Development of time integration schemes and advanced boundary ...

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Development of time integration schemes and

advanced boundary conditions for dynamic

geotechnical analysis

A thesis submitted to the University of London

for the degree of Doctor of Philosophy and for the Diploma of

the Imperial College of Science, Technology and Medicine

By

Stavroula Kontoe

Department of Civil and Environmental Engineering

Imperial College of Science, Technology and Medicine

London, SW7 2BU

May 2006

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ABSTRACT

This thesis details the three major developments that were undertaken to extend

the dynamic capabilities of the finite element program ICFEP (Imperial College

Finite Element Program).

The first development concerns the implementation of a new time integration

scheme. This was chosen to be the generalized-α algorithm (CH) of Chung &

Hulbert (1993), which has to date only been used in the field of structural

dynamics. This scheme is unconditionally stable, second order accurate and

possesses controllable numerical dissipation. The CH algorithm was further

developed to deal with dynamic coupled consolidation problems. The

implementation of the scheme was verified with closed form solutions for both

uncoupled and coupled consolidation problems. The behaviour of the CH scheme

was also compared with more commonly used schemes in a nonlinear problem of

a deep foundation subjected to various dynamic loadings.

The second development involves the incorporation of absorbing boundary

conditions, which can model the radiation of energy towards infinity in a

truncated domain. Hence, the standard viscous boundary of Lysmer and

Kuhlemeyer (1969) and the cone boundary of Kellezi (1998, 2000) were

implemented and validated with closed form solutions and numerical examples

from the literature.

The last development concerns the implementation of the domain reduction

method (DRM). The DRM is a two-step procedure that aims at reducing the

domain that has to be modelled numerically by a change of variables. The

seismic excitation is introduced directly into the computational domain and the

artificial boundary is needed only to absorb the scattered energy of the system.

The method was further developed to deal with dynamic coupled consolidation

problems. In addition, the performance of the absorbing boundary conditions

under the DRM framework is examined.

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The final topic of the thesis concerns dynamic and quasi static FE analyses of a

highway tunnel response during the 1999 Duzce earthquake in Turkey. The

analyses were performed using measured strong ground motion for two cross-

sections and the results were compared with observed behaviour and simplified

analytical methods of analysis.

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ACKNOWLEDGEMENTS

First and foremost I would like to express my sincere gratitude to my supervisors

Dr. L. Zdravković and Prof. D.M. Potts. Their guidance, encouragement and

assistance have been unfailing throughout the period of this work.

The research presented in this thesis was funded by the Soil Mechanics section at

Imperial College. This support is gratefully acknowledged.

Many thanks are due to Dr. S.J. Hardy for his help and guidance during the early

stages of this project. I am also grateful to Dr. F. Strasser for giving me helpful

advice and providing me with seismic records. I would also like to thank Dr.

C.O. Menkiti, from the Geotechnical Consulting Group, for providing data for

the Bolu tunnel case study and valuable discussions.

Being part of the research group in the Soil Mechanics section at Imperial

College has been a great experience. Thanks are due to all academic staff and

students in the section for creating such an enjoyable and unique working

environment. Special thanks are due to my colleagues in the numerical group for

sharing happy as well as anxious times.

Finally, my deepest gratitude is kept for the people closest to me. Without their

encouragement, love and understanding I would not have been able to complete

this thesis.

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TABLE OF CONTENTS

ABSTRACT........................................................................................2

ACKNOWLEDGEMENTS...............................................................4

TABLE OF CONTENTS...................................................................5

LIST OF FIGURES .........................................................................11

LIST OF TABLES ...........................................................................23

LIST OF SYMBOLS .......................................................................25

Chapter 1: .........................................................................................35

INTRODUCTION............................................................................35

1.1 General ................................................................................................35

1.2 Scope of research................................................................................36

1.3 Layout of thesis...................................................................................37

Chapter 2: .........................................................................................40

FINITE ELEMENT THEORY.......................................................40

2.1 Introduction ........................................................................................40

2.2 The Finite Element Method for Static Problems ............................41

2.2.1 Element Discretisation .................................................................41

2.2.2 Primary variable approximation...................................................42

2.2.3 Element Equations .......................................................................44

2.2.4 Global equations ..........................................................................50

2.2.5 Boundary conditions ....................................................................51

2.2.6 Solution of the global equations...................................................51

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2.2.7 Nonlinear finite element theory ...................................................51

2.2.8 Consolidation theory ....................................................................54

2.3 Summary.............................................................................................58

Chapter 3: .........................................................................................60

DYNAMIC FINITE ELEMENT FORMULATION....................60

3.1 Introduction ........................................................................................60

3.2 Finite element formulation of the equation of motion ....................60

3.2.1 Constitutive soil models...............................................................67

3.2.2 Spatial discretization ....................................................................68

3.3 Direct integration method .................................................................69

3.3.1 Characteristics of integration schemes.........................................70

3.3.2 Houbolt method............................................................................76

3.3.3 Park method .................................................................................77

3.3.4 Newmark method.........................................................................77

3.3.5 Quadratic acceleration method.....................................................81

3.3.6 Wilson θ-method..........................................................................82

3.3.7 Collocation method ......................................................................83

3.3.8 HHT method ................................................................................84

3.3.9 WBZ method................................................................................86

3.3.10 Generalized-α method ..................................................................87

3.3.11 Other schemes ..............................................................................90

3.3.12 Comparative study of integration schemes ..................................90

3.3.13 The generalized-α method in dynamic nonlinear analysis...........97

3.4 Dynamic consolidation theory...........................................................99

3.4.1 Dynamic finite element formulation for coupled problems.......100

3.4.2 Implementation of the CH method for coupled problems .........105

3.5 Summary...........................................................................................107

Chapter 4: .......................................................................................108

NUMERICAL INVESTIGATION OF THE CH METHOD ....108

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4.1 Introduction ......................................................................................108

4.2 Validation Exercises.........................................................................109

4.2.1 Harmonically forced single degree of freedom system..............109

4.2.2 Consolidating elastic soil layer subjected to cyclic loading.......115

4.2.3 Consolidating elastic soil layer subjected to a step load............120

4.3 Performance of the CH method in a boundary value problem ...127

4.3.1 Description of the numerical model...........................................128

4.3.2 Input ground motion...................................................................130

4.3.3 Parametric study of the CH algorithm .......................................131

4.3.4 Analyses for various excitations ................................................134

4.3.5 Analyses for various soil properties...........................................140

4.3.6 Computational cost.....................................................................141

4.4 Summary...........................................................................................142

Chapter 5: .......................................................................................145

ABSORBING BOUNDARY CONDITIONS...............................145

5.1 Introduction ......................................................................................145

5.2 Literature review..............................................................................146

5.2.1 Statement of the problem ...........................................................146

5.2.2 Local boundaries ........................................................................152

5.2.3 Consistent boundaries ................................................................169

5.3 Standard viscous boundary.............................................................170

5.3.1 Theory ........................................................................................170

5.3.2 Implementation ..........................................................................174

5.4 Cone Boundary.................................................................................176

5.4.1 Theory ........................................................................................176

5.4.2 Implementation ..........................................................................182

5.5 Verification and validation of absorbing boundary conditions ...185

5.5.1 Plane strain analysis ...................................................................185

5.5.2 Axisymmetric analysis ...............................................................196

5.5.3 Rayleigh wave absorption ..........................................................204

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5.5.4 Soil layer with vertically varying stiffness.................................213

5.5.5 Nonlinear waves.........................................................................218

5.6 Conclusions .......................................................................................219

Chapter 6: .......................................................................................221

DOMAIN REDUCTION METHOD............................................221

6.1 Introduction ......................................................................................221

6.2 Theoretical background to the method ..........................................222

6.2.1 Literature Review.......................................................................222

6.2.2 Formulation of the Domain Reduction Method.........................224

6.3 Formulation of the DRM for dynamic coupled consolidation

analysis ..............................................................................................230

6.4 Verification and validation of the DRM.........................................237

6.4.1 Verification of the DRM formulation for dynamic coupled

consolidation linear analysis ......................................................................237

6.4.2 Verification of the DRM formulation for dynamic coupled

consolidation nonlinear analysis ................................................................243

6.5 Performance of absorbing boundary conditions in the DRM......246

6.5.1 Application of the cone boundary in the step II model of the DRM

246

6.5.2 Numerical results and discussions .............................................248

6.6 Summary...........................................................................................256

Chapter 7: .......................................................................................258

