DISPLACEMENT BASED APPROACH FOR SEISMIC STABILITY OF ...
Transcript of 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
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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
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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 Analytical methods ......................................................................................... 67
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
2.5.1.2 Numerical methods ......................................................................................... 75
2.5.1.3 Experimental methods ..................................................................................... 78
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
2.5.2.1 Analytical methods ......................................................................................... 88
List of Contents
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2.5.2.1.1 One-block methods ................................................................................... 88
2.5.2.1.2 Two-block methods ................................................................................... 90
2.5.2.2 Numerical methods ......................................................................................... 91
2.5.2.3 Experimental methods ..................................................................................... 94
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
2.5.3.1 Analytical 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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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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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
5.5.2.1 Acceleration response ................................................................................... 188
5.5.2.2 Relative horizontal displacement of the retaining wall ................................. 190
5.5.2.3 Seismic earth pressure force ......................................................................... 192
5.5.2.4 Relationship between seismic earth pressure and displacement of retaining
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
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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
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6.4.1 Effect of earthquake characteristics .................................................................. 233
6.4.1.1 Acceleration response ................................................................................... 233
6.4.1.2 Seismic earth pressure ................................................................................... 235
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
6.4.2.1 Acceleration response ................................................................................... 239
6.4.2.2 Seismic earth pressure force Pstem ................................................................. 241
6.4.2.3 Seismic earth pressure force Pvp .................................................................... 243
6.4.2.4 Shear force and bending moment .................................................................. 244
6.4.2.5 Relative horizontal displacement of retaining wall W-F ............................... 246
6.4.3 Effect of relative density of soil ........................................................................ 247
6.4.3.1 Acceleration response ................................................................................... 248
6.4.3.2 Seismic earth pressure ................................................................................... 249
6.4.3.3 Shear force and bending moment .................................................................. 250
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
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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
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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
Figure 4.14: a) Sinusoidal wave applied at the base of the Nakamura (2006) test and the FE
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
Figure 5.3: Real earthquake time history of the 1989 Loma Prieta earthquake: a) acceleration ,
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 6.15: Acceleration response at the top of retaining wall and backfill soil for different
amplitudes and frequency content of the applied earthquake acceleration ............................... 235
Figure 6.16: Seismic earth pressure force behind the stem and along the virtual line for different
amplitudes and frequency content of the applied earthquake acceleration ............................... 236
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
amplitudes and frequency content of the applied earthquake acceleration ............................... 238
Figure 6.19: Acceleration response at the top of retaining wall and backfill soil for different
amplitudes and frequency content of the applied earthquake acceleration ............................... 241
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
Chapter 1: Introduction
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)
Chapter 1: Introduction
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;
Chapter 1: Introduction
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.
Chapter 1: Introduction
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,
Chapter 1: Introduction
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
Chapter 2: Literature Review
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
Chapter 2: Literature Review
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
Chapter 2: Literature Review
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.
Chapter 2: Literature Review
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)
Chapter 2: Literature Review
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
Chapter 2: Literature Review
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.
Chapter 2: Literature Review
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.
Chapter 2: Literature Review
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
Chapter 2: Literature Review
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
Chapter 2: Literature Review
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
Chapter 2: Literature Review
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
Chapter 2: Literature Review
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.
Chapter 2: Literature Review
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.
Chapter 2: Literature Review
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
Chapter 2: Literature Review
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
Chapter 2: Literature Review
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)
Chapter 2: Literature Review
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.
Chapter 2: Literature Review
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.
Chapter 2: Literature Review
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)
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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.
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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.
Chapter 2: Literature Review
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).
Chapter 2: Literature Review
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)
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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
Chapter 2: Literature Review
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.
Chapter 2: Literature Review
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
Chapter 2: Literature Review
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.
Chapter 2: Literature Review
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
Chapter 2: Literature Review
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.
2.5.1.1 Analytical methods
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.
Chapter 2: Literature Review
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)
Chapter 2: Literature Review
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)
Chapter 2: Literature Review
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
Chapter 2: Literature Review
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-
Chapter 2: Literature Review
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:
Chapter 2: Literature Review
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)
Chapter 2: Literature Review
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
Chapter 2: Literature Review
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.
2.5.1.2 Numerical methods
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.
Chapter 2: Literature Review
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.
Chapter 2: Literature Review
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).
Chapter 2: Literature Review
78
2.5.1.3 Experimental methods
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.
Chapter 2: Literature Review
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
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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.
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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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82
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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83
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)
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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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85
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
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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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87
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
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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.
2.5.2.1 Analytical methods
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:
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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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90
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.
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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.
2.5.2.2 Numerical methods
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
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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.
