centrifuge test, the scaling law is (N = 50). Table 4.1 shows

examples of some dimensions in centrifuge tests and they are converted to actual

dimensions of the prototype, where, N = scaling factor.

Wall

Dry silica sand

Shaking direction

10.8cm

38cm

28.9cm

5.2cm12.3cm 1.7cm

0.44cm

Dry silica sand

0.44cm

Shaking direction

Wall

Dry silica sand

Dry silica sand 21.6cm

0.7cm

38cm

10.6cm 3.4cm 10.4cm0.7cm

21.6cm

(a) (b)


143

Table 4.1: Centrifuge and prototype model dimensions for Saito (1999), Nakamura (2006) and

Jo et al. (2014) test model

Saito (1999) Nakamura (2006) Jo et al. (2014)

model (a)

Jo et al. (2014)

model (b)

scaling law

(N = 30)

scaling law

(N = 30)

scaling law

(N = 50)

scaling law

(N = 50)

centrifuge

(cm)

prototype

(m)

centrifuge

(cm)

prototype

(m)

centrifuge

(cm)

prototype

(m)

centrifuge

(cm)

prototype

(m)

wall

height 30 cm 9 m 30 cm 9 m 10.8 cm 5.4 m 21.6 cm 10.8 m

base

width 15 cm 4.5 m 12.5 cm 3.75 m 7.34 cm 3.67 m 14.5 cm 7.25 m

Once the dimensions of the prototype have been determined, a 2D plane strain FE

model is prepared in PLAXIS2D. For all the FE models, the following is also to be

noted:

the total number of FE elements in the discretised domain is decided after carrying

out a sensitivity analysis of the mesh size (more details in Section 4.5);

the boundary conditions have been chosen to be same as described in Chapter 3;

Chapter 3; and

a horizontal acceleration time history is applied at the base of the FE model.

4.3.1 Saito (1999) test

As per the scaling laws as mentioned above, a 9 m high retaining wall with a 4.5 m base

width is modelled in this analysis. The backfill soil is modelled with the same height of

the retaining wall and extends horizontally to 24 m while the foundation soil is

modelled with a 3 m height and width of 45 m (see Figure 4.4). The retaining wall and

soil are modelled by 150 and 1356 elements respectively. The interface between the

retaining wall and backfill soil is defined by 18 interface elements while the interface

between the retaining wall and foundation soil is defined by 9 interface elements.


144

Figure 4.4: Finite element model of Saito (1999) centrifuge test


A 9 m high retaining wall is modelled with a 3.75 m base width is modelled in this

analysis. The backfill soil is modelled with the same height of the retaining wall and

extends horizontally to 24.9 m while the foundation layer is modelled with a height of 3

m and width of 45 m, as shown in Figure 4.5. The retaining wall and soil are modelled

by 158 and 1492 elements respectively. The interface between the retaining wall and

backfill soil is defined by 21 interface elements while the interface between the

retaining wall and foundation soil is defined by 7 interface elements.

Figure 4.5: Finite element model of Nakamura (2006) centrifuge test

4.3.3 Jo et al. (2014) test

A 5.4 m high retaining wall with a 3.67 m base slab width is modelled in this analysis.

The backfill layer extends 14.1 m horizontally. The foundation layer is 19 m in height

and 24.45 m in width, as shown Figure 4.6. The retaining wall is modelled by 15 plate

element while the backfill and foundation soil are modelled by 3128 solid elements. The

24 m4.5 m16.5 m

9 m

3 m

24 m1.5 m

3 m

24 m3.75 m16.5 m

9 m

1.5 m

24 m1.5 m

3 m


145

interface between the retaining wall and backfill soil is defined by 13 interface elements

while the interface between the retaining wall and foundation soil is defined by 7

interface element.

Figure 4.6: Finite element model of Jo et al. (2014) centrifuge test

4.3.4 Material parameters

HSsmall model is chosen to simulate the stress-strain behaviour of the backfill and

foundation soils for the FE model of Saito (1999), Nakamura (2006) and Jo (2014), and

for these the material parameters were chosen from available literature. The retaining

wall material for the above FE models was modelled using a linear viscoelastic

constitutive law. Table 4.2 shows the parameters required to run the FE model

simulations for the 3 centrifuge tests.

2.6 m 14.25 m

17.05 m

5.4 m

1.15 m6.15 m

Dry Silica sand

Stem

Base slab

19 m


146

Table 4.2: Parameters required to run the FE model simulations for the 3 centrifuge tests

Parameters

Unit Centrifuge test

Nakamura

(2006)

Saito

(1999)

Jo et al

(2014)

Soil

Dr % 88 82 78

γ kN/m3 16 16 14.23

' o 41 41 40

50

refE MPa 66.7 62.4 46.8

ref

OedE MPa 66.7 62.4 46.8

ref

urE MPa 200 187.2 140.4

o 11 11 10

ur - 0.2 0.2 0.2

y - 0.5 0.5 0.5

50

refG MPa 150 144 113

γ0.7 - 0.0003 0.0003 0.0002

pref kN/m

2 100 100 100

% 3 3 3

Rf - 0.9 0.9 0.9

Retaining wall

E MPa 30000 30000 68000

- 0.15 0.15 0.334

kN/m3 18 18 26.6

% 3 3 3

m4 - - 8.873 10

-4

4.4 NATURAL FREQUENCY AND MODE SHAPES FOR THE 3

CENTRIFUGE TESTS

As described in Chapter 3, ABAQUS was used to find the first 2 natural frequencies of

the wall-soil system. The same approach has also been adopted here for determining

the natural frequency and mode shapes of the FE models of the centrifuge tests.

4.4.1 Saito (1999) test

The first 2 natural frequencies of the Saito test were found to be 3.54 Hz and 4.83 Hz,

and Figure 4.7a and b show the first 2 mode shapes predicted by ABAQUS software.


147

From Figure 4.7a it is observed 1st mode shape (fn1 = 3.54 Hz) that a maximum

displacement of the wall-soil system is concentrated at the top of the wall as well as the

top of backfill, which is in contact with the retaining wall, at frequency 3.54 Hz.

However, for the 2nd

mode shape ((fn2 = 4.83 Hz), a maximum displacement is observed

at the point at the top of backfill soil located away from the retaining wall.

Figure 4.7: Mode shapes for Saito (1999) centrifuge test model obtained from the current finite

element study: a) 1st mode, b) 2

nd mode


The first 2 natural frequencies of Nakamura (2006) centrifuge test are also numerically

computed by using ABAQUS software, and they are 3.37 Hz of the first natural

frequency and 4.721 Hz of the second natural frequency. Figure 4.8a and b show the

first 2 shape modes. It can note from the Figure 4.8a and b that the mode shapes of 1st

and 2nd

second natural frequency of the wall-soil system are similar to the Saito (1999)

centrifuge model mode shapes described in previous section.

(a)

(b)


148

Figure 4.8: Mode shapes for Nakamura (2006) centrifuge test model obtained from the current

finite element study: a) 1st mode, b) 2

nd mode

4.4.3 Critical discussion on the natural frequency of the wall-soil system

In Sections 4.4.1 and 4.4.2, the natural frequency of the wall-soil system has been

computed by using ABAQUS software for two centrifuge models taking into account

the effect of backfill soil, foundation soil, and the interaction between the retaining wall

and soil. Further investigation has been made in order to clarify the effect of wall-soil

interaction on the natural frequency of the retaining wall. The natural frequency of the

retaining wall and backfill soil have been predicted individually and then the results are

compared with results obtained for the wall-soil system and also the results obtained

from previous analytical solutions available in the literature.

A critical review of the literature shows that limited research has been done to study the

natural frequency of the retaining wall. Ghanbari et al. (2012) proposed analytical

solution to compute the natural frequency of the retaining wall considering the effect of

backfill soil behind the retaining wall only and the retaining wall was assumed to have a

fixed base. Ramezani et al. (2017) proposed analytical solution to compute the natural

frequency of the retaining wall taking into account the rigid mode of deformation and

flexural deformation mode. The effect of foundation soil was modelled by torsional and

(a)

(b)


149

translational massless springs to consider the rotation and sliding of the retaining wall.

The effect of backfill soil was modelled by massless translation spring.

Table 4.3 shows the results of the natural frequency of three different models predicted

in the present study: 1) backfill soil layer model; 2) retaining wall model; and 3)

retaining wall-backfill-foundation system model, and they are compared with results

obtained from the previous studies. It can be noted from Table 4.3 that the natural

frequency of the backfill soil layer in Saito (1999) centrifuge model predicted in present

study is 2.654 Hz and it matches with the natural frequency of backfill soil layer

predicted by using Gazetas (1982) analytical solution (f1 = vs/4H = 2.8709 Hz). It is also

observed that if the retaining wall in Saito (1999) centrifuge model is modelled without

considering the effect of foundation soil in present study, the natural frequency of the

retaining wall will be increased to 34 Hz. The natural frequency of same retaining wall

model without considering the effect of foundation soil is computed by Ghanbari et al.

(2012) analytical solution and it is 28.9 Hz. As discussed above, Ghanbari et al. (2012)

model ignored the effect of the foundation soil below the retaining wall. Hence, a

similar trend has been observed when the result of present study of the retaining wall

model without considering the effect of foundation soil is compared with the result

obtained from Ghanbari et al. (2012) analytical solution. However, it can be observed

from Table 4.3 that the natural frequency of the retaining wall-backfill foundation

system is 3.54 Hz. The natural frequency of the retaining wall-backfill foundation

system obtained from Ramezani et al. (2017) analytical is 7.1 Hz. As discussed

previously the Ramezani et al. (2017) model takes into account the effect of foundation

and backfill soil to compute the natural frequency of the retaining wall. It can be

concluded that the natural frequency of retaining wall is highly affected by the wall-soil

interaction and the natural frequency of the retaining wall will be reduced when the

effect of foundation and backfill soil will be taken into account in the analysis. The

result obtained in present study is on a good agreement with result obtained by previous

analytical model available in the literature.


150

Table 4.3: Comparison of natural frequency of three different models predicted in present study

with results of natural frequency obtained from the previous studies

Model (Saito (1999)

centrifuge)

Natural frequency, f1 (Hz)

Present study

(ABAQUS) Previous studies

Backfill soil layer 2.645 2.8709 ( Gazetas (1982) analytical solution)

Retaining wall 34 28.9 (Ghanbari et al. (2012) analytical solution)

Wall-backfill-foundation

system 3.54 7.1 (Ramezani et al. (2017) analytical solution)

4.5 MESH SIZE SENSITIVITY ANALYSIS

For the FE models of the 3 centrifuge tests a static FE analysis is performed to decide

upon the size of the FE mesh. To do this a mesh size sensitivity analysis was carried

out until a fairly convergent solution was achieved. The total static earth pressure force

was chosen to be the parameter for the sensitivity analysis. The results of the sensitivity

analysis are discussed below.

Figure 4.9a and b show that with a reduction in the FE size the total static earth pressure

force increases very rapidly until about the FE size is 0.5m. When the element size is

reduced further, there is no appreciable increase in the total static earth pressure force.

An element size of 0.5 m also complies with the recommendations of Kuhlemeyer and

Lysmer (1973), as deduced by Equation 3.1. Hence, for further analysis the minimum

size of the elements has been chosen to be 0.5 m. It is to be note that the total static

earth pressure force was also also compared with the results obtained from the

Coulomb’s and Rankine’s earth pressure solutions and the optimum FE mesh size of 0.5

m gives results which compare very well with the results obtained by using the

Coulomb’s and Rankine’s earth pressure theories.


151

2.5 2.0 1.5 1.0 0.580

90

100

110

120

130

140

150

Sta

tic e

art

h p

ressu

re

forc

e (

kN

/m)

Element size (m)

Rankine theory

Coulomb theory

FE model (Nakamura , 2006)

FE model (Saito ,1999)

(a)

2.5 2.0 1.5 1.0 0.525

30

35

40

45

50

55

60

65

Sta

tic e

art

h p

ressu

re

forc

e (

kN

/m)

Element size (m)

Rankine theory

Coulomb theory

FE model (Jo et al. ,2014)

(b)

Figure 4.9: Finite element mesh sensitivity analysis for modelling (a) the Saito (1999) and

Nakamura (2006) centrifuge tests, (b) the Jo et al. (2014) centrifuge test

4.6 VALIDATION OF FE RESULTS

This section presents the results obtained from the current FE models, and they are

compared with results recorded from the centrifuge tests conducted by Saito (1999),

Nakamura (2006), and Jo et al. (2014).

4.6.1 Saito (1999) test

For the FE model of the Saito test an equivalent horizontal sinusoidal acceleration time

history with amplitude of 0.4 g and frequency 1.5 Hz for 25 s (see Figure 4.10a and b)


152

was applied at its base to model a seismic load, and horizontal displacement for the base

of retaining wall were predicted. A comparison of the Saito actual centrifuge results and

those obtained by the FE model is shown in Figure 4.10c. It is very clearly observed

that there is an excellent agreement between the centrifuge FE model results.

0 5 10 15 20 25

-0.4

-0.2

0.0

0.2

0.4A

cce

lera

tio

n,

a(g

)

Time, t (sec)

f = 1.5 Hz

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Acce

lera

tio

n,

a(g

)

Frequency, f (Hz)

(b)


153

0 5 10 15 20 25

0.0

0.4

0.8

1.2

1.6

Dis

pla

ce

me

nt,

(m

)

Time, t (sec)

FE model (present study)

Centrifuge test (Saito ,1999)

(c)

Figure 4.10: a) Sinusoidal wave applied at the base of the Saito (1999) test and the FE model, b)

Frequency content, c) Horizontal displacement at the base of the wall, recorded by test and

obtained from the current FE study

The deformation shape comparison of the Saito centrifuge test and the one obtained

from the FE model is also shown in Figure 4.11. It is clearly observed that the FE

model very clearly captures both the residual horizontal displacement and rotation about

toe of the wall. Thus, from these observations it can be confidently said that the FE

model developed to simulate the Saito test is working very well and hence gets

validated.

Figure 4.11: Residual deformation of the wall-soil system after the end of the eartquake shaking

a) Experimental results of the Saito (1999) centrifuge, b) Current results of FE model

(a) (b)


154


4.6.2.1 Horizontal displacement and rotation

For the FE model of the Nakamura test an equivalent horizontal sinusoidal acceleration

time history was applied at its base to model a seismic load with a frequency of 2 Hz for

21 s and increasing amplitude of 0.014 g per wave (see Figure 4.12a and b) to predict

horizontal displacement for the base of retaining wall. A comparison of the Nakamura

actual centrifuge results for the horizontal displacement at the top of the retaining wall,

which was loaded dynamically between time t = 12 sec to 21 sec, and those obtained by

the FE model is shown in Figure 4.12c. It is very clearly observed that there is an

excellent agreement between the centrifuge and FE model results.

12 14 16 18 20-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

Acce

lera

tio

n,

a(g

)

Time, t (sec)

f = 2 Hz

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Acce

lera

tio

n,

a(g

)

Frequency, f (Hz)

(b)


155

12 14 16 18 20

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Dis

pla

ce

me

nt,

(m

)

Time, t (sec)

FE model (present study)

Centrifuge test (Nakamura (2006))

(c)


model, b) Frequency content, c) Horizontal displacement at the top of the wall, recorded by test

and obtained from the current FE study

Figure 4.13a shows the residual deformation shape of the wall-soil system after the end

of the dynamic test in the centrifuge model, while Figure 4.13b shows the residual

deformation shape of the wall-soil system at the end of seismic numerical simulation.

The comparison between the experimental and numerical results shows that the same

deformation shape of the wall-soil system is obtained. It can be noted that the retaining

wall at the end of shaking event of both centrifuge test and current FE model is moving

horizontally and rotating about the toe away from the backfill layer.


156

Figure 4.13: Residual deformation of the wall-soil system after the end of the earthquake

shaking a) Experimental results of the Nakamura (2006) centrifuge, b) Current results of FE

model

4.6.2.2 Seismic earth pressure

Seismic earth pressure was also made for the results of the Nakamura test and the

corresponding FE model. For this purpose, a sinusoidal acceleration time history with

amplitude 0.6 g and frequency content 4 Hz for 9 sec (see Figure 4.14a and b) is applied

at the base of FE model to compute total seismic earth pressure force increment.

Figure 4.14c shows that the total seismic earth pressure force increment predicted by FE

model while Figure 4.14d shows the total seismic earth pressure force obtained from the

Nakamura (2006) centrifuge test. It can be seen that there is an excellent agreement

between the Nakamura centrifuge test results and the ones obtained by the FE model.

This is true for both the minimum and maximum values of seismic earth pressure force

increment and the residual seismic earth pressure increment.

To compare the distribution of the seismic earth pressure along the height of the

retaining wall, another FE analysis is also performed and the results are compared with

the Nakamura centrifuge results both for the active and passive states. For this, a

sinusoidal acceleration time history was applied at its base with amplitude of 0.6g and

frequency of 2 Hz for 16 sec (see Figure 4.15a and b). Figure 4.16a shows the seismic

active earth pressure distribution along the height of the retaining wall, as reported by

the Nakamura test and the one obtained by the FE model for time t = 8.34 sec; while

Figure 4.16b shows the same for a passive case for time t = 8.58 sec. It is clear from

both Figures 4.15a and b that a very good agreement is achieved between the Nakamura

(a) (b)


157

centrifuge test and FE model results, both in terms of the magnitude and distribution

type (which is nonlinear along the height of the retaining wall).

2 4 6 8-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8A

cce

lera

tio

n,

a(g

)

Time, t (sec)

f = 4 Hz

0 1 2 3 4 5 6 7 80.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Acce

lera

tio

n,

a(g

)

Frequency, f (Hz)

2 4 6 8-50

0

50

100

150

200

250

Incre

me

nt

of

tota

l e

art

h

pre

ssu

re (

kN

/m)

Time, t (sec)


model, b) Frequency content, c) Total seismic earth pressure force increment recorded by test ,

d) Total seismic earth pressure force increment obtained from the current FE study

(a)

(c)

(b)

(d)


158

0 2 4 6 8 10 12 14 16-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.1

0.2

0.3

0.4

0.5

0.6

t=8.34 sec [active state]

Acce

lera

tion

, a(g

)

Time, t (sec)

t=8.58 sec [passive state]f = 2 Hz

Accele

ratio

n,

a(g

)

Frequency, f (Hz)

(b)(a)

Figure 4.15: Sinusoidal wave applied at the base of the Nakamura (2006) test and the FE model

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 60 700.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100 120 140

(b) FE model (present study)

Centrifuge (Nakamura ,2006)

Seismic active earth pressure (kN/m2)

[t = 8.34 sec]

No

rma

lise

d w

all

he

igh

t (z

/H)

(a) FE model (present study)

Centrifuge (Nakamura ,2006)

Seismic passive earth pressure (kN/m2)

[t = 8.58 sec]

No

rma

lise

d w

all

he

igh

t (z

/H)

Figure 4.16: Distribution of seismic earth pressure along the height of the wall recorded by test

and obtained from the current FE study: a) active state at t = 8.34 sec, b) passive state at t =

8.58 sec

4.6.3 Jo (2014) test

4.6.3.1 Simulation of construction process

To simulate the construction sequence of the cantilever retaining wall, a static analysis

is first carried out. The retaining wall is assumed to be constructed in 6 stages, in which


159

the first stage relates to the initial condition of placing the foundation soil layer and

installing the retaining wall. The remaining 5 stages simulate the placement of the

backfill soil in lifts of thickness 0.2H for each stage. In all the 5 stages, the retaining

wall will get deformed, and this is shown in Figure 4.17. It is noted from Figure 4.17

that during the placement of the backfill soil in lifts of 0.2H thickness, the stem and as

well as the base slab of the retaining wall rotate as a rigid body towards the backfill soil;

this continues until stage 4, while in stage 5, the stem of the retaining wall appears to

move away from the backfill soil, perhaps because of the development of the earth

pressure thereby causing an elastic deformation of the stem. This deformation

behaviour of the retaining wall matches with what has been observed in real field

observations reported by Bentler and Labuz (2006).

Figure 4.17: Deformation shape of a cantilever retaining wall during its construction process

4.6.3.2 Static earth pressure

After the simulation of the construction stages, the static earth pressures are computed

by the FE model, both behind the stem, pstem and along the virtual plane, pvp, which are

then compared with the Rankine solution and as well as the centrifuge test results of Jo

et al. (2014). This comparison is shown in Figure 4.18a and b for 2 retaining walls of

heights 5.4 m and 10.8 m, respectively. From Figure 4.18a and b, it is observed that the

FE model predictions are in an excellent agreement with the centrifuge test results.

Also, it is observed that pstem and pvp values obtained by the FE model in the top ¾H of

the retaining wall are very close to the static active earth pressure, obtained by the

Rankine’s theory, while for the bottom ¼H of the retaining wall these values are


160

between the static active value (obtained by the Rankine’s theory) and at-rest value.

From Figure 4.18a and b, it is very interesting to note that all along the height of the

retaining wall, pstem and pvp are very close to each other.

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 60

Static earth pressure (kN/m2)

z/H

Rankine active earth pressure

At-rest earth pressure

Centrifuge test (Jo et al., 2014)

FE present study pstem

FE present study pvp

Rankine active earth pressure

At-rest earth pressure

Centrifuge test (Jo et al., 2014)

FE present study pstem

FE present study pvp

Fig. 6. Comparison of static earth pressure predicted by FE method with

Rankine method and centrifuge test

(b): H = 10.8 m

Static earth pressure (kN/m2)

z/H

(a): H=5.4 m

Figure 4.18: Distribution of static earth pressures along the height of the wall for: a) H = 5.4 m,

b) H = 10 m

4.7 SUMMARY

This chapter included the validation of the FE model against three centrifuge tests

conducted by Saito (1999), Nakamura (2006) and Jo et al. (2014). The analysis

compared the horizontal displacement and deformation shapes as well as the seismic

earth pressure predicted by the current FE model and those measured by centrifuge

tests. In comparing the FE results with the centrifuge test results, the FE models are

successful in the replication of the seismic behaviour of the retaining wall in the

centrifuge tests. However, in spite of this agreement, it can be noted that there are some

variations in the results between the experimental and numerical models.

It can be noted that the horizontal displacements computed by the FE model are slightly

higher than the horizontal displacement recorded by the centrifuge tests, as shown in

Figures 4.9b and 4.11b. It is also noted that the seismic earth pressures at some certain

retaining wall heights are also slightly higher than seismic earth pressures measured by


161

the physical model, as shown in Figure 4.15a and b. These variations in the results

could be justified by considering some limitations Related to the FE models and

centrifuge tests.

Relating to the FE models, the hardening soil with small strain model has some

limitations in replicating the real complex behaviour of the soil. The second reason,

which contributed, to the variations between the results is that approximations are

used to evaluate the parameters of the hardening soil with small strain constitutive

model. Another justification is that the simplification assumed to model the interface

behaviour between the retaining wall and soil may lead to some diversity between

the results.

According to centrifuge tests, the uncertainty of measurement devices like load cells,

accelerometers and laser displacement transducers, scaling laws effects, and

modelling of the boundaries may also reflect some errors in the measurement of the

displacements and seismic earth pressures during the tests.

In spite of the limitations discussed above, it can be said that the proposed FE models

are capable of predicting the seismic behaviour of retaining walls and can be used to

investigate the seismic behaviour of the rigid and cantilever retaining wall. The next

chapter will discuss the problem of a rigid retaining wall under seismic loading by using

FE analysis, which has already been verified with experimental results in the present

chapter.

Chapter 5: Finite Element Analysis of A Rigid Retaining Wall

162

CHAPTER 5

FINITE ELEMENT ANALYSIS OF A RIGID

RETAINING WALL

This Chapter presents the Finite Element modelling and analysis of rigid retaining wall

to assess its seismic performance. It starts with a brief description of the problem under

investigation, followed by the details of the FE model and the material properties. A

critical analysis of the results obtained from the seismic analysis using the performance-

based method is presented. A parametric study is also presented in order to draw a

comprehensive understanding about the seismic behaviour of a rigid retaining wall, and

to produce a relationship between the seismic earth pressure and displacement of a

retaining wall. The parametric study presented in this Chapter includes investigating

the effect of retaining wall height, earthquake characteristics (amplitude and frequency

content), and relative density of backfill and foundation soils. This Chapter ends with

summary highlighting new findings from the current study.

5.1 PROBLEM DESCRIPTION

A typical rigid retaining wall of height H, retaining dry cohesionless backfill soil to its

full height and seated on a dry foundation of thickness h, is shown in Figure 5.1. Under

static and seismic conditions, the retaining wall will be subject to static and seismic

earth pressure, respectively. For the seismic earth pressure case, the retaining wall will

undergo (lateral) movement in the horizontal direction (along the x-axis) and/or rotation

(in the x-y plane, see Figure 5.1). Depending upon the direction in which the retaining

wall moves and/or rotates the soil behind the retaining wall will be either in an active

state or in passive state of earth pressure. This study will cover the seismic performance

of a rigid retaining wall by considering:

the deformation mechanisms of the retaining wall-soil system;

the phase difference between the wall inertia force and seismic earth pressure;


163

the relationship between the seismic earth pressure and displacement of retaining

wall by considering the effect of the retaining wall height, and amplitude and

frequency content of the earthquake acceleration;

the effect of relative density of the backfill and foundation soil on the seismic

response of retaining wall.

Figure 5.1: Sketch of a gravity retaining wall showing seismic earth pressure, wall inertia

forces, direction of wall movement and important locations of interest.

5.2 FE MODELLING AND MATERIAL PROPERTIES

PLAXIS2D has been used to develop the FE model of the retaining wall and is shown

in Figure 5.2. The model consists of a 4 m high trapezoidal cross-section retaining wall

with a top width of 1.5 m and base width of 3 m. The retaining wall is resting on a 10 m

thick foundation soil and retains a dry cohesionless soil to its full height. It is assumed

that a 2D plane-strain condition exists by considering that the length of the retaining

wall is significantly large in comparison to the extents in the x- and y-directions. To

ensure that the boundary effects are minimized from the analysis, a large domain

extending 25 m to the right and 20 m to the left of the back-face of the retaining wall

has been considered. The elastic boundaries, element type, mesh design, the interaction

between the wall and soil, free vibration analysis, and absorbing boundaries adopted in

the present study.


164

Figure 5.2: FE model of the gravity retaining wall

The soil and retaining wall properties and other parameters required to define the

HSsmall constitutive model and run the PLAXIS2D simulation for the proposed FE

model were chosen from literature (Benz, 2007). It is to be noted that Benz (2007)

reports that these parameters have been derived from tests on real soils and and

calibrated with triaxial and odometer tests. These soil and retainign wall parameters

chosen for this study are shown in Table 5.1.

5.3 SEISMIC LOADING

The earthquake effect on the retaining wall-soil system has been simulated by applying

horizontal earthquake acceleration at the base of the retaining wall-soil FE model. In

the present study, 2 type of horizontal earthquake accelerations are applied at the base

of the FE model – (1) a real earthquake acceleration time history of the Loma Prieta

1989 earthquake (Database, 2015), and (2) an equivalent sinusoidal acceleration time

history of varying amplitudes and frequency contents. For a real earthquake

acceleration, the Loma Prieta (1989) earthquake, having a peak ground acceleration of

0.264 g (see Figure 5.3(a)) and dominant frequencies of 0.7 Hz and 2.5 Hz (see

Figure 5.3(b)).

1.5m

17m 3m

10m

4m

25m

Foundation soil

Backfill soil

Retaining Wall6-noded triangular element Interface element

25m



165

Table 5.1: Soil and retaining wall parameters chosen for the present study

Parameters Unit Backfill soil Foundation soil

Soil

γ kN/m3 19 19

' o

35 38

50

refE MPa 45 105

ref

OedE MPa 45 105

ref

urE MPa 180 315

o 5 6

ur - 0.2 0.2

y - 0.55 0.55

50

refG MPa 168.75 -

γ0.7 - 0.0002 -

pref

kN/m2 100 100

% 3 3

Rf - 0.9 0.9

Retaining wall

E MPa 30000

- 0.15

kN/m3 18

% 3

0 5 10 15 20 25 30

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0 2 4 6 8 10 12 14 16 18 20

0.00

0.05

0.10

0.15

0.20

(b)

Acce

lera

tio

n,

a(g

) (m

/se

c2)

Time, t (sec)

a(max)

= 0.256g

(a)

f = 2.5Hz

f = 0.7Hz

Fo

urie

r a

cce

lera

tio

n

am

plit

ud

e,

a(g

) (m

/se

c2)

Frequency, f (Hz)


b) frequency domain


166

5.4 RESULTS AND DISCUSSION

After conducting the static analysis in order to define the initial stresses in the FE

domain, the seismic analysis has been conducting by applying the real earthquake

acceleration time history at the base of FE model. Through the seismic analysis, the

following are obtained:

An acceleration response of the soil-retaining wall system and as well as the wall

seismic inertia force;

A deformation mechanism of the wall-soil system which includes prediction of: (i)

total horizontal displacement response at different locations for the wall-soil system,

(ii) relative displacement between the retaining wall and foundation soil , (iii)

rotation of the retaining wall and (iv) relative displacement between the backfill soil

and foundation soil.

Seismic earth pressure force developed behind the retaining wall.

It is important to highlight that the acceleration, horizontal displacement and/or rotation

and the wall seismic inertia force of the retaining wall can have positive and negative

senses, as shown in Figure 5.4. A positive horizontal displacement and/or rotation

means that the retaining wall moves towards the backfill soil, while a negative

displacement and/or rotation means that the retaining wall moves away from the

backfill. Likewise, a positive wall seismic inertia force of the retaining wall will act

towards the backfill soil, while a negative wall seismic inertia force of the retaining wall

act away from the backfill soil. It is also important to highlight that all abovementioned

results will be presented in time profile in order to study the seismic performance of a

rigid retaining wall not only at the end of the earthquake but also during the earthquake

activity.