CASE STUDY ON SEISMIC TUNNEL RESPONSE................258

7.1 Introduction ......................................................................................258

7.2 Project description ...........................................................................259

7.2.1 Background ................................................................................259

7.2.2 Construction details....................................................................260

7.3 The 1999 Duzce earthquake ............................................................262

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7.4 Post-earthquake field observations ................................................265

7.5 Ground conditions............................................................................267

7.6 Earthquake effects on tunnels.........................................................269

7.7 Finite element analyses ....................................................................276

7.7.1 Spatial discretization ..................................................................276

7.7.2 Input ground motion...................................................................278

7.7.3 Construction sequence ...............................................................281

7.7.4 Discussion on the boundary conditions and mesh width ...........283

7.7.5 Constitutive models used in the analyses...................................291

7.7.6 1D nonlinear dynamic analyses .................................................296

7.7.7 2D nonlinear static analyses at chainage 62+850 ......................304

7.7.8 2D nonlinear dynamic analyses at chainage 62+850 .................309

7.7.9 2D static and dynamic analyses with the MCCJ model.............314

7.7.10 Quasi-static analyses ..................................................................318

7.7.11 Comparison with analytical solutions ........................................320

7.7.12 2D nonlinear analyses at chainage 62+870................................322

7.8 Conclusions .......................................................................................327

Chapter 8: .......................................................................................331

CONCLUSIONS AND RECOMMENDATIONS.......................331

8.1 Introduction ......................................................................................331

8.2 Direct integration method ...............................................................332

8.2.1 Selection and implementation of time integration scheme ........333

8.2.2 Validation...................................................................................334

8.2.3 Conclusions from the deep foundation analysis.........................334

8.3 Modelling the unbounded medium.................................................336

8.3.1 Conclusions from the investigation and implementation of

absorbing boundary conditions ..................................................................336

8.3.2 Conclusions from the implementation and validation of the

Domain Reduction Method........................................................................338

8.4 Case study on seismic tunnel response ...........................................339

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8.4.1 Lessons learned from the numerical investigation of the case

study 340

8.4.2 Conclusions regarding the comparison of the finite element

analyses with the post-earthquake field observations ................................342

8.5 Recommendations for Further Research.......................................343

8.5.1 Time integration.........................................................................344

8.5.2 Modelling the unbounded medium ............................................344

8.5.3 Case study on seismic tunnel response ......................................345

References .......................................................................................347

Appendix A: Spectral stability analysis of the CH method .......363

Appendix B: Material parameters ...............................................369

B.1 Modified Cam Clay parameters...........................................................369

B.2 Small strain stiffness model parameters..............................................371

B.3 Two-surface kinematic hardening model parameters .......................373

B.4 Equivalent linear elastic model parameters........................................373

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LIST OF FIGURES

Figure 2.1: 8 noded isoparametric element (after Potts and Zdravković, 1999)..43

Figure 2.2: 3 noded beam element (after Potts and Zdravković, 1999)...............43

Figure 2.3: The Modified Newton Raphson method (after Potts and Zdravković,

1999) ............................................................................................................53

Figure 3.1: Relationship between Rayleigh damping parameters and damping

ratio (after Zerwer et al, 2002).....................................................................67

Figure 3.2: Illustration of period and amplitude error in numerical solution.......74

Figure 3.3: CAA with and without viscous damping...........................................75

Figure 3.4: Interpolation of acceleration and interpretation of the Newmark

parameters α and δ (after Argyris and Mlejnek 1991). ................................79

Figure 3.5: Linear acceleration assumption of the Wilson θ-method. .................83

Figure 3.6: Evaluation of the various terms of the equilibrium equation of motion

at different points within a time interval with the HHT algorithm. .............86


at different points within a time interval with the CH algorithm.................88

Figure 3.8: Spectral radii for NMK1, NMK2, linear acceleration and quadratic

acceleration methods....................................................................................91

Figure 3.9: Algorithmic damping ratios (a) and period elongation (b) for NMK1,

NMK2, linear acceleration and quadratic acceleration methods. ................91

Figure 3.10: Spectral radii for HHT (α-method), collocation, Houbolt and Park

methods (from Hilber and Hughes, 1978). ..................................................93

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Figure 3.11: Algorithmic damping ratios for HHT (α-method), collocation,

Houbolt and Park methods (from Hilber and Hughes, 1978). .....................93

Figure 3.12: Period elongation for HHT (α-method), collocation, Houbolt and

Park methods (from Hilber and Hughes, 1978). ..........................................93

Figure 3.13: Spectral radii for NMK1, NMK2, HHT, WBZ and CH methods. ..94

Figure 3.14: Algorithmic damping ratios (a) and period elongation (b) for

NMK1, NMK2, HHT, WBZ and CH methods. ...........................................95

Figure 3.15: Spectral radii the CH (ρ∞=0.0, 0.42, 0.6, 0.818), NMK1 and NMK2

methods. .......................................................................................................96

Figure 3.16: Algorithmic damping ratios (a) and period elongation (b) for the CH

(ρ∞=0.0, 0.42, 0.6, 0.818), NMK1 and NMK2 methods. .............................96

Figure 4.1: Single degree of freedom system ....................................................110

Figure 4.2: Sketch of the FE model (with solid elements) for the SDOF problem

....................................................................................................................110

Figure 4.3: SDOF undamped response modelled with solid elements ..............112

Figure 4.4: SDOF damped response modelled with solid elements (ξ=5%) .....113

Figure 4.5: SDOF undamped response modelled with beam elements .............114

Figure 4.6: SDOF damped response modelled with beam elements (ξ=5%) ....114

Figure 4.7: Analysis arrangement for 1-D consolidation examples...................117

Figure 4.8 Zones of sufficient accuracy for various approximations (after

Zienkiewicz et al, 1980a)...........................................................................118

Figure 4.9: ICFEP results compared to closed form solution for Π1=0.1..........119

Figure 4.10: ICFEP results compared to closed form solution for Π1=1.0........120

Figure 4.11: ICFEP results compared to closed form solution ..........................120

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Figure 4.12: Load level versus normalised settlement for an elastic consolidating

soil layer (from Meroi et al, 1995).............................................................122

Figure 4.13: FE model for 1-D consolidation of Kim et al (1993) ....................123

Figure 4.14: Comparison of surface settlement history predictions of ICFEP with

Kim et al (1993) .........................................................................................124

Figure 4.15: Comparison of pore pressure history predictions of ICFEP with Kim

et al (1993) .................................................................................................125

Figure 4.16: Comparison of pore pressure history predictions of ICFEP with

Terzaghi’s solution using a finer mesh ......................................................126

Figure 4.17: Comparison surface settlement history predictions of ICFEP with

Meroi et al (1995) ......................................................................................127

Figure 4.18: Mesh and boundary conditions assumed in dynamic analyses .....128

Figure 4.19 Filtered accelerograms, obtained from Ambraseys et al (2004).....131

Figure 4.20: Elastic acceleration response spectra.............................................131

Figure 4.21: Settlement history of foundation base for various values of ρ∞ for

the TITO recording ....................................................................................132

Figure 4.22: Fourier amplitude spectra of the horizontal acceleration time history

at the foundation base.................................................................................134

Figure 4.23: Settlement history of foundation base for the TITO, VELS and

PETO recordings........................................................................................136

Figure 4.24: Percentage deviation from the NMK1 for the NMK2 (a) and the

HHT (b)......................................................................................................136

Figure 4.25: Percentage deviation from the NMK1 for various values of ρ∞ ....137

Figure 4.26: Horizontal acceleration time history of foundation base (for the

VELS record) for NMK1, NMK2, CH and HHT ......................................138

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at the foundation base (for the VELS record) ............................................139

Figure 5.1: Incidence of a P-wave on a free surface ..........................................147

Figure 5.2: Shear wave vertically propagating through a layered soil deposit ..148

Figure 5.3 : Multiple reflections and refraction of an incoming wave due to the

presence of a structure................................................................................149

Figure 5.4: Wave propagation at infinity (from Meek and Wolf, 1993)............150

Figure 5.5: Dynamic models of unbounded medium: Substructure method (a) and

Direct method (b) (from Kellezi, 2000) .....................................................150

Figure 5.6: Illustration of the modified Smith boundary (after Wolf, 1988) .....154

Figure 5.7: Elastic half – space subjected to an out-of-plane shear wave (SH)

(after Kausel, 1988)....................................................................................155