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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,
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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 Experimental methods
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
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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.
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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
Chapter 2: Literature Review
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.
2.5.3.1 Analytical methods
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
Chapter 2: Literature Review
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
Chapter 2: Literature Review
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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100
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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101
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.
Chapter 2: Literature Review
102
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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103
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.
2.5.3.2 Numerical methods
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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104
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.
Chapter 2: Literature Review
105
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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106
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.
2.5.3.3 Experimental methods
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:
Chapter 2: Literature Review
107
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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108
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
Chapter 2: Literature Review
109
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.
Chapter 2: Literature Review
110
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
Chapter 2: Literature Review
111
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:
Chapter 2: Literature Review
112
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)
Chapter 2: Literature Review
113
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)
Chapter 2: Literature Review
114
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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115
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.
Chapter 2: Literature Review
116
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
Chapter 2: Literature Review
117
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
118
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)
Chapter 3: Finite Element Modelling Methodology
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
Chapter 3: Finite Element Modelling Methodology
120
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
Chapter 3: Finite Element Modelling Methodology
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
Earthquake acceleration
Stem
Base slab
6-noded triangularelement
Interfaceelement
(a)
(b)
Chapter 3: Finite Element Modelling Methodology
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.
Chapter 3: Finite Element Modelling Methodology
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)
Chapter 3: Finite Element Modelling Methodology
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
Chapter 3: Finite Element Modelling Methodology
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,
Chapter 3: Finite Element Modelling Methodology
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
Chapter 3: Finite Element Modelling Methodology
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.
Chapter 3: Finite Element Modelling Methodology
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
‗
‗
Chapter 3: Finite Element Modelling Methodology
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.
Chapter 3: Finite Element Modelling Methodology
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)
Chapter 3: Finite Element Modelling Methodology
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.
Chapter 3: Finite Element Modelling Methodology
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.,
Chapter 3: Finite Element Modelling Methodology
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 -
Chapter 3: Finite Element Modelling Methodology
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
Chapter 3: Finite Element Modelling Methodology
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
Chapter 3: Finite Element Modelling Methodology
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)
Figure 3.12: Real earthquake time history of the 1989 Loma Prieta earthquake: a) acceleration ,
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:
Chapter 3: Finite Element Modelling Methodology
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.
Chapter 3: Finite Element Modelling Methodology
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
Chapter 4: Validation of FE Model
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
Chapter 4: Validation of FE Model
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
Chapter 4: Validation of FE Model
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)
Chapter 4: Validation of FE Model
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.
Chapter 4: Validation of FE Model
144
Figure 4.4: Finite element model of Saito (1999) centrifuge test
4.3.2 Nakamura (2006) 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
Chapter 4: Validation of FE Model
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
Chapter 4: Validation of FE Model
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.
Chapter 4: Validation of FE Model
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
4.4.2 Nakamura (2006) test
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)
Chapter 4: Validation of FE Model
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)
Chapter 4: Validation of FE Model
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.
Chapter 4: Validation of FE Model
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.
Chapter 4: Validation of FE Model
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)
Chapter 4: Validation of FE Model
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)
Chapter 4: Validation of FE Model
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)
Chapter 4: Validation of FE Model
154
4.6.2 Nakamura (2006) test
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)
Chapter 4: Validation of FE Model
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)
Figure 4.12: a) Sinusoidal wave applied at the base of the Nakamura (2006) test and the FE
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.
Chapter 4: Validation of FE Model
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)
Chapter 4: Validation of FE Model
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)
Figure 4.14: a) Sinusoidal wave applied at the base of the Nakamura (2006) test and the FE
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)
Chapter 4: Validation of FE Model
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
Chapter 4: Validation of FE Model
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
Chapter 4: Validation of FE Model
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
Chapter 4: Validation of FE Model
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;
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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.
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
Earthquake acceleration
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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)
Figure 5.3: Real earthquake time history of the 1989 Loma Prieta earthquake: a) acceleration ,
b) frequency domain
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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.
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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)
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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.
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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)
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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.
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
Chapter 5: Finite Element Analysis of A Rigid 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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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.
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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.
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
Earth pressure (kN/m2)
No
rma
lise
d h
eig
ht
(z/H
)
Earth pressure (kN/m2)
No
rma
lise
d h
eig
ht
(z/H
)
Coulomb's theory
At-rest state po
M-O method pae
Finite element ppe
Earth pressure (kN/m2)
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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.
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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).
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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.
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
5.5.1.2 Relationship between seismic earth pressure and displacement of retaining
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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.
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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.
5.5.2.1 Acceleration response
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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.