167

Figure 5.4: Acceleration, wall seismic inertia force and wall and soil displacement directions

5.4.1 Acceleration response of the soil-retaining wall system

For the Loma Prieta (1989) applied earthquake acceleration (Figure 5.3a), the

acceleration response of the soil-retaining wall system is predicted for the following

four locations of the FE model as shown in Figure 5.1:

Centre of gravity of the retaining wall (w_CG – Figure 5.1);

Base of the FE model;

Top of the retaining wall (wall_top – Figure 5.1); and

Top of the backfill soil (top_soil – Figure 5.1);

Figure 5.5shows the acceleration-time history predictions for the abovementioned

locations for a duration between 3 - 7 sec. The time window of 3 – 7 sec has been

chosen because during this time, the intensity of the applied earthquake acceleration

had a maximum concentration as already shown in Figure 5.3a It is observed from

Figure 5.5 that for any particular time, the acceleration response for the top of the

backfill (top_soil – Figure 5.1) is lagging behind the acceleration response for the top of

the retaining wall (top_wall – Figure 5.1). Moreover, the acceleration response for both

locations top_soil and top_wall is lagging behind the acceleration predicted at the base

of the FE model. This implies that there is a phase difference in the acceleration

response between locations top_soil, top_wall and base of FE model. Another important

observation from the acceleration amplitude is that the acceleration for the top of the

retaining wall (top_wall) and top of the backfill soil (top_soil) is more than the

acceleration recorded at the base of the FE model. As an illustration, at time t = 4.5 sec,

the acceleration at the top of the backfill (top_soil) reaches its maximum value 0.67g,

(+) Horizontal displacement

(+) Horizontal seismic inertia

force of the retaining wall

(+) Acceleration

(+) Rotation


168

while the acceleration at the base of the FE model is only about 0.25g, which is

approximately 40% less than the acceleration at the top of the FE model (locations

top_soil) and top_wall). This highlights that the acceleration is being amplified towards

the top of the model.

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4A

cce

lera

tio

n,

a(g

) (m

/se

c2)

Time, t (sec)

base of FE model

w_ CG

top_wall

top_soil

Figure 5.5: Acceleration response at different locations in the wall-soil system

5.4.2 Horizontal displacement

By performing the abovementioned seismic analysis the horizontal displacement,

relative horizontal displacement and rotation of the wall soil system at different

locations are extracted and analysed as below:

5.4.2.1 Horizontal displacement of the wall-soil system

Figure 5.6 shows the horizontal displacement predictions for (i) the centre of gravity of

the retaining wall w_CG; (ii) top of retaining wall (top_wall); (iii) top of backfill layer

(top_soil); and (iv) at a point 0.5 m in the foundation soil below the base of the retaining

wall (point P2 – Figure 5.1).

Figure 5.6should be read in conjunction with the earthquake acceleration (Figure 5.3a).

From Figure 5.6, it is observed that the retaining wall initially moves towards the

backfill soil, thereby creating a passive earth pressure condition for the backfill. After

about time t = 3.3 sec, the retaining wall starts to move away from the backfill soil; and

in the process the retaining wall first comes back to its original (at-rest) position, and


169

then starts to move away from the backfill soil until it attains a maximum horizontal

displacement in the active direction at about time t = 3.8 sec (see point dactive in

Figure 5.6). The maximum displacement in the active direction almost corresponds to

the maximum (positive) acceleration of the earthquake acceleration at time t = 3.8 sec

(see Figure 5.3a). As shown in Figure 5.3a, for time t > 3.8 sec, the earthquake

acceleration again changes direction, thereby correspondingly forcing the retaining wall

to move towards the backfill soil. The retaining wall continues to push towards the

backfill soil until it has been displaced by the maximum amount (see point dpassive in

Figure 5.6), which, like for the active case, corresponds to the earthquake acceleration at

about time t = 4.5 sec. After time t = 4.5 sec, the retaining wall keeps on moving

towards and away from the backfill, until about time t = 30 sec – a time at which the

earthquake acceleration almost diminishes to zero (Figure 5.3a). Figure 5.6 shows that

the horizontal displacement predicted for the locations top_wall, top_soil and P2 are all

following the trend of the horizontal displacement predicted at the centre of gravity of

the retaining wall (location w_CG).

0 5 10 15 20 25 30

-0.15

-0.12

-0.09

-0.06

-0.03

0.00

0.03

0.06dpassive, [t=4.5 sec]

Ho

rizon

tal dis

pla

ce

men

t, (

m)

Time, t (sec)

P1

w_ GC

top_wall

top_soil

dactive, [t=3.8sec]

Figure 5.6: Horizontal displacement at different locations in the wall-soil system


170

5.4.2.2 Relative horizontal displacement of the retaining wall with respect to

foundation soil

The most important component in the seismic design of retaining wall by using

displacement-based approach is the relative horizontal displacement of the retaining

wall. Figure 5.7 shows the relative horizontal displacement, which is computed by

taking the difference between the total horizontal displacement for locations w_CG and

P2. It can be noted from the Figure 5.7 that the maximum sliding away from the backfill

soil occurs when the applied earthquake acceleration has a maximum value and is

applied towards the backfill soil (t = 3.8 sec – see Figure 5.3a). However, it is also

observed that the retaining wall experienced sliding towards the backfill soil when the

applied earthquake acceleration has a maximum value and is applied away from the

backfill soil (t = 4.5 sec – see Figure 5.3a). The sliding towards the backfill soil is

observed to be much smaller than the sliding away from the backfill soil. At time t = 30

sec, i.e., at the end of the seismic analysis (t = 30 sec), the retaining wall has a

permanent sliding away from the backfill soil.

5.4.2.3 Comparison with Newmark sliding block method (Newmark, 1965)

The predicted relative horizontal displacement from the current FE results is compared

with that computed by using conventional Newmark sliding block method (Newmark,

1965). The most important step in using the Newmark sliding method is evaluating the

yield acceleration. The yield acceleration is defined as the average acceleration to

produce a wall seismic inertia force, which will be required to overcome friction

resistance between the base of retaining wall and foundation soil, so that the retaining

wall starts to slide away from the backfill soil (Kramer, 1996). The yield acceleration

can be computed by using the pseudo-static analysis. By considering the equilibrium of

horizontal forces which are acting on the wall-soil system (see Figure 2.34) can be

written as:

cos sin tany W AE W ae bk W P W P (5.1)


171

where, yk = yield acceleration coefficient, WW = weight of the retaining wall, aeP =

seismic earth pressure force, determined by the M-O method (Richards and Elms,

1979), = wall friction angle , and b = base friction.

MATLAB code has been developed to perform numerical integration of relative

acceleration of retaining wall, which is computed by ( ) ( ) ( )r yk g a g k g ,in order to

compute the relative horizontal displacement of the retaining wall. Where ( )a g =

acceleration response of the retaining wall and it is predicted by present FE analysis at

the centre of the wall (Location w_GC) in order to compare the results obtained from

Newmark method with those predicted by present FE analysis. A comparison between

Newmark method and presents FE analysis shows that the Newmark sliding block

method overestimates the relative horizontal displacement. Possible explanations for

overestimation of relative horizontal displacement of the retaining wall by Newmark

sliding block analysis are:

The Newmark sliding block analysis does not take into account for:

The real behaviour of seismic earth pressure force during the time of the earthquake

as the seismic earth pressure force in the Newmark sliding block method is computed

by using the pseudo-static method (see section 5.4.4).

The foundation soil deformability for the duration of the earthquake; and in doing so,

it does not take in to account the effect of the retaining wall rotating about its toe.

The relative horizontal displacement towards the backfill soil (t = 4.5 sec – see

Figure 5.7) when the earthquake acceleration is applied way from the backfill soil,

and that could cause overestimation of the relative horizontal displacement computed

by Newmark sliding block analysis.


172

0 5 10 15 20 25 30-0.025

-0.020

-0.015

-0.010

-0.005

0.000

0.005

Re

lative

dis

pla

ce

me

nt,

(

)

Time, t (sec)

Current FE study

Newmark sliding block method

Figure 5.7: Comparison between relative horizontal displacement predicted by the present FE

analysis and Newmark sliding block method

5.4.2.4 Comparison with the Eurocode 8

The gravity retaining wall is modelled in the current FE analysis by assuming the

retaining wall is rigid and free. Hence, according to the Eurocode 8 the retaining wall

can accept a displacement dr < 300 S (mm), where; S = maximum acceleration

amplitude of seismic loading, and current study is equal to the 0.264g. Hence, the

gravity retaining wall can accept displacement about 7.92 cm. This value is higher than

the displacement predicted by current FE analysis (2 cm) and lower than the

displacement computed by the Newmark sliding block method (7.6 m).

5.4.2.5 Rotation of the retaining wall about its toe

Another deformation mechanism of the retaining wall is the rotation of retaining wall

under the effect of seismic loading. In the current numerical study, the rotation of

retaining wall is computed by using:

_ _1tantop wall base wall

wH

(5.2)


173

where, θw = rotation of the retaining wall about an axis passing through location

(wall_base Figure 5.1), _top wall = total horizontal displacement at the top of the

retaining wall, _base wall = total horizontal displacement response at the base of the

retaining wall. Figure 5.8 shows the rotation of the retaining wall. A similar trend is

observed between the sliding and rotation of the retaining wall: ie, the retaining wall

gets rotated by a maximum amount when the maximum earthquake acceleration is

applied towards the backfill soil (t = 3.8 sec), while it rotates a maximum amount when

the maximum earthquake acceleration is applied away from the backfill soil. It can be

observed that the retaining wall experiences permanent rotation about its toe away from

the backfill soil at the end of the earthquake (t = 30 sec).

0 5 10 15 20 25 30-0.15

-0.12

-0.09

-0.06

-0.03

0.00

0.03

Ro

tation, (d

egre

e)

Time, t (sec)

[t=4.5sec]: rotaion towards backfill soil layer

[t=3.8sec]: rotaion away from backfill soil layer

Figure 5.8: Rotation of the retaining wall

5.4.3 Wall seismic inertia force Fw

To analyse the seismic response of a retaining wall on the development of seismic earth

pressure and provide an in-depth understanding of the wall stability, the wall seismic

inertia force is predicted. Figure 5.9 shows the wall seismic inertia force, determined by

using the procedure as outlined in Chapter 3 (see section 3.13.2). It can be noted from

the Figure 5.9that the maximum wall seismic inertia force (Fwa) is about 75 kN/m acting

away from the backfill soil at t = 3.8 sec, while the same (Fwp) is about 121 kN/m acting


174

towards the backfill soil at t = 4.5 sec. It can also be noted that the direction of the wall

seismic inertia force is acting in opposite direction to the applied earthquake

acceleration.

0 5 10 15 20 25 30-100

-50

0

50

100

150

Maximum Fwp

[t = 4.5 sec]

Wa

ll se

ism

ic in

ert

ia fo

rce

, F

w (

kN

/m)

Time, t (sec)

Maximum Fwa

[t= 3.8 sec]

Figure 5.9: Wall seismic inertia force

5.4.4 Seismic earth pressure force P

In this section, seismic earth pressure force profiles predicted by FE model are

presented. A comparison between the predicted seismic earth pressure force by FE

model with the results obtained from the pseudo-static M-O method (Mononobe and

Matsuo, 1929) is also presented. The last part of this section discusses the distribution

of seismic earth pressure along the height of the retaining wall.

5.4.4.1 Seismic earth pressure force time history

The seismic earth pressure force time history is computed by using the procedure

outlined in Chapter 3 (see section 3.13.3). Figure 5.10a shows the variation of seismic

earth pressure force P with time t, while Figure 5.10b is a simplified version of the

seismic earth pressure force P variation with the time between the time t = 0 – 10 sec.


175

0 5 10 15 20 25 30-110

-100

-90

-80

-70

-60

-50

-40

Se

ism

ic e

art

h p

ressu

re f

orc

e,

P (

kN

/m)

Time, t (sec)

(a)

0 2 4 6 8 10-110

-100

-90

-80

-70

-60

-50

-40

Pre

Ppe

[t = 4.5 sec]

Pae

[t = 3.8 sec]

Pa

Seis

mic

Seis

mic

ea

rth

pre

ssu

re fo

rce,

P (

kN

/m)

Time, t (sec)

Sta

tic

Po

(b)

Figure 5.10: Seismic earth pressure force P: a) obtained from the FE model , b) simplified

version of (a)

It is observed that during the static analysis (i.e. t < 0 sec), the retaining wall displaces

away from the backfill from its at-rest position to a partially active position. The earth

pressure force at at-rest condition (Po) is about 64.8 kN/m, which gradually reduces to a

minimum value (Pa) of about 45.85 kN/m (Figure 5.10b). With the start of the dynamic

analysis (at t = 0 sec), the static earth pressure force (Pa) of 45.85 kN/m further reduces

until it attains a minimum value of about 42.36 kN/m (at about t = 3.8 sec). From

Figure 5.6 it is clear that at time t = 3.8 sec, the retaining wall moved away from the

backfill soil, thereby creating a state of active earth pressure behind the retaining wall


176

(Pae). However, With continued acceleration and at time t = 4.5 sec the retaining wall

starts to move towards the backfill soil (Figure 5.6), thereby developing a state of

passive earth pressure behind the retaining wall and in the process gradually

incrementing the earth pressure until it attains a maximum value (Ppe) of about 103.6

kN/m (Figure 5.10b). The total earth pressure force increment ∆P is approximately

equal to 103.6 – 42.36 = 61.24 kN/m. At the end of the seismic analysis, it can be noted

there is a residual seismic earth pressure (Pre) about (65.23 kN/m) and this value is close

to the value of at-rest earth pressure force.

5.4.4.2 Comparison with M-O theory

M-O theory (Mononobe and Matsuo, 1929) is based on the Coulomb’s earth pressure

theory, and it is widely used in the seismic design of a gravity-type retaining wall. The

most important assumption in the M-O theory is the seismic force can be converted to

pseudo-static force. Therefore, for active state, it is assumed that the inertia force

developed in the backfill soil ΔPae can be added to the static earth pressure force Pa to

produce the seismic active earth pressure force Pae as below

ae a aeP P P (5.3)

The maximum acceleration applied at the base of the FE model is about 0.264g causing

the active condition in the wall-soil system. So, the kh used in M-O method is measured

at the mid-height of the backfill soil and is equal to (0.3) because of the amplification of

acceleration response towards the top (see Figure 5.5). By using the Equation 2.12, the

coefficient of seismic active earth pressure is about (0.484). The total seismic earth

pressure force can be computed by using Equation 2.11, and it is equal to (73.64 kN/m).

It can be noted from the Figure 5.10b that at the time of maximum acceleration is

applied towards the backfill soil (t = 3.8 sec), the seismic active earth pressure force

(Pae) is about 42.36 kN/m, and it is close to the static earth pressure force value (45.85

kN/m –see Figure 5.10b). Hence, one can say that:

ae aP P (5.4)

Equation 5.4 suggests that for the active earth pressure case, the seismic earth pressure

force Pae is either close to or less than the static active earth pressure force Pa. In other


177

words, it can be said that there is no contribution of the seismic inertia forces of the

backfill soil on the seismic active earth pressure force Pae. This observation is in

contrast with the conventional M-O theory, according to Equation (5.4),

The above observation was also found to be valid in previous centrifuge modelling

studies like Nakamura (2006) and Al Atik and Sitar (2008).

For the same earthquake loading, at the time when maximum acceleration is applied

away from the backfill soil (t = 4.5 sec – see Figure 5.3a), it can noted from the Figure

5.10b that the maximum passive earth pressure developed behind the wall (Ppe) is about

103 kN/m, and it can be written be as below

pe a peP P P (5.5)

where, peP = increment of seismic earth pressure force in passive direction, and it

developed because the development of inertia forces in retaining wall and backfill soil

in passive direction. Like active case, the above observation was also found to be valid

in previous centrifuge modelling studies like Nakamura (2006) and Al Atik and Sitar

(2008). The above results are extremely important as they reveal very interesting facts

about the M-O theory.

5.4.4.3 Comparison with Eurocode 8

As discussed in Section 2.6.2, the horizontal acceleration coefficient kh can be computed

according to Eurocode 8 based on the allowable displacement of the retaining wall (see

Equation 2.32). The parameter r in Equation 2.32 can be selected based on the type of

the retaining wall and allowable displacement of retaining wall. For the case in current

study, the retaining wall was modelled as free gravity wall; and hence, the parameter r

is equal to 2 (see section 2.6.2). Therefore, the horizontal acceleration coefficient kh will

be equal to 0.123. By applying M-O theory as recommended by Eurocode 8, and using

Equations 2.11 and 2.12, the seismic active earth pressure force Pae is equal to 63.31

kN/m. The seismic active earth pressure force Pae predicted by the current FE analysis is

equal to 42.36 kN/m. Hence, it can be observed that the Eurocode 8 is also

overestimating the seismic active earth pressure force Pae.


178

5.4.4.4 Distribution of seismic earth pressure

Figure 5.11 shows the variation of the lateral earth pressures with a height of the

retaining wall at a different time during the static and seismic analysis.

As shown in Figure 5.11a, the static earth pressure from the FE model is very close to

the static earth pressure estimated by using the Coulomb’s earth pressure theory. Also,

the distribution of the earth pressure from the FE model is observed to be nonlinear,

especially in the lower z/H= 0.25. Figure 5.11b shows the variation of the seismic active

earth pressure with the normalised height of the retaining wall (z/H) at time 3.8 sec. It is

observed that the seismic active earth pressure pae is very close to the Coulomb’s active

earth pressure, and clearly, it is significantly less than the earth pressure obtained by

using the M-O theory. Further, the variation of the seismic active earth pressure with the

normalised height of the retaining wall (z/H) (Figure 5.11b) is similar to the variation of

active earth pressure obtained using the FE model (Figure 5.11a). This validates the

findings of Equation 5.4 and emphasises that the M-O theory significantly

overestimates the seismic active earth pressure.


179

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 30 35 400.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 60

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40

(d)(c)

(b)

Earth pressure (kN/m2)

No

rma

lise

d h

eig

ht

(z/H

)

Coulomb's theory

At-rest state po

Finite element pa

(a) Coulomb's theory

At-rest state po

M-O method pae

Finite element pae


No

rma

lise

d h

eig

ht

(z/H

)


No

rma

lise

d h

eig

ht

(z/H

)

Coulomb's theory

At-rest state po

M-O method pae

Finite element ppe


No

rma

lise

d h

eig

ht

(z/H

)

Coulomb's theory

At-rest state po

M-O method pae

Finite element pre

Figure 5.11: Distribution of seismic earth pressure along the height of the retaining wall

Similarly, the distribution of the passive earth pressure with a normalised height of the

retaining wall (z/H) is nonlinear as shown in Figure 5.11c. It can also be noted that the

seismic passive earth pressure is much higher than the at-rest earth pressure and this

validates the findings of Equation 5.5. Figure 5.11d shows the variation of the seismic

earth pressure with the normalised height of the retaining wall (z/H) at the end of the

earthquake (t = 30 sec). It can be noted that the distribution of earth pressure is also

nonlinear and it is so similar to what was observed for the passive case (Figure 5.11c).

5.4.5 Effect of wall seismic inertia force Fw on the earth pressure force increment

∆P

The wall seismic inertia force Fw and earth pressure force increment ∆P (∆P= P –Pa)

are combined together as shown in Figure 5.12, and they are presented of time between

3 sec -7 sec. It is observed that when the maximum wall seismic inertia force F acts in


180

the active direction at time t = 3.8 sec, the seismic earth pressure force increment ∆P is

close to zero. It is also observed that the maximum seismic earth pressure force

increment ∆P occurs at time t = 4.5 sec when the wall seismic inertia force Fw acts in

the passive direction. This indicates that the seismic earth pressure force and the wall

seismic inertia force are acting out of phase during the application of the earthquake

acceleration.

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0

-80

-40

0

40

80

120

Fw a

nd

P

, (k

N/m

)

Time, t (sec)

Fw

P

Figure 5.12: Phase difference between seismic earth pressure force increment and wall seismic

inertia force

Thus, it can be said that for the active case, the wall seismic inertia force

Fw along with the static active earth pressure force Pa which are controlling the

displacement and/or the rotation of retaining wall. In other words, the total earth

pressure force increment ∆P does not contribute to the displacement and/or rotation of

the retaining wall. This observation matches very well with what has been discussed in

section 5.4 above and consequently what has been proved by Equation 5.4. On the other

hand, for the seismic passive earth pressure case, the displacement and/or rotation of the

retaining wall is affected by both, i.e., the wall seismic inertia force Fw and the total

earth pressure force increment ∆P. So, it can be said that the displacement and/or

rotation of the retaining wall is predicted by the wall seismic inertia force while they is

resisted by the total seismic passive earth pressure force. Like the active case, this


181

observation matches well with what has been mentioned in section 5.4 and consequently

proved by Equation 5.5.

5.4.6 Effect of wall displacement on the wall seismic inertia force Fw

On comparing Figure 5.9 with Figure 5.6, it is observed that the horizontal displacement

of the retaining wall follows the trend of the wall seismic inertia force Fw. Similarly, it

is also observed that the retaining wall experiences maximum displacement away from

the backfill when the maximum wall seismic inertia force Fw acts in the same direction

as the direction of the retaining wall displacement.

5.4.7 Effect of wall displacement on seismic earth pressure force P

Studying Figure 5.6 in conjunction with Figure 5.10 helps us to decipher a relationship

between the horizontal displacement and/or rotation of the retaining wall and the

seismic earth pressure force P. It is observed from Figure 5.6 that the retaining wall and

backfill soil move to a maximum amount in the active direction at time t = 3.8 sec

(point dactive in Figure 5.6), while at time t = 4.5 sec the retaining wall and backfill soil

move to a maximum amount in the passive direction (point dpassive in Figure 5.6).

However, from Figure 5.13, it can be noted that the relative horizontal displacement

between the retaining wall and backfill soil, which is predicted at H= 4 m, has a

maximum value at time t = 3.8 sec in the active direction, thereby implying that the

retaining wall is undergoing a larger displacement (in the active direction) than the

backfill soil up to time t = 3.8 sec. As the retaining wall gets displaced more than the

backfill soil, a state of seismic active earth pressure is developed inside the backfill soil.

The same is shown in Figure 5.10 at time t = 3.8 sec. Also, from Figure 5.13 it can be

observed that the relative horizontal displacement between the retaining wall (top_wall)

and backfill soil (top_soil) has a maximum value at time t = 4.5 sec in the passive

direction, thereby implying that the retaining wall is undergoing a larger displacement

(in the passive direction) than the backfill soil up to time t = 4.5 sec. Further, as the

retaining wall gets displaced more than the backfill soil towards the backfill, a state of

seismic passive earth pressure is developed inside the backfill soil. The same is shown

in Figure 5.10 at time t = 4.5 sec.


182

0 5 10 15 20 25 30-0.0025

-0.0020

-0.0015

-0.0010

-0.0005

0.0000

Maximum W-B

[t = 4.5 sec]

Re

lative d

ispla

cem

ent, (

m)

Time, t (sec)

Maximum W-B

[t = 3.8 sec]

Figure 5.13: Relative horizontal displacement between the retaining wall and backfill soil

5.5 PARAMETRIC STUDY

As observed above, the development of the seismic earth pressure force P is

significantly affected by the magnitude and direction of the horizontal displacement and

rotation of the retaining wall. In addition, it is observed that the seismic active earth

pressure is overestimated by using the M-O method. To investigate the effect of various

factors involved in the above analysis, a parametric study has been carried out by

varying, height of the retaining wall, acceleration level of the earthquake acceleration,

and frequency content of the seismic loading as well as the relative density of the

backfill and foundation soil

For this purpose the results from above FE model have been obtained by varying the

abovementioned parameters to capture the following:

Maximum horizontal displacement at the top of the retaining wall (top_wall);

Relative horizontal displacement between the centre of gravity of the retaining wall

(location w_CG) and a point 0.5 m below the base of the retaining wall (P2);

Rotation of the retaining wall θ;

Residual rotation;

Maximum acceleration at the centre of gravity of the retaining wall (w_CG);

Maximum wall seismic inertia force; and

Total seismic active, passive and residual earth pressure force (Pae, Ppe, and Pre).


183

5.5.1 Effect of the earthquake acceleration level and retaining wall height

The earthquake acceleration level is simulated by using the real earthquake-time history

of the Loma Prieta (1989) earthquake (see Figure 5.3a) – for studying its effect; its

amplitude is scaled to vary between 0 to 0.6g. These acceleration levels are applied for

three heights of the retaining wall (4 m, 8 m, and 12 m). The results obtained from the

parametric study presented above are listed in Table A.1 in the Appendix A. Figure 5.14

shows the effect of retaining wall height on the acceleration response at the top of the

retaining wall (top_wall), relative horizontal displacement between the retaining wall

and foundation layer W-F, and seismic earth pressure force P considering three

acceleration amplitudes (0.2g. 0.4g, and 0.6g).

5.5.1.1 Acceleration response

Figure 5.14a, b, and c show the acceleration response at the top of the retaining wall

(top_wall) of retaining wall heights 4m, 8m, and 12m respectively. The acceleration

response is presented in Figure 5.14a, b, and c for time t = 3 - 7 sec. It is observed from

Figure 5.14 a, b, and c that when the acceleration is applied towards the backfill soil, the

rate of amplification of retaining wall acceleration response increases drastically for

acceleration levels up to about 0.4g for all retaining wall heights. For strong earthquake

motions, i.e., for acceleration levels between 0.4g to 0.6g, the rate of amplification of

the retaining wall acceleration response does not increase at the same rate. This may be

because of de-amplification of the acceleration of strong earthquake motion. This

observation of de-amplification of the acceleration for strong earthquake motion

matches well with previous studies like Athanasopoulos-Zekkos et al. (2013), Griffiths

et al. (2016) and Stamati et al. (2016).

5.5.1.2 Relative horizontal displacement

Figure 5.14d, e, and f show the relative horizontal displacement between the retaining

wall and foundation layer W-F, for retaining wall heights 4 m, 8 m, and 12 m,

respectively. The relative horizontal displacement is presented in Figure 5.14d, e, f for

the time t = 0-14 sec where a maximum relative horizontal displacement of retaining

wall is accumulated in this time period. It can be noted from Figures d, e, and f that the

relative horizontal displacement of retaining wall increases significantly with an


184

increase in the acceleration level for all the three retaining wall heights. Additional

observations have been indicated from the Table A.1. It can be noted from Table A.1

that the residual rotation of retaining wall increases with increasing amplitude of

earthquake acceleration for different retaining wall heights. The maximum residual

rotation of retaining wall (of 2.528°) is observed for a retaining wall height H = 12

m while the minimum residual rotation of retaining wall (of 0.891°) is predicted for

a retaining wall height H = 8 m.

5.5.1.1 Seismic earth pressure force

Figure 5.14g, h, and k show the seismic earth pressure force P predicted of retaining

wall heights 4m, 8m, and 12m respectively. The seismic earth pressure force is

presented in Figure 5.14g, h , k for time t = 3 - 7 sec where the minimum seismic active

earth pressure force and maximum seismic passive earth pressure forces are developed

in between time 3 – 7 sec. It is interesting to note from Figure 5.14g, h, and k that for all

the acceleration levels and retaining wall heights, the seismic active earth pressure force

Pae remains almost constant – and its value remains very close to the static active earth

pressure force Pa (t ≈ 3.9 sec – 4.2 sec). As already discussed, this is in contrast to the

conventional M-O theory, which inherently assumes that the seismic active earth

pressure force Pae increases with an increase in the acceleration level. However, for the

passive case, the total seismic passive earth pressure force Ppe for all retaining wall

heights increases with increasing acceleration levels up to 0.4g, and after that, the rate

of the increment for the total seismic passive earth pressure force Ppe is reduced. This

could be because both the horizontal displacement of the top of the retaining wall

(top_wall) and the maximum passive seismic earth pressure force Ppe are significantly

influenced by the local site effects – amplification of low and moderate earthquake

acceleration and de-amplification of the strong earthquake.


185

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0-1.5

-1.0

-0.5

0.0

0.5

1.0

0 2 4 6 8 10 12 14-0.4

-0.3

-0.2

-0.1

0.0

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0-160

-140

-120

-100

-80

-60

-40

-20

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0-1.5

-1.0

-0.5

0.0

0.5

1.0

0 2 4 6 8 10 12 14-0.5

-0.4

-0.3

-0.2

-0.1

0.0

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0

-440

-400

-360

-320

-280

-240

-200

-160

-120

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0-1.5

-1.0

-0.5

0.0

0.5

1.0

0 2 4 6 8 10 12 14-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0

-1000

-900

-800

-700

-600

-500

-400

-300

-200

a(g

),(m

/sec

2)

Time, t (sec)

a(g)=0.2 a(g)=0.4 a(g)=0.6

H= 4m

(k)

(h)

(g)

(f)

(e)

(d)

(c)

(b)

H= 4m

W

-F (

m)

Time, t (sec)

(a)

H= 4m

P (

kN

/m)

Time, t (sec)

H= 8m

a(g

) (m

/sec

2)

Time, t (sec)

H= 8m

W

-F (

m)

Time, t (sec)

H= 8m

P (

kN

/m)

Time, t (sec)

H= 12m

a(g

) (m

/sec

2)

Time, t (sec)

H= 12m

W

-F (

m)

Time, t (sec)

H= 12m

P (

kN

/m)

Time, t (sec)

Figure 5.14: Effect of retaining wall height on seismic response of wall-soil system considering

different amplitudes of the applied earthquake acceleration


wall considering different retaining wall heights and acceleration levels

From the above results presented in Table A.1, and Figure 5.14, a relationship between

the seismic earth pressure and the displacement of the top of retaining wall has been

developed as shown in Figure 5.15. It is important to observe that as discussed in

Section 5.7, the seismic earth pressure is significantly affected by the relative horizontal

displacement between the retaining wall and backfill soil; however, as (1) this relative

horizontal displacement is very small (as an example see Figure 5.13); and (2) difficult

to record during the laboratory experiments and as well as in field, the relationship

between seismic earth pressure and displacement has been developed considering the

total horizontal displacement response at the top of retaining wall in the proposed

design chart (Figure 5.15). The displacement of retaining wall is measured at the time of

minimum seismic active earth pressure (t = 3.8 sec) as well as at the time of maximum


186

seismic passive earth pressure (t = 4.5 sec) developed behind the retaining wall –

mention times as well for these max and min values.