Figure 5.8 : Inclined scalar wave at an artificial boundary with apparent velocity

in perpendicular direction (after Wolf and Song, 1996) ............................159

Figure 5.9 : FE mesh up to the artificial boundary of a semi-infinite rod on elastic

foundation (after Wolf and Song, 1996) ....................................................161

Figure 5.10: Comparison of various boundaries for a semi-infinite rod on elastic

foundation (after Wolf and Song, 1996) ....................................................162

Figure 5.11 : Two inclined mutually perpendicular boundaries AB and AC (from

Naimi et al, 2001) ......................................................................................165

Figure 5.12: Typical wave envelope model (from Astley, 1994) ......................167

Figure 5.13: A PML adjacent to a truncated domain (from Basu and Chopra,

2004) ..........................................................................................................169

Figure 5.14: Semi-infinite rod model.................................................................171

Figure 5.15: 4 noded isoparametric element ......................................................176

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Figure 5.16: Semi infinite conical rod model ....................................................177

Figure 5.17: Application of the cone boundary on a homogeneous model with

rectangular boundary..................................................................................184

Figure 5.18: FE models for the extended and small meshes (from Kellezi, 2000)

....................................................................................................................186

Figure 5.19 : Delta function type loads and their Fourier transforms (from

Kellezi, 2000).............................................................................................187

Figure 5.20: Comparison of the response at surface points (E, B) for vertical

excitation, Tp=0.4sec (from Kellezi, 2000)................................................188

Figure 5.21: Comparison of the response at surface points for vertical excitation

and Tp=0.4sec (ICFEP results)...................................................................189

Figure 5.22: Comparison of the stress response for vertical excitation and

Tp=0.4sec (ICFEP results)..........................................................................190

Figure 5.23: Comparison of the response for M10x10 and M15x15 for vertical

excitation, Tp=0.2s (from Kellezi, 2000) ...................................................191


excitation, Tp=0.2s (ICFEP results) ...........................................................192


Tp=0.2sec (ICFEP results)..........................................................................193

Figure 5.26: Comparison of the response at surface points for horizontal

excitation and Tp = 0.1 sec. (from Kellezi, 2000) ......................................194


excitation and Tp = 0.1 sec. (ICFEP results) ..............................................195

Figure 5.28: FE model for the cavity problem...................................................198

Figure 5.29: Displacement normal to the cavity (ρ∞ =0.8, D=10m) ..................199

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Figure 5.31: Displacement normal to the cavity at points A, C (ρ∞ =0.4, D=10m)

....................................................................................................................201


Figure 5.33: Exponential decay functions..........................................................202

Figure 5.34: Displacement normal to the cavity for 1α0 = ................................203

Figure 5.35: Displacement normal to the cavity for 5α0 = ...............................203

Figure 5.36: Displacement normal to the cavity for 50α0 = .............................204

Figure 5.37: Horizontal and vertical amplitudes of Rayleigh waves .................205

Figure 5.38: FE models subjected to Rayleigh wave excitation ........................206

Figure 5.39: Response for Rayleigh wave loading of To=0.1s ..........................207

Figure 5.40: Response for Rayleigh wave loading of To=0.4s. .........................208

Figure 5.41: Response for Rayleigh wave loading of To=1.0s. .........................209

Figure 5.42: Response for Rayleigh wave loading of To=2.0s ..........................210

Figure 5.43: Response for Rayleigh wave loading of To=0.4s and ν= 0.33.......211

Figure 5.44: Response for Rayleigh wave loading of To=0.4s and ν= 0.4.........212

Figure 5.45: Comparison of the displacement response for vertical excitation,

Tp=0.2sec....................................................................................................215

Figure 5.46: Comparison of the stress response for vertical excitation, Tp=0.2sec

....................................................................................................................216

Figure 5.47: Comparison of the displacement response for horizontal excitation,

Tp=0.2sec....................................................................................................217

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Figure 5.48: Comparison of the stress response for horizontal excitation,

Tp=0.2sec....................................................................................................218

Figure 6.1: (a) Model of soil in natural state and (b) model of soil-structure

system (after Bielak and Christiano 1984).................................................222

Figure 6.2: (a) Initial complete model (b) background model ...........................225

Figure 6.3: Summary of the two steps of DRM (after Bielak et al 2003)..........229

Figure 6.4: (a) Initial complete model (b) background model for coupled

consolidation problems ..............................................................................230

Figure 6.5: Background model (a) and reduced model (b) of the verification

example ......................................................................................................239

Figure 6.6: Filtered Montenegro 1979 earthquake record .................................240

Figure 6.8: Comparison of horizontal displacements of nodes A, B for linear

analyses ......................................................................................................242

Figure 6.9: Comparison of horizontal accelerations of nodes A, B for linear

analyses ......................................................................................................242

Figure 6.10: Comparison of pore pressures of integration points E, F for linear

analyses ......................................................................................................243

Figure 6.11: Comparison of horizontal displacements of nodes A, B for nonlinear

analyses ......................................................................................................244

Figure 6.12: Comparison of horizontal accelerations of nodes A, B for nonlinear

analyses ......................................................................................................245

Figure 6.13: Comparison of pore pressures of integration points E, F for

nonlinear analyses ......................................................................................246

Figure 6.15: Comparison of the displacement response at nodes C, D for a pulse

of To =1.0s..................................................................................................250

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of To =2.0s..................................................................................................251


of To=4.0s...................................................................................................252

Figure 6.18: Comparison of the acceleration response at nodes C, D for pulses of

To=2.0, 4.0s. ...............................................................................................253

Figure 6.19: Comparison of the displacement response at node G for pulses of

To=2.0s ,4.0s...............................................................................................254

Figure 6.21: Comparison of the displacement response of nodes C, D. ............256

Figure 7.1: Layout of a 27.3 km part of the Gumusova-Gerede motorway (from

Menkiti et al, 2001b)..................................................................................260

Figure 7.2: Longitudinal section of the left tunnel (from Menkiti et al, 2001b)260

Figure 7.3: Design solution for the thick zones of fault gouge clay (after Menkiti

et al 2001b) ................................................................................................261

Figure 7.4: Unmodified strong motion records of the Bolu Station (from

Ambraseys et al, 2004) ..............................................................................263

Figure 7.5: The surface rupture of the November 1999 Düzce earthquake and

active faults around Bolu (from Akyüz et al, 2002) ..................................264

Figure 7.6: The collapsed LBPT after it had been re-excavated and back filled

with foam concrete (Menkiti 2005, personal communication) ..................266

Figure 7.7: Plan view of the Asarsuyu left tunnel..............................................267

Figure 7.8 Ground profile at chainage 62+850 (cross-section AB)...................267

Figure 7.9: Ground profile at chainage 62+870 (cross-section CD)..................268

Figure 7.10: Axial (a) and bending (b) deformation along the tunnel axis (after

Owen and Scoll, 1981)...............................................................................271

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Figure 7.11: Ovaling deformation of a circular tunnel’s cross section (after Owen

and Scoll, 1981) .........................................................................................272

Figure 7.12: Forces and moments induced by seismic waves (from Power et al

1996) ..........................................................................................................274

Figure 7.13: FE mesh configuration for chainage 62+850 after the excavation of

the tunnels ..................................................................................................277

Figure 7.14: Acceleration (a), velocity (b) and displacement (c) time histories of

the E-W component of the Bolu record .....................................................279

Figure 7.15: Fourier amplitude spectrum (a) and elastic acceleration response

spectrum (b) of the E-W component of the Bolu record............................279

Figure 7.16: Scaled and truncated accelerogram used in the FE analyses.........280

Figure 7.17: FE mesh with boundary conditions ...............................................284

Figure 7.18: Shear strain time history (a) and maximum shear strain profile (b)

computed with the 2D model (with viscous boundary conditions), 1D model

and EERA ..................................................................................................286

Figure 7.19: Cross-section view of the 3D FE mesh and the boundary conditions

used in the analyses of Brown et al (2001) and Maheshwari et al (2004).286

Figure 7.20: Free-field acceleration time histories obtained by 3D and 1D models

(after Brown et al 2001).............................................................................287


computed with the 2D model (with tied degrees of freedom boundary

conditions), 1D model and EERA..............................................................288

Figure 7.22: FE mesh with boundary conditions used in the step II DRM analysis

....................................................................................................................289


computed with the 2D model (with DRM and viscous boundary conditions),

1D model and EERA..................................................................................290

20

Figure 7.24: Yield surface (from Potts and Zdravković, 1999) .........................292