Chapter 5: Finite Element Analysis of A Rigid 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)
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
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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):Towards the backfill - a(g): Away from the backfill
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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.
5.5.2.4 Relationship between seismic earth pressure and displacement of retaining
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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.
Chapter 5: Finite Element Analysis of A Rigid 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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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.
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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)
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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.
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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.
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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%).
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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%).
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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
Chapter 5: Finite Element Analysis of A Rigid Retaining Wall
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.
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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)
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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.
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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)
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
Earthquake acceleration
Stem
Base slab
6-noded triangularelement
Interfaceelement
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
Earth pressure (kN/m2)
Norm
alis
ed
heig
ht (z
/H)
t = 0 sec t = 3.9 sec
(d)(c)
(b)
Earth pressure (kN/m2)
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
Earth pressure (kN/m2)
Norm
alis
ed
heig
ht (z
/H)
t = 30 sec
Earth pressure (kN/m2)
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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.
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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:
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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.
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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.
The stem rotates by 0.017°
away from the backfill soil,
while the base slab toe
rotates by 0.01° in to the
foundation soil. Also, the
retaining wall slides away
from the backfill soil about
0.017 m.
The stem rotates by 0.204°
away from the backfill soil,
while the base slab toe
rotates by 0.038° in to the
foundation soil. Also, the
retaining wall slides away
from the backfill soil about
0.035 m.
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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.
6.4.1.1 Acceleration response
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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).
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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 6.15: Acceleration response at the top of retaining wall and backfill soil for different
amplitudes and frequency content of the applied earthquake acceleration
6.4.1.2 Seismic earth pressure
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
amplitudes and frequency content of the applied earthquake acceleration
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
+ a(g):Towards the backfill - a(g): Away from the backfill
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
amplitudes and frequency content of the applied earthquake acceleration
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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.
6.4.2.1 Acceleration response
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-
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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.
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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)
Wall height (H) = 5.4 m (fwn
= 4.5 Hz)
Figure 6.19: Acceleration response at the top of retaining wall and backfill soil for different
amplitudes and frequency content of the applied earthquake acceleration
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.
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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.
Chapter 6: Finite Element Analysis of A Cantilever 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
Wall height (H) = 10.8 m (fwn
= 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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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).
6.4.2.4 Shear force and bending moment
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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.
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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.
6.4.3.1 Acceleration response
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
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
249
6.4.3.2 Seismic earth pressure
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.
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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
6.4.3.3 Shear force and bending moment
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.
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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.
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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.
Chapter 6: Finite Element Analysis of A Cantilever Retaining Wall
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.
Chapter 7: Analytical Methods
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.
Chapter 7: Analytical Methods
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.
Chapter 7: Analytical Methods
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.
Chapter 7: Analytical Methods
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
Chapter 7: Analytical Methods
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
Chapter 7: Analytical Methods
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
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 -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
* negative values indicate Fwa, while all others are Fwp.
Chapter 7: Analytical Methods
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
Chapter 7: Analytical Methods
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
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 -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
* negative values indicate Fwa, while all others are Fwp.
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
Chapter 7: Analytical Methods
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
Chapter 7: Analytical Methods
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
Chapter 7: Analytical Methods
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:
Chapter 7: Analytical Methods
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)
Chapter 7: Analytical Methods
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
Chapter 7: Analytical Methods
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.
Chapter 7: Analytical Methods
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:
Chapter 7: Analytical Methods
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)
Chapter 7: Analytical Methods
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)
Chapter 7: Analytical Methods
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
Chapter 7: Analytical Methods
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.
Chapter 7: Analytical Methods
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
Chapter 7: Analytical Methods
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
Chapter 8: Conclusions and Recommendations For Future Research
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.
Chapter 8: Conclusions and Recommendations For Future Research
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.
Chapter 8: Conclusions and Recommendations For Future Research
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.
Chapter 8: Conclusions and Recommendations For Future Research
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:
Chapter 8: Conclusions and Recommendations For Future Research
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
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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
Retaining wall height, H = 8 m
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
Retaining wall height, H = 12 m
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
Seismic earth pressure 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
earthquake acceleration amplitude , a = 0.4g
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
earthquake acceleration amplitude , a = 0.6g
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
Horizontal displacement
(m) Rotation (degree)
Acceleration and wall seismic
inertia force
Seismic earth pressure force
FE model
(kN/m)
M-O theory
(kN/m)
top_wal
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
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
Horizontal displacement
(m) Rotation (degree)
Acceleration and wall seismic inertia
force
Seismic earth pressure 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
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
Horizontal displacement
(m) Rotation (degree)
Acceleration and wall seismic inertia
force
Seismic earth pressure 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
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');