Figure 5.15, shows the relationship between the seismic active and passive earth

pressure force and the total displacement of the top of the retaining wall for different

acceleration amplitudes (0.1g - 0.6g) considering three retaining wall heights (4m, 8m,

and 12m). It is observed from Figure 5.15 that the relationship between the seismic

active and passive earth pressure force and total displacement of the top of the retaining

wall is proportional to the height of the retaining wall. An increase in the height of the

retaining wall leads to an increase in the seismic active and passive earth pressure with

the same displacement response recorded at the top of the retaining wall. The unique

design chart as shown in Figure 5.15 considers the effect of seismic response of a rigid

retaining wall on the development of seismic earth pressure causing the development of

active and passive conditions under the same seismic scenario. The development of

active and passive as shown in Figure 5.15 is in contrast to many studies available in

literature like Matsuo (1941), Matsuo and Ohara (1960), Sherif et al. (1982), Bolton M.

D. and Steedman (1982), Sherif and Fang (1984b), Steedman (1984 ), Bolton and

Steedman (1985), Ishibashi and Fang (1987), Green et al. (2003), Ortiz et al. (1983),

Bolton and Steedman (1985), Zeng (1990;Steedman and Zeng (1991), Stadler (1996),

and Dewoolkar et al. (2001) Siddharthan and Maragakis (1989), Athanasopoulos-

Zekkos et al. (2013) Geraili et al. (2016), Candia et al. (2016) where they assumed that

the development of active state during the seismic scenario. It can also be observed

from Figure 5.15 for all retaining wall heights that the seismic active earth pressure

force is independent of the displacement of retaining wall while the seismic passive

earth pressure force is observed to be highly dependent on the displacement of the

retaining wall in passive direction. From Figure 5.16, which presents a relationship

between the permanent displacement (sliding) of retaining wall and acceleration

amplitude for different retaining wall heights (4 m, 8 m, and 12 m), it is observed that

an increase in the height of the retaining wall leads to a significant increase in the

permanent displacement of the retaining wall under the same acceleration amplitude.


187

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.30

200

400

600

800

1000

Passive

H =12 m

H = 8 m

H = 4 m

Se

ism

ic e

art

h p

ressu

re

forc

e,

P (

kN

/m)

Displacement at the top of the wall (m)

Active

Figure 5.15: Design chart demonstrating the relationship between seismic earth pressure and

wall displacement for different retaining wall heights

0.1 0.2 0.3 0.4 0.5 0.6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Perm

ane

nt

dis

pla

ce

men

t o

f

th

e w

all,

W

-F (

m)

Acceleration, a(g) (m/sec2)

H = 12 m

H = 8 m

H =4 m

Figure 5.16: Variation of relative horizontal displacement between wall and foundation soil with

acceleration levels for different retaining wall heights

5.5.2 Effect of the frequency content of the earthquake acceleration

As highlighted in Chapter 2 previous researchers have focused on the calculation of

seismic earth pressure considering the amplitude of earthquake acceleration only like

pseudo-static methods; however, limited research has been done to study the effect of

frequency content of earthquake acceleration, which is a very critical parameter and


188

may influence the design, on the seismic behaviour of the wall-soil system. Therefore,

the current parametric study is set out to explore the influence of amplitude and

frequency content on the development of seismic behaviour of the wall-soil system. The

frequency content of the earthquake acceleration is investigated by applying a uniform

sinusoidal acceleration at the base of the FE model. Each uniform sinusoidal

acceleration-time history has eight cycles. They are scaled to achieve three different

amplitudes of 0.2g, 0.4g, and 0.6g. For the same amplitude, five uniform acceleration

time histories are defined by five frequencies, viz., 0.3 Hz, 0.6 Hz, 1 Hz, 2 Hz, and 3

Hz. Detailed results obtained from this parametric study are presented in Table A.2 in

Appendix A. The subsections below will discuss the effect of earthquake characteristics

(amplitude and frequency content) on the acceleration response of the retaining wall and

backfill soil, seismic earth pressure force, and relative horizontal displacement of the

retaining wall. After that, a design chart will be produced to correlate the seismic earth

pressure with the displacement of the retaining wall considering the effect of the

earthquake characteristics.


In order to investigate the impact of applied earthquake acceleration characteristic

(amplitude and frequency content) on the acceleration response of the retaining wall and

backfill soil, the acceleration response is predicted at the top of retaining wall (top_wall)

and at the top of the backfill soil layer (top_soil). It can be indicated from the

Figure 5.17a that the acceleration responses at the top of the retaining wall and backfill

soil are not amplified when the earthquake is applied with amplitude (0.2g) and

frequency content 0.6Hz, and they have the same amplitude of the earthquake

acceleration (0.2g – see Figure 5.17a ). However, Figure 5.17b and c show that the

acceleration responses at the top of the retaining wall and backfill soil when the

earthquake accelerations are applied with the frequency content of 0.6Hz and

amplitudes 0.4g and 0.6g respectively. The amplitude of both acceleration responses

(a(g)≈ 0.3) is smaller than the amplitude of applied earthquake acceleration when it is

applied away from the backfill soil (see Figure 5.17b and c). This de-amplification in

the acceleration response is because the highly nonlinear behaviour of the soil behaviour

tends to de-amplify the strong earthquake. This observation of de-amplification of the


189

acceleration for strong earthquake motion matches well with previous studies like

Athanasopoulos-Zekkos et al. (2013), Griffiths et al. (2016) and Stamati et al. (2016).

It can also be noted that the retaining wall is highly influenced by the acceleration

response of the backfill soil layer, and the acceleration response at the top of the

retaining wall (a(g)≈ 0.3) is also smaller than the amplitude of applied earthquake

acceleration with frequency content 0.5 Hz and amplitude >0.4g (see Figure 5.17b and

c). Figure 5.17d, e, and f show the acceleration response at the top of the retaining wall

and backfill soil when the earthquake acceleration is applied with a frequency content of

2 Hz and amplitude 0.2g, 0.4g, and 0.6g, respectively. It can be found that for all

earthquake acceleration amplitudes, the acceleration response at the top of the retaining

wall and backfill soil is amplified when the earthquake acceleration is applied away

from the backfill soil (a(g)≈ 0.3)- Figure 5.17d, (a(g)≈ 0.5-Figure 5.17e), (a(g)≈ 0.9-

Figure 5.17f). However, the acceleration response seems to de-amplify when the

acceleration of earthquake acceleration is applied towards the backfill soil (a(g)≈ 0.3)-

Figure 5.17d, (a(g)≈ 0.5-Figure 5.17e), (a(g)≈ 0.9-Figure 5.17f). Figure 5.17g, h, and k

show the acceleration response at the top of the retaining wall and backfill soil when the

earthquake acceleration is applied with a frequency content 3Hz and amplitude 0.2g,

0.4g, and 0.6g respectively. It can be noted that the acceleration response at the top of

the retaining wall and the top of backfill soil is amplified when earthquake acceleration

is applied away from the backfill soil (a(g)≈ 0.25) - Figure 5.17g, (a(g)≈ 0.8 -

Figure 5.17h, (a(g)≈ 0.8 - Figure 5.17k). However, it is also observed that the

acceleration response at the top of the retaining wall and the top of backfill soil is

amplified when earthquake acceleration is applied towards the backfill soil (a(g)≈ 0.35)-

Figure 5.17g, (a(g)≈ 0.85-Figure 5.17h), (a(g)≈ 1.2-Figure 5.17k). This amplification in

acceleration response is because the earthquake acceleration is applied with frequency

content 3 Hz, and this frequency content is very close of the natural frequency content

of the wall-soil system, which is predicted by using ABAQUS software, and it is equal

to 4.5 Hz. It can be noted that the acceleration response at the top of the retaining wall is

higher than the acceleration response predicted at the top of the backfill soil when the

earthquake acceleration is applied with frequency content 3 Hz and amplitude >0.4g (

see Figure 5.17h and k). This could because the frequency content of applied earthquake

acceleration becomes very close to natural frequency of the retaining wall.


190

0 5 10 15 20-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0 5 10 15 20-0.6

-0.4

-0.2

0.0

0.2

0.4

0 5 10 15 20-0.6

-0.4

-0.2

0.0

0.2

0.4

0 2 4 6 8-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0 2 4 6 8-0.6

-0.4

-0.2

0.0

0.2

0.4

0 2 4 6 8-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 1 2 3 4 5

-0.4

-0.2

0.0

0.2

0.4

0 1 2 3 4 5

-0.8

-0.4

0.0

0.4

0.8

0 1 2 3 4 5-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

f =0.6 Hz, a(g)=0.2

a(g

) (m

/sec

2)

Time, t (sec)

(f)

(c)

(h)

(e)

f =0.6 Hz, a(g)=0.4

a(g

) (m

/sec

2)

Time, t (sec)

f =0.6 Hz, a(g)=0.6

a(g

) (m

/sec

2)

Time, t (sec)

(b)

(g)

f =2 Hz, a(g)=0.2

a(g

) (m

/sec

2)

Time, t (sec)

f =2 Hz, a(g)=0.4

a(g

) (m

/sec

2)

Time, t (sec)

(k)

f =2 Hz, a(g)=0.6

a(g

) (m

/sec

2)

Time, t (sec)

f =3 Hz, a(g)=0.2

a(g

) (m

/sec

2)

Time, t (sec)

f =3 Hz, a(g)=0.4

a(g

) (m

/sec

2)

Time, t (sec)

(d)

(a)

f =3 Hz, a(g)=0.6

(+) sign:Towards the backfill (-) sign: Away from backfill Wall (top_wall) Backfill (top_soil)

a(g

) (m

/sec

2)

Time, t (sec)


amplitudes and frequency content of the applied earthquake acceleration

5.5.2.2 Relative horizontal displacement of the retaining wall

Figure 5.18b, d, and f show the relative horizontal displacement of the retaining wall

(sliding - W-F) computed when the earthquake acceleration is applied with frequency

contents of 0.6 Hz, 2 Hz, and 3 Hz respectively (see Figure 5.18a, c, and e) considering

three acceleration amplitudes; 0.2g, 0.4g, and 0.6g. It can be noted that the relative

horizontal displacement of the retaining wall W-F remarkably increases with increasing

amplitude of earthquake acceleration for different frequency contents. However, it can

be seen that the relative horizontal displacement of the retaining wall W-F reduces when

the frequency content of the earthquake acceleration is increased from the 0.6 Hz to 2

Hz (see Figure 5.18b and d) under the same acceleration amplitude. For example, when

the earthquake acceleration is applied with a frequency content of 0.6 Hz and amplitude

0.6g (see Figure 5.18a), the maximum relative horizontal displacement of retaining wall

(/H) is equal to 0.275 (see Figure 5.18b). However, when the earthquake acceleration

is applied with frequency content 2 Hz and amplitude 0.6g (see Figure 5.18c), the


191

maximum relative horizontal displacement of retaining wall (/H) is equal to 0.09 (see

Figure 5.18d). A possible explanation for these results might be that the driving forces

causing the sliding of the retaining wall (wall seismic inertia force and seismic earth

pressure force) will push the retaining wall to slide longer time with applying the

earthquake acceleration with a low frequency content.

Another observation from Figure 5.18d and f is that the relative horizontal displacement

of the retaining wall W-F is almost still the same when the frequency content of

earthquake acceleration increased from 2 Hz to 3 Hz. For example, when the

earthquake acceleration is applied with frequency content 2 Hz and amplitude 0.6g (see

Figure 5.18c), the maximum relative horizontal displacement of retaining wall (/H) is

equal to 0.09 (see Figure 5.18d). However, when the earthquake acceleration is applied

with frequency content 3 Hz and amplitude 0.6g (see Figure 5.18e), the maximum

relative horizontal displacement of retaining wall (/H) is equal to 0.0875 (see

Figure 5.18c). These results may reflect the effect of de-amplification of acceleration

response in the retaining wall and backfill soil when the earthquake acceleration is

applied with higher frequency content like 2Hz and 3Hz.

It is interesting to indicate that for all earthquake accelerations , which are applied with

a variety of frequency contents and amplitudes, the amplitude of relative horizontal

displacement of retaining wall W-F is sensitive to the number of acceleration cycles of

applied earthquake acceleration s. So, it can be found from the Figure 5.18b, d, and f

that the amplitude of relative horizontal displacement of retaining wall W-F increass

when the number of acceleration cycles of applied earthquake acceleration s is

increased. Additional observations have been noted from Table A.2 related to the

rotation of the retaining wall. It can be indicated from the Table A.2 that the residual

rotation of retaining wall is highly influenced by the earthquake acceleration

characteristics. It can be noted that the residual rotation of retaining wall increases

when the frequency content of earthquake acceleration is decreased, and in the same

time, the amplitude of earthquake acceleration is increased. For example, the residual

rotation of retaining wall is equal to 11.247° when the earthquake acceleration is

applied with frequency content 0.33 Hz and amplitude 0.6g. However, the residual


192

rotation of retaining wall is equal to 0.011° when the earthquake acceleration is

applied with frequency content 3 Hz and amplitude 0.6g (see Table A.2).

0 5 10 15 20

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 5 10 15 20-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0 2 4 6 8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 2 4 6 8-0.125

-0.100

-0.075

-0.050

-0.025

0.000

0 1 2 3 4 5

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 1 2 3 4 5-0.100

-0.075

-0.050

-0.025

0.000

a(g

) (m

/se

c2)

Time, t (sec)

a(g)=0.2

a(g)=0.4

a(g)=0.6

W

-F (

/H)

W

-F (

/H)

a(g)=0.2 a(g)=0.4 a(g)=0.6

+ a(g):Towards the backfill - a(g): Away from the backfill

W

-F (

/H)

Time, t (sec)

a(g

) (m

/se

c2)

Time, t (sec)

a(g)=0.2

a(g)=0.4

a(g)=0.6

Time, t (sec)

a(g

) (m

/se

c2)

Time, t (sec)

a(g)=0.2

a(g)=0.4

a(g)=0.6

Time, t (sec)

Figure 5.18: Relative horizontal displacement between retaining wall and foundation soil for

different amplitudes and frequency content of the applied earthquake acceleration

5.5.2.3 Seismic earth pressure force

Figure 5.19b shows the seismic earth pressure force, which is predicted when the

earthquake acceleration is applied with frequency content 0.6 Hz and amplitude of 0.2g,

0.4g, and 0.6g (see Figure 5.19a). It can be seen from the Figure 5.19b that the seismic

active earth pressure (Pae≈ 50 kN/m) is close to the static earth pressure force even

when the amplitude of earthquake acceleration is increased, while the seismic passive

earth pressure force Ppe significantly increases with increasing earthquake acceleration

amplitude. Figure 5.19d, and f show the seismic earth pressure forces, which are


193

predicted when the earthquake acceleration s are applied with the frequency content 2

Hz and 3 Hz respectively (see Figure 5.19c and e) with an amplitude of 0.2g, 0.4g, and

0.6g. It can be indicated that for both earthquake accelerations with the frequency

content of 2 Hz and 3 Hz, the seismic active earth pressure force Pae is reduced with

increasing of the earthquake acceleration amplitude. However, the seismic passive earth

pressure force Ppe is highly increased with increasing the amplitude of earthquake

acceleration from 0.2g to 0.4g, while the rate of increment of seismic passive earth

pressure force Ppe is reduced with increasing the amplitude of earthquake acceleration

from 0.4g to 0.6g. A possible explanation of above result is that the seismic earth

pressure forces at low frequency content of earthquake acceleration are almost acting in

the same phase along the height of the retaining wall. However, when earthquake

acceleration is applied with high frequency content, the seismic earth pressure force is

not acting in the same phase along the height of the retaining wall. So, for earthquake

acceleration with low frequency, the maximum amplitudes of seismic passive earth

pressure forces Ppe along the retaining wall height could coincide with each other

thereby causing an increase of the total seismic passive earth pressure amplitude.

Based on the result in Figure 5.19b, d, and f, it can generally be found that the seismic

active earth pressure force Pae is close to the static earth pressure force Pa when the

frequency content increased from the 0.6 Hz to 3 Hz. For example, the seismic active

earth pressure is close to 50 kN/m when the earthquake acceleration is applied

amplitude 0.4g and frequency content 0.6 Hz and 3 Hz (see Figure 5.19b and f).

However, the seismic passive earth pressure force Ppe is decreased with increasing of

the frequency content of earthquake acceleration from 0.6 Hz to 3 Hz. For example, the

seismic passive earth pressure is close to 180 kN/m when the earthquake acceleration is

applied amplitude 0.6g and frequency content 0.6 Hz (see Figure 5.19b), while it is close

to 140 kN/m when the earthquake acceleration is applied amplitude 0.6g and frequency

content 3 Hz (see Figure 5.19f).

It is interesting to note that for all earthquake acceleration s, which are applied with a

variety of frequency contents and amplitudes, the amplitude of seismic active earth

pressure force Pae and passive earth pressure force Ppe is not sensitive to the number of

acceleration cycles of applied earthquake acceleration s. So, it can be seen from the


194

Figure 5.19b, d, and f that the seismic active Pae and passive Ppe earth pressure force

amplitudes are almost the same at each acceleration cycle of applied earthquake

acceleration s (see Figure 5.19a, c, and e).

0 5 10 15 20

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 5 10 15 20-200

-180

-160

-140

-120

-100

-80

-60

-40

0 2 4 6 8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 2 4 6 8

-120

-100

-80

-60

-40

0 1 2 3 4 5

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 1 2 3 4 5

-160

-140

-120

-100

-80

-60

-40

-20

0

a(g

) (m

/sec

2)

Time, t (sec)

f = 0.6 Hz f = 0.6 Hz

P (

kN

/m)

Time, t (sec)

f = 2 Hz

a(g

) (m

/sec

2)

Time, t (sec)

f = 2 Hz

P (

kN

/m)

Time, t (sec)

f = 3 Hz

(f)(e)

(d)(c)

(b)

a(g

) (m

/sec

2)

Time, t (sec)

(a)

f = 3 Hz


a(g)=0.2 a(g)=0.4 a(g)=0.6

P (

kN

/m)

Time, t (sec)

Figure 5.19: Seismic earth pressure force for different amplitudes and frequency content of the

applied earthquake acceleration

Additional observations have been obtained from Table A.2 related to the seismic earth

pressure force. It can be noted from the Table A.2 that the M-O method overestimates

the seismic active earth pressure force and seismic passive earth pressure force for a

variety of amplitudes and frequency content of earthquake acceleration. It can also

indicated that the frequency content of earthquake acceleration is a critical parameter

that affected the development of seismic passive earth pressure force, and it has been

already not considered in a pseudo-static method like M-O method. However, it can

also be indicated that seismic active earth pressure force has been not affected by


195

frequency content of earthquake acceleration, and this is in opposite to what has been

predicted by using a pseudo-dynamic method like Steed-Zeng method. So, the pseudo-

dynamic method has shown that the seismic active earth pressure force is reduced with

increasing of the frequency content of earthquake acceleration.


wall considering different amplitudes and frequency content of earthquake

acceleration

From the above parametric study results, the relationship between the seismic earth

pressure and the displacement of the top of retaining wall considering the effect of

amplitude and frequency content of earthquake acceleration as shown in Figure 5.20.

According to the considerations as already discussed in section 5.5.1.2, the relationship

between seismic earth pressure and displacement has been developed considering the

total displacement response at the top of retaining wall. Furthermore, it was observed

from sections 5.5.2.2 and section 5.5.2.3 that the amplitude of seismic active and

passive earth pressure forces is not sensitive to the number of acceleration cycles of

earthquake acceleration. So, the relationship between the seismic earth pressure force

and total displacement response at the top of the retaining wall is designed for one

acceleration cycle of earthquake acceleration.

It can be observed from Figure 5.20 that for the active earth pressure case; it appears

that both the acceleration amplitude and frequency content of the earthquake

acceleration do not affect the seismic earth pressure versus wall displacement

relationship. On the other hand, for the passive case, both the amplitude and frequency

of the earthquake acceleration significantly affects the displacement of the top of the

retaining wall and hence the seismic earth pressure. From Figure 5.20, it is observed

that a decrease of frequency content leads to the development of a higher seismic

passive earth pressure and larger displacement for the same acceleration amplitude.

Figure 5.21 shows a relationship between the permanent displacement (sliding) of the

retaining wall amplitude for different frequency content of the earthquake acceleration.

It was observed from the section 5.5.2.3 that the relative horizontal displacement it so

sensitive to a number of acceleration cycles of earthquake acceleration, but the rate of

increment of the relative horizontal displacement is almost the same of each


196

acceleration cycle. Hence, the relationship between the relative horizontal displacement

and earthquake acceleration characteristics is designed for one acceleration cycle. It is

also clear from Figure 5.20 that the permanent displacement of the retaining wall is

remarkably affected by the frequency content of the earthquake acceleration.

Figure 5.20 and Figure 5.21 are unique design charts developed by considering the

seismic response of retaining wall under different earthquake characteristics. It has been

indicated that both active and passive states have been developed for a variety of

earthquake characteristics, and this is in contrast to many studies available in literature

like Matsuo (1941), Matsuo and Ohara (1960), Sherif et al. (1982), Bolton M. D. and

Steedman (1982), Sherif and Fang (1984b), Steedman (1984 ), Bolton and Steedman

(1985), Ishibashi and Fang (1987), Green et al. (2003), Ortiz et al. (1983), Bolton and

Steedman (1985), Zeng (1990;Steedman and Zeng (1991), Stadler (1996), and

Dewoolkar et al. (2001) Siddharthan and Maragakis (1989), Athanasopoulos-Zekkos et

al. (2013) Geraili et al. (2016), Candia et al. (2016), where they assumed that the

seismic active state is only developed under the effect a variety of earthquake

characteristic. Further, the above design charts show the effect of frequency content

parameter on the development of seismic passive earth pressure force and relative

horizontal displacement of retaining wall, and the effect of earthquake duration on the

relative horizontal displacement of retaining wall.


197

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

30

60

90

120

150

180

210

240

Passive

Seis

mic

ea

rth

pre

ssu

re

forc

e,

P (

kN

/m)

Displacement at the top of retaining wall, (m)

f = 0.33Hz

f = 0.66 Hz

f = 1 Hz

f = 2 Hz

f = 3 Hz

Active

Figure 5.20: Relationship between seismic earth pressure and displacement of the retaining wall

for different amplitudes and frequency content of the applied earthquake acceleration

0.2 0.3 0.4 0.5 0.6

0.0

0.1

0.2

0.3

0.4 f = 0.33 Hz

f = 0.66 Hz

f = 1 Hz

f = 2 Hz

f = 3 Hz

Rela

tive d

isp

lacem

ent

pe

r o

ne

acce

lera

tio

n c

ycle

(m

)

Acceleration , a(g) (m/sec2)

Figure 5.21: Relationship between relative horizontal displacement and acceleration amplitude

for different frequency content of the applied earthquake acceleration


198

5.5.3 Effect of the relative density of the soil material

Despite the fact that the pseudo-static and have indicated that the seismic earth pressure

force is highly influenced by the relative density of soil material (as discussed in

Chapter 2 –section 2.5.1.1.1) , limited research has been done to investigate the effect of

relative density of soil material on the seismic behaviour of the wall-soil system by

using performance-based methods. Hence, the current parametric study is set out to

investigate the impact of the relative density of soil material on the seismic response of

the wall-soil system. The effect of relative density of soil material is examined by

choosing three relative densities; relatively loose soil (Dr = 40%), relatively medium-

dense soil (Dr = 65%), and relatively stiff soil (Dr = 85%). The material properties,

which are used to run the FE models, are presented in Table 5.2 for three relative

densities of soil materials. The effect of the relative density of soil material is simulated

by using 3 combinations:

1st combination includes that the same material is used to construct the backfill soil,

and foundation soil layer. The material properties are simulated by using three

relative densities of soil material; 40%, 65%, and 85% as shown in Figure 5.22a.

2nd

combination includes that the foundation layer is simulated by relative density

equal to 65% while the backfill soil is simulated by three relative densities 40%,

65%, and 85% (see Figure 5.22b) in order to study the effect of the relative density

of backfill soil layer only on the seismic response of the wall-soil system.

3rd

combination includes that the backfill soil layer is simulated by relative density

65%, while the relative density of foundation layer is simulated by three relative

densities 40%, 65%, and 85% (see Figure 5.22c) in order to investigate the effect of

the relative density of foundation layer on the seismic response of the wall-soil

system.

The seismic loading is simulated by applying a uniform sinusoidal earthquake

acceleration at the base of the FE model with amplitude 0.3g and frequency content 2Hz

in the current parametric study, as shown in Figure 5.23, in order to invrstigate the

effect of the reltive density of soil material on the seismic reponse of wall-soil system.


199

Figure 5.22: Different combinations of relative densities of backfill and foundation soil (a) 1st

combination, (b) 2nd

combination and (c) 3rd

combination

0 2 4 6 8-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Acce

lera

tio

n a

(g)

(m/s

ec

2)

Time, t (sec)

Figure 5.23: Earthquake acceleration applied at the base of FE model to investigate the effect of

relative density of soil materials on the seismic response of wall-soil system

Dr = 40%

Dr = 40%

Dr = 65%

Dr = 65%

Dr = 85%

Dr = 85%

Dr = 40% Dr = 65% Dr = 85%

Dr = 65% Dr = 65% Dr = 65%

Dr = 65% Dr = 65% Dr = 65%

Dr = 40% Dr = 65% Dr = 85%

(a)

(b)

(c)


200

Table 5.2: Parameters required for running FE model considering different relative densities of

soil material

Parameters Unit Soil

Loose medium stiff

Soil

Dr % 40 65 85

γ kN/m3 16 17.6 18.6

' o

33 36.13 40

𝐸50𝑟𝑒𝑓

MPa 24 39 54

𝐸𝑜𝑒𝑑𝑟𝑒𝑓

MPa 24 39 54

𝐸𝑢𝑟𝑟𝑒𝑓

MPa 72 117 162

o 3 6.125 10

ur - 0.2 0.2 0.2

y - 0.57 0.497 0.4188

𝐺𝑜𝑟𝑒𝑓

MPa 87 104.2 121.2

γ0.7 - 0.00016 0.00014 0.00011

pref

kN/m2 100 100 100

% 3 3 3

Rf - 0.95 0.91 0.89

Retaining wall

E MPa 30000

- 0.15

kN/m3 18

% 3

5.5.3.1 Effect of soil material (1st combination)

The parametric study is carried out to investigate the effect of soil material on the

seismic response of the wall-soil system. The analysis in this parametric study is

adopted by using first procedure mentioned in section 5.5.3. Results of the study are

presented in Table A.3 in Appendix A. Figure 5.24 shows the effect of relative density

of soil material on the acceleration response at the top of the retaining wall (top_wall)

and backfill soil (top_soil), the relative horizontal displacement of the retaining wall W-

F, and seismic earth pressure force P.


201

Figure 5.24a shows the effect of relative density of soil material on the acceleration

response at the top of the wall (top_wall), while Figure 5.24b shows the effect of

relative density of soil material on the acceleration response at the top of backfill soil

(top_soil). From Figure 5.24a and b, it can be seen that the maximum amplification of

acceleration response (a(g)= 0.45) is observed when the soil has a relatively low density

(Dr = 40%), and the rate of amplification of acceleration response is reduced when the

relative density of soil material is increased from 40% to 85%. It can be also noted

there is a phase difference in the acceleration response when the soil material is

simulated with different relative densities. So, it can be indicated that the maximum lag

in acceleration response is that when the soil material is simulated with the relatively

loose material (Dr = 40%). The rate of lag in acceleration response is reduced with

increasing relative density of soil materials. A possible explanation for these results

might be that when the backfill soil has a high relative density, the stiffness of the soil is

increased, and consequently, the shear velocity of backfill soil is increased. So, the

seismic wave will propagate faster in soil layer towards the top., it can also be noted

that the acceleration response of retaining wall is significantly affected by the relative

density of soil material.

Figure 5.24c shows the influence of the relative density of the soil material on the

relative displacement of the retaining wallW-F. It can be seen that the relative

displacement of the retaining wall (/H) is reduced from 0.035 to 0.01 m as shown in

Figure 5.24c when the relative density of soil materials is increased from 40% to 85%.

These results may be explained by the fact that the maximum amplification of

acceleration response at the top of the retaining wall is observed when the soil material

is simulated by relatively loose density (see Figure 5.24a). Maximum wall seismic

inertia force will be developed in the retaining wall causing maximum relative

horizontal displacement of the retaining wall. There is another possible explanation for

this result is that the stiffness parameters of interface elements, which connect the base

of the retaining wall with foundation soil, are highly affected by stiffness parameters of

surrounding soil (see chapter 3 – section 3.4.3). So, the friction force between the base

of retaining wall and foundation soil will be reduced when the soil material is simulated

with loose relative density, and maximum relative horizontal displacement of the

retaining wall will be predicted. Additional observations have been predicted from the


202

Table A.3 related to the rotation of retaining wall. It can be noted from Table A.3 that

the residual rotation of the retaining wall is reduced from 0.503° to 0.222° when the

relative density of soil material is increased from 40% to 85%.