Figure 7.25: Two-surface kinematic hardening model (after Potts and

Zdravković, 1999) ......................................................................................295

Figure 7.26: Iteration of shear modulus (a) and damping ratio (b) with shear

strain in equivalent line analysis. ...............................................................296

Figure 7.27: Maximum shear strain profile computed with the MCCJ, the M2-

SKH models and EERA.............................................................................297

Figure 7.28: Representative strain time histories for the two clays layers (i.e.

layer 2 (a) and layer 4 (b))..........................................................................298

Figure 7.29: Comparison of strain time histories at a depth of z=157.5m.........298

Figure 7.30: Shear strain time history (a), shear stress-strain curve (b) and p΄-J

stress path (c) of an integration point at a depth z=157.5m computed with

MCCJ model for the first 11.38sec of the earthquake. ..............................299

Figure 7.31: p΄-J stress path (a) and shear stress-strain curve (b) of an integration

point at a depth of z=157.5m computed with the MCCJ model for the whole

duration of the earthquake..........................................................................300

Figure 7.32: Relative horizontal displacement (a) and horizontal acceleration (b)

time histories at a depth of z=163.5m ........................................................301


stress path (c) of an integration point at a depth z=157.5m computed with

the M2-SKH model for the first 6.48sec of the earthquake. ......................302


point at a depth of z=157.5m computed with the M2-SKH model for the

whole duration of the earthquake...............................................................302

Figure 7.35: Representative strain time histories for the rock layers (i.e. layer 1

(a), layer 3 (b) and layer 5 (c)) ...................................................................303

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Figure 7.36: Horizontal displacement (a) and acceleration time histories (b) at a

depth of z=163.5m for ∆t=0.005sec and ∆t=0.01sec.................................304

Figure 7.37: Mesh configuration around the tunnels at the end of the static

analysis.......................................................................................................305

Figure 7.38: Accumulated thrust (a), bending moment (b) and hoop stress (c)

distribution around the tunnels’ lining at the end of the static analysis .....307

Figure 7.39: Contours of pore pressure distribution around the tunnels at the end

of the static analysis. ..................................................................................308

Figure 7.40: Contours of plastic shear strain around the tunnels at the end of the

static analysis. ............................................................................................308

Figure 7.41: Maximum shear strain profile computed with the M2-SKH model

for 1D and 2D analyses ..............................................................................309

Figure 7.42: Enlarged view of the deformed mesh at t=8.0sec..........................310

Figure 7.43: Snapshots (at t=5.0, 6.0, 7.0 and 8.0sec) of deviatoric stress (J)

contours in the vicinity of the tunnels (for the area indicated in Figure 7.42)

....................................................................................................................310

Figure 7.44: Pore water pressure (a) and shear strain (b) time histories for

integration points adjacent to the crowns of the BPTs...............................311


distribution around the tunnels’ lining at t=10.0sec...................................313

Figure 7.46: Thrust (a) and bending moment (b) time histories at θ=137˚ for both

BPTs...........................................................................................................313


distribution around the tunnels’ lining at the end of the static analysis

computed with MCCJ model .....................................................................315

22


integration points adjacent to the crowns of the BPTs computed with the

MCCJ model ..............................................................................................316

Figure 7.49: Thrust (a) and bending moment (b) time histories at θ=137˚ of both

BPTs computed with the MCCJ model......................................................317

Figure 7.50: Schematic representation of FE mesh configuration in quasi-static

analysis.......................................................................................................318


distribution around the tunnels’ lining at the end of the quasi-static analysis

....................................................................................................................319


the tunnel....................................................................................................323

Figure 7.53: Maximum shear strain profile computed with the 2BPTs-AB, the

1BPT-AB and the 1BPT-CD model at x=70.0m (a) and at x=0.0m (b) ....325

Figure 7.54: Shear strain time history computed with the 2BPTs-AB, the 1BPT-

AB and the 1BPT-CD model at integration point R ..................................325


distribution around the tunnels’ lining at t=10.0sec computed with the

2BPTs-AB, the 1BPT-AB and the 1BPT-CD model .................................326

Figure A1: Classification of the CH, HHT, WBZ methods in fm αα − space (after

Chung and Hulbert, 1993)..........................................................................368

23

LIST OF TABLES

Table 4.1: Single degree of freedom finite element analysis parameters ..........111

Table 4.2: Equivalent material properties of the pendulum...............................111

Table 4.3: Parameters for the FE analysis of the soil layer subjected to cyclic

load.............................................................................................................118

Table 4.4: Parameters for the FE analysis of the soil layer subjected to a step load

....................................................................................................................121

Table 4.5: Variable time step for the FE analysis of a soil layer subjected to a

step load .....................................................................................................123

Table 4.6: Material properties for the FE analysis of a soil layer subjected to a

step load .....................................................................................................123

Table 4.7: Material properties for foundation analyses .....................................129

Table 4.8: Summary of analyses undertaken at different fundamental frequencies

....................................................................................................................140

Table 4.9: Summary of results for various fundamental frequencies ................141

Table 4.10: Comparison of computational cost .................................................142

Table 5.1: Summary of constitutive damping and stiffness matrices ................184

Table 6.1: Parameters used in the small strain stiffness model..........................244

Table 7.1: Summary of ground motion records from Duzce and Bolu stations

(from Menkiti et al, 2001a)........................................................................262

Table 7.2: Shear wave velocity profile at the Bolu station (Menkiti 2005,

Personal communication)...........................................................................264

24

Table 7.3: Geotechnical description and index properties .................................268

Table 7.4: Estimated strength and stiffness parameters .....................................269

Table 7.5: Summary of estimated minimum shear wave velocity and resulting

maximum element side length ...................................................................277

Table 7.6: Strength and stiffness properties of the BPTs at the time of earthquake

at chainage 62+850 ....................................................................................281

Table 7.7: Geometrical and material properties of tunnel linings......................282

Table 7.8: Material properties used in elastic analyses......................................284

Table 7.9 :Summary of the diametral movements and strains after the static

analysis.......................................................................................................305

Table 7.10: Maximum hoop stress at shoulder and knee locations of the BPTs’

lining computed with the M2-SKH model.................................................314


lining computed with the MCCJ model .....................................................317

Table 7.12: Analytical methods parameters.......................................................320

Table 7.13: Summary of analytical results for the LBPT ..................................321

Table 7.14: Summary of analytical results for the RBPT ..................................321

Table 7.15: Maximum hoop stress developed at the LBPT for various analyses

....................................................................................................................327

25

LIST OF SYMBOLS

A and B Parameters used in evaluating the Rayleigh damping matrix.

[ ]A Amplification matrix of integration.

[ ]B Matrix containing the derivatives of the shape functions.

c Constant representing the damping characteristics of the

material.

c’ Cohesion intercept of a soil.

cv Consolidation factor.

C Compressibility ratio of tunnel lining.

[ ]EC Elemental damping matrix.

[ ]GC Global damping matrix.

D Bulk modulus of soil skeleton.

[ ]D Total stress constitutive matrix.

[ ]epD Elasto-plastic constitutive matrix.

[ ]fD Pore fluid stiffness matrix.

[ ]D' Effective stress constitutive matrix.

e Void ratio.

E Young’s modulus.

26

[ ]E Vector containing the derivatives of the pore fluid shape

functions.

fo Frequency of harmonic loading or predominant a frequency

of transient loading.

fn Natural frequency of soil column on a rigid base.

F Flexibility ratio of tunnel lining.

g(θ) Gradient of the yield function in the J- p΄ plane, as a

function of Lode’s angle.

gpp(θ) Gradient of the plastic potential function in the J- p΄ plane,

as a function of Lode’s angle.

G Shear modulus.

[ ]GG Global pore fluid inertia matrix.

[ ]h Hydraulic head.

Gi Vector defining the direction of gravity.

[ ]I Identity matrix.

I Moment of inertia.

J Deviatoric stress.

[ ]J Jacobian matrix, used to transform parent coordinate system

to the global coordinate system.

J Determinant of Jacobian matrix

[ ]k Matrix of soil permeability.

k Soil permeability.

27

k Stiffness of a single degree of freedom system.

K0 Coefficient of earth pressure at rest.

Ke Equivalent bulk modulus of pore fluid.

Kf Bulk modulus of pore fluid.

Ks and Kn Shear and normal stiffness of interface elements.

Ks Bulk modulus of solid soil particles.

[ ]GK Effective global stiffness matrix, including contributions

from the global mass and damping matrices.

[ ]EK Elemental stiffness matrix.

[ ]GK Global stiffness matrix.

[ ]GL Global off diagonal sub-matrix in consolidation stiffness

matrix.

m Mass of a single degree of freedom system.

[ ]Tm Multiplying vector equal to [ ]000111 .

M Bending moment in tunnel lining.

Mmax Maximum bending moment in tunnel lining.

[ ]EM Elemental mass matrix.

[ ]tEM Sub-matrix of a beam element’s mass matrix referring to

translational degrees of freedom.

[ ]rEM Sub-matrix of a beam element’s mass matrix referring to

rotational degrees of freedom.

[ ]GM Global mass matrix.

28

n Soil porosity.

[ ]Gn Global right hand side load vector for pore fluid equilibrium

equation.

[ ]pN Matrix of pore fluid interpolation functions.

N Isoparametric shape function.

[ ]N Matrix of all element interpolation functions.

N Substitute shape function for beam element.

p Pore fluid pressure.

p΄ Mean effective stress.

P(t) Forcing function.

Po Amplitude of harmonic forcing function.

R Tunnel lining-soil racking ratio.

[ ]GS Global matrix of pore fluid compressibility.

S, T Natural coordinates.

t Current time in any analysis.

t Thickness of element.

t Thickness of tunnel lining.

T Undamped natural period of a single degree of freedom

system.

T′ Undamped natural period of a single degree of freedom

system obtained by a numerical solution.

T Thrust force in tunnel lining.

29

Tmax Maximum thrust in tunnel lining.

To Period of harmonic loading or predominant period of

transient loading.

Tv Normalised time given by tcT vv = .

u and v Displacement components for an isoparametric element.

vx, vy Components of pore fluid velocity in Cartesian coordinate

directions.

Vc Compression wave velocity in water.

Vmin Lowest considered velocity of wave propagation.

VP Dilatational wave velocity of propagation.

VR Rayleigh wave velocity of propagation.

VS Shear wave velocity of propagation.

w Velocity of the pore fluid relative to the solid component.

α and δ Newmark parameters (after Bathe, 1996), equivalent to the

parameters β and γ respectively, introduced by Newmark

(1959).

αf Parameter of the HHT and CH integration schemes that

specifies the time instant within the increment that all but

inertia terms are evaluated.

αm Parameter of the WBZ and CH integration schemes that

specifies the time instant within the increment that the

inertia terms are evaluated.

β Integration parameter introduced to indicate how the pore

pressure is assumed to vary during an increment.

γ Algorithmic parameter of the quadratic acceleration method.

30

γ Bulk unit weight of soil.

γf Bulk unit weight of the pore fluid.

γmax Maximum free-field shear strain.

Γ Boundary between the internal (Ω) and the external area

( +Ω ) in the domain reduction method.

+Γ Outer boundary of the external area ( +Ω ) in the domain

reduction method.

Γ Outer boundary of the external area ( +Ω ) of the reduced

model in the domain reduction method.

eΓ Boundary within the external area of the background model

in the domain reduction method defining a strip of elements

between eΓ and Γ .

T∆d Vector of incremental displacement components given by

∆v∆u,∆dT =

∆DE Incremental elemental damping energy.

∆EE Incremental elemental total potential energy

TF∆ Vector of incremental body forces given by

yx

T∆F,∆F∆F = .

∆IE Incremental elemental inertia energy.

∆l Length of an element side.

∆LE Incremental work done within an element by any applied

loads and/or body forces

∆p Incremental pore fluid pressure.

31

∆P Incremental right hand side load vector

∆Q Represents any sinks and/or sources of flow within an

increment.

E∆R Incremental elemental right hand side load vector.

G∆R Incremental global right hand side load vector.

GR∆ Effective global right hand side vector including known

values from the previous time step.

TT∆ Vector of incremental surface tractions given by

yx

T∆T,∆T∆T = .

∆t Incremental time step.

∆tcr Critical time step required to ensure a time scheme remains

stable.

u∆ Incremental displacement.

u∆ Vector of incremental displacement components in the

domain reduction method, consistent with the notation of

Bielak et al (2003).

u&∆ Incremental velocity.

u&&∆ Incremental acceleration.

∆WE Incremental elemental strain energy

∆ε Vector of incremental strain.

∆εv Incremental volumetric strain.

∆εx, ∆εy and ∆γxy Horizontal, vertical and shear strain components within an

element.

32

∆σ Vector of change in total stress.

σ∆ ′ Vector of change in total stress.

f∆σ Vector of change in pore fluid pressure.

θ Rotational degree of freedom.

θ Algorithmic parameter of the Wilson θ-method and the

collocation method.

θ Lode’s angle.

κ, β, Π1 and Π2 Dimensionless parameters in Zienkiewicz et al (1980a)

analytical solution for consolidating elastic soil layer

subjected to cyclic loading

λ Solution of the characteristic equation of the amplification

matrix [ ]A .

λmin Wavelength associated with the highest considered

frequency of the input wave.

λP Wavelength of dilatational wave associated with the

predominant period of excitation.

λR Wavelength of Rayleigh wave associated with the

predominant period of excitation.

λS Wavelength of shear wave associated with the predominant

period of excitation.

ν, v Poisson’s ratio.

ξ Damping ratio for single degree of freedom problem.

ξt Target damping ratio.

ξ′ Algorithmic damping ratio.

33

ρ(A) Spectral radius of the amplification matrix [ ]A .

∞ρ Value of spectral radius at inifinity.

fρ Fluid density.

ρ Material density.

1σ′ , 2σ′ and 3σ′ Principal effective stresses.

φ’ Angle of internal shearing resistance of a soil.

[ ]GΦ Global permeability sub-matrix in consolidation stiffness

matrix.

ψ Angle of dilation.

ψ Vector of residual load.

ω Angular frequency of a single of single degree of freedom

system.

ωD Damped natural frequency of a single degree of freedom

problem.

Ω′ Undamped natural frequency of a single degree of freedom

system obtained by a numerical solution.

Ωo Angular frequency of harmonic loading or predominant

angular frequency of transient loading.

Ω Internal area of both the reduced and the background models

in the domain reduction method.

+Ω External area of both the reduced and the background

models in the domain reduction method.

0Ω Internal area of the background model in the domain

reduction method.

34

+Ω External area of the reduced model in the domain reduction

method.

35

Chapter 1:

INTRODUCTION

1.1 General

Problems related to dynamic loading of soils and earth structures are

often encountered by a geotechnical engineer. Some illustrative examples include

the design of geotechnical structures against vibration effects of vehicles and pile

driving, the foundation design of offshore foundations and more importantly the

design of geotechnical structures against earthquake induced dynamic loading.

To achieve a rigorous design procedure for these problems, an understanding of

the behaviour of both the soil and the structure under both static and dynamic

loading conditions is required. In engineering practice, due to the complexity of

dynamic soil-structure interaction phenomena, simplified analytical methods are

traditionally adopted for design. On the other hand, the finite element method has

been developing rapidly over the last 30 years and is nowadays an indispensable

analysis tool. Consequently the use of finite element analysis has been gaining

popularity in the field of dynamic soil-structure interaction problems.

In dynamic finite element analysis of soil-structure interaction problems

three distinct methodologies can be identified: modal analysis, frequency domain

analysis and direct integration. Classical modal analysis and frequency domain

analysis have severe limitations, as they are not directly applicable to nonlinear

systems. The direct integration method is generally more time consuming, but is

a powerful approach which has become more attractive over the last decade due

to its increased ability to analyse realistic problems and achieve accurate

predictions.

In a similar fashion to static finite element analysis, the accuracy of the

predictions largely depends on the adoption of appropriate constitutive

relationships that can realistically model the soil behaviour. In addition, when

36

analysing a dynamic phenomenon in the time domain, the accuracy of the

predictions significantly relies on the adopted time marching scheme. Hence

depending on the features of the time integration algorithm the efficiency and

accuracy of the method can considerably improve or deteriorate.

Another major challenge that arises in dynamic finite element analysis of

soil-structure interaction problems is to model accurately and economically the

far-field medium. The most common way is to restrict the theoretically infinite

computational domain to a finite one with artificial boundaries. The reduction of

the solution domain makes the computation feasible, but special care is needed to

absorb spurious reflections from the artificial boundaries that can seriously affect

the accuracy of the results.

1.2 Scope of research

The aim of this research was to further develop the existing dynamic

capabilities of the geotechnical finite element program ICFEP and then to apply

them to a geotechnical earthquake engineering case study.