Figure 5.24d shows the impact of relative density of backfill soil on the development of

seismic earth pressure P. It can be noted from Figure 5.24d that the minimum seismic

earth pressure force (Pae≈ 40kN/m) is developed when the soil material is simulated

with relatively loose density (Dr = 40%), while the maximum seismic passive earth

pressure force (Ppe≈ 105kN/m) is developed when the soil material is simulated with

relatively stiff density Dr = 85%. There are several possible factors could explain these

results. Firstly, based on the static earth pressure theory, as discussed in Chapter 2 –

section 2.4.2, the passive earth pressure force is directly proportional to the soil unit

weight and angle of shear resistance, and they are increased with increasing of the

relative density of soil material. Secondly, when the soil material is simulated with the

relatively stiff material of Dr = 85%, the maximum seismic passive earth pressure forces

along the height of the retaining wall is almost acting in the same phase when the soil

material is simulated with a relative density of Dr = 85% because the stiff soil has a

larger shear velocity. So, the amplitude of total seismic earth pressure force becomes at

its maximum value when the soil material is simulated with a relative density Dr = 85%.

When the soil material is simulated with a smaller relative density like Dr = 40%, the

phase difference between seismic passive earth pressure forces along the height of the

retaining wall becomes large, and this could reduce the amplitude of total seismic

passive earth pressure force Ppe.

Additional observations have been predicted from the Table A.3 related to the seismic

earth pressure force. It can be noted from Table A.3 that the M-O method overestimates

the seismic active and passive earth pressure forces for all relative densities of soil

materials.


203

0 2 4 6 8-0.6

-0.4

-0.2

0.0

0.2

0.4

0 2 4 6 8-0.6

-0.4

-0.2

0.0

0.2

0.4

0 2 4 6 8-0.04

-0.03

-0.02

-0.01

0.00

0 2 4 6 8

-100

-80

-60

-40

-20

a(g

) (m

/sec

2)

Time, t (sec)

a(g

) (m

/sec

2)

Time, t (sec)

(

/H)

Time, t (sec)

(d)(c)

(b)

Dr=40% Dr=65% Dr=85% f = 2 Hz

P (

m)

Time, t (sec)

(a)

Figure 5.24: Effect of soil material relative density on the seismic response of wall-soil system

5.5.3.2 Effect of the relative density backfill soil layer (2nd

combination)

The current parametric study is set out to study the effect of relative density of backfill

soil layer on the seismic response of the wall-soil system. The simulation of soil

material is adopted by using the second procedure mentioned in section 5.5.3. The

results of the parametric study are presented in Table A.4 in Appendix A. Figure 5.25

shows the influence of the relative density of backfill soil layer on the acceleration

response at the top of the retaining wall (top_wall) and backfill soil (top_soil), the

relative horizontal displacement of the retaining wall W-F, and seismic earth pressure

force P.

Figure 5.25a and b show the impact of the relative density of backfill soil layer on the

acceleration response at the top of the retaining wall (top_wall) and backfill soil layer

(top_soil) respectively. The results shown in Figure 5.25a and b indicate that the

maximum amplification (a(g)= 0.8) and phase difference is observed when the backfill

soil is simulated with loose soil material (Dr = 40%).


204

Figure 5.25c shows the effect of the relative density of backfill soil layer on the relative

horizontal displacement of the retaining wall W-F. It can be seen that the maxium

relative horizontal displacement of the retaining wall (/H = 0.095) is predicted when

the backfill soil is simulated with loose material (Dr = 40%) despite the fact that the

stiffness parameters of interface elements, which connect the base of the retaining wall

with the foundation layer, is kept the same in current parametric study. So, the results

may be explained by the fact that the acceleration response of the retaining wall

retaining wall is highly affected by the acceleration response of the backfill soil, and the

later is amplified to a maximum value (a(g)= 0.8) when the backfill soil layer is

simulated with loose material (Dr = 40%) as shown in Figure 5.25a, and b. Based on

this, it is clear that a larger amplitude of wall seismic inertia forces will be developed in

the retaining wall causing the retaining wall to move a larger distance.

Additional observations have been predicted from the Table A.4 related to the rotation

of retaining wall. It is interesting to note from Table A.4 that the residual rotation of the

retaining wall is increased from 0.007° to 0.349° on increasing the relative density of

soil material from 40% to 85%. A possible explanation for this trend is that when the

backfill soil is simulated with a soil of relatively low relative density (40%), the

retaining wall can rotate with larger amplitude towards the backfill soil (0.259°).

However, when the backfill soil is simulated with a soil of relatively high relative

density (85%), the retaining wall rotates with a small amplitude towards the backfill soil

(0.107°). A possible explanation of this trend is that the retaining wall will be strongly

resisted by backfill soil layer with high relative density when it rotates towards the

backfill soil layer (see Table A.4 – rotation of retaining wall away from and towards the

backfill soil layer).

Figure 5.25d shows the effect of relative density of the backfill soil layer on the

development of seismic earth pressure force P. It can be noted from Figure 5.25d that

the minimum seismic active earth pressure force (Pae = 38 kN/m) is developed when the

backfill soil layer is simulated with relatively loose density (Dr = 40%), while the

maximum seismic passive earth pressure force (Ppe = 110 kN/m) is developed when the

backfill soil layer is simulated with relatively stiff backfill soil density (Dr = 85%).


205

Additional observations have been predicted from the Table A.4 related to the seismic

earth pressure force. It can be noted from Table A.3 that the M-O method is

overestimated the seismic active and passive earth pressure forces for all relative

densities of backfill soil layer.

0 2 4 6 8-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 2 4 6 8-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 2 4 6 8-0.125

-0.100

-0.075

-0.050

-0.025

0.000

0 2 4 6 8-120

-100

-80

-60

-40

a(g

) (m

/sec

2)

Time, t (sec)a

(g)

(m/s

ec

2)

Time, t (sec)

(

)

Time, t (sec)

(d)

(b)

(c)

(a)

Dr=40% Dr=65% Dr=85% f = 2 Hz

P (

kN

/m)

Time, t (sec)

Figure 5.25: Effect of backfill soil relative density on the seismic response of wall-soil system

5.5.3.3 Effect of the foundation soil material (3rd

combination)

The pseudo-static and pseudo-dynamic methods, as well as the research methods

available in the literature, do not take into account the effect of relative density of

foundation soil on the seismic response of the wall-soil system. Limited studies have

been carried out to consider the effect of the stiffness of the foundation layer on the

seismic earth pressure. Therefore, in order to explore the effect of the relative density of

the foundation soil on the seismic response of the wall-soil system, a parametric study is

conducted. The soil materials are simulated by using the 3rd

procedure mentioned in

section 5.5.3. Results of the parametric study are presented in Table A.5 in Appendix

A. Figure 5.26 shows the effect of relative density of foundation soil on the


206

acceleration response at the top of the retaining wall (top_wall) and backfill soil

(top_wall), the relative horizontal displacement of the retaining wall W-F, and seismic

earth pressure force P.

Figure 5.26a, and b show the influence of the relative density of foundation layer on the

acceleration response at the top of the retaining wall and backfill soil layer, respectively.

It can be noted that the maximum rate of amplification (a(g)= 0.7) and phase lag of the

acceleration response at the top of retaining wall and backfill soil is that when the

foundation layer is simulated with the relatively loose material (Dr = 40%).

Figure 5.26c shows the effect of relative density of the foundation soil on the relative

horizontal displacement of the retaining wall W-F. It can be seen that the maximum

relative horizontal displacement of the retaining wall (/H = 0.0875) is predicted when

the foundation soil is also simulated with a soil of relatively low relative density

material (Dr = 40%). These results may be explained by the fact that the acceleration

response of the retaining wall is also highly affected by the acceleration response of

backfill soil; and the use of a relatively loose soil in the foundation causes higher

earthquake amplification effects both in the backfill soil and retaining wall, and larger

wall seismic inertia force. Another possible explanation is that the stiffness parameters

of the interface elements, which connect the base of the retaining wall with foundation

soil are highly influenced by the stiffness parameters of the foundation soil itself. This

could cause a reduction in the friction force between the base of the retaining wall and

foundation soil when the foundation soil is simulated by a soil of relatively low relative

density Dr = 40%), thereby causing prediction a higher relative horizontal displacement

between the retaining wall and foundation layer. Additional observations have been

predicted from the Table A.5. It can be noted from Table A.5 that the residual rotation

of the retaining wall is reduced from 1.439° to 0.276° when the relative density of

foundation layer is increased from 40% to 85%. Figure 5.26d shows the effect of the

relative density of the foundation layer on the development of seismic earth pressure

force P. It is interesting to indicate that the minimum seismic active earth pressure force

(Pae = 38 kN/m) and maximum seismic passive earth pressure force (Ppe = 128 kN/m) is

predicted when the foundation layer is simulated with relatively loose material (Dr =

40%). This can be understood by looking at the effect of the rate of amplification of the


207

acceleration response of the backfill soil and retaining wall when the foundation layer is

simulated with the relatively loose material (Dr = 40% - see Figures 5.28a and b). This

could cause the development of minimum seismic active earth pressure force Pae and

also maximum seismic passive earth pressure force Ppe.

Additional observations have been predicted from the Table A.5. It can be noted from

Table A.3 that the M-O method overestimates the seismic active and passive earth

pressure forces for all relative densities of backfill soil layer. The M-O method does not

take into account the effect of relative density of foundation layer, while the current

performance-based analysis has shown that the seismic active earth pressure force and

seismic passive earth pressure force is affected by the relative density of foundation

layer (see Figure 5.28d and Table A.5).

0 2 4 6 8-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 2 4 6 8-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8-0.100

-0.075

-0.050

-0.025

0.000

0 2 4 6 8-140

-120

-100

-80

-60

-40

-20

a(g

) (m

/sec

2)

Time, t (sec)

a(g

) (m

/sec

2)

Time, t (sec)

(d)(c)

(b)(a)

(

)

Time, t (sec)

Dr=40% Dr=65% Dr=85% f = 2 HzP

(kN

/m)

Time, t (sec)

Figure 5.26: Effect of foundation relative density on the seismic response of wall-soil system


208

5.6 SUMMARY

After validation of the FE model with experimental results obtained from centrifuge

tests available in the literature, this chapter presented a critical analysis of the seismic

response of a rigid retaining wall by using an innovative performance-based method.

The deformation mechanism of the wall-soil system has been discussed in detail. The

results have shown that the retaining wall, backfill soil layer, and foundation layer are

moving at the same time in the active and passive direction under the effect of seismic

loading. The results of the FE analysis indicated that the Newmark sliding block method

overestimates the relative horizontal displacement of the retaining wall. The results of

FE analysis have also proven the development of seismic active and passive earth

pressure state under the effect of seismic loading. A critical analysis of seismic earth

pressure force time history, obtained from the FE analysis, has shown that the M-O

method overestimates the seismic active earth pressure force. Further, the seismic earth

pressure force has been found to be highly affected by the seismic response of retaining

wall – a very unique contribution of this research, and is something which was not

addressed by the existing pseudo-static and pseudo-dynamic methods. A comprehensive

parametric study has been carried out in this study in order to produce a relationship

between the seismic earth pressure force and wall displacement. Unique design charts

have been developed to correlate the seismic earth pressure force and the displacement

of retaining wall by considering the effect of retaining wall height and earthquake

characteristics (amplitude and frequency content). It has been observed that the seismic

active earth pressure is not dependent on the wall displacement while, on the other hand,

the seismic passive earth pressure has been found to be highly influenced by the wall

displacement. The seismic active earth pressure force is also observed not to be

sensitive to the amplitude and frequency content of earthquake acceleration, while the

seismic passive earth pressure force is found to be highly influenced by both the

amplitude and frequency content of the earthquake acceleration. It has been noted that

the maximum seismic passive earth pressure force is exerted behind the retaining wall

when the ground earthquake acceleration is applied with minimum frequency content

and maximum amplitude. The relative horizontal displacement between a rigid retaining

wall and foundation layer is found to be highly affected by the earthquake characteristic.

The critical scenario of the relative horizontal displacement of the retaining wall is that


209

when earthquake acceleration is applied with minimum frequency content and

maximum amplitude, and it is highly affected by the duration of the earthquake in

contrast to what has been observed for the seismic earth pressure force. Generally, it

was found that no relationship between the frequency content of earthquake acceleration

to natural frequency of a rigid retaining wall-soil system was observed.

The effect of relative density of soil material on the seismic response of wall-soil

system has also been investigated in the present chapter.

Studying the seismic stability of a rigid retaining wall by using performance-based

method has shown that the retaining wall is sliding or rotating away from the backfill

soil layer under the effect of its seismic inertia force and static earth pressure only. It

has also been noted that according to special geometry of a rigid retaining wall, it can

resist the maximum seismic passive earth pressure force developed during a seismic

scenario. However, the case of maximum passive earth pressure force may be critical

for other types of retaining wall like a cantilever-type retaining wall. Chapter 6 of this

Thesis will present a critical analysis of the seismic behaviour of a cantilever retaining

wall by using the performance-based method in order to study the effect of the

development of seismic earth pressure and consequent effect on the stability of a

cantilever-type retaining wall.

Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall

210

CHAPTER 6

FINITE ELEMENT MODELLING AND ANALYSIS OF A

CANTILEVER RETAINING WALL

This chapter critically discusses the seismic performance of a cantilever-type retaining

wall. It begins with the description of the problem of cantilever retaining wall. Then, FE

method and material properties used in the current study are briefly discussed. After

that, a critical analysis of the results obtained from seismic analysis of a cantilever

retaining wall using the performance-based method is presented. Following that, a

parametric study is presented in order to draw a comprehensive understanding of the

structural integrity and global stability of a cantilever retaining wall during the seismic

scenario. The parametric study is set out in this chapter to investigate the effect of

earthquake characteristics (amplitude and frequency content), retaining wall height, and

relative density of backfill soil. This chapter ends with the summary highlighting the

new outcome of the current study.

6.1 PROBLEM DESCRIPTION

For a typical cantilever-type retaining wall, like the one shown in Figure 6.1a – which is

one of the most common types of retaining structures – the most important design load

that these walls need to be designed for comes from the earth pressure (static or

seismic). For design purposes a cantilever-type retaining wall is considered as a

flexible structure and a design must address the strength integrity and global stability,

arising because of the earth pressure. As shown in Figure 6.1b the earth pressure will

create a shear force, Nw, and bending moment, Mw, on the stem of the retaining wall and

also tend the base slab to slide relatively to the foundation layer and rotate about the toe,

thereby overturning the wall (Figure 6.1c). For structural integrity, the stem of the wall

should be designed to resist the shear force and bending moment; while for the global

stability, the wall should be designed to resist sliding and overturning.


211

Figure 6.1: a) Sketch of a cantilever retaining wall showing important locations of interest, b)

Development of shear force and bending moment in the stem, c) Sliding of base slab relatively

to foundation soil and rotation of the wall about its toe

6.1.1 Structural integrity

For assessing the seismic structural integrity of the retaining wall, the retaining wall will

be subjected to (Figure 6.2a): (1) the total seismic earth pressure force, coming from the

backfill soil, which is assumed to act behind the stem, and denoted as Pstem; and (2) the

wall seismic inertia force, Fwa and Fwp. For Pstem and Fwa, Fwp, the following needs to

be noted: (i) Depending upon the direction of the applied earthquake acceleration – i.e.,

whether it is acting towards the backfill soil or away from it, the wall seismic inertia

force will also either be acting towards the backfill soil, Fwp, or away from the backfill

soil, Fwa (Figure 6.2a), (ii) As the earthquake amplitude will vary with time, Pstem, Fwa

and Fwp will also be time-varying, and (iii) Pstem, Fwp and Fwa will produce shear force,

Nw, and bending moment, Mw, on the stem of the retaining wall. This chapter presents a

(a)

(b) (c)


212

methodology to: accurately predict Nw and Mw; estimate the relative contributions of

Pstem, Fwa and Fwp on Nw and Mw; and identify a critical case for the structural integrity

of the retaining wall which causes a maximum load case for the stem of the retaining

wall during an earthquake.

6.1.2 Global stability

For the seismic global stability analysis of the retaining wall, the wall is considered to

be subjected to (Figure 6.2b): (1) the total seismic earth pressure force, coming from the

backfill soil, which is assumed to act at a vertical virtual plane passing through the heel

of the wall, and denoted as Pvp; (2) the backfill seismic inertia force of the backfill soil

located above the base slab, Fsa and Fsp, and (3) the wall seismic inertia force, Fwa and

Fwp. Like Fwa and Fwp, depending upon the direction of the applied earthquake

acceleration, the backfill seismic inertia force will also either be acting towards the

backfill soil, Fsp, or away from the backfill soil, Fsa (Figure 6.2b). It is important to

highlight that seismic earth pressure, Pvp, is computed along the virtual plane because

the global stability of the cantilever-type retaining wall is maintained by the weight of

the backfill soil above the base slab in addition to the weight of the cantilever retaining

wall itself. This chapter presents a methodology to: predict the deformation mechanism

of the retaining wall so as to compute the relative horizontal displacement between its

base slab and foundation soil; estimate the contribution of Pvp to the abovementioned

relative horizontal displacement; and to identify a critical scenario with regards to the

global stability of the wall.

For both the structural integrity and global stability analyses, the effects of earthquake

characteristics (i.e., its amplitude and frequency content), natural frequency of the wall-

soil system, and relative density of soil material have been studied.


213

Figure 6.2: Forces acting on the cantilever retaining wall for: a) structural integrity analysis, and

b) global stability analysis

6.2 FE MODEL AND MATERIAL PROPERTIES

A FE model has been developed by using the PLAXIS2D software (Brinkgreve et al.

2016). To validate the FE model and compare the results, the dimensions of the

retaining wall model and as well as the material properties were chosen similar to the

one used by Jo et al. (2014). The FE model is shown in Figure 6.3, in which the

retaining wall has a height of 5.4 m, which sits on a 9 m thick foundation soil. The

confines of the model in the horizontal and vertical direction are large enough so as to

exclude the boundary effects. For the FE model the backfill and foundation soil are

modelled using 6-noded triangular elements while the retaining wall is modelled using

plate elements (Brinkgreve et al., 2016). The interaction between the cantilever-type

retaining wall and backfill soil was modelled by using the 6-noded interface elements,

available in the PLAXIS 2D library (Brinkgreve et al., 2016).

The soil and retaining wall properties and other parameters required to define the

HSsmall constitutive model and run the PLAXIS2D simulation for the proposed FE

were chosen the same to the parameters used in centrifuge test proposed by Jo et al.

(2014). Other stiffness parameters are computed by using empirical equation proposed

by Brinkgreve et al., (2010). Table 6.1 shows the parameters of backfill soil and

cantilever retaining wall required to run the FE model in the current study.

(a) (b)


214

Figure 6.3: FE model of the cantilever retaining wall

6.2.1 Seismic loading

For The above FE model is subjected to a seismic loading, which, for this study

comprises of a real acceleration-time history of the Loma Prieta (1989) earthquake,

having a peak ground acceleration of 0.264 g (see Figure 5.3a) and dominant

frequencies of 0.7 Hz and 2.5 Hz (see Figure 5.3b). The acceleration-time history is

applied at the base of the FE model (Figure 6.3). To investigate the effects of

earthquake amplitude and its frequency content on the seismic response of the wall-soil

system, a scaled uniform sinusoidal acceleration-time history is also chosen with 3

different amplitudes of 0.2 g, 0.4 g and 0.6 g, and scaled frequencies of 0.5 Hz, 2 Hz,

and 4 Hz.

9 m

2.6 m 40.25m

42.85 m

5.4 m

1.15 m16m

Backfill soil

Foundation soil


Stem

Base slab

6-noded triangularelement

Interfaceelement


215

Table 6.1: Parameters of soil and retaining wall used to run the FE model

Parameters Unit Value

Soil

γ kN/m3 14.23

' o

40

50

refE MPa 46.8

ref

OedE MPa 46.8

ref

urE MPa 140.4

o 10

ur - 0.2

y - 0.5

50

refG MPa 113

γ0.7 - 0.0002

pref

kN/m2 100

% 3

Rf - 0.9

Retaining wall

E MPa 68000

I m4 8.873 10

-4

v - 0.334

γ kN/m3 26.6

ξ % 3

6.3 SEISMIC ANALYSIS

After obtaining the results of the static analysis, the FE model is subjected to a seismic

analysis, which, as mentioned above, is carried out by applying a seismic loading in the

form of acceleration-time history at its base. Through the seismic analysis, apart from

obtaining Pstem, Pvp, Fwa, Fwp, Fsa, Fwp, Nw, and Mw, the acceleration and sliding response

of the retaining wall-soil system is also obtained.

It is important to highlight that the acceleration, horizontal displacement, and the wall

and backfill seismic inertia forces can have positive and negative senses, as discussed in

chapter 5-section 5.4. A positive horizontal displacement means that the retaining wall

moves towards the backfill soil, while a negative displacement means that the retaining

wall moves away from the backfill. Likewise, a positive horizontal seismic inertia force


216

of the retaining wall will act towards the backfill soil, while a negative horizontal

seismic inertia force of the retaining wall act away from the backfill soil (Figure 5.4).

6.3.1 Acceleration response of the retaining wall-soil system

The acceleration response of the retaining wall-soil system is shown in Figure 6.4 for

the time duration of 3 to 7 sec. This is the duration in which the intensity of the applied

earthquake acceleration is concentrated (Figure 5.3a) and hence it was chosen for

presenting the results of the analysis. From Figure 6.4, it is observed that the

acceleration response for the top of the stem and top of the backfill soil (points top_stem

and top_soil, respectively in Figure 6.1a) match each other. This implies that at the top

of the FE model, the stem of the wall and backfill soil move together, and can be said to

be in-phase. It is also observed that the acceleration of the top of the stem and backfill

soil (points top_stem and top_soil, respectively in Figure 6.1a) is higher than the

acceleration at the base of the retaining wall (point base_stem in Figure 6.1a), thereby

implying a possible amplification of acceleration towards the top of the FE model.

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Acce

lera

tio

n,

a(g

) (m

/se

c2)

Time, t (sec)

top_stem (Figure 6.1a)

base_stem (Figure 6.1a)

top_soil (Figure 6.1a)

base of the FE model

Figure 6.4: Acceleration response at different locations in the wall-soil system


217

6.3.2 Wall and backfill seismic inertia forces

To understand the possible cause of the acceleration amplification for the top of the

retaining wall and backfill soil, wall and backfill seismic inertia forces are estimated

by using the procedure as mentioned in Chapter 3 – section 3.13. As shown in

Figure 6.5 Fw and Fs are dependent upon the applied earthquake acceleration, including,

its amplitudes and frequency content. It can be noted from Figure 6.5 that at the time of

the maximum value of earthquake acceleration is applied towards the backfill soil (t =

3.9 sec – see Figure 5.3a), the maximum wall seismic inertia force (Fwa =20 kN/m) and

backfill seismic inertia force (Fsa = 54 kN/m) are acting in active direction. However,

when the maximum value of earthquake acceleration is applied away from the backfill

soil (t = 4.5 sec – see Figure 5.3a), the maximum wall seismic inertia force (Fwp = 34

kN/m) and backfill seismic inertia force soil (Fsp = 88 kN/m) are acting in passive

direction. It can be also noted that the backfill seismic inertia force (Fs) has higher

amplitude than of wall seismic inertia force (Fw) in both active and passive direction. It

is observed from Figure 6.5 that Fw and Fs are in-phase, which implies that the retaining

wall and backfill soil move as one entity. This finding will significantly affect the

development the active state in the wall-soil system when the earthquake acceleration

towards the backfill soil as it will be discussed in next section.

0 5 10 15 20 25 30

-60

-40

-20

0

20

40

60

80

100

Fwa

[@ t = 3.9 sec]

Fsa

[@t = 3.9 sec]

Fwp

[@ t = 4.5 sec]

Seis

mic

ine

rtia

fo

rce

, F

(kN

/m)

Time, t (sec)

Backfil seismic inertia force, Fs

Wall seismic inertia force, Fw

Fig. 9. Seismic inertia force of: Retaining wall (stem + base) FW ,

and backfill above the heel Fs

Fsp

[@ t = 4.5 sec]

Figure 6.5: Wall and backfill seismic inertia forces


218

6.3.3 Seismic earth pressure force

The seismic earth pressure force behind the stem Pstem and the total seismic earth

pressure force at the virtual plane Pab have been estimated by adopting the following

procedure as mentioned in Chapter 2 – section 3.13.3. Pstem and Pvp, and their variation

with time is shown in Figure 6.6.

6.3.3.1 Seismic earth pressure behind the stem Pstem

From Figure 6.6a, at the beginning of the seismic analysis (t = 0 sec), Pstem is about 53

kN/m, which is between the static active and at-rest earth pressure force; at time t = 3.9

sec, when the applied earthquake acceleration has a maximum value and is applied

towards the backfill soil, Pstem has a maximum value of 112 kN/m, while it attains a

minimum value of about 45 kN/m when the applied earthquake acceleration has a

maximum value but is applied away from the backfill soil at t = 4.5 sec. As discussed

above, the cantilever-type retaining walls are designed by considering the same

concepts as used for the design of rigid retaining walls; but the above-noted present

study results, which are in contrast with the observation of Nakamura (2006), who

observed that for a rigid retaining wall, Pstem is developed when the applied earthquake

acceleration is maximum but applied away from the backfill soil, show that Pstem is

maximum when the applied acceleration is applied towards the backfill soil. Thus, an

active state is not developed behind the stem despite the fact that the acceleration is

applied towards the backfill soil, and consequently the retaining wall moves away from

backfill soil. The present study observations are in contrast to what was observed for

the behaviour of a cantilever retaining wall modelled via a numerical model by Green et

al. (2008) and via an experimental work by Kloukinas et al. (2015) – both reported that

a maximum value of Pstem is the same to that is observed for a rigid retaining wall.

6.3.3.2 Seismic earth pressure behind the virtual plane Pvp

Figure 6.6b shows the variation of Pvp with time. It is observed that at the beginning of

the seismic analysis (t = 0 sec), Pvp is about 60 kN/m, which, like Pstem, is between the

static active and at-rest state earth pressure force; at time t = 3.9 sec, when the applied

earthquake acceleration has a maximum value and is applied towards the backfill soil,

Pvp has a minimum value of 61 kN/m, while it attains a maximum value of about 165


219

kN/m when the applied earthquake acceleration has a maximum value but is applied

away from the backfill soil at t = 4.5 sec.

0 5 10 15 20 25 30-120

-110

-100

-90

-80

-70

-60

-50

-40

0 5 10 15 20 25 30

-160

-140

-120

-100

-80

-60

Maximum Pstem [@ t= 3.9 sec]

Seis

mic

eart

h p

ressure

fo

rce,

Pste

m (

kN

/m)

Time, t (sec)

Minimum Pstem [@ t= 4.5 sec]

Maximum Pvp [@ t= 4.5 sec]

Minimum Pvp [@ t= 3.9 sec]

Seis

mic

eart

h p

ressure

fo

rce,

Pvp (k

N/m

)

Time, t (sec)

(b)

Fig. 10. Seismic earth pressure force predicted by finite element model: (a)behind

the stem Pstem ,(b) along virtual plane ab Pvp - (Fig. 1)

(a)

Figure 6.6: Seismic earth pressure force: a) behind the stem, Pstem, b) along the xx Pstem

These observations are similar to the observations of a rigid retaining wall as reported

by centrifuge test carried out by Nakamura (2006). Thus, it can be said that at time t =

3.9 sec, when the applied earthquake acceleration has a maximum value and is applied

towards the backfill soil, a maximum load case is developed behind the stem of the

wall, while a minimum load case is developed at the vertical virtual plane; and on the

other hand, at time t = 4.5 sec, when the applied earthquake acceleration has a

maximum value and is applied away from the backfill soil, a minimum load case is


220

developed behind the stem, while a maximum load case is developed at the vertical

virtual plane.

6.3.3.3 Comparison between Pstem and Pvp

A comparison between the seismic earth pressure force behind the stem Pstem and the

seismic earth pressure force at the vertical virtual plane Pvp reveals clearly points to the

fact that Pstem and Pvp peak and attain a minimum value at different times, thus

suggesting that there is a phase difference between these 2 quantities. Thus, for the

purpose of structural integrity and global stability, they have to be assessed individually.

It is also observed from Figure 6.6a and b, that at time t = 30 sec, i.e., at the end of the

seismic analysis, there is a residual Pstem of about 88 kN/m and Pvp of about 100 kN/m –

these residual seismic earth pressure forces could be because of the densification of

backfill soil during the earthquake.

6.3.3.4 Distribution of seismic earth pressure

Figure 6.7 shows the variation of the lateral earth pressure behind the stem pstem and

along the virtual plane pvp with the normalised height of the wall-soil system (z/H) at

after the static analysis and different durations during the seismic analysis. The

distribution of earth pressures are compared with the static active earth pressure

estimated by using the Rankine’s earth pressure theory, static earth pressure at-rest state,

and seismic active earth pressure estimated by using M-O method. As shown in

Figure 6.7a, the static earth pressures behind the stem pstem(static) and along the virtual

plane ab pvp(static) obtained from FE model are very close to the static active earth

pressure at the normalised height larger than z/H = 0.25. However, for the normalised

height lower than z/H = 0.25, the static earth pressure behind the stem pstem(static) and

along the virtual plane ab pvp is between active and at-rest state. Figure 6.7b shows the

distribution of seismic earth pressure behind the stem pstem and along virtual plane ab pvp

at time t = 3.9 sec when the maximum value of the earthquake acceleration is applied

towards the backfill soil (see Figure 5.3a). It can be noted from Figure 6.7b that the

distribution of both seismic earth pressure behind the stem pstem and along the virtual

plane ab pvp is close to the distribution of seismic earth pressure computed by M-O


221

theory and static earth pressure predicted in the at-rest state of the normalised height

larger than z/H = 0.5. However, for the normalised height between lowe than z/H = 0.5,

the distribution seismic earth pressure force behind the stem pstem is higher than

distribution that predicted by M-O theory and at rest state, while distribution of the

seismic earth pressure along the virtual plane ab pab is lower than the distribution of

earth pressure computed by M-O theory and at-rest state.