Hence the first objective was the identification of an efficient time

integration scheme that is able to perform accurately and economically dynamic

finite element analyses. For this purpose the generalized-α algorithm of Chung &

Hulbert (1993), which has to date only been used in the field of structural

dynamics, was chosen. This method was extended to deal with coupled

consolidation problems and was then implemented into ICFEP. Subsequently,

the newly implemented algorithm was validated and evaluated in a geotechnical

boundary value problem.

The second development involved the incorporation of absorbing

boundary conditions, which can model the radiation of energy towards infinity in

a truncated domain. After reviewing the available boundary conditions for

solving wave propagation problems in unbounded domains, two well-established

absorbing boundary conditions (i.e. the standard viscous boundary and the cone

boundary) were chosen for implementation into ICFEP. The validation of the

37

newly implemented boundaries identified their limitations. To overcome the

identified shortcomings it is proposed to use these methods in conjunction with

the domain reduction method (DRM). The DRM has a dual role as it not only

reduces the domain that has to be modelled numerically, but in conjunction with

the standard viscous boundary or the cone boundary also serves as an advanced

absorbing boundary condition. This method was extended to deal with coupled

consolidation problems, was then implemented into ICFEP and was finally

validated in a boundary value problem.

The final task of the thesis was to use the modified dynamic version of

ICFEP to analyse a case study. Hence the case of the Bolu highway twin tunnels

that experienced a wide range of damage severity during the 1999 Duzce

earthquake in Turkey was considered. The Bolu tunnels establish a well-

documented case, as there is information available regarding the ground

conditions, the design of the tunnels, the ground motion and the earthquake

induced damage. The first objective of this study was the investigation of the

theoretical issues of dynamic finite element method like spatial discretization,

absorbing boundary conditions, time integration and constitutive modelling on a

practical application. The second objective was to qualitatively and quantitatively

compare the finite element analysis results with simplified analytical methods

and with post-earthquake field observations.

1.3 Layout of thesis

The work presented in this thesis is divided into the following chapters:

Chapter 2 discusses aspects of the finite element theory which are necessary for

the analysis of static geotechnical problems. This includes an overview of the

basic steps of the finite element formulation, a description of the technique used

to solve the non-linear finite element equations and a presentation of the finite

element governing equations for coupled consolidation analysis.

Chapter 3: presents the extensions that are required to the static finite element

formulation to perform dynamic analyses. This includes a comparative study of

38

some of the most popular time integration methods which are used for the

solution of the dynamic equilibrium equation. Particular emphasis is placed on

the implementation of the chosen integration scheme (i.e. the generalized-α

algorithm) into ICFEP and on its development to deal with dynamic coupled

consolidation problems.

Chapter 4 initially details a series of validation exercises which were used to

verify the implementation of the generalized-α algorithm into ICFEP. A closed

form solution was used to verify the uncoupled dynamic formulation of ICFEP

for solid and beam elements. Besides, a number of published numerical examples

and an analytical solution were employed to verify the dynamic coupled

formulation of ICFEP. The final part of the chapter compares the behaviour of

the generalized-α algorithm with more commonly used time integration schemes

in a boundary value problem of a deep foundation subjected to various seismic

excitations.

Chapter 5 at first reviews some of the most popular boundary conditions for

solving wave propagation problems in unbounded domains. Particular emphasis

is placed on the two boundary conditions (i.e. the standard viscous boundary and

the cone boundary) that were chosen to be implemented into ICFEP. The

implementation of the two methods is then validated for two dimensional plane

strain and axisymmetric analyses. The effectiveness of the newly implemented

boundaries for the cases of soil layers with vertically varying stiffness and with

Rayleigh wave propagation is also considered.

Chapter 6 presents the domain reduction method (DMR) which is a two-step

procedure that aims at reducing the domain that has to be modelled numerically

by a change of governing variables. The chapter also illustrates the development

of the method to deal with dynamic coupled consolidation problems and its

implementation into ICFEP. Numerical tests are then presented which verify the

development and implementation of the DRM into ICFEP. The final part of the

chapter highlights another important aspect of the DRM which is the application

of the method as a boundary condition in unbounded domains. As part of this, a

methodology is suggested which allows the use of cone boundary in conjunction

with the DRM in earthquake engineering problems.

39

Chapter 7 presents a case study on the Bolu highway twin tunnels that

experienced a wide range of damage severity during the 1999 Duzce earthquake

in Turkey. The first part of this chapter details a description of the case study.

This includes an overview of the Bolu tunnels project, a description of the

ground conditions and a summary of construction issues for the analysed

sections. The seismicity of the Bolu area is also briefly discussed, while more

emphasis is placed on the description of the 1999 Duzce earthquake.

Furthermore, post-earthquake field observations of the damage are presented and

linked to a general discussion regarding the seismic hazards associated with

underground structures. In the second part of the chapter a thorough discussion

on the adopted numerical model is given. Theoretical issues presented in

previous chapters regarding spatial discretization, absorbing boundary

conditions, time integration and constitutive modelling are investigated and

applied to the case study. Finally results of dynamic and quasi static FE analyses

are presented and compared qualitatively and quantitatively with simplified

analytical methods and with post-earthquake damage observations.

Chapter 8 gives a summary of the main conclusions reached in the previous

chapters and makes recommendations for related further research.

40

Chapter 2:

FINITE ELEMENT THEORY

2.1 Introduction

The finite element method has been widely used the last forty years to

solve boundary-value problems in many fields of engineering practice (e.g.

structural mechanics, fluid mechanics, geotechnics). In contrast to various

analytical approaches (e.g. limit equilibrium, limit analysis), the FE method

fulfils all the requirements of a true theoretical solution. In general the four basic

requirements for a theoretical solution to be correct are: equilibrium,

compatibility, material constitutive behaviour and boundary conditions. There

are two additional requirements for coupled consolidation analysis of

geotechnical problems: continuity of flow and validity of generalised Darcy’s

law.

All the developments and analyses presented in this thesis were

performed with the finite element program ICFEP. ICFEP employs a

displacement based finite element method and it has been specifically developed

for the analysis of geotechnical problems. It should be noted that ICFEP is

capable of performing two-dimensional (plane strain, plane stress and

axisymmetric), full three-dimensional and Fourier series aided three-dimensional

analyses. Only plane strain and axisymmetric conditions were considered in this

study. For simplicity, all the derivations in this chapter refer to plane strain

conditions.

This chapter gives a brief description of the fundamental aspects of the

finite element method in static domain. The full range of ICFEP features for

static analyses can be found in Potts and Zdravković (1999).

41

2.2 The Finite Element Method for Static Problems

The general formulation of the finite element method consists of the

following steps: (i) element discretisation, (ii) primary variable approximation,

(iii) formulation of element equations, (iv) assembly of the global equations and

(v) the solution of the global equations. When the soil behaviour is described by

nonlinear constitutive relationships the solution strategy of the global equations

becomes more complicated and it is therefore separately discussed in Section

2.2.7. Furthermore, when the pore fluid pressure is also considered as a primary

unknown (in addition to displacement), a second set of equations governing the

flow of pore fluid through the soil skeleton is needed. This case is addressed in

Section 2.2.8.

2.2.1 Element Discretisation

The first step in the FE procedure is to approximate the geometry of the

problem to be analyzed with an equivalent FE mesh. The mesh comprises of

discrete small regions called elements. The elements are geometrically defined

by the coordinates of their nodes. In general, for elements with straight sides the

nodes are located at the corners, while to define elements with curved sides

additional, usually mid-side, nodes are required. For two dimensional analyses,

the elements employed to model the soil behaviour are typically triangular or

quadrilateral in shape.

The number of elements in the FE mesh controls both the accuracy and

the run time of the analysis. Therefore, optimum mesh design requires having as

few elements as possible to reduce the computational cost and refining the mesh

at locations of stress concentration to give an accurate result. In many cases the

mesh design can be determined by carrying out numerical tests and comparing

the results with analytical solutions, although more commonly the discretization

is based on previous experience.