Figure 6.7c shows the distribution of seismic earth pressure behind the stem pstem and

along the virtual plane ab pvp when the maximum value of the earthquake acceleration is

applied away from the backfill soil (see Figure 5.3a). It can be observed from

Figure 6.7c that distribution of the seismic earth pressure behind the stem pstem is higher

than that computed by M-O theory and at-rest state for the normalised height higher

than z/H = 0.5, while it seems to be smaller than the that computed by M-O theory and

at-rest state for the normalised height lower than z/H = 0.5. However, the seismic earth

pressure along the virtual plane ab pvp is observed higher than the seismic earth pressure

predicted by M-O theory and at-rest state static earth pressure along the entire the

normalised height of wall-soil system (0<z/H<1).

Figure 6.7d shows the variation of the residual earth pressure behind the stem pstem and

along the virtual plane ab pvp at the end of seismic analysis (t = 30 sec). It can be noted

from Figure 6.7d that the distribution of the residual earth pressure behind the stem pstem

and along the virtual plane pvp is approximately close to the distribution of static earth

pressure in the at-rest state.

It can be indicated from Figure 6.7 that the distribution of static and seismic earth

pressure is nonlinear. It can also be indicated that the abovementioned observations

related to the seismic earth pressure behind the stem pstem are in contrast to the

observations of a rigid retaining wall discussed in Chapter 5-section 5.4.4. The

abovementioned observations will critically affect the evaluation of structural integrity

of a cantilever wall and identify the critical load case causing the maximum shear force

and bending moment at the entire height of the stem. However. The observations

related to the seismic earth pressure along the virtual plane pvp are similar to the

observations of a rigid retaining wall, which were discussed in Chapter 5-section 5.4.4.

It can be noted that these observations will also critically influence the assessment of


222

the global stability of a cantilever retaining wall, and estimation of the permanent

displacement of the wall-soil system.

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 60

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40


Norm

alis

ed

heig

ht (z

/H)

t = 0 sec t = 3.9 sec

(d)(c)

(b)


Norm

alis

ed

heig

ht (z

/H)

At-rest Rankine method M-O method

pstem

(current FE) pvp

(current FE)

(a)

t = 4.5 sec


Norm

alis

ed

heig

ht (z

/H)

t = 30 sec


Norm

alis

ed

heig

ht (z

/H)

Figure 6.7: Distribution of seismic earth pressures along the height of the wall-soil system: a)

Immediately before the seismic analysis at t = 0 sec, b) At t = 3.9 sec of earthquake acceleration,

c) At t = 4.5 sec of earthquake acceleration, d) At the end of seismic analysis (t = 30 sec)

6.3.4 Total seismic earth pressure force increments, Pstem, Pvp and wall and

backfill seismic inertia forces Fwa, Fwp, Fsa, Fsp

This section details the phase-difference between various forces acting on the retaining

wall under seismic conditions. in order to clearly understand the total seismic earth

pressure force, it is studied in terms of the total seismic earth pressure force increments,

Pstem and Pvp, respectively defined as: Pstem = Pstem – Pstem(static) and Pvp = Pvp –

Pvp(static), where Pstem(static) = total static earth pressure force acting at the stem, and


223

Pvp(static) = total static earth pressure force acting at vertical virtual plane.. Figure 6.8a, b,

c, and d show the variation of total seismic earth pressure force increments Pstem, Pvp

and wall and backfill seismic inertia forces Fwa, Fwp Fsa, Fsp for the top ⅓H and bottom

⅓H of the retaining wall. From Figure 6.8a and b it is observed that the total seismic

earth pressure force increment Pstem and wall seismic inertia force, Fwa, Fwp, for the top

⅓H of the wall are out of phase from each other, while for the bottom ⅓H of the

retaining wall, Pstem is in-phase with the wall seismic inertia forces. The reason for

this disparity could be because of the fact that the stem of the retaining wall is

monolithically fixed with the base slab, thereby not allowing any relative horizontal

displacement between the stem and the backfill soil. Similarly, from Figure 6.8c and d

it is observed that the total seismic earth pressure force increment Pvp and soil seismic

inertia forces Fsa, Fsp, do not act in-phase.

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0-5

-4

-3

-2

-1

0

1

2

3

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0-60

-50

-40

-30

-20

-10

0

10

20

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0-20

-15

-10

-5

0

5

10

15

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0-30

-20

-10

0

10

20

Pvp

[@ t = 3.9 sec]

Pvp

[@ t = 4.5 sec]

Pvp

[@ t = 4.5 sec]F

sa [@ t = 3.9 sec]

Fsa

[@ t = 3.9 sec] Fsp

[@ t = 4.5 sec]

Fsp

[@ t = 4.5 sec]

Pstem

[@ t = 3.9 sec]

Fwa

[@ t = 3.9 sec]

Pstem

[@ t = 4.5 sec]

Fwp

[@ t = 4.5 sec]

Pstem

[@ t = 3.9 sec]

Pstem

[@ t = 4.5 sec]

Fwa

[@ t = 3.9 sec]Fw

a,

Fw

p,

Pste

m (

kN

/m)

Time, t (sec)

Fwa

, Fwp

, Fsa

, Fsp

Pstem

, Pvp

Fwp

[@ t = 4.5 sec]

Pvp

[@ t = 3.9 sec]

Fsa,

Fsp,

Pvp (

kN

/m)

Time, t (sec)

Fw

a,

Fw

p,

Pste

m (

kN

/m)

Time, t (sec)

(d)

(c)

(b)

Along the virtual plane

Fsa,

Fsp,

Pvp (

kN

/m)

Time, t (sec)

Behind the stem

(a)

Fig. 11. Phase difference between : a) Pstem

and Fwa

, Fwp

for the top 1/3H of the wall, b) Pstem

and Fwa

, Fwp

for the bottom 1/3H of the wall, ) Pvp

and Fsa

, Fsp

for the top 1/3H of the wall, at the bottom of the wall, d) Pstem

and Fwa

, Fwp

for

the bottom 1/3H of the wall

Figure 6.8: Total seismic earth pressure force increments, Pstem, Pvp and wall and backfill

seismic inertia forces Fwa, Fwp, Fsa, Fsp


224

6.3.5 Shear force Nw and bending moment Mw

Figure 6.9a and b respectively show the shear force Nw- and bending moment Mw-time

history predicted at the base of the stem (point base_stem –Figure 6.1a) of the retaining

wall. Studying Figure 6.9a and b in conjunction with Figure 6.6, it can be noted that the

shear force Nw and bending moment Mw time histories have the same trend as the

seismic earth pressure force behind the stem Pstem time history (Figure 6.6). The

maximum shear force Nw of about 120 kN/m and bending moment Mw 220 kN.m/m are

both predicted at t = 3.9 sec which corresponds with the time when the earthquake

acceleration has its maximum value and is acting towards the backfill soil (Figure 5.3a).

0 5 10 15 20 25 30-140

-120

-100

-80

-60

-40

-20

0 5 10 15 20 25 30-240

-220

-200

-180

-160

-140

-120

-100

-80

-60

Maximum Mw [@ t = 3.9 sec]

Maximum Nw [@ t = 3.9 sec]

Minimum Mw [@ t = 4.5 sec]

Sh

ea

r fo

rce

, N

w (

kN

/m)

Time, t (sec)

Minimum Nw [@ t = 4.5 sec]

(b)

Be

nd

ing

mo

me

nt,

Mw (

kN

.m/m

)

Time, t (sec)

(a)

Fig. 12. a) shear force Nw, and b) bending moment M

w, (Location j - Fig. 1(a))

Figure 6.9: Variation of a) Shear force, b) Bending moment at the base of the stem


225

On the other hand, the minimum shear force Nw of about 25 kN/m and bending moment

Mw of about 64 kN.m/m are predicted at time t = 4.5 sec when the earthquake

acceleration is applied away from the backfill soil and has its maximum value.

Also, it is observed that at the end of the end of earthquake (i.e., at time t > 30 sec),

there are residual shear force and bending moment. From the above, it can be argued

that the critical case for the structural integrity of a cantilever retaining wall is when the

maximum acceleration is applied towards the backfill soil.

6.3.6 Relative horizontal displacement of the wall and backfill soil with respect to

the foundation soil

This section critically discusses the deformation mechanism of the cantilever retaining

wall-soil system. Different deformation patterns are presented in order to understand

behaviour of a cantilever retaining wall under the effect of seismic loading. The total

horizontal displacement is predicted at the different locations in the wall-soil system.

The relative horizontal displacement between the cantilever retaining wall and

foundation, the rotation of the stem and footing slab are also predicted.

6.3.6.1 Total displacement response

Figure 6.10 shows the horizontal displacement predictions for different locations in the

wall-soil system:

Top of retaining wall (top_wall – Figure 6.1a);

Top of backfill soil (top_soil – Figure 6.1a);

Bottom of retaining wall (base_stem – Figure 6.1a); and

At a point 0.5 m in the foundation soil below the base of the retaining wall (point P1

– Figure 6.1a).

It is noted from Figure 6.10 that the retaining wall, backfill soil, and foundation layer

move together towards the backfill soil about (0.03m) at time t = 3.3 sec. After that,

when the maximum value of the warthquake acceleration (positive) is applied towards

the backfill soil at time t = 3.9 sec (see Figure 5.3a), the retaining wall, backfill soil, and

foundation move maximum displacement (negative) together away from the backfill

soil but at different amplitudes (see Figure 6.10). When the maximum value of


226

earthquake acceleration (negative) is applied away from the backfill soil as shown in

Figure 5.3a at time t = 4.5 sec, the retaining wall, backfill soil, and foundation soil also

move together towards the backfill soil but also at different amplitudes. After time t =

4.5 sec, the retaining wall, backfill, and foundation layer keep on moving towards and

away from the backfill, until about time t = 30 sec – a time at which the seismic input

acceleration almost diminishes to zero (Figure 5.3a).

0 5 10 15 20 25 30

-0.15

-0.10

-0.05

0.00

0.05

Ho

rizo

nta

l d

isp

lace

me

nt (m

)

Time, t (sec)

top_stem (Figure 6.1a)

top_soil (Figure 6.1a)

base_stem (Figure 6.1a)

point P1 (Figure 6.1a)

Figure 6.10: Horizontal displacement at different locations in the wall-soil system

These observations could help to understand the deformation mechanism of the wall-

soil system under the effect of seismic loading and the development of seismic earth

pressures. It can be indicated that the total displacement response at foundation layer

represents the ground displacement response. However, the total displacement response

at the top of retaining wall represents: 1) the retaining wall displacement response

because body displacement response; 2) the sliding of retaining wall relatively to

foundation layer; 3) the elastic deflection of stem because of the increment of bending

moment; 4) the rotation of footing slab about the toe of retaining wall. Hence, next

section will present a critical analysis of abovementioned components of displacement

response of the wall-soil system.


227

6.3.6.2 Relative horizontal displacement of the wall and backfill soil with respect to

the foundation soil

For the wall and backfill soil, relative horizontal sliding displacement profiles were

constructed for the following pairs: (1) base of the stem (base_stem – Figure 6.1a) and

foundation soil (a point 0.5 m below the base of the wall (point P2 – Figure 6.1a), and

(2) centre of gravity of the backfill soil (point s_CG – Figure 6.1a) and foundation soil

(a point 0.5 m below the base of the wall (point P2 – Figure 6.1a), and these are shown

in Figure 6.11. It is observed from Figure 6.11 that a maximum relative horizontal

sliding displacement (of about 0.035 m) between the stem and foundation is for t = 3.9

sec, which is the same time at which Pvp is minimum; similarly, the relative horizontal

sliding displacement between the backfill soil and foundation soil also achieves its

maximum value of about 0.025 m for the t = 3.9 sec, and remains constant until the end

of the seismic analysis. Thus, from the above 2 observations, it can be said that the

retaining wall and backfill soil move as a single entity.

The predicted relative horizontal sliding displacement of the wall-soil system from the

present FE analysis is compared with relative horizontal displacement computed by

using conventional Newmark sliding block method. The first step in Newmark sliding

block method is that estimation of the yield acceleration. The yield acceleration can be

computed by using a pseudo-static analysis. The equilibrium of horizontal forces are

acting in wall-soil system at time of sliding of retaining wall can be given by

( ) cos sin tany W S ae W S ae bk W W P W W P (6.1)

where, yk = yield acceleration coefficient, WW = weight of the cantilever retaining wall,

WS = weight of backfill soil above footing slab aeP = seismic earth pressure force and

can be determined by the M-O method along the virtual plane, = friction angle

between the wall and backfill soil, and b = friction angle between the base of the wall

and foundation layer.

A computer program, written in MATLAB has been developed to perform numerical

integration of relative acceleration of the wall-soil, computed by:


228

( ) ( ) ( )r yk g a g k g (6.2)

where ( )a g = acceleration response of the wall-soil system, predicted by the current FE

model at the middle of the backfill soil above the footing slab (point s_CG – Figure

6.1a). Figure 6.11 shows the comparison between the present FE MODEL and

Newmark sliding block method results. It can be noted that the Newmark sliding block

method overestimates the relative horizontal sliding displacement. Possible explanations

for overestimation of relative horizontal sliding displacement of the wall-soil system by

Newmark sliding block analysis are :

0 5 10 15 20 25 30-0.020

-0.015

-0.010

-0.005

0.000

Cantilever retaining wall, W-F

Backfill soil above footing slab, S-F

Newmark sliding block method

Rela

tive d

ispla

cem

ent

(/H

)

Time, t (sec)

Figure 6.11: Relative horizontal displacement between the wall and foundation soil as well as

between the backfill soil above base slab and foundation soil

The seismic earth pressure force is computed in the Newmark sliding block

method by using the pseudo-static method. The results obtained from current the

FE model show that the pseudo-static method overestimates the seismic earth

pressure force behind the virtual plane when the wall-soil system moves away

from the backfill.

Newmark sliding block method does not account for the deformation of the

foundation soil during the earthquake, so the retaining wall rotates about its toe


229

because the deformation of foundation soil is causing additional sliding

resistance.

Newmark sliding block method does not account for the relative horizontal

sliding displacement towards the backfill soil (t = 4.5 sec – see Figure 6.11)

when the earthquake acceleration is applied way from the backfill soil, which

could cause overestimation of the relative displacement computed by this

method.

6.3.6.3 Rotation of stem

The rotation of the stem is the other deformation mechanism, which needs to be

considered in the FE modelling. The stem rotation has been computed about a vertical

axis, passing through the centre-line of the stem, by using :

1tantop stem base stem

stemH

(6.3)

where, θstem = rotation of the stem , top stem = total horizontal displacement at the top of

the stem, base stem = total horizontal displacement at the bottom of the stem. Figure 6.12

shows the rotation of the stem of the cantilever retaining wall . It is observed that the

stem gets rotated by a maximum amount away from the backfill soil when the applied

earthquake acceleration has its maximum value and is applied towards the backfill soil

(t = 3.8 sec – see Figure 5.3a) in which the maximum seismic earth pressure force Pstem,

shear force Nw, and bending moment Mw are predicted as shown in Figures 6.6, 6.9a and

b, respectively. However, when the maximum value of earthquake acceleration is

applied away from the backfill soil at time t = 4.5 sec, the stem rotates: a minimum

amount away from the backfill soil relatively to its orginal position at the beginning of

seismic analysis (t = 0 sec). It is also observed that the stem experiences permanent

rotation away from the backfill soil at the end of the seismic analysis t = 30 sec.

It is important to point out that the total rotation of the stem about the vertical axis is

accumulated from two sources: 1) the elastic deflection of the stem because of the

development of a maximum value of bending moment, and 2) the rotation of cantilever

retaining wall about the toe as a rigid body because of foundation deformability.


230

0 5 10 15 20 25 30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

Ro

tatio

n o

f ste

m (

de

gre

e)

Time, t (sec)

Figure 6.12: Rotation of the stem

6.3.6.4 Rotation of the base slab

In order to clearly understand the deformation mechanism of a cantilever retaining wall

during the applied seismic loading, the rotation of the base slab about its toe and into the

foundation soil is computed by:

( ) ( )1tan

toe y heel y

slabb

(6.4)

where, θslab = rotation of the base slab about the toe, ( )toe y = total vertical displacement

at the toe predicted from the FE model, ( )heel y = total vertical displacement response at

the heel predicted from the FE model, and b = width of base the slab. The main

assumption used in Equation 6.4 is that the base slab deforms as a rigid body. Figure

6.13 shows the rotation of the base slab of the cantilever retaining about the toe.

It is observed that at the end of the static analysis (t = 0 sec) , the base slab rotates away

from the foundation soil (Figure 6.13). The base slab rotates by a maximum amount

about the toe towards the foundation layer (0.065° + 0.01° = 0.075°), as shown in

Figure 6.13) when the maximum value of earthquake acceleration is applied towards the

backfill soil (t = 3.9 sec – see Figure 5.3a). However, when the maximum value of

earthquake acceleration is applied away from the backfill soil at time t = 4.5 sec, the


231

base slab rotates about 0.035° away from the foundation layer relatively to the point of

maximum rotation predicted at time t = 3.8 sec. It is also observed that the base slab

experiences residual rotation about the toe towards the foundation soil at the end of

seismic analysis t = 30 sec, and this is in contrast to trend at the beginning of seismic

analysis (t = 0 sec) where the footing slab rotates away from the foundation soil.

0 5 10 15 20 25 30-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08R

ota

tio

n o

f fo

otin

g s

lab

(d

eg

ree

)

Time, t (sec)

Figure 6.13: Rotation of base slab about the toe

6.3.6.5 Deformation shape of a cantilever retaining wall

Figure 6.14 shows the deformation shape of the retaining wall at time t = 0 sec (i.e.,

start of the seismic analysis), t = 3.9 sec, 4.5 sec and t = 30 sec (i.e., at the end of the

seismic analysis). It is important to highlight that the deformation shape of the stem and

base slab shown in Figure 6.14 is measured relatively to its original position. Figure

6.14a shows that at the start of the seismic analysis (time t = 0 sec), the stem has a

rotation of 0.02° away from the backfill soil while the base slab heel has a rotation of

0.065° in to the foundation soil – thereby suggesting that the stem and base slab were

rotating in opposite directions to each other. However, at time t = 3.9 sec when the

earthquake acceleration has its maximum value and is applied towards the backfill soil,

the stem rotates by 0.217° away from the backfill soil, while the base slab toe rotates by

0.014° in to the foundation soil as shown in Figure 6.14b – thereby suggesting that both

the stem and base slab rotate in the same direction. Also, the retaining wall slides as a

rigid body away from the backfill soil by about 0.025 m. At time t = 4.5 sec, when the


232

earthquake acceleration has its maximum value and is applied away from the backfill

soil, the stem rotates back towards the backfill but is still away from its original position

by about 0.017° while the base slab toe rotates back but is still having rotation in to the

foundation soil by 0.01° as compared with its original position. The retaining wall slides

towards the backfill soil; however, it is still away from the backfill soil by about 0.017m

as compared with its original positon, as shown in Figure 6.14c. At the end of the

seismic analysis at time t = 30 sec, the stem has a permanent rotation of 0.204° relative

to its original position and the base slab toe has a permanent rotation of 0.038° in to the

foundation soil as well as the retaining wall has a residual sliding away from the backfill

soil of about 0.035 m at the end of the seismic analysis, as shown in Figure 6.14d.

Figure 6.14: Deformation shapes of the stem and base slab at different durations during the

earthquake

(a): at t = 0 sec (b): at t = 3.9 sec

(c): at t = 4.5 sec (d): at t = 30 sec

The stem has a rotation of

0.02° away from the

backfill soil while the base

slab heel has a rotation of

0.065° in to the foundation

soil.

The stem rotates by 0.217°

away from the backfill soil,

while the base slab toe

rotates by 0.014° in to the

foundation soil. Also, the

retaining wall slides away

from the backfill soil about

0.025 m.








0.017 m.








0.035 m.


233

6.4 PARAMETRIC STUDY

The current parametric study is conducted in order to investigate the effect of

earthquake characteristics, the natural frequency of the cantilever retaining wall, and the

relative density of soil on the structural integrity and global stability of the cantilever

retaining wall during the seismic scenario.

6.4.1 Effect of earthquake characteristics

To investigate the effect of earthquake characteristics (amplitude and frequency content)

on the seismic performance of the cantilever retaining wall, a variety of earthquake

accelerations are applied at the base of FE model. Three groups of ground motions are

applied at the base of the FE model with frequency content 0.5Hz, 2Hz, and 4Hz

respectively. The amplitude of earthquake acceleration in each group is simulated by

0.2g, 0.4g, and 0.6g. However, the subsections will discuss the effect of earthquake

characteristics on the acceleration response of wall-soil system, seismic earth pressure

forces, shear force, bending moment, and relative horizontal displacement of the

cantilever retaining wall.


The acceleration response of the retaining wall-soil system, when it is subjected to a

uniform sinusoidal acceleration-time history of different amplitudes and frequency

contents is shown in Figure 6.15. From Figure 6.15a it is observed that when the

amplitude of the applied earthquake acceleration is 0.2 g with frequency content of 0.5

Hz, the amplitude of the acceleration response for both the top of the retaining wall and

backfill soil matches with the amplitude of the applied earthquake acceleration itself.

However, when the frequency content of the applied earthquake acceleration is

increased 4 times to 2 Hz, while the amplitude of the applied acceleration is kept same

as 0.2 g, the amplitude of the acceleration response for the top of the retaining wall and

backfill soil amplifies to a value close to 0.4 g as shown in Figure 6.15b. On a further

increase of the frequency content to 4 Hz, with the amplitude of the applied acceleration

remaining same as 0.2 g, the amplitude of the acceleration response for the top of the

retaining wall is much higher than that for the backfill soil (Figure 6.15c). Similarly,

from Figure 6.15d and g it is observed that for an applied acceleration with 0.5 Hz


234

frequency, the amplitudes of acceleration for the top of the retaining wall and backfill

soil have the same amplitude as the applied acceleration amplitudes of 0.4 g and 0.6 g,

respectively. However, for an applied earthquake acceleration with a frequency content

of 2 Hz, as shown in Figure 6.15e and h, the amplitude of acceleration for the top of the

retaining wall is higher than the top of the backfill soil. When the frequency content of

the applied acceleration is further increased to 4 Hz for applied acceleration amplitudes

of 0.4 g and 0.6 g, as shown in Figure 6.15f and j, respectively, it is observed that the

amplitude of acceleration for the retaining wall amplifies to a maximum value 1.8 g

(i.e., it becomes more than the amplitude of the applied acceleration). On the other

hand, the amplitude of acceleration for the top of the backfill soil seems to deamplify

and its maximum value becomes less than the amplitude of the applied earthquake

acceleration. This behaviour – of acceleration amplification for the top and a de-

amplification for the bottom of the FE model – possibly reflect a non-linear soil

behaviour which de-amplifies a strong earthquake, resulting in a higher dissipation of

the seismic energy. It is to be noted that similar de-amplification behaviour of soil for

strong earthquakes was also reported by Griffiths et al. (2016) and Stamati et al. (2016).

However, for an input acceleration frequency of 2 Hz – Figure 6.15e and h – the

acceleration response for the top of the retaining wall is higher than the response

predicted for the top of the backfill soil. When the frequency content of the input

acceleration is further increased to 4 Hz for input amplitudes of 0.4 g and 0.6 g -

Figure 6.15f and j, it is observed that the retaining wall response amplifies to a

maximum value (1.8 g). However, it can be observed that the acceleration response at

the top of the backfill soil seems to de-amplify less than the amplitude of input

acceleration. This behaviour can reflect the nonlinear site characteristics of soil material

which de-amplifies strong earthquake. For a strong earthquake, the highly nonlinear

behaviour of backfill soil leads to higher dissipation of the seismic energy. It is to be

noted that a similar de-amplification for strong earthquake has also been reported by

Griffiths et al. (2016) and Stamati et al. (2016).


235

0 5 10 15 20 25 30-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0 1 2 3 4 5 6 7 8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0 1 2 3 4-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0 1 2 3 4 5 6 7 8-1.2

-0.9

-0.6

-0.3

0.0

0.3

0.6

0.9

0 1 2 3 4-1.5

-1.0

-0.5

0.0

0.5

1.0

0 5 10 15 20 25 30-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 1 2 3 4 5 6 7 8-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

0 1 2 3 4-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

(j)

(h)

(g)

(f)

(e)

(d)

(c)

(b)

Wall (top_stem) Backfill (top_soil)

a(g

)

Time, t (sec)

(a) f =0.5 Hz, a(g)=0.2

(+) sign:Towards the backfill (-) sign: Away from backfill

f =2 Hz, a(g)=0.2

a(g

)

Time, t (sec)

f =4 Hz, a(g)=0.2

a(g

)

Time, t (sec)

f =0.5 Hz, a(g)=0.4

a(g

)

Time, t (sec)

f =2 Hz, a(g)=0.4

a(g

)

Time, t (sec)

f =4 Hz, a(g)=0.4

a(g

)

Time, t (sec)

f =0.5 Hz, a(g)=0.6

a(g

)

Time, t (sec)

f =2 Hz, a(g)=0.6

a(g

)

Time, t (sec)

f =4 Hz, a(g)=0.6

a(g

)

Time, t (sec)




Figure 6.16(a, b, c) show the effect of varying earthquake amplitude and frequency

content on Pstem; while the same for Pvp is shown in Figure 6.16 (d, e, f). From

Figure 6.16 (a - f), it is observed that for all amplitudes and frequencies of the applied

earthquake, Pstem is maximum when the earthquake acceleration was applied towards

the backfill soil while Pstem is minimum when the earthquake acceleration was applied

away from the backfill soil; on the other hand, Pvp is maximum when the earthquake

acceleration was applied away from the backfill soil, and Pvp is minimum when the

earthquake acceleration was applied towards the backfill soil. It is further observed that

Pstem and Pvp increase signifcantly when the amplitude of the applied earthquake is

increased from 0.2 g to 0.4 g, while on further increasing the amplitude to 0.6 g, Pstem

and Pvp do not change with the same proportion as before – again indicating a possible


236

de-amplification of the acceleration response of the backfill soil for a strong earthquake.

Both Pstem and Pvp are not significantly sensitive to the number of acceleration cycles.

On the other hand, Pstem is highly sensitive to the natural frequency of the wall, while

Pvp does appear to be significantly affected by the natural frequency of the wall.

0 5 10 15 20 25 30

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 5 10 15 20 25 30

-160

-140

-120

-100

-80

-60

-40

-20

0 5 10 15 20 25 30-180

-160

-140

-120

-100

-80

-60

-40

0 1 2 3 4 5 6 7 8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 1 2 3 4 5 6 7 8

-160

-140

-120

-100

-80

-60

-40

-20

0

0 1 2 3 4 5 6 7 8-240

-220

-200

-180

-160

-140

-120

-100

-80

-60

-40

0 1 2 3 4

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 1 2 3 4-220

-200

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

0 1 2 3 4-180

-160

-140

-120

-100

-80

-60

-40

f = 0.5 Hz

a(g

) (m

/se

c2)

Time, t (sec)

a=0.2(g) a=0.4(g) a=0.6(g) + a(g):Towards the backfill - a(g): Away from the backfill

(f)

(e)

(d)

(c)

(b)

(a)

(III)

(II)

f = 0.5 Hz

(I)

Fig. 10. Seismic earth pressure at:behind the stem Pstem

and along virtual plane ab

P vp

for different amplitudes and frequency contents of input motion

Pste

m (

kN

/m)

Time, t (sec)

P v

p (

kN

/m)

Time, t (sec)

f = 0.5 Hz

f = 2 Hz

a(g

) (m

/se

c2)

Time, t (sec)

f = 2 Hz

Pste

m (

kN

/m)

Time, t (sec)

f = 2 Hz

P v

p (

kN

/m)

Time, t (sec)

f = 4 Hz

a(g

) (m

/se

c2)

Time, t (sec)

f = 4 Hz

Pste

m (

kN

/m)

Time, t (sec)

f = 4 Hz

Seismic earth pressure force,

Pvp

(kN/m)

Seismic earth pressure force,

Pstem

(kN/m)

P v

p (

kN

/m)

Time, t (sec)

Earthquake acceleration,

a (g) (m/sec2)

Figure 6.16: Seismic earth pressure force behind the stem and along the virtual line for different


6.4.1.3 Shear force and bending moment

Figure 6.17(a, b, c) and Figure 6.17(d, e, f) show the effect of varying earthquake

amplitude and frequency content on Nw and Mw. It is observed that Nw and Mw show the

same trends as were observed for the Pstem as discussed in previous section, and also,

they are highly sensitive to the ampltiude of the applied earthquake when its value is

between 0.2 g - 0.4 g. For values of ampltidue of applied earthquake more than 0.4 g,

Nw and Mw do not remain as sensitive as before – again concreting the fact that de-

amplification effects creep in for strong earthquakes. It can also be noted that Nw and


237

Mw, like Pstem and Pvp, are not senstive to the number of acceleration cycles where the

maximum vlaues of shear force and bending moment are still having the same rate with

increasing of acceleration cycles.