42

2.2.2 Primary variable approximation

In the displacement based finite element method, the main approximation

is to assume a polynomial form for the variation of displacement components

over the computational domain. The order of the polynomial form depends on

the number of nodes in the element and it should satisfy the conditions of

compatibility. In two dimensional plane strain analyses, the displacement field is

characterized by two global displacements u and v in the x and y coordinate

directions respectively. The displacement components of each element can be

conveniently expressed in terms of their values at the element nodes:

[ ] [ ]nodes

T

nn2211v

uNv,uv,u,v,uN

v

u

==

KK 2.1

where [ ]N is the matrix of the displacement interpolation functions, known as

the matrix of shape functions and the subscript n denotes the number of nodes in

the element. In this way the number of degrees of freedom becomes finite, equal

to the number of nodal displacements. Any other displacement within an element

can be interpolated using the shape functions and the known nodal values. The

variation of displacement is linear for triangular 3-noded and quadrilateral 4-

noded elements and quadratic for triangular 6-noded and quadrilateral 8-noded

elements. In the present study 4-noded and 8-noded quadrilateral isoparametric

elements were employed to model soil behaviour. For an isoparametric element,

the global element (Figure 2.1b) is mapped on to a parent element (Figure 2.1a)

which has the same number of nodes, but it is expressed in terms of natural

coordinates (-1≤S≤1 and -1≤T≤1). The term isoparametric refers to the fact that

the interpolation functions that are used to approximate the displacement

variation across the element are also employed to map the element geometry

from the natural to the global coordinates. The global coordinates x, y of a point

in an element can be expressed in terms of the global coordinates of the element

nodes xi, yi:

∑∑==

==n

1i

ii

n

1i

ii yNy,xNx 2.2

43

where n denotes the number of nodes in the element and iN are the interpolation

functions which constitute the shape function matrix [ ]N in Equation 2.1. The

interpolation functions for the 8-noded isoparametric element of Figure 2.1 are

given by the following formulas (Potts and Zdravković, 1999):

( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( ) 854

2

8

763

2

7

652

2

6

851

2

5

N21N21T1S141N T1S121N

N21N21T1S141N T1S121N

N21N21T1S141N T1S121N

N21N21T1S141N T1S121N

nodesCorner nodessideMid

−−−−=−−=

−−++=+−=

−−−+=−+=

−−−−=−−=

−

2.3

Figure 2.1: 8 noded isoparametric element (after Potts and Zdravković, 1999)

Furthermore two types of special element were used during the research for this

thesis. The first type concerns 2 and 3-noded line (beam) elements that can

represent structural components (e.g. retaining walls, tunnel lining). An example

of such a 3-noded isoparametric beam element is shown in Figure 2.2.

y, v

x, u

w1

u1α

12

3

a) Global coordinates b) Natural coordinates

23

-1 0 +1

S1

Figure 2.2: 3 noded beam element (after Potts and Zdravković, 1999)

44

Standard two dimensional elements can also model structural components, but

they often result in uneconomical meshes (i.e. a large number of elements) or

unreasonable aspect ratios of the element. The beam elements in the ICFEP

element library were developed by Day (1990) and Day and Potts (1990) to be

compatible with the above-mentioned two dimensional elements. Hence the same

quadratic interpolation functions are used, but each node has three degrees of

freedom, two displacements and one rotation. The interpolation functions for the

3 noded beam element of Figure 2.2 are given by the following formulas (Potts

and Zdravković, 1999):

( )

( )

( )23

2

1

S-1N

1SS2

1N

1SS2

1N

=

+=

−=

2.4

The second type of special element used is a zero thickness interface (or

joint) element which can model soil-structure interfaces. Whilst nodal

compatibility does not allow continuum elements to model differential

movement of the soil and the structure, these zero thickness elements can model

discontinuities. Isoparametric interface elements with 4 and 6 nodes are available

in the ICFEP element library and, like the solid elements, each node has two

displacement degrees of freedom. A complete description of the implementation

and the performance of interface elements in ICFEP can be found in Day (1990)

and Potts and Zdravković (1999).

2.2.3 Element Equations

The derivation of the equations that govern the deformational behaviour

of each element is based on compatibility, equilibrium and constitutive

conditions. Given that the soil usually behaves nonlinearly, it is preferable to

formulate all equations in an incremental form. Consequently, the primary

variable approximation of displacements given by Equation 2.1 can be also

written as:

45

[ ] [ ] nn

∆dN∆v

∆uN

∆v

∆u∆d =

=

= 2.5

where n∆d contains all the nodal displacements for a single element.

According to the compatibility condition, the strains corresponding to the above

displacements are defined as:

( ) ( ) ( ) ( )

Txyzyx

T

zyxzz

xyyx

∆γ,∆ε,∆ε,∆ε∆ε;0∆γ∆γ∆ε

x

∆v

y

∆u∆γ;

y

∆v∆ε;

x

∆u∆ε

====

∂∂

−∂

∂−=

∂∂

−=∂

∂−=

2.6

Combining Equations 2.5 and 2.6 for an element with n nodes the strains can be

expressed in terms of nodal displacements:

[ ] n∆dB∆ε = 2.7

where [ ]B contains the derivatives of the shape functions iN .

Furthermore the stresses can be determined employing the material’s

constitutive relationship:

[ ] ∆εD∆σ = 2.8

where xyzyx

T∆τ∆σ∆σ∆σ∆σ = . For an isotropic linear elastic material

the constitutive matrix [ ]D takes the form:

[ ]( )( )

( )( )

( )( )

−

−

−

−

−+=

ν21000

0ν1νν

0νν1ν

0ννν1

2ν1ν1

ED 2.9

where E is the Young’s modulus and ν is the Poisson’s ratio.

Unlike other engineering materials (e.g. steel), saturated soil is composed

of two separate phases, the soil skeleton and the fluid which fills the pores

between the individual soil particles. Depending on the rate of loading and the

46

permeability of the soil (see Section 2.2.8), two extreme classes of problems can

be identified in geotechnical engineering. The first case assumes fully drained

conditions in which there is no change in pore fluid pressure. In this case [D] is

constructed using effective stress parameters. This is permissible as there is no

change in pore water pressure and therefore ∆σ = ∆σ΄ in Equation 2.8. The

second is the undrained case in which there is no overall volume change for fully

saturated conditions. In this case the constitutive behaviour can either be

expressed in terms of total or effective stresses. When it is expressed in terms of

effective stresses, it is necessary to consider the principle of effective stress.

Hence, the incremental total stress is split into effective stress and pore fluid

pressure components:

f∆σ∆σ'∆σ += 2.10

where 0∆p∆p∆p∆σT

f = , ∆σ is the change in total stress, ∆σ' is

the change in effective stress, and ∆p is the change in pore fluid pressure.

Consequently in Equation 2.8 the stiffness of the pore fluid [ ]fD must be added

to the stiffness of the soil skeleton [ ]D′ to create a new constitutive matrix [ ]D :

[ ] [ ] [ ]fDDD +′= 2.11

The pore fluid stiffness matrix [ ]fD is related to the bulk modulus of the pore

fluid Kf and it has the form:

[ ]

=

00

0IKD

3

ef 2.12

where I3 is a 3 x 3 matrix all elements of which are 1 and Ke is the equivalent

bulk modulus of the pore fluid and it is related to both the pore fluid (Kf) and the

solid soil particles (Ks) bulk moduli by the following relationship:

( )

sf

e

K

n1

K

n

1K

−+

= 2.13

47

where n is the soil porosity. Potts and Zdravković (1999) showed that the Ks is

always much greater than Kf and that both of them (Ks, Kf) are much larger than

the soil skeleton stiffness (Kskel). Thus, the above expression reduces to Ke ≈ Kf.

Furthermore Potts and Zdravković (1999) suggest that in undrained analyses Ke

must be assigned to a high value compared with Kskel. However, too high values

lead to numerical instability as the equivalent undrained Poisson's ratio νu

approaches 0.5 (Potts and Zdravković, 1999). Within this thesis a value of Ke =

1000Kskel was adopted for undrained analyses and a zero value was assigned to

Ke for drained analyses. For situations between these two extreme cases of fully

drained and undrained soil behaviour it is necessary to consider the equations

governing the flow of the pore fluid (see Section 2.2.8).