0 5 10 15 20 25 30

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 5 10 15 20 25 30

-160

-140

-120

-100

-80

-60

-40

-20

0 5 10 15 20 25 30-280

-260

-240

-220

-200

-180

-160

-140

-120

-100

-80

0 1 2 3 4 5 6 7 8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 1 2 3 4 5 6 7 8

-160

-140

-120

-100

-80

-60

-40

-20

0

0 1 2 3 4 5 6 7 8-300

-250

-200

-150

-100

-50

0

0 1 2 3 4

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 1 2 3 4-220

-200

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

0 1 2 3 4-350

-300

-250

-200

-150

-100

-50


f = 0.5 Hz f = 0.5 Hz

Bending moment, Mw (kN.m/m)Shear force, N

w (kN/m) Earthquake acceleration, a (g) (m/sec

2)

a(g

) (m

/sec

2)

Time, t (sec)

a= 0.2(g) a= 0.4(g) a= 0.6(g)

f = 2 Hz

Nw (

kN

/m)

Time, t (sec)

Fig. 13. Shear force and bending moment at the base of the stem for different amplitudes and

frequency contents: (a, b,c) shear force, and (d, e, f) bending moment

(f)

(e)

(d)

(c)

(b)

(a)

(III)

(II)

(I) f = 0.5 Hz

Mw (

kN

.m/m

)

Time, t (sec)

f = 2 Hzf = 2 Hz

a(g

) (m

/sec

2)

Time, t (sec)

Nw (

kN

/m)

Time, t (sec)

f = 4 Hzf = 4 Hz

Mw

(kN

.m/m

)

Time, t (sec)

f = 0.5 Hz

a(g

) (m

/sec

2)

Time, t (sec)

Nw (

kN

/m)

Time, t (sec)

Mw (

kN

.m/m

)

Time, t (sec)

Figure 6.17: Shear force and bending moment at the base of the stem for different amplitudes

and frequency content of the applied earthquake acceleration

6.4.1.4 Relative horizontal displacement

Figure 6.18 shows the effect of amplitude of the applied earthquake and its frequency

content on the horizontal sliding displacement of the wall-soil system. As the amplitude

of the applied earthquake acceleration increases from 0.2 g to 0.6 g, horizontal sliding

displacement of the wall-soil system increases, while with an increase in the frequency

content of the applied earthquake acceleration from 0.5 Hz to 4 Hz, the relative

horizontal displacement of the wall-soil system reduces. This is in contrast to what has

been observed for the bending moment and shear force, which, as described in the

preceding sections, attain maximum values when both the frequency content and the


238

amplitude of the applied earthquake are maximum (see Figure 6.17). It is also

interesting to note that the wall-soil system slides by about 0.2 m for an applied

earthquake ampltiude of 0.6 g and a frequency content of 4 Hz (Figure 6.18c), while the

wall-soil system slides by about 0.25 m for for an applied earthquake ampltiude of 0.4 g

and a frequency content of 2 Hz (Figure 6.18b), thereby suggesting that the frequency

content of the applied earthquake is a more dominating factor than its amplitude

contributing to the sliding displacement of the retaining wall.

0 2 4 6 8-0.125

-0.100

-0.075

-0.050

-0.025

0.000

0 1 2 3 4

-0.04

-0.03

-0.02

-0.01

0.00

0 5 10 15 20 25 30

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 1 2 3 4 5 6 7 8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 1 2 3 4

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 5 10 15 20 25 30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

f= 0.5 Hz

f= 2 Hz

W

-F (

/H)

Time, t (sec)

f= 4 Hz

W

-F (

/H)

Time, t (sec)

(a)

(III)

(II)

a(g

)

Time, t (sec)

(c)

(b)

f= 2 Hz

a(g

)

Time, t (sec)

f= 4 Hz

a(g

)

Time, t (sec)

0.2g 0.4g 0.6g

f= 0.5 Hz

(I)

W

-F (

/H)

Time, t (sec)

Relative displacement, W-F

(/H)Earthquake acceleration, a(g) (m/sec2)

Figure 6.18: Relative horizontal displacement of the cantilever retaining wall for different



239

From the above discussion, it can be argued that for a global stability of a cantilever

retaining wall a low frequency content of the applied earthquake causes a critical case

scenario, while for the structural integrity of the cantilever retaining wall a high

frequency content of applied earthquake creates a critical case scenario. The results also

show that the sliding of the wall-soil system is highly sensitive to the number of

acceleration cycles (duration of the applied earthquake acceleration), which is in

contrast to what has been observed for the structural integrity where the bending

moment and shear force are not sensitive to the acceleration cycles.

6.4.2 Effect of the natural frequency a cantilever retaining wall (height)

In order to study the influence of the height and thereby the natural frequency of the

retaining wall on the structural integrity and global stability under seismic loading, a

cantilever-type retaining wall of a different height of H = 10.8 m is also analysed by

using the abovementioned FE model. The new model has been prepared, and the

dimension of a new cantilever wall (H= 10.8 m) and material properties were selected

similar to the one used by Jo et al. (2014) for their centrifuge tests (see Table 6.1). In the

current analysis, the height of the retaining wall (H – Figure 6.1a) is equal to 10.8 m,

and the footing slab width (b – Figure 6.1a) is assumed equal to 7.35 m. The seismic

loading is applied at the base of the FE model by using a sinusoidal input motion with

amplitude equal to 0.4g, and with three different frequency contents (0.5Hz, 2Hz, and

4Hz) in order to evaluate the effect of the ratio between the natural frequency of a

cantilever retaining wall and frequency content of earthquake acceleration on the

structural integrity and global stability. The comparison is conducted between the

results of a new cantilever wall (H= 10.8 m) and previous cantilever wall (H= 5.4 m) for

the accelration response, seismic earth pressure, shear force and bending moment, and

relative displacement.


The acceleration response is predicted at the top of retaining wall and at the top of the

backfill soil for 2 retaining wall heights (H = 5.4 m and H= 10.8 m) considering a

variety of frequency contents of earthquake acceleration. It can be indicated from the

Figure 6.19a and b that the acceleration response at the top of the retaining wall is de-


240

amplified (a(g) = 0.3) when the earthquake acceleration is applied with frequency

content of 0.5 Hz for both retaining wall height models H= 5.4 m and H= 10.8 m.

However, when the earthquake acceleration is applied with a frequency content of 2 Hz,

the acceleration response at the top of retaining wall is amplified to a(g) = 1), for H= 5.4

m (Figure 6.19c) and a(g) = 1.2, for H = 10. 8 m (Figure 6.19d). It can be noted that the

maximum amplification is observed at the top of retaining the wall (H= 10.8 m) (see

Figure 6.19d) when the earthquake acceleration is applied with a frequency content of

2Hz because the natural frequency of the retaining the wall (H= 10.8 m - fwn= 2.15 Hz)

is close to the frequency content of the ground input motion (2 Hz). When the frequency

content of earthquake acceleration is applied with (4 Hz), it can be observed that the

acceleration response at the top of retaining wall (H= 5.4 m) is (a(g) = 1.5) – see Figure

6.19e) while the acceleration response at the top of retaining wall (H=10.8 m) is (a(g) =

0.6) – see Figure 6.19f). It can also be noted that the maximum amplification at the top

of retaining wall is that when the retaining wall is modelled with H = 5.4 m (see Figure

6.19e) because the natural frequency of retaining wall (H= 5.4 m - fwn= 6 Hz) is close to

the frequency content of ground input motion (4 Hz). Hence, it can be said that the

acceleration response of the retaining wall is highly influenced by the natural frequency

of the retaining wall in addition to the frequency content of the earthquake acceleration.

It can also be noted that the acceleration response at the top of backfill soil is highly

affected by the acceleration response of retaining wall as shown in Figure 6.19a, b, c, d,

e, and f. For example, the acceleration response at the top of backfill soil is amplified to

a maximum value when the natural frequency of retaining wall is close to the frequency

content of input motion as shown in Figure 6.19d and e.


241

0 5 10 15 20 25 30-0.6

-0.4

-0.2

0.0

0.2

0.4

0 5 10 15 20 25 30-0.6

-0.4

-0.2

0.0

0.2

0.4

0 1 2 3 4 5 6 7 8-1.2

-0.8

-0.4

0.0

0.4

0.8

0 1 2 3 4 5 6 7 8

-1.5

-1.0

-0.5

0.0

0.5

1.0

0 1 2 3 4

-1.5

-1.0

-0.5

0.0

0.5

1.0

0 1 2 3 4-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

(b)

(e)

(c)

(a)

a(g) = 0.4 top_stem top_ soil

a(g

) (m

/sec

2)

Time, t (sec)

f = 0.5 Hz f = 0.5 Hza(g

) (m

/sec

2)

Time, t (sec)

f = 2 Hza(g

) (m

/sec

2)

Time, t (sec)

(f)

(d)

f = 2 Hz

a(g

) (m

/sec

2)

Time, t (sec)

f = 4 Hz

a(g

) (m

/sec

2)

Time, t (sec)

f = 4 Hz

Wall height (H) = 10.8 m (fwn

= 2.15 Hz)

a(g

) (m

/sec

2)

Time, t (sec)


= 4.5 Hz)



6.4.2.2 Seismic earth pressure force Pstem

In order to study the influence of the natural frequency of retaining wall on the

development of Pstem, Pstem is predicted for two different heights of retaining wall H=

5.4 m and H= 10.8 m. Figure 6.20a, b, and c show the seismic earth pressure behind

stem (H= 5.4 m) when the earthquake acceleration is applied with a variety of frequency

contents. However, Figure 6.20d, e, and f show Pstem of retaining wall (H=10.8 m) when

the earthquake acceleration is applied with frequency contents 0.5 Hz, 2 Hz, and 4 Hz

respectively.


242

0 5 10 15 20 25 30

-140

-120

-100

-80

-60

-40

0 5 10 15 20 25 30-500

-450

-400

-350

-300

-250

-200

0 1 2 3 4 5 6 7 8

-140

-120

-100

-80

-60

-40

0 1 2 3 4 5 6 7 8

-800

-700

-600

-500

-400

-300

-200

-100

0 1 2 3 4

-200-180-160-140-120-100-80-60-40-20

0 1 2 3 4-550

-500

-450

-400

-350

-300

-250

-200

-150

f = 0.5 Hz

Pste

m (

kN

/m)

Time, t (sec)

f = 0.5 Hz

Pste

m (

kN

/m)

Time, t (sec)

f = 2 Hz

Pste

m (

kN

/m)

Time, t (sec)

f = 2 Hz

Pste

m (

kN

/m)

Time, t (sec)

f = 4 Hz

Pste

m (

kN

/m)

Time, t (sec)

(f)

(e)

(d)

(c)

(b)

f = 4 Hz

Pste

m (

kN

/m)

Time, t (sec)

(a)

a(g)= 0.4

Wall height (H)= 10.8 m (fwn

= 2.15 Hz)Wall height (H)= 5.4 m (fwn

= 4.5 Hz)

Figure 6.20: Effect of the natural frequency of the retaining wall on the seismic earth pressure

behind the stem

It can be indicated from the Figure 6.20 that the maximum value of Pstem (H= 5.4 m -

180 kN/m) is developed when the earthquake acceleration is applied with a frequency

content of 4 Hz, and the frequency content is close to the natural frequency of retaining

wall (fwn= 6 Hz) as shown in Figure 6.20c. However, for retaining wall height H= 10.8

m, the maximum value of Pstem (750 kN/m) is developed when the earthquake

acceleration is applied with a frequency content of 2 Hz and the frequency content is

close to the natural frequency of retaining wall (fwn= 2.15 Hz) as shown in Figure 6.20e.

Hence, it can be said that the maximum value of Pstem is developed when the earthquake

acceleration is applied with a frequency content close to the natural frequency of

retaining wall.


243

6.4.2.3 Seismic earth pressure force Pvp

Figure 6.21a, b, and c show Pvp for wall-soil system (H= 5.4 m) when the earthquake is

applied with frequency contents of 0.5 Hz, 2 Hz, and 4 Hz respectively. However,

Figure 6.21d, e, and f show Pvp for retaining wall H= 10.8 m when the earthquake

acceleration is applied with frequency contents 0.5 Hz, 2 Hz, and 4 Hz respectively.

0 5 10 15 20 25 30

-160

-140

-120

-100

-80

-60

0 5 10 15 20 25 30-800

-700

-600

-500

-400

-300

-200

0 1 2 3 4 5 6 7 8-180

-150

-120

-90

-60

-30

0 1 2 3 4 5 6 7 8

-800

-700

-600

-500

-400

-300

-200

0 1 2 3 4

-140

-120

-100

-80

-60

-40

0 1 2 3 4

-600

-500

-400

-300

-200(f)

(e)

(d)

(c)

(b)

(a)

f = 0.5 Hz

Pvp (

kN

/m)

Time, t (sec)

f = 0.5 Hz

Pvp (

kN

/m)

Time, t (sec)

f = 2 Hz

Pvp (

kN

/m)

Time, t (sec)

f = 2 Hz

Pvp (

kN

/m)

Time, t (sec)

a(g) = 0.4


= 2.15 Hz)Wall height (H) = 5.4 m (fwn

= 4.5 Hz)

f = 4 Hz

Pvp (

kN

/m)

Time, t (sec)

f = 4 Hz

Pvp (

kN

/m)

Time, t (sec)

Figure 6.21: Effect of the natural frequency of retaining wall on the seismic earth pressure force

along virtual plane

It can be indicated from the Figure 6.21 that the maximum value of Pvp for both

retaining wall heights H= 5.4 m and H= 10.8 m does not change when the earthquake

acceleration is applied with a variety of frequency contents 0.5 Hz, 2 Hz, and 4 Hz. It

can also be observed that the minimum value of Pvp is close or less than the static earth


244

pressures for both retaining wall heights H= 5.4 m and H=10.8 m. Hence, it can be said

that the maximum and minimum value of Pvp are not affected by the natural frequency

of retaining wall. This trend is in contrast to the trend of Pstem, which has been found to

be sensitive to the natural frequency of retaining wall (height of retaining wall).


Figure 6.22 and Figure 6.23show the effect of the natural frequency of retaining wall on

the Nw and Mw developed at the base of the stem (base_stem –Figure 6.1a) respectively.

0 5 10 15 20 25 30

-140

-120

-100

-80

-60

-40

0 5 10 15 20 25 30

-400

-350

-300

-250

-200

-150

0 1 2 3 4 5 6 7 8-160

-140

-120

-100

-80

-60

-40

-20

0

0 1 2 3 4 5 6 7 8

-600

-500

-400

-300

-200

-100

0 1 2 3 4-210

-180

-150

-120

-90

-60

-30

0 1 2 3 4-360

-330

-300

-270

-240

-210

-180

-150

f = 0.5 Hz

NW

(kN

/m)

Time, t (sec)

f = 0.5 Hz

NW

(kN

/m)

Time, t (sec)

(f)

(e)

(d)

f =2 Hz

NW

(kN

/m)

Time, t (sec)

f = 2 Hz

NW

(kN

/m)

Time, t (sec)

(c)

(b)

(a)

f = 4 Hz

NW

(kN

/m)

Time, t (sec)

f = 4 Hz

a(g) = 0.4Wall height (H)= 10.8 m (f

wn= 2.15 Hz)Wall height (H) = 5.4 m (f

wn= 4.5 Hz)

NW

(kN

/m)

Time, t (sec)

Figure 6.22: Effect of the natural frequency of the retaining wall on the development of shear

force predicted at the base of stem


245

0 5 10 15 20 25 30

-240

-220-200

-180

-160

-140

-120-100

-80

0 1 2 3 4 5 6 7 8-300

-250

-200

-150

-100

-50

0 1 2 3 4-400

-350

-300

-250

-200

-150

-100

-50

0 5 10 15 20 25 30-1400

-1200

-1000

-800

-600

-400

0 1 2 3 4 5 6 7 8-1800

-1600

-1400

-1200

-1000

-800

-600

-400

0 1 2 3 4-1200

-1100

-1000

-900

-800

-700

-600

-500

-400

Mw (

kN

.m/m

)

Time, t (sec)

f = 0.5 Hz

f = 2 Hz

Mw (

kN

.m/m

)

Time, t (sec)

f = 4 Hz

Mw (

kN

.m/m

)

Time, t (sec)

f =0.5 Hz

Mw (

kN

.m/m

)

Time, t (sec)

f = 2 Hz

Mw (

kN

.m/m

)Time, t (sec)

a(g)= 0.4Wall height (H)= 10.8 m (f

wn= 2.15 Hz)Wall height (H)= 5.4 m (f

wn= 4.5 Hz)

f = 4 Hz

(f)

(e)

(d)

(c)

(b)

(a)

Mw (

kN

.m/m

)

Time, t (sec)

Figure 6.23: Effect of the natural frequency of the retaining wall on the development of bending

moment predicted at the base of the stem

It can be indicated from Figure 6.22 and Figure 6.23 that for retaining wall height H=

5.4 m, the maximum values of Nw (200 kN/m – Figure 6.22c) and and Mw (330 kN.m –

Figure 6.23c) are predicted when the earthquake acceleration has frequency content 4

Hz, and this frequency content is close to the natural frequency of the retaining wall (6

Hz). However, it can be noted from Figure 6.22 and Figure 6.23 that for retaining wall

height (H= 10.8 m), maximum values of Nw (580 kN/m - Figure 6.22e) and Nw (1750

kN.m – Figure 6.23e) are predicted when the earthquake accelerationis applied with

frequency content (2 Hz), and the value of frequency content is also close to the natural

frequency of the retaining wall (2.14 Hz). Therefore, it can be said that the shear force

Nw and bending moment Mw have the same trend of Pstem as discussed in in previous

sections. Hence, it can be indicated that the critical case of the structural integrity is that


246

when the earthquake acceleration is applied towards the backfill soil, and its frequency

content is close to the natural frequency of the retaining wall.

6.4.2.5 Relative horizontal displacement of retaining wall W-F

Figure 6.24 shows the effect of the natural frequency of the retaining wall on W-F.

Figure 6.24a, b, c show W-F of retaining wall (H= 5.4 m) is predicted when the

earthquake acceleration is applied with frequency content 0.5 Hz, 2 Hz, and 4 Hz

respectively. It can be noted from Figure 6.24a, b, and c that for retaining wall height

(H=5.4m), the maximum value of W-F (0.35m – Figure 6.24a) is predicted when the

earthquake acceleration has frequency content of 0.5 Hz. Also with increasing the

frequency content of earthquake acceleration, W-F reduces. However, Figure 6.24d, e,

and f show W-F of retaining wall H= 10.8 m when the earthquake acceleration is

applied with the frequency content 0.5 Hz, 2 Hz, and 4 Hz respectively. It can be noted

from Figure 6.24d, e, and f that for retaining wall height (H= 10.8 m), the maximum

value of W-F (1.4 m – Figure 6.24d) is predicted when the earthquake acceleration is

also applied with frequency content (0.5 Hz), and also with increasing of frequency

content the of the earthquake acceleration, the relative horizontal displacement is

reduced.

Hence, it can be said that the W-F is not sensitive to the natural frequency of the

retaining wall, but it is sensitive to the frequency content of the earthquake acceleration.

It can be indicated that the for both retaining wall heights H= 5.4 m and H= 10.8 m, the

W-F increases when the frequency content of the earthquake acceleration reduces. It can

also be noted that W-F of both retaining wall heights H= 5.4 m and H=10.8 m is

sensitive to the number of cycles of the earthquake acceleration.


247

0 5 10 15 20 25 30

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0 1 2 3 4 5 6 7 8

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0 1 2 3 4-0.20

-0.15

-0.10

-0.05

0.00

0 5 10 15 20 25 30-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

0.00

0 1 2 3 4 5 6 7 8-0.90

-0.75

-0.60

-0.45

-0.30

-0.15

0.00

0 1 2 3 4-0.150

-0.125

-0.100

-0.075

-0.050

-0.025

0.000

W-F

(m)

W

-F(m

)

W

-F(m

)

W-F

(m)

(f)

(e)

(d)

(c)

(a)

W

-F(m

)

Time, t (sec)

f = 0.5 Hz

(b)

f = 2 Hz

Time, t (sec)

f = 4 Hz

Time, t (sec)

f = 0.5 Hz

Time, t (sec)

f = 2 Hz

W

-F(m

)Time, t (sec)

a(g)= 0.4Wall height (H)= 10.8 m (f

wn= 2.15 Hz)Wall height (H)= 5.4 m (f

wn= 4.5 Hz)

f = 4 Hz

Time, t (sec)

Figure 6.24: Effect of the natural frequency of the cantilever retaining wall on the relative

horizontal displacement of retaining wall

6.4.3 Effect of relative density of soil

The present parametric study is conducted in order to investigate the effect of relative

density of soil material on the seismic behaviour of a cantilever retaining wall. The

results obtained from the present parametric study critically discuss the impact of the

relative density of soil material on the acceleration response of the wall-soil system, the

seismic earth pressure behind the stem, Pstem, seismic earth pressure along the virtual

plane Pvp, shear force Nw, bending moment Mw, and relative horizontal displacement of

the retaining wall, W-F. The effect of relative density of soil material is examined by

choosing three relative densities; relative loose soil (Dr = 40%), relatively medium-

dense soil (Dr = 65%), and relatively dense soil (Dr = 85%). The material properties,

which are used to run the FE models, are presented in Table 5.2 considering three


248

relative densities. The seismic loading is simulated by applying a uniform sinusoidal

acceleration time history at the base of FE models with amplitude of 0.4g and frequency

content of 2 Hz.


Figure 5.23a shows the effect of relative density of soil on the acceleration response at

the top of the wall, while Figure 5.23b shows the effect of relative density of soil on the

acceleration response at the top of the backfill soil. From Figure 5.23a and b, it can be

noted there is a phase difference in the acceleration response when the soil is simulated

with different relative densities. So, it can be indicated that the maximum lag in

acceleration response is that when the soil is simulated with a relatively loose soil of Dr

= 40%. The rate of lag in acceleration response is reduced with increasing the relative

density of soil to Dr = 85%. A possible explanation for these results might be that when

the soil has a high relative density, the acceleration wave will be travel faster towards

the top of the wall-soil system because the shear velocity of soil increases when the

relative density of soil increases. It can also be noted that the rate of amplification of

acceleration response reduces when the soil is simulated with a relatively high relative

density. It can be noted from Figure 5.23a, and b the minimum rate of amplification is

that when the soil is simulated with a soil of relatively high density Dr = 85%.

0 1 2 3 4 5 6 7 8

-1.5

-1.0

-0.5

0.0

0.5

1.0

0 1 2 3 4 5 6 7 8

-0.9

-0.6

-0.3

0.0

0.3

0.6

a(g

) (m

/se

c2)

Time, t (sec)

Dr=40%

Dr=65%

Dr=85%

a(g

) (m

/se

c2)

Time, t (sec)

Dr=40%

Dr=65%

Dr=85%

(b)(a)

Figure 6.25: Effect of soil relative density of soil on the acceleration response at the top of: a)

the retaining wall and b) backfill soil


249


Figure 6.26a and b show the effect of relative density of soil on Pstem and Pvp

respectively. It can be noted from Figure 6.26a and b that the static earth pressure force

behind the stem and static earth pressure force along virtual plane at t = 0 sec are

affected by the relative density of the backfill soil. It can be indicated that both the static

active earth pressure forces reduce when the relative density is increased from Dr = 40%

to Dr = 85%. The trend of static active earth pressure forces against the relative density

of backfill soil is similar to the trend of the static active earth pressure predicted by

Coulomb method or Rankine earth pressure theories. However, it can be noted from

Figure 6.26a and b that Pstem and Pvp is influenced by the static earth pressure force

predicted at t = 0 sec. In order further sunderstand the seismic earth pressure forces,

Pstem and Pvp are presented by the seismic earth pressure force increment, Pstem and

Pvp as shown in Figure 6.26c and d respectively. The seismic earth pressure force

increments can be computed as below:

( )stem stem stem staticP P P (6.5)

( )vp vp vp staticP P P (6.6)

where, stemP = seismic earth pressure force increment behind the stem, stemP = seismic

earth pressure force behind the stem, ( )stem staticP = static earth pressure force behind the

stem, vpP = seismic earth pressure force increment along virtual plane, vpP = seismic

earth pressure force along virtual plane, and ( )vp staticP = static earth pressure force along

virtual plane.

It can be noted from Figure 6.26c and d that the minimum value of stemP and vpP

reduce when the relative density of backfill soil is reduced from Dr = 85% to Dr = 40%.

However, it can be noted that the maximum value of stemP and vpP increase when the

relative density of backfill soil is increased from Dr = 40% to Dr = 85%. This trend is in

contrast to the trend of static earth pressure force as discussed above.


250

0 1 2 3 4 5 6 7 8-200

-180

-160

-140

-120

-100

-80

-60

-40

-20

0 1 2 3 4 5 6 7 8

-240

-220

-200

-180

-160

-140

-120

-100

-80

-60

-40

0 1 2 3 4 5 6 7 8-120

-100

-80

-60

-40

-20

0

20

40

60

0 1 2 3 4 5 6 7 8-180-160-140-120-100

-80-60-40-20

02040

Pste

m (

kN

/m)

Time, t (sec)

Dr=40% Dr=65% Dr=85%

Pvp (

kN

/m)

Time, t (sec)

(d)

(b)

(c)

(a)

P

ste

m (

kN

/m)

Time, t (sec)

P

vp (

kN

/m)

Time, t (sec)

Figure 6.26: Effect of soil relative density of soil on the seismic earth pressure forces behind the

stem and along the virtual plane


Figure 6.27a and b show the effect of relative density of backfill soil on the Nw and Mw

while Figure 6.27c and d shows the effect of relative density on the Nw and Mw.

where, Nw = shear force increment and Mw = bending moment increment.

However, it can be indicated from Figure 6.27c and d that the Nw and Mw increase

when the relative density is increased from Dr = 40% to Dr = 85%. This trend is similar

to the trend of stemP discussed in the previous section. Hence, in contrast to the static

state, it can be said that when backfill soil is simulated with a higher relative density,

the shear force and bending moment increases at the base of the stem.


251

0 1 2 3 4 5 6 7 8

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

0 1 2 3 4 5 6 7 8-350

-300

-250

-200

-150

-100

-50

0

0 1 2 3 4 5 6 7 8-120

-100

-80

-60

-40

-20

0

20

40

60

80

0 1 2 3 4 5 6 7 8

-200

-150

-100

-50

0

50

100(d)(c)

(b)

NW

(kN

/m)

Time, t (sec)

Dr=40% Dr=65% Dr=85%

(a)

MW

(kN

.m/m

)

Time, t (sec)

N

W (

kN

/m)

Time, t (sec)

M

W (

kN

.m/m

)

Time, t (sec)

Figure 6.27: Effect of soil relative density of soil on the shear force and bending moment

6.4.3.4 Relative horizontal displacement of the wall

Figure 6.28 shows the impact of relative density of soil on W-F. It can be indicated from

Figure 6.28 that W-F is highly affected by the relative density of soil. When the relative

density is increased from Dr = 40% to Dr = 85%, W-F (decreases from 0.204 to

0.05. A possible explanation of this trend is that when the relative density of backfill

soil is reduced, the stiffness parameters of interface element connected with the base of

the cantilever retaining wall by foundation soil also reduced. Another explanation is that

the amplification of acceleration response in the wall-soil system reduces when the

relative density increases from Dr = 40% to Dr = 85% as discussed in section 6.4.3.1.

Consequently, less amplitude of wall and backfill seismic inertia forces will be

developed in the wall-soil system causing development of a minimum W-F.


252

0 2 4 6 8

-0.20

-0.15

-0.10

-0.05

0.00

W

-F (

/H)

Time, t (sec)

Dr = 40%

Dr = 65%

Dr = 85%

Figure 6.28: Effect of relative density of soil on the relative horizontal displacement of the

cantilever retaining wall

6.5 SUMMARY

A performance-based FE seismic analysis method is presented to evaluate a critical

scenario for the structural integrity and global stability of a cantilever-type retaining

wall-soil system by considering the seismic earth pressure, computed at the stem (Pstem)

and as well as along a vertical virtual plane (Pvp), and wall and backfill seismic inertia

forces. Pstem contributes to the structural integrity, while Pvp contributes to the global

stability. It is observed that Pstem and Pvp are out of phase during the entire duration of

the earthquake and hence, the structural integrity and global stability should be

evaluated individually. A critical case for the structural integrity is observed when the

earthquake acceleration is applied towards the backfill soil and has frequency content

close to the natural frequency of the retaining wall. Also, the wall seismic inertia force

has a significant effect on the structural integrity only for the top of the stem while the

bottom of the stem does not get affected significantly. A critical case for the global

stability is observed when the earthquake acceleration has maximum amplitude and is

applied towards the backfill soil with minimum frequency content. Also, it is

significantly affected by the wall and as well as soil seismic inertia forces. The number

of acceleration cycles of the applied earthquake acceleration does not affect the

structural integrity while the global stability is observed to be highly sensitive to this.


253

It is also observed that the relative density of backfill soil has a considerable effect on

the structural integrity and global stability of a cantilever retaining wall. The results

obtained from the FE model show that the structural integrity of a cantilever retaining

will be reduced when the backfill soil has high relative density, while the global

stability will be increased when the backfill soil has high relative density during the

seismic scenario.

Next chapter will present simplified analytical approach in order to evaluate the

contribution of wall seismic inertia force to the total shear force and bending moment.

Also, the next chapter includes modifying the Newmark sliding block method to

compute the relative horizontal displacement of a rigid and as well as a cantilever

retaining wall.

Chapter 7: Analytical Methods

254

CHAPTER 7

ANALYTICAL METHODS

Chapters 3, 4, 5 and 6 have discussed in detail the FE-based modelling of a rigid and

cantilever-type retaining wall. In this Chapter an analytical solution development

methodology is presented for both the rigid and cantilever type retaining wall. The first

part of the Chapter details a simplified procedure proposed to assess, analyse and

evaluate the contribution of wall seismic inertia force to the shear force and bending

moment developed in the stem of a cantilever-type retaining wall, while the second part

of the Chapter presents a simplified procedure proposed to modify the Newmark sliding

block method with an aim of computing relative horizontal displacement of a rigid-type

and cantilever-type retaining walls. Finally, worked examples are presented to validate

the previous findings of the FE model and also demonstrate the applicability of the

proposed analytical method.

7.1 CONTRIBUTION OF WALL SEISMIC INERTIA FORCE ON THE

TOTAL SHEAR FORCE AND BENDING MOMENT

As discussed in Chapter 6, the seismic response of a cantilever-type retaining wall has a

significant effect on the development of seismic earth pressure behind the stem, Pstem

and shear force, Nw and bending moment, Mw. However, the traditional methods like

pseudo-static methods (Mononobe and Matsuo, 1929) and pseudo-dynamic methods

((Steedman and Zeng, 1990b)), used for the seismic analysis and design of a cantilever

retaining wall assume that it is only the seismic earth pressure which contributes to the

development of shear force, Nw and bending moment, Mw . This clearly suggests that

there were conflicting views about it, and hence, to concrete this, it was decided to

evaluate the contribution of wall inertia force on the development of the shear force Nw

and bending moment Mw with an overarching aim to provide a more accurate

assessment of the structural integrity of the cantilever retaining wall.


255

7.1.1 Problem definition

Under the effect of an earthquake acceleration, the shear force Nw and bending moment

Mw will be developed in the stem of a cantilever retaining wall (see Figure 6.2a). As per

the free body diagram as shown in Figure 6.2a the components which contribute to the

shear force Nw and bending moment Mw during an earthquake are (1) total seismic earth

pressure force (Pstem) considered to be acting behind the stem of the wall and (2) wall

seismic inertia force Fwa and Fwp as shown in Figure 6.2a. Pstem will be a function of

time and will vary over the duration of the earthquake and the height of the stem. The

objective here is to evaluate the contribution of wall seismic inertia force on the shear

force (Nw) and bending moment (Mw). To properly analyse the problem, the following

times of applied earthquake acceleration are identified:

At the beginning of the seismic analysis (i.e., at time t = 0 sec);

At a time when the applied earthquake acceleration has its maximum value and is

applied towards the backfill soil (see Figure 5.3a), and correspondingly the retaining

wall moves away from the backfill soil (i.e., at time t = 3.8 sec);

At a time when the applied earthquake acceleration has its maximum value and is

applied away from the backfill soil (see Figure 5.3a), and correspondingly the

retaining wall moves towards the backfill soil (i.e., at time t = 4.5 sec);

At the end of the seismic analysis (i.e., at time t =30 sec).

7.1.2 Assumptions made in the simplified procedure

Figure 7.1 shows the free body diagram of the external forces acting on the stem of the

cantilever retaining wall for a seismic scenario. Shear force Nw(z,t) and bending

moment Mw(z,t) vary with the height of the stem, z and the time of the earthquake, t.

The shear force Nw(z,t) and bending moment Mw(z,t) will be predicted herein from two

sources: the seismic earth pressure behind the stem Pstem(z,t) and wall seismic inertia

force, Fwa(z,t) and Fwp(z,t) The seismic earth pressure Pstem increases from the top

towards the bottom of the stem as observed in Chapter 6, while the wall seismic inertia

force increases from the bottom towards the top of the stem because of the amplification

of the acceleration response in the stem towards the top.


256

Figure 7.1: Free body diagram of external forces acting on the stem of the wall during the

earthquake, producing shear force and bending moment

It is assumed that the stem of the retaining wall behaves like a cantilever beam and it

has a fixed connection with the base slab. Hence, the shear force Nw(z,t) and bending

moment Mw(z,t) can be computed by using the dynamic Euler–Bernoulli beam theory

as shown in Equation (7.1) and (7.2), respectively:

3 2

3 200

( , ) ( ) ( , ) ( , )stem

tH

w

x xN z t EI t m z t pstem z t dzdt

z t

(7.1)

2 22

2 200

( , ) ( ) ( , ) ( , )stem

stem

HtH

w

o

x xM z t EI t m z t pstem z t dz dt

z t

(7.2)

where Nw(z, t), Mw(z, t) = shear force and bending moment for time t and at location z

along the height of the stem, measured from the top of the base slab (Figure 7.1); EI =

Young’s modulus multiplied by the second moment of inertia of the stem of the

cantilever wall section per metre length; x = elastic deflection of the stem in x-axis; 2

2

x

t

= predicted acceleration at depth z of the stem in x-axis and it is equal to 𝑎(𝑧, 𝑡); and

𝑝𝑠𝑡𝑒𝑚(𝑧, 𝑡) = seismic earth pressure at location z along the height of the stem, measured

from the top of the base slab.


257

If n is the total number of stem elements, then Equations (7.1) and (7.2) could be re-

written as:

_

1 1

( ) ( ) ( )n n

w n n stem nN t m a t P t (7.3)

_

1 1

( ) ( ) ( ) ( ) ( )n n

w n n n stem n nM t m a t z P t z (7.4)

where an(t) = acceleration for the nth

element; mn = mass of the nth

element; 𝑧𝑛 = vertical

distance between the nth

element and the base of the cantilever retaining wall; and

Pstem_n(t) = total seismic earth pressure force computed at the centre of gravity of the nth

element. Equations (7.3) and (7.4) can be further simplified as:

1 _

1 1

( ) ( ) ( )n n

w w w n n n stem nN t t z z a t P t (7.5)

1 _

1 1

( ) ( ) ( )n n

w w w n n n n stem n nM t t z z a t z P t z (7.6)

where 𝛾𝑤 = unit weight of the wall material; tw = thickness of the stem; and 1n nz z =

height of the nth

element. From Equations (7.5) and (7.6), it is clear that the shear force

and bending moment depend upon: (i) the wall seismic inertia force, and (ii) the seismic

earth pressure force. Their effects are discussed next for top and bottom ⅓H of the wall

and as well as middle of the height of the wall.

7.1.3 Effect of wall seismic inertia force for the top ⅓H of the stem on Nw and Mw

Figure 7.2a and b respectively show the shear force Nw and bending moment Nw

computed by using Equations (7.5) and (7.6) for the top 1/3H of the stem, and they are

separated into 2 components; the wall seismic inertia force and seismic earth pressure

force . However, Figure 7.2c and d show the shear force and bending moment computed

Equations (7.5) and (7.6) for the top 1/3H of the stem respectively, and they are

predicted by sum of abovementioned 2 components; the wall seismic inertia force and

seismic earth pressure force.


258

0 5 10 15 20 25 30-6

-4

-2

0

2

0 5 10 15 20 25 30-30

-20

-10

0

10

20

0 5 10 15 20 25 30-5

-4

-3

-2

-1

0

0 5 10 15 20 25 30-25

-20

-15

-10

-5

0(d)(c)

Nw (

kN

/m)

Time, t (sec)

Nw and M

w predicted by F

W N

w and M

w predicted by P

stem)

t = 3.8 sec

t = 4.5 sec

t =3.9 sec

t = 4.5 sec

t = 3.9 sec

t = 3.9 sec

t = 4.5 sec

t = 4.5 sec

t = 3.9 sec

t = 3.9 sec

t = 4.5 sec

t = 4.5 sec

Mw (

kN

.m/m

)

Time, t (sec)

(b)(a)N

w (

kN

/m)

Time, t (sec)

Nw and M

w predicted by F

W and P

stem

Mw (

kN

.m/m

)

Time, t (sec)

Figure 7.2: Effect of wall seismic inertia force on Nw and Mw for the top ⅓H of the stem

Table 7.1 shows the estimated shear force and bending moment for different time

durations of the earthquake. As shown in Table 7.1, the shear force and bending

moment are produced from 3 components: Static earth pressure force (Pstatic); Wall

seismic Inertia force (Fw); and Seismic earth pressure force increment (ΔPstem = Pstem -

Pstatic). A closer look at Table 7.1 suggests the following:, total static earth pressure

force Pstatic, wall seismic inertia force Fwa, Fwp, and total seismic earth pressure force

increment ΔPstem (= Pstem - Pstatic) contribute to Nw and Mw. At the beginning of the

seismic analysis (t = 0 sec), only Pstatic causes Nw and Mw; however, at t = 3.9 sec, when

the applied earthquake acceleration has a maximum value and is applied towards the

backfill soil, Pstatic, Fwa, and ΔPstem all contribute to Nw and Mw and all of these

quantities produce Nw and Mw which act away from the backfill soil (ie in a direction

opposite to the direction of the applied earthquake). When the applied earthquake

acceleration has a maximum value but is applied away from the backfill soil at time t =

4.5 sec, Pstatic, Fwp, and ΔPstem all contribute to Nw and Mw. But, unlike the previous

case, Pstatic and ΔPstem produce Nw and Mw in the same as the direction of the applied


259

earthquake acceleration, while the Fwp produce Nw and Mw in a direction opposite to the

direction of the applied earthquake.

Table 7.1: Effect of wall seismic inertia force on Nw and Mw for the top ⅓H of the stem

Time, t

Contribution to NW (kN/m) from: Contribution to MW (kN.m/m) from:

Static

(Pstatic)

*Inertia

Fwa/Fwp

Increment

(ΔPstem)

Total

(NW)

Static

(Pstatic)

*Inertia

Fwa/Fwp

Increment

(ΔPstem)

Total

(MW)

t = 0 sec -0.72 0.006 0.00 -0.72 -3.68 0.02 0.00 -3.68

t = 3.9 sec -0.72 -1.08 -0.85 -2.65 -3.68 -8.89 -1.93 -14.5

t = 4.5 sec -0.72 3.72 -3.88 -0.88 -3.68 17.22 -21.22 -7.68

t = 30 sec -0.72 0.00 -1.48 -2.2 -3.68 0.00 -9.43 -13.11

* negative values indicate Fwa, while all others are Fwp.

7.1.4 Effect of wall seismic inertia force for the mid-height of the stem on Nw and

Mw

Figure 7.3a and b respectively show the shear force and bending moment computed by

using Equations (7.5) and (7.6) for the mid-height of the stem and they are separated

into 2 components; the wall seismic inertia force and seismic earth pressure force.

However, Figure 7.3c and d show the shear force and bending moment computed

Equations (7.5) and (7.6) for the mid-height of the stem respectively, and they are

predicted by sum of abovementioned 2 components; the wall seismic inertia force and

seismic earth pressure force. . Table 7.2 shows the effect of total static earth pressure

force Pstatic, wall seismic inertia force Fwa, Fwp, and total seismic earth pressure force

increment ΔPstem on the Nw and Mw. At the beginning of the seismic analysis (t = 0 sec)

and similarly at the top ⅓H of the stem, only Pstatic causes Nw and Mw; however, at t =

3.9 sec, when the applied earthquake acceleration has a maximum value and is applied

towards the backfill soil, Pstatic, Fwa, and ΔPstem all contribute to Nw and Mw but the

effect of Fwa on Nw and Mw is less than at the top ⅓H of the stem. Also, all of these

quantities act away from the backfill soil (ie in a direction opposite to the direction of

the applied earthquake). When the applied earthquake acceleration has a maximum

value but is applied away from the backfill soil at time t = 4.5 sec, like at the top ⅓H of

the stem, Pstatic, Fwp, and ΔPstem all contribute to Nw and Mw. But, Pstatic and ΔPstem


260

produce Nw and Mw in the same as the direction of the applied earthquake acceleration,

while the Fwp produce Nw and Mw in a direction opposite to the direction of the applied

earthquake.

0 5 10 15 20 25 30

-21

-14

-7

0

7

14

0 5 10 15 20 25 30-120

-90

-60

-30

0

30

60

0 5 10 15 20 25 30-32

-28

-24

-20

-16

-12

-8

-4

0 5 10 15 20 25 30-140

-120

-100

-80

-60

-40

-20

t = 3.9 sec

t = 4.5 sec

t = 3.9 sec

t = 4.5 sec

Nw (

kN

/m)

Time, t (sec)

(a)

(d)(c)

(b) t = 4.5 sec

t = 3.9 sec

t = 3.9 sec

Mw (

kN

.m/m

)

Time, t (sec)

t = 3.9 sec

t = 4.5 sec

Nw (

kN

/m)

Time, t (sec)

Nw and M

w predicted by F

W N

w and M

w predicted by P

stem)

Nw and M

w predicted by F

W and P

stem

t = 3.9 sec

t=4.5 sec

Mw (

kN

.m/m

)

Time, t (sec)

Figure 7.3: Effect of wall seismic inertia force on Nw and Mw for the mid-height of the stem

Table 7.2: Effect of wall seismic inertia force on Nw and Mw for the mid-height of the stem

Time, t


Static

(Pstatic)

*Inertia

Fwa/Fwp

Increment

(ΔPstem)

Total

(NW)

Static

(Pstatic)

*Inertia

Fwa/Fwp

Increment

(ΔPstem)

Total

(MW)

t = 0 sec -9.45 0.02 0.00 -9.45 -36.4 0.08 0.00 -36.32

t = 3.9 sec -9.45 -4.68 -12.68 -26.79 -36.4 -24.79 -49.49 -110.68

t = 4.5 sec -9.45 11.33 -12.39 -10.51 -36.4 47.94 -54.15 -42.61

t = 30 sec -9.45 0.00 -13.52 -22.7 -36.4 0.00 -52.55 -88.95



261

7.1.5 Effect of wall seismic inertia force for the bottom ⅓H of the stem on Nw and

Mw

Figure 7.4a and b respectively show the shear force Nw and bending moment Nw

computed by using Equations (7.5) and (7.6) for the bottom ⅓H of the stem, and they

are separated into 2 components; the wall seismic inertia force and seismic earth

pressure force . However, Figure 7.4c and d show the shear force and bending moment

computed Equations (7.5) and (7.6) for the bottom ⅓H of the stem respectively, and

they are predicted by sum of abovementioned 2 components; the wall seismic inertia

force and seismic earth pressure force.

0 5 10 15 20 25 30-120

-100

-80

-60

-40

-20

0

20

0 5 10 15 20 25 30

-200

-150

-100

-50

0

50

100

0 5 10 15 20 25 30-250

-225

-200

-175

-150

-125

-100

-75

-50

0 5 10 15 20 25 30

-120

-100

-80

-60

-40

-20

t= 3.9 sec

t = 4.5 sec

t = 4.5sec

t=3.8sec

t = 4.5sec

t = 3.9 sec

Nw (

kN

/m)

Time, t (sec)

t= 4.5 sec

t= 3.9 sec

t= 3.8 sec

t= 4.5 secM

w (

kN

.m/m

)

Time, t (sec)

(d)(c)

(b)(a)

Nw and M

w predicted by F

W and P

stem

Nw and M

w predicted by F

W N

w and M

w predicted by P

stem)

t = 3.9 sec

t = 4.5 sec

Mw (

kN

.m/m

)

Time, t (sec)

Nw (

kN

/m)

Time, t (sec)

Figure 7.4: Effect of wall seismic inertia force on Nw and Mw for the bottom ⅓H of the stem

Table 7.3 shows the effect of total static earth pressure force Pstatic, wall seismic inertia

force Fwa, Fwp, and total seismic earth pressure force increment ΔPstem on the Nw and Mw

for the bottom ⅓H of the stem. At the beginning of the seismic analysis (t = 0 sec) and

similarly at the top ⅓H of the and the mid-height of the stem, only Pstatic causes Nw and

Mw; however, at t = 3.9 sec, when the applied earthquake acceleration has a maximum


262

value and is applied towards the backfill soil, the contribution of Fwa to Nw and Mw is

very small comparing with effect of Pstatic and ΔPstem. Also, all of these quantities act

away from the backfill soil. When the applied earthquake acceleration has a maximum

value but is applied away from the backfill soil at time t = 4.5 sec, similar observations

at the top ⅓H and mid-height of the stem are predicted at the bottom the top ⅓H.

Table 7.3: Effect of wall seismic inertia force on Nw and Mw for the bottom ⅓H of the stem

Time, t


Static

(Pstatic)

*Inertia

Fwa/Fwp

Increment

(ΔPstem)

Total

(NW)

Static

(Pstatic)

*Inertia

Fwa/Fwp

Increment

(ΔPstem)

Total

(MW)

t = 0 sec -53.1 0.04 0.00 -52.97 -92.45 0.11 0.00 -92.34

t = 3.9 sec -53.1 -11.86 -58.2 -122.9 -92.45 -29.34 -95.78 -217.6

t = 4.5 sec -53.1 21.75 3.18 -28.17 -92.45 64.53 -35.98 -63.9

t = 30 sec -53.1 0.00 -53.46 -88.56 -92.45 0.00 -81.45 -173.9


From the above, it can be concluded that when the earthquake acceleration is applied

towards the backfill soil then for the top half of the retaining wall the wall seismic

inertia force has a major contribution to Nw and Mw, while for the bottom half of the

retaining wall it is the total static earth pressure force and total seismic earth pressure

force increment which contribute to Nw and Mw. When the earthquake acceleration is

applied away from the backfill soil, inertia force of the stem produces shear force and

bending moment in opposite direction of the static earth pressure and the increment of

seismic earth pressure causing the shear force and bending moment attain a minimum

value, and it was less than the static value.

7.2 MODIFICATION OF NEWMARK SLIDING BLOCK METHOD

Details of the seismic behaviour of a rigid and cantilever retaining wall, which were

presented in Chapter 5 and 6, indicated that a minimum seismic earth pressure was

developed when the retaining walls moved away from the backfill soil, while a

maximum seismic earth pressure was developed when the retaining walls moved

towards the backfill soil. The effect of the later on the global stability of a rigid and

cantilever retaining wall is small because as highlighted in Ch5 and 6 it was found to be


263

out-of-phase with respect to the maximum wall seismic inerita force causing the sliding

of the wall away from the backfill soil . Hence, these retaining walls may fail by a rigid

body movement (ductile failure mechanism) during the earthquake in which they slide

or overturn away from the backfill soil. Newmark sliding block method is a simple yet

robust method that has been used to estimate the permanent seismic displacement of

retaining walls. This method was initially developed by Newmark (1965) to analyse the

seismic stability of earth dams and embankments and then it was developed for

analysing the seismic stability of retaining wall. Newmark sliding block method can be

used to compute the seismic permanent displacement of retaining walls to provide a

measure of expected damage. A key component of this method is that the definition of

yield acceleration. The yield acceleration is the threshold value in which the rigid body

starts accumulated permanent movement considering the factor of safety equal to one.

The twice integration of the acceleration time history exceeds the yield acceleration will

produce the permanent displacement of the retaining walls.

Despite the fact that Newmark sliding methods is a powerful tool used to estimate the

permanent displacement of retaining walls, it has some limitations when applied to the

problem of retaining walls, as they are listed below:

Newmarks sliding block method only accounts the base excitation and do not

consider the effect of amplification of backfill soil acceleration response. It has been

observed in Chapter 5 and 6 that by considering the soil-structure interaction effect,

the acceleration response of retaining wall is highly affected by the amplification of

acceleration response of backfill soil.

Newmark sliding block method initially requires the definition of yield acceleration.

The pseudo-static methods are usually used to calculate the yield acceleration. As

discussed in Chapter 5 and 6, the pseudo-static methods have some limitations, and

they do not represent the real seismic behaviour of retaining walls. They are

overestimating the seismic earth pressure developed behind the retaining walls.

Hence, a simplified approach is proposed in the present study in order to modify

Newmarks sliding block method and cope with its limitations described above. The

current procedure is based on the observations obtained from FE analysis in Chapter 5


264

and 6. Several assumptions are made in the current procedure to produce a more

accurate estimation of permanent displacement of a rigid and cantilever retaining wall,

and they presented below

The backfill soil is assumed dry;

The rigid and cantilever retaining walls set on a rigid (non-deformable foundation

layer);

The rigid retaining wall, cantilever retaining and backfill soil above the footing slab

is assumed a ductile system;

The seismic active earth pressure acting behind the retaining wall will be assumed

equal to static earth pressure force when the retaining walls move away from the

backfill soil. However, the seismic passive earth pressure is acting out of phase

when retaining walls move away from the backfill soil. Hence, for simplicity, the

effect of seismic passive earth pressure is ignored in the present study.

The acceleration response in the retaining wall is assumed amplifying linearly from

the base towards the top of the retaining wall. The amplification effect is defined by

the amplification factor, which is highly affected by earthquake characteristics and

backfill soil properties;

The overturning of retaining walls do not consider in the proposed approach; and

It is not required to calculate the yield acceleration in the current approach. Hence, it

is not required to use the pseudo-static method.

7.2.1 Modified Newmark sliding block method applied to rigid retaining walls

As discussed in Chapter 5, the permanent displacement of a rigid retaining wall during

an earthquake depends on: the amplitude and frequency content of earthquake

acceleration; the weight of the retaining wall; the frictional resistance between the base

of the retaining wall and foundation soil; the amplification of acceleration response in

the retaining wall; and the relative density of the soil material. A rigid retaining wall is

shown in Figure 7.5. This retaining wall will slide away from the backfill soil if the


265

total horizontal driving force acting on the retaining wall is greater than the base

frictional resistance force.

Figure 7.5: Forces acting in the wall-soil system causing sliding of the wall

( ) ( )driving RF t F t (7.7)

where, ( )drivingF t = total horizontal driving force; and RF = base frictional resistance

force, in which

( ) ( ) ( )driving w aeF t F t P t (7.8)

with ( )wF t = wall seismic inertia force, ( )aeP t = seismic earth pressure force acting

behind the retaining wall. The total friction resistance can be computed as below:

( )( ) tanR w ae v bF t W P (7.9)

where, wW = weight of the retaining wall, ( )ae vP = vertical component of the total seismic

earth pressure force and b = friction angle between the base of the retaining wall and

foundation soil layer.

The total seismic earth pressure force will be assumed to be equal to the static earth

pressure force when the retaining wall slides away from the backfill soil, i.e.,

( )ae aP t P (7-10)

Hence, Equation (7.7) can be written as:


266

( )( ) tanw a w a v bF t P W P (7-11)

During the time of the earthquake in which the total horizontal driving force exceeds the

base frictional resistance, the retaining wall will accumulate a relative horizontal

displacement. The relative acceleration arel(t) of the wall-soil system causing the

relative horizontal displacement of the retaining wall can be computed by dividing the

total horizontal driving force exceeding the frictional resistance force by the mass of the

retaining wall as:

( )( ) tan 0( )

w a w a v b

rel

w

F t P W Pa t

m

(7.12)

where, mw= mass of the retaining wall.

Equation (7.12) can also be expressed as:

2

( )( ) 0.5 tan 0( )

w w s a w a v b

rel

w

m a t K H W Pa t

m

(7.13)

where, aw(t) = acceleration of the retaining wall, γs = unit weight of the backfill soil, H =

height of the retaining wall, Ka = static active earth pressure coefficient, computed using

the Coulomb’s earth pressure theory. To account for the effect of the amplification and

de-amplification of the relative horizontal displacement of the rigid retaining wall,

Nimbalkar and Choudhury (2008b) approach is used to estimate the wall seismic inertia

force as:

1

( ) ( ) 1 1n

nw n wn a

zF t m a t f

H

(7.14)

By substituting Equation 7.14 in 7.13, the relative acceleration can be computed by:

2

( )

1

( ) 1 1 0.5 tan 0

( )

nn

n wn a s a w a v b

rel

w

zm a t f K H W P

Ha t

m

(7.15)


267

A double integration of the relative acceleration obtained from Equation (7.15) will

produce relative horizontal displacement of the retaining wall. For this purpose a code

was developed using MATLAB code and the results are discussed next via a worked

example.

7.2.2 Worked example and numerical validation

The same rigid retaining wall, which was used in Chapter 5, is considered in the present

worked example. The wall-soil system consists of a 4 m high trapezoidal cross-section

retaining wall with a top width of 1.5 m and base width of 3 m. The retaining wall is

assumed to be resting on a rigid foundation soil. The retaining wall retains a dry

cohesionless soil to its full height. A real earthquake-time history of the Loma Prieta

(1989) earthquake (as shown in Figure 5.3a) is used to simulate the seismic loading.

The acceleration response, which was predicted at the base of the retaining wall during

the FE analysis, will be used herein as base excitation. The friction angle between the

retaining wall and foundation layer is assumed to be equal to 0.5 , which is similar to

the one used in the FE analysis. The comparison between the acceleration response

predicted at the top and bottom of the retaining wall by the FE analysis in Chapter 5

shows that the acceleration is amplified by 1.2 times. The same amplification factor is

used in the current analysis. Figure 7.6(b) shows the relative horizontal displacement

predicted by the approach, and its comparison with those computed via the FE analysis

(Chapter 5) and the conventional Newmark sliding block method (Newmark, 1965). It

is noted that the relative horizontal displacement (/H) calculated by the proposed

analytical approach (0.01) is closer to the FE analysis results (0.005), while too far off

from the Newmark sliding block method result (0.02). For further validation of the

proposed analytical method, two uniform sinusoidal acceleration time histories with

different amplitudes and frequency contents were used to predict the relative horizontal

displacement of a rigid retaining wall. The first example includes applying a uniform

sinusoidal acceleration time history at the base of retaining wall with amplitude 0.4g

and frequency content 0.66 Hz as shown in Figure 7.6(c); while the second example

includes applying a uniform sinusoidal acceleration time history at the base of retaining

wall with amplitude 0.3g and frequency content 2 Hz as shown in Figure 7.6e. It can be

noted for the first example (see Figure 7.6d) that the relative horizontal displacement


268

(/H) predicted by current approach is about 0.31 while the relative horizontal

displacement (/H) predicted by the FE analysis and Newmark sliding block method is

about 0.25 and 0.575, respectively. Figure 7.6f shows the relative horizontal

displacements for the later acceleration time history, and it is observed that by the

present analytical method the relative horizontal displacement (/H) is about 0.055

while the one predicted by the FE analysis was it was predicted by FE analysis (0.325)

as well as Newmark method (0.08). The comparison between the relative horizontal

displacement obtained from the current simplified procedure and FE result as well as

Newmark sliding block method in two example (see Figure 7.6d and f) shows that the

results obtained from the current simplified procedure are more reasonable than those

predicted by Newmark sliding block method.


269

0 5 10 15 20 25 30-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 5 10 15 20 25 30-0.030

-0.025

-0.020

-0.015

-0.010

-0.005

0.000

0 5 10 15 20-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 5 10 15 20-0.750

-0.625

-0.500

-0.375

-0.250

-0.125

0.000

0 1 2 3 4 5 6 7 8-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 1 2 3 4 5 6 7 8-0.090

-0.075

-0.060

-0.045

-0.030

-0.015

0.000

a(g

) (m

/se

c2)

Time, t (sec)

W

-F (

/H)

Time, t (sec)

FE (Chapter 5) Current modified method Newmark method

f= 0.66 Hz

a(g

) (m

/se

c2)

Time, t (sec)

W

-F (

/H)

Time, t (sec)

f= 0.66 Hz

a(g

) (m

/se

c2)

Time, t (sec)

f= 2 Hz

W

-F (

/H)

Time, t (sec)

f= 0.5 Hz

Figure 7.6: Relative horizontal displacement of the rigid wall comparison between the modified

Newmark procedure, current study FE results and the classic Newmark sliding block method

7.2.3 Cantilever retaining wall

As discussed in Chapter 6, a cantilever retaining wall maintains its stability to resist the

sliding from the weight of backfill soil above the footing slab in addition to its self-

weight. Hence, under the effect of seismic loading, the inertia forces are developed in

the cantilever retaining wall as well as the backfill soil above the base slab.

The current approach is proposed in order to cope with the limitations of Newmark

sliding block method. As shown in Figure 7.7, the wall-soil system slides away from the

backfill soil when the total horizontal driving force acting on the wall-soil is greater

than the frictional resistance force between the footing slab and foundation soil layer as

below:


270

Figure 7.7: Forces acting on the cantilever wall-soil system causing the sliding of the retaining

wall

( ) ( )driving RF t F t (7.16)

where; ( )drivingF t = total horizontal driving force, and RF = base frictional resistance

force.

( ) ( ) ( ) ( )driving w s aeF t F t F t P t (7.17)

where, ( )wF t = wall seismic inertia force, ( )sF t = backfill seismic inertia force, ( )aeP t =

seismic earth pressure force acting along the virtual plane extending from the heel to

ground surface as shown in Figure (7.7).

( ) tanR w s ae v bF W W P (7.18)

where, wW = weight of retaining wall, sW = weight of the backfill soil above the base

slab and b = friction angle between the base of the retaining wall and foundation soil

layer.

The total seismic earth pressure force along virtual plane will be assumed to be equal to

the static earth pressure force when the retaining wall slides away from the backfill soil,

i.e.,

( )ae aP t P (7.19)


271

Hence, Equation 7.16 can be written as below:

( )( ) ( ) tanw s a w s a v bF t F t P W W P (7.20)

When the total horizontal driving force exceeds the frictional resistance force between

the base of retaining wall and foundation layer, the retaining wall starts accumulating

relative horizontal displacement. The relative acceleration of retaining wall arel(t)

causing the relative horizontal displacement of the retaining wall can be computed by

dividing the total horizontal driving force exceeding the frictional resistance force by

the mass of the wall-soil system as below:

( )( ) ( ) tan 0( )

w s a w s a v b

rel

w s

F t F t P W W Pa t

m m

(7.21)

where, mw= mass of the cantilever retaining wall and ms= mass of the backfill soil above

the footing slab

Equation 7.20 can also be expressed as below:

2

( )( ) ( ) 0.5 tan 0( )

w w s s s a w s a v

rel

w s

m a t m a t K H W W Pa t

m m

(7.22)

where, aw(t)=acceleration response of the retaining wall, as(t)= acceleration response of

the backfill above base slab, γs= unit weight of backfill soil, H=height of the retaining

wall, Ka= static active earth pressure coefficient, computed using the Rankine’s earth

pressure theory. However, the mathematical expression proposed by Nimbalkar and

Choudhury (2008b) will be used herein in order to account the effect of the

amplification of acceleration response on the relative horizontal displacement of the

wall-soil. The wall and backfill seismic inertia force inertia forces can be computed as

below:

1

( ) ( ) 1 1n

nw wn wn aw

zF t m a t f

H

(7.23)

1

( ) ( ) 1 1n

ns sn sn as

zF t m a t f

H

(7.24)


272

1 1

2

( )

( ) 1 1 ( )

0

1 1 0.5 tan

( )

n nn

wn wn aw sn sn

nas s a w s a v

rel

w s

zm a t f m a t

H

zf K H W W P

Ha t

m m

(7.25)

The double integration of the relative acceleration obtained from Equation (7.25) will

produce the relative horizontal displacement of the cantilever retaining wall-soil system.

For this purpose a code was developed using MATLAB code and the results are

discussed next via a worked example.

7.2.4 Worked example and numerical validation

The current worked example is carried out to compute the relative horizontal

displacement of the cantilever wall-soil system. In order to compare the results

predicted from the current simplified approach with the result obtained from FE

analysis, the current worked example is proposed to predict the relative horizontal

displacement of same the cantilever retaining wall-soil system proposed in Chapter 6.

The cantilever retaining wall consists a 5.4 m stem height and a 3.9 m of base slab

length. The retaining wall is also assumed resting on a rigid foundation layer. The

retaining wall retains a dry cohesionless soil to its full height. A real earthquake-time

history of the Loma Prieta (1989) earthquake (as shown in Figure 5.3a) is used to

simulate the seismic loading. The acceleration response, which was predicted at the base

of the retaining wall during the FE analysis, will be used herein as base excitation.

The friction angle between the retaining wall and foundation layer is assumed to be

equal to 0.5 , which is similar to the one used in the FE analysis. The comparison

between the acceleration response predicted at the top of the retaining wall and backfill

soil and bottom of the retaining wall by FE analysis in Chapter 6 show that the

acceleration response at the top of the wall is amplified by (faw = 2.2). However, the

acceleration response is amplified at the top of backfill soil about (fas = 1.9). The same

amplification factors, faw and fas are used in the current worked example.

Figure 7.8b shows the relative horizontal displacement predicted by the approach, and

its comparison with those computed via the FE analysis (Chapter 6) and the


273

conventional Newmark sliding block method (Newmark, 1965). It is noted that the

relative horizontal displacement (/H) calculated by the proposed analytical approach

(0.0083) is closer to the FE analysis results (0.0056), while too far off from the

Newmark sliding block method result (0.0204). For further validation of the proposed

analytical method, two uniform sinusoidal acceleration time histories with different

amplitudes and frequency contents were used to predict the relative horizontal

displacement of a rigid retaining wall. The first example includes applying a uniform

sinusoidal acceleration time history at the base of retaining wall with amplitude 0.4g

and frequency content 0.5 Hz as shown in Figure 7.8 c; while the second example

includes applying a uniform sinusoidal acceleration time history at the base of retaining

wall with amplitude 0.4g and frequency content 2 Hz as shown in Figure 7.8e. It can be

noted for the first example (see Figure 7.8d) that the relative horizontal displacement

(/H) predicted by current approach is about 0.315 while the relative horizontal

displacement (/H) predicted by the FE analysis and Newmark sliding block method is

about 0.297 and 0.5, respectively. Figure 7.8f shows the relative horizontal

displacements for the later acceleration time history, and it is observed that by the

present analytical method, the relative horizontal displacement (/H) is about 0.059

while the one predicted by the FE analysis was (0.067) but by Newmark method, it was

(0.1074). The comparison between the relative horizontal displacement obtained from

the current simplified procedure and FE result as well as Newmark sliding block

method in two example (see Figure 7.8d and f) shows that the results obtained from the

current simplified procedure are more reasonable than those predicted by Newmark

sliding block method.

The comparison between the result obtained from the current simplified approaches and

those predicted numerically of the rigid retaining wall and the cantilever retaining wall

reveals that the current simplified procedure overestimates the relative horizontal

displacement. This could be justify by fact that the current simplified procedure ignores

the effect of sliding towards the backfill soil as observed in FE analysis as well as the

current simplified approach ignores the deformability of foundation soil in which the

retaining walls experience further resistance to the sliding when they are sliding away

from the backfill soil and their toe embedded in the foundation soil layer.


274

0 5 10 15 20 25 30-0.6

-0.4

-0.2

0.0

0.2

0.4

0 5 10 15 20 25 30-0.025

-0.020

-0.015

-0.010

-0.005

0.000

0 5 10 15 20 25 30-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 5 10 15 20 25 30-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0 1 2 3 4 5 6 7 8-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 1 2 3 4 5 6 7 8

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

a(g

) (m

/sec

2)

Time, t (sec)

W

-F (

/H)

Time, t (sec)

FE (Chapter 6) Current modified method Newmark method

a(g

) (m

/sec

2)

Time, t (sec)

f= 0.5 Hzf= 0.5 Hz

W

-F (

/H)

Time, t (sec)

f= 2 Hza(g

) (m

/sec

2)

Time, t (sec)

f= 2 HzW

-F (

/H)

Time, t (sec)

Figure 7.8: Relative horizontal displacement of the cantilever wall comparison between the

modified Newmark procedure, current study FE results and the classic Newmark sliding block

method

7.3 SUMMARY

The first part of this chapter presented a simplified procedure to evaluate the

contribution of wall seismic inertia force to total shear force and bending moment

developed in the stem of the wall. The results obtained from this analysis shows that the

wall inertia force has a significant contribution to the total shear force and bending

moment at the upper half of the height of the stem while for the lower half of the height

of the stem, a very small contribution of the wall seismic inertia force to total shear

force and bending moment was observed. The second part of this chapter included

modifying the Newmark sliding block method in order to compute the relative

horizontal displacement of a rigid and cantilever retaining walls precisely. The

assumptions made in current modified procedure were based on the observations

obtained from FE analysis in previous chapters. The results obtained from current


275

modified procedures are more reasonable than those computed by Newmark sliding

block method when they are compared with the results obtained from the FE methods

Chapter 8: Conclusions and Recommendations For Future Research

276

CHAPTER 8

CONCLUSIONS AND RECOMMENDATIONS FOR

FUTURE RESEARCH

The main aim of this thesis was to investigate the seismic performance of a rigid and

cantilever retaining wall. In order to achieve this aim, a comprehensive numerical

programme was developed by using the geotechnical FE PLAXIS2D software. To gain

further insight into the seismic behaviour of a rigid and a cantilever retaining wall under

focus, the research also included developing an analytical procedure to estimate the

contribution of wall seismic inertia force to the total shear force and bending moment in

the stem of the cantilever retaining wall. Newmark sliding block method was modified

to compute the relative horizontal displacement of a rigid and cantilever retaining wall.

Overall, this research demonstrated that a rigid retaining wall shows different seismic

performance than a cantilever retaining wall. Hence, it is not appropriate to use the same

traditional methods to analyse a rigid and a cantilever retaining wall. A unique design

chart was proposed to correlate the seismic earth pressure with the displacement of a

rigid retaining wall.

8.1 CONCLUSIONS OF THIS RESEARCH

The following conclusions can be drawn from the undertaken research tasks:

8.1.1 FE modelling of a rigid retaining wall

After validation of the FE model with experimental results obtained from centrifuge

tests available in the literature, a critical analysis of the seismic response of a rigid

retaining wall was performed by using an innovative performance-based method.

Relating to deformation mechanism of the wall-soil system, it was observed that the

retaining wall, backfill soil, and foundation move at the same time in the active and

passive direction under the effect of earthquake acceleration. The results of the FE


277

analysis indicated that the Newmark sliding block method overestimates the relative

displacement of the retaining wall.

The results of FE analysis have also proven the development of seismic active and

passive earth pressure states under the effect of earthquake acceleration.

The M-O method overestimates the seismic active earth pressure force. Further, the

seismic earth pressure force has been found to be highly affected by the seismic

response of retaining wall – a very unique contribution of this research, and is

something which was not addressed by the existing pseudo-static and pseudo-

dynamic methods.

Unique design charts have been developed to correlate the seismic earth pressure

force and the displacement of retaining wall by considering the effect of retaining

wall height and earthquake characteristics (amplitude and frequency content).

The distribution of seismic active and passive earth pressure was observed nonlinear.

The acceleration response at the top of backfill soil is found to be amplified when the

earthquake acceleration is applied with low level while it is de-amplified when the

earthquake acceleration is applied with a high level which larger than 0.4g. The

seismic passive earth pressure is found to be highly affected by this trend.

The seismic active earth pressure is not dependent on the wall displacement while,

on the other hand, the seismic passive earth pressure has been found to be highly

influenced by the wall displacement.

The seismic active earth pressure force is also observed not to be sensitive to the

amplitude and frequency content of earthquake acceleration, while the seismic

passive earth pressure force is found to be highly influenced by both the amplitude

and frequency content of the earthquake acceleration.

The maximum seismic passive earth pressure force is exerted behind the retaining

wall when the earthquake acceleration is applied with minimum frequency content

and maximum amplitude.


278

The critical scenario of the relative horizontal displacement of the retaining wall is

that when earthquake acceleration is applied with minimum frequency content and

maximum amplitude, and it is highly affected by the duration of the earthquake in

contrast to what has been observed for the seismic earth pressure force.

No relationship was observed between the frequency content of earthquake

acceleration and natural frequency of a rigid retaining wall-soil system.

The maximum amplification of acceleration response in the retaining wall was

observed when the backfill and foundation soil has a lower relative density.

The horizontal relative displacement of the retaining wall reduces when the relative

density of backfill and foundation soil is increased.

The seismic active and passive earth pressure increase when the relative density of

backfill soil is increased.

The seismic active earth pressure reduces when the relative density of foundation soil

reduces while the seismic passive earth pressure increases when the relative density

of foundation soil reduces.

8.1.2 FE modelling of a cantilever retaining wall

For the seismic response of a cantilever retaining wall, an innovative

performance-based method was performed in order to evaluate the structural

integrity and global stability of a cantilever retaining wall under the effect of

earthquake acceleration.

The structural integrity and global stability of a cantilever retaining wall have

been investigated by considering the seismic earth pressure, computed at the

stem (Pstem) and as well as along a vertical virtual plane (Pvp), and wall and

backfill seismic inertia forces. Pstem contributes to the structural integrity, while

Pvp contributes to the global stability.

It is observed that Pstem and Pvp are out of phase during the entire duration of the

earthquake and hence, the structural integrity and global stability should be

evaluated individually.


279

The shear force and bending moment are developed in the same trend of seismic

earth pressure behind the stem.

A critical case for the structural integrity is observed when the earthquake

acceleration is applied towards the backfill soil and has frequency content close

to the natural frequency of the retaining wall.

A critical case for the global stability is observed when the earthquake acceleration

has a maximum amplitude and is applied towards the backfill soil with minimum

frequency content. In addition, it is significantly affected by the wall and as well as

soil seismic inertia forces while the seismic earth pressure along the virtual plane is

close to the static earth pressure value.

The results obtained from numerical simulation have shown that the cantilever

retaining wall and backfill soil above base slab move as in single entity.

The number of acceleration cycles of the applied earthquake acceleration does not

affect the seismic earth pressure behind the stem as well as the shear force and

bending moment while the relative displacement between the wall-soil system and

foundation soil is observed to be highly sensitive to this.

It is also observed that the relative density of backfill soil has a considerable

effect on the structural integrity and global stability of a cantilever retaining

wall.

The structural integrity of a cantilever retaining will be reduced when the

backfill soil has high relative density, while the global stability will be increased

when the backfill soil has high relative density during the seismic scenario.

In contrast with the rigid retaining wall, the structural integrity of the cantilever

retaining wall is found to be highly dependent on the ratio of the frequency

content of earthquake acceleration and natural frequency of a cantilever

retaining wall.


280

8.1.3 Analytical methods

Analytical methods have been developed in order to concrete the findings of the

numerical models. A simplified procedure has been proposed to evaluate the

contribution of wall seismic inertia force to total shear force and bending moment

developed in the stem of the wall.

It has been observed that at the time of critical case for the structural integrity of

retaining wall, the wall seismic inertia force has a significant contribution to the

total shear force and bending moment at the upper half of the height of the stem.

However, for the lower half of the height of the stem, a very small contribution

of the wall seismic inertia force to total shear force and bending moment was

observed while the seismic earth pressure has a significant contribution to the

total shear force and bending moment.

When the maximum value of earthquake acceleration is applied away from the

backfill soil, the wall seismic inertia force acts in opposite direction of the seismic

earth pressure causing the reduction of the total shear force and bending moment less

than their static values.

A simplified procedure was also proposed to modify the classic Newmark sliding

block method to compute the relative horizontal displacement between a rigid and

cantilever retaining wall and foundation soil. In contrast to the Richard-Elams

method, no iterative procedure is required in present modified Newmark sliding

block method to compute the relative horizontal displacement of the walls.

The results obtained from present modified Newmark sliding block method are found

to be more reasonable than the results obtained from classic Newmark sliding block

method when compared with the results obtained from FE models.

8.2 RECOMMENDATIONS FOR FUTURE RESEARCH

Not all aspects related to the research were covered during the PhD project. The PhD

project could be extended by:


281

The current study is limited to the simulation of a dry cohesionless soil of backfill

and foundation layer. The effects of saturated soil and liquefied soil and as well as

the cohesive soil on the seismic response of retaining wall can be investigated.

The current study is limited to the investigation of the seismic behaviour of a rigid

and a cantilever retaining wall. The seismic response of other retaining walls like

embedded retaining wall and bridge abutment may also be investigated.

The seismic active and passive earth pressure are found highly affected by the wall

seismic inertia force - a very unique contribution of this research, and is something

which was not addressed by the existing pseudo-static and pseudo-dynamic methods.

Hence, analytical methods may be derived to compute the seismic earth pressure in

order to account the effect of the wall seismic inertia force

For a cantilever retaining wall, it was found that the seismic earth pressure behind the

stem is highly affected by the ratio between the frequency content of earthquake

acceleration to the natural frequency of the retaining wall and wall seismic inertia

force. Efforts will be required to derive methods to compute the seismic earth

pressure behind the stem taking into account the effect of these important parameters.

Development of a simplified procedure by using the extensive parametric study with

to include the retaining wall, relative density and earthquake characteristics effect for

the amplification factor proposed in Chapter 7 to investigate any further

improvement could be provided for modified Newmark sliding block method.

References

282

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Appendix A

290

APPENDIX A

RESULT OF THE FINITE ELEMENT ANALYSIS OF A RIGID RETAINING WALL

Table A.1: Effect of retaining wall height on seismic response of wall-soil system considering different amplitudes of the applied earthquake acceleration

a (g)

Wall displacement (m) Wall rotation (degree) Acceleration (a(g)) and

wall seismic inertia force (kN/m)

Seismic earth pressure force

FE model

(kN/m)

M-O theory

(kN/m)

top_wall

active top_wall

passive Sliding active passive residual aactive Fa apassive Fp Pa Pae

Ppe

Pre

Pae

Ppe

Retaining wall height, H = 4 m

0.1g 0.064 0.047 0.003 0.048 0.022 0.035 0.22 47.01 0.25 53.25 43.28 44.13 85.32 62.34 46.45 1031

0.2g 0.129 0.091 0.011 0.074 0.045 0.096 0.33 72.21 0.44 95.04 43.28 42.11 93.11 65.35 57.72 941.9

0.3g 0.209 0.137 0.029 0.199 0.074 0.271 0.41 90.23 0.76 164.2 43.28 40.42 136.5 69.32 72.09 849.5

0.4g 0.297 0.168 0.071 0.374 0.094 0.575 0.48 96.7 0.91 196.6 43.28 37.47 148.9 72.38 91.077 752.6

0.5g 0.391 0.169 0.162 0.665 0.137 1.221 0.53 114.5 0.95 205.2 43.28 37.22 153.2 67.89 117.66 648.1

0.6g 0.493 0.165 0.291 1.206 0.149 2.156 0.61 131.8 0.93 200.9 43.28 37.17 133.9 67.51 159.79 527.9


0.1g 0.074 0.054 0.015 0.081 0.044 0.116 0.149 92.98 0.245 152.9 187.3 192.2 258.3 240.7 185.8 4124

Appendix A

291

0.2g 0.141 0.096 0.038 0.172 0.057 0.228 0.228 142.3 0.417 260.2 187.3 188.1 322.2 240.9 230.9 3767

0.3g 0.221 0.146 0.089 0.211 0.131 0.366 0.289 180.3 0.583 363.8 187.6 173.2 366.9 267.9 288.39 3398

0.4g 0.284 0.173 0.175 0.194 0.148 0.511 0.379 236.5 0.659 411.2 187.4 175.4 384.9 298.1 364.3 3010

0.5g 0.374 0.217 0.291 0.236 0.201 0.701 0.402 250.9 0.784 489.2 187.4 165.8 402.2 302.2 470.63 2592

0.6g 0.428 0.2147 0.408 0.307 0.203 0.891 0.419 261.5 0.868 541.6 187.4 184.5 406.5 304.4 639.2 2111


0.1g 0.083 0.052 0.064 0.092 0.027 0.367 0.116 125.3 0.222 239.8 359.2 351.6 558.1 409.7 418.1 9280

0.2g 0.165 0.089 0.136 0.182 0.067 0.721 0.207 223.6 0.371 400.7 359.6 352.4 672.8 433.1 519.5 8477

0.3g 0.259 0.138 0.225 0.316 0.112 1.152 0.274 295.9 0.499 538.9 359.4 334.8 768.5 439.4 648.9 7646

0.4g 0.339 0.152 0.332 0.416 0.134 1.596 0.324 349.9 0.582 628.6 359.6 324.3 853.9 484.5 819.7 6773

0.5g 0.437 0.204 0.461 0.572 0.229 2.029 0.387 417.9 0.691 746.3 359.6 322.4 897.4 519.2 1058.9 5832

0.6g 0.489 0.208 0.613 0.503 0.228 2.528 0.429 463.3 0.723 780.8 359.6 321.1 910.6 529.1 1438.2 4751

Appendix A

292

Table A.2 : Effect of frequency content of earthquake acceleration with different acceleration amplitude.

f (Hz)

Horizontal displacement

(m) Rotation (degree)

Acceleration and wall seismic inertia

force


FE model

(kN/m)

M-O theory

(kN/m)

top_wall

active

top_wall

passive

Sliding

(/H) active passive residual

a(g)

CG

active

Fa

kN/m

a(g)

CG

passive

Fp

kN/m Pa Pae

Ppe

Pre

Pae

Ppe

earthquake acceleration amplitude , a = 0.2g

0.33 0.533 0.799 0.040 0.246 0.028 1.6 0.211 45.58 0.223 50.33 45.26 41.6 138.6 68.25 57.72 941.95

0.66 0.139 0.2371 0.003 0.037 0.026 0.13 0.213 46.11 0.251 54.22 45.26 55.17 107.5 75.75 57.72 941.95

1 0.067 0.104 0.003 0.039 0.027 0.14 0.231 49.89 0.231 49.89 45.26 51.50 88.74 73.16 57.72 941.95

2 0.021 0.031 0.002 0.038 0.031 0.081 0.236 50.98 0.215 46.44 45.26 47.36 86.48 71.23 57.72 941.95

3 0.017 0.017 0.008 0.037 0.027 0.11 0.232 50.11 0.236 50.98 45.26 51.48 94.37 58.23 57.72 941.95


0.33 1.05 1.17 0.340 0.862 0.077 9.012 0.304 65.66 0.495 106.9 45.26 51.68 153.1 74.16 57.72 941.95

0.66 0.289 0.385 0.088 0.256 0.058 1.726 0.312 67.39 0.472 101.9 45.26 49.82 130.8 70.54 57.72 941.95

1 0.137 0.188 0.054 0.123 0.054 0.606 0.313 67.61 0.405 87.48 45.26 50.91 106.1 66.08 57.72 941.95

Appendix A

293

2 0.048 0.0513 0.338 0.091 0.077 0.175 0.319 68.91 0.489 105.6 45.26 36.66 103.3 64.67 57.72 941.95

3 0.042 0.019 0.056 0.105 0.091 0.134 0.445 96.12 0.722 155.9 45.26 65.52 143.8 83.69 57.72 941.95


0.33 1.464 1.517 0.813 1.861 0.51 11.247 0.392 84.67 0.758 163.7 45.26 51.68 153.1 74.16 57.72 941.95

0.66 0.434 0.448 0.251 0.469 0.159 3.488 0.394 85.11 0.709 153.1 45.26 49.82 130.8 70.54 57.72 941.95

1 0.218 0.222 0.178 0.149 0.095 1.221 0.382 82.51 0.658 142.2 45.26 50.91 106.1 66.08 57.72 941.95

2 0.077 0.056 0.088 0.129 0.122 0.124 0.348 75.17 0.754 162.9 45.26 36.66 103.3 64.67 57.72 941.95

3 0.051 0.019 0.088 0.146 0.131 0.011 0.438 94.61 0.742 160.3 45.26 65.52 143.8 83.69 57.72 941.95

Appendix A

294

Table A.3: Effect of soil material relative density on the seismic response of wall-soil system

Dr



Acceleration and wall seismic

inertia force


FE model

(kN/m)

M-O theory

(kN/m)

top_wal

active

top_wall

passive

Sliding


a(g)

CG

active

Fa

kN/m

a(g)

CG

passive

Fp

kN/m Pa Pae

Ppe

Pre

Pae

Ppe

40 % 0.051 0.051 0.034 0.412 0.3636 0.503 0.312 67.39 0.422 91.15 38.06 35.71 91.88 54.62 66.76 510.3

65 % 0.039 0.038 0.021 0.141 0.111 0.321 0.307 66.32 0.419 90.51 39.02 41.88 97.64 57.89 65.62 745.3

85 % 0.031 0.031 0.009 0.084 0.056 0.222 0.302 65.23 0.347 47.95 40.25 47.25 103.87 67.51 60.67 1299

Appendix A

295

Table A.4: Effect of backfill soil relative density on the seismic response of wall-soil system

Dr




force


FE model

(kN/m)

M-O theory

(kN/m)

top_wall

active

top_wall

passive

Sliding


a(g)

CG

active

Fa

kN/m

a(g)

CG

passive

Fp

kN/m Pa Pae

Ppe

Pre

Pae

Ppe

40 % 0.067 0.029 0.096 0.275 0.259 0.007 0.545 117.7 0.6433 138.9 45.64 38.81 71.94 48.75 66.76 510.3

65 % 0.039 0.038 0.021 0.141 0.111 0.321 0.307 66.32 0.419 90.51 39.02 41.88 97.64 57.89 65.62 745.3

85 % 0.034 0.034 0.019 0.151 0.107 0.349 0.295 63.72 0.404 87.62 38.93 45.03 109.48 67.99 60.67 1299

Appendix A

296

Table A.5: Effect of foundation soil relative density on the seismic response of wall-soil system

Dr




force


FE model

(kN/m)

M-O theory

(kN/m)

top_wall

active

top_wall

passive

Sliding


a(g)

CG

active

Fa

kN/m

a(g)

CG

passive

Fp

kN/m Pa Pae

Ppe

Pre

Pae

Ppe

40 % 0.081 0.036 0.086 0.353 0.206 1.439 0.502 108.4 0.633 136.7 38.668 36.04 128.25 68.32 65.62 745.3

65 % 0.039 0.038 0.021 0.141 0.111 0.321 0.307 66.32 0.419 90.51 39.02 41.88 97.64 57.89 65.62 745.3

85 % 0.036 0.042 0.016 0.129 0.088 0.276 0.296 63.93 0.369 79.71 42.56 49.25 104.04 64.67 65.62 745.3

Appendix B

297

APPENDIX B

MATLAB PROGRAMS

B.1 MATLAB PROGRAM FOR COMPUTING THE RELATIVE

HORIZONTAL DISPLACEMENT OF A RIGID RETAINING WALL

…………………………………………………………………………………………….

d=xlsread('base acceleration_rigid wall.xlsx',1,'A1:D6000'); for i=1:y t(i)=d(i,1); % time a(i)=d(i,2); % base acceleration FR(i)=d(i,3); % base resistance force Pa(i)=d(i,4); % static earth pressure force end

%=====================================================================

n=8; % number of the parts of retaining wall height b1=3; % base width of retaining wall b2=1.5; % to width of the retaining wall H=4; fa=1; % amplification factor

gamma wall= 24; % unit weight of the wall g=9.81;

mw= (b1+b2)*0.5* H /g

rel_a = [0]; % relative acceleration rel_vel = [0]; % relative velocity rel_dis = [0]; % relative displacement t = [0]; % time

for i=1:6000 for w=1:n f(i,w)=gamma wall*((((b1-b2)/H)*(H-(H/n)*w)+((b1-b2)/H)*(H-

(H/n)*(w-1)))*(0.5*H/n)+b2*(H/n))*(a(i)*(1+((((H/n)*w)/H)*(fa-1))));

end end F=sum(f,2); % wall seismic inertia force

for i=2:6000

time(i) = tm(i); delt = time(i)-time(i-1); % time step rel_a(i)=((F(i)+ (Pa(i)-FR(i))))/(mw); % relative acceleration

rel_vel(i) = rel_vel(i-1) + 0.5*1*(rel_a(i-1)+rel_a(i))*delt;

% relative velocity

if(rel_a(i)<0); rel_vel(i)=0;

Appendix B

298

rel_a(i)=0; end

rel_dis(i) = rel_dis(i-1) + rel_vel(i-1)*delt +(2*rel_acc(i-

1)+rel_acc(i))*delt*delt/6; % Intergrating displacement to 3rd order FT(i)= Fh(i)+Pa(i); end figure(1); plot(time,a,'-k','LineWidth',2); grid on;grid minor; xlabel('Time [sec]'); ylabel('Earthquake Acceleration of Base (a_Database ()) [g]'); title ('Given Acceleration Time History');

figure(2); plot(time,rel_ac,'-k','LineWidth',2); grid on;grid minor; xlabel('Time [sec]'); ylabel('Relative Acceleration (a_{rel}) [g]'); title ('Relative Acceleratio');

figure(3); plot(time,rel_vel,'-k','LineWidth',2); grid on;grid minor; xlabel('Time [sec]'); ylabel('Relative Velocity(v_{rel}) [m/sec]'); title ('Relative Velocity ');

figure(4); plot(time,rel_dis,'-k','LineWidth',2); grid on;grid minor; xlabel('Time [sec]'); ylabel('Relative Displacement(\delta_{rel}) [m]'); title ('Relative Displacement ');

Appendix B

299

B.2 MATLAB PROGRAM FOR COMPUTING THE RELATIVE

HORIZONTAL DISPLACEMENT OF A CANTILEVER RETAINING WALL

…………………………………………………………………………………………….

d=xlsread('base acceleration_cantilever wall.xlsx',1,'A1:D6000'); for i=1:6000 tm(i)=d(i,1); % time a(i)=d(i,2); % base acceleration FR(i)=d(i,3); % base resistance force Pa(i)=d(i,4); % static earth pressure force end

%=====================================================================

g = 9.81; % gravity constant H=5.4; % height of the wall t1=0.22; % thickness of the stem b1=3.9; % length of base slab b2=2.6; % width of backfill soil above base slab t2=0.22; % thickness of base slab gw=26.6; % unit weight of the wall gs=14.23; % unit weight of the soil

Ww=H*t1*gw; % weight of the stem Wb=b1*t2*gw; % weight of the base slab Ws=H*b2*gs; % weight of backfill soil above base slab Wt =Ww+Wb+Ws; % total weight of the wall-soil system faw=1.84; % amplification factor of the wall fas=1.84; % amplification factor of the soil

a = [0]; % base acceleration rel_a = [0]; % relative acceleration rel_vel = [0]; % relative velocity rel_dis = [0]; % relative displacement time = [0]; % time

for i=1:6000 n=9; % number of parts of stem height for y=1:n fw(i,y)=(Ww/n)*(a(i)*(1+((((H/n)*y)/H)*(faw-1)))); fs(i,y)=(Ws/n)*(a(i)*(1+((((H/n)*y)/H)*(fas-1)))); end Fb(i)=Wb*a(i); end Fw=sum(fw,2); % wall seismic inertia force Fs=sum(fs,2); % backfill seismic inertia force for i=2:6000 time(i) = tm(i); delt = time(i)-time(i-1);

rel_acc(i) = (Fw(i)+Fb(i)+Fs(i)+Pa(i)-FR(i))/(254);

Appendix B

300

rel_vel(i) = rel_vel(i-1) + 0.5*g*(rel_acc(i-

1)+rel_acc(i))*delt; if (rel_vel(i)<0); rel_acc(i)=0; % relative acceleration rel_vel(i)=0; % relative velocity end

rel_dis(i) = rel_dis(i-1) + rel_vel(i-1)*delt +(2*rel_acc(i-

1)+rel_acc(i))*delt*delt/6; % Intergrating displacement to 3rd order

end figure(1); plot(time,a,'-k','LineWidth',2); grid on;grid minor; xlabel('Time [s]'); ylabel('Earthquake Acceleration of Base (a_Database ()) [g]'); title ('Given Acceleration Time History');

figure(2); plot(time,rel_a,'-k','LineWidth',2); grid on;grid minor; xlabel('Time [sec]'); ylabel('Relative Acceleration (a_{rel}) [g]'); title ('Relative Acceleration);

figure(3); plot(time,rel_vel,'-k','LineWidth',2); grid on;grid minor; xlabel('Time [sec]'); ylabel('Relative Velocity (v_{rel}) [m/sec]'); title ('Relative Velocity');

figure(4); plot(time,rel_dis,'-k','LineWidth',2); grid on;grid minor; xlabel('Time [sec]'); ylabel('Relative Displacement (\delta_{rel}) [m]'); title ('Relative Displacement');

DISPLACEMENT BASED APPROACH FOR SEISMIC STABILITY OF ...

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