Once the constitutive relationship is specified, the element equations are

determined by invoking the principle of minimum potential energy. According to

this principle, the static equilibrium position of a loaded body is the one that

minimizes the total potential energy. Hence the equilibrium of an element in

incremental form is expressed as:

EEE ∆L∆W∆E −= 2.14

where E∆E is the total potential energy, E∆W is the strain energy and E∆L is the

work done by any applied loads and/or body forces. The incremental strain

energy in an element and the incremental work done by the external loads are

given by Equations 2.15 and 2.16 respectively:

∫=Vol

T

E dVol∆σ∆ε2

1∆W 2.15

∫∫ +=Surface

T

Vol

T

E dSurface∆T∆ddVol∆F∆d∆L 2.16

where ∆v∆u,∆dT = = displacements;

yx

T∆F,∆F∆F = = body forces;

yx

T∆T,∆T∆T = = surface tractions;

48

the volume integral is over the volume of the element and the surface integral is

over that part of the element boundary over which the surface tractions are

applied. Substituting Equations 2.15 and 2.16 into Equation 2.14, expressing the

displacement variation in terms of nodal values and then summing the potential

energies of the separate elements, leads to the following global equation:

[ ] [ ][ ] [ ] ( )

[ ] i

N

1i

Surf

TT

n

Vol

TT

nn

TT

n

dSurf∆TN∆d

dVol∆FN∆d2∆dBDB∆d2

1

∆E ∑∫

∫

=

−−

=

2.17

To satisfy the principle of minimum potential energy the potential energy ∆E is

differentiated with respect to the incremental nodal displacements and the result

set to zero. This latest expression, when written in the form of Equation 2.18,

represents the governing equation for the finite element method:

[ ] ( ) ∑ ∑= =

=N

1i

N

1i

EiniE ∆R∆dK 2.18

where [ ] [ ] [ ][ ]∫ ==Vol

T

E dVolBDBK Element stiffness matrix;

[ ] [ ] ∫ ∫ =+=Vol Surf

TT

E dSurf∆TNdVol∆FN∆R Right hand side load vector.

The integrals of Equation 2.18 are formulated in terms of the global coordinates

x, y. The matrix of shape functions [ ]N and consequently its derivatives are

however expressed in terms of the natural coordinates S and T. Therefore, the

volume and surface integrals need to be evaluated using the natural coordinate

system of the parent element, invoking the determinant of the Jacobian matrix

J . The isoparametric coordinate transformation for the volume integrals, for

example, gives:

dTdSJtdydxtdVolVol

==∫ 2.19

49

where t is the element thickness (which is taken as unity for plane strain

problems) and the Jacobian matrix is given by Equation 2.20:

[ ]

∂∂

∂∂

∂∂

∂∂

=

T

y

T

xS

y

S

x

J 2.20

Accordingly the stiffness matrix of a single element can be calculated in terms of

the natural coordinates S and T as:

[ ] [ ] [ ][ ]∫ ∫− −

=1

1

1

1

T

E dTdSJBDBtK 2.21

The right hand side load vector can be transformed into the natural coordinate

system in a similar way. The integrals in Equation 2.18 cannot generally be

calculated explicitly and it is therefore convenient to evaluate them numerically.

Numerical integration is usually performed by replacing the integral of a function

by a weighted sum of the function evaluated at a number of integration points.

The number of integration points determines the integration order. ICFEP

employs Gaussian integration, for which the optimum integration order depends

on the type of element being used. Potts and Zdravković (1999) suggest that for

an 8-noded isoparametric solid element either a second order (reduced) or a third

order (full) integration should be used.

It should be noted that in the cases of beam and interface elements, the

volume and surface integrals of Equation 2.18 reduce to one-dimensional

integrals. Hence the element stiffness matrix is given by:

[ ] [ ] [ ][ ]∫=length

T

E dlBDBK 2.22

where l is the length of the element. The integral is evaluated in the natural

ordinate system in Equation 2.23.

[ ] [ ] [ ][ ]∫−

=1

1

T

E dSJBDBK 2.23

50

where J is the determinant of the Jacobian matrix given by Equation 2.24.

2

1

22

dS

dy

dS

dxJ

+

= 2.24

Furthermore, the use of third order integration in beam elements can lead to

numerical instability, as membrane and shear force locking occurs. To overcome

this problem Day (1990) introduced substitute shape functions for some of the

terms in the strain equations. These substitute functions take the form:

3

2=

+=

−=

3

2

1

N

S3

1

2

1N

S3

1

2

1N

2.25

It should be noted that the substitute shape functions coincide with the usual

isoparametric shape functions of Equations 2.4 at the reduced Gaussian

integration points.

2.2.4 Global equations

Once the element stiffness matrices and right hand side vectors have been

formulated they can be assembled, as indicated by Equation 2.18, to give the

governing equation, which can be expressed as:

[ ] GnGG ∆R∆dK = 2.26

where [ ]GK is the global stiffness matrix, nG∆d is the vector of degrees of

freedom for the entire finite element mesh and G∆R is the global right hand

side load vector. The direct stiffness method is employed to perform the

assembly process (Potts and Zdravković, 1999). The main idea of this method is

that the global terms are obtained by summing the individual element

contributions and considering the degree of freedom which are common between

51

elements. Therefore, the resulting global stiffness matrix has generally many zero

terms (sparse) with the non-zero terms concentrated along the main diagonal

(banded). This band width can be minimized to reduce the computer storage by

using an appropriate node numbering scheme.

2.2.5 Boundary conditions

Before the system of global equations can be solved any applied

boundary conditions need to be considered. Load boundary conditions affect the

right hand side vector G∆R of the global system of equations. Examples of

such conditions are body forces, point loads, surcharges and forces from

excavated and constructed elements. Displacement boundary conditions affect

the displacement vector on the left hand side of Equation 2.26. It has always to

be ensured that enough displacements are prescribed so as to prevent rigid body

motions, such as translations and rotations, of the whole problem domain. If

insufficient displacements are prescribed, the global stiffness matrix becomes

singular.

2.2.6 Solution of the global equations

In the final stage of the finite element method the system of global

equations must be solved to give the unknown nodal displacements. Both direct

and iterative mathematical algorithms have been developed to solve the system

of global equations. Direct algorithms based on Gaussian elimination are most

commonly used (Potts and Zdravković, 1999). Once the primary variables, the

nodal displacements, have been found, the secondary variables, stresses and

strains, can be calculated using Equations 2.7 and 2.8.

2.2.7 Nonlinear finite element theory

Real soil behaviour is highly nonlinear as both the strength and the

stiffness depend on stress and strain levels. Incorporation of such behaviour into

the finite element method leads to a constitutive matrix [ ]D which is not

constant, but depends on stress and/or strain level. Furthermore, to introduce the

52

plasticity facets of soil behaviour, the constitutive matrix [ ]D needs to be

replaced by the elasto-plastic one [ ]epD . Details on the derivation of the elasto-

plastic matrix [ ]epD can be found in Potts and Zdravković (1999). The

governing finite element equation for nonlinear problems can be written as:

[ ] iG

i

nG

i

G ∆R∆dK = 2.27

where [ ]iGK is the incremental global stiffness matrix, inG∆d is the vector of

incremental nodal displacements, iG∆R is the vector of incremental nodal

forces and i is the increment number. The final solution is obtained by summing

the results of each increment. As the global stiffness matrix is dependent on the

current stress and strain levels, it not only varies between increments but also

during each increment. There are several techniques available in ICFEP to solve

the above nonlinear global equilibrium equation: the tangent stiffness method,

the visco-plastic method, the Newton Raphson method and the Modified Newton

Raphson method (MNR). According to Potts and Zdravković (1999) the

Modified Newton Raphson scheme is relatively insensitive to the increment size

and it is, among the afore-mentioned methods, the most robust and economical.

Therefore, this scheme was employed throughout this thesis and it is briefly

described in this section.

Figure 2.3 illustrates the MNR method for the solution of a non-linear

system with a single degree of freedom. The MNR scheme invokes an iterative

technique to solve Equation 2.27. In the first iteration the stiffness matrix, [ ]iGK ,

calculated from the stresses and strains at the end of the previous increment, is

used to obtain a first estimate of the incremental nodal displacements ∆d1. It is

however recognised that this solution is in error, as the stiffness matrix is

changing during an increment. So, the predicted displacements from the first

iteration are then used to calculate the incremental strains at each integration

point and the constitutive model is then integrated along the incremental strain

paths to obtain an estimate of the stress changes. These stress changes are added

to the stresses at the beginning of the increment and are then integrated to give

53

the equivalent nodal forces. The difference between these forces and the

externally applied load increment iG∆R gives the residual load vector, ψ1.

[K ]Gi

∆d1 ∆d2

∆d i

∆R i

LoadTrue solution

Displacement

ψ 1 ψ 2

Figure 2.3: The Modified Newton Raphson method (after Potts and Zdravković,

1999)

In the next step, Equation 2.27 is solved again with the residual load ψ1,

also known as the out-of-balance force, forming the right hand side vector:

[ ] ( ) 1-jji

nG

i

G ψ∆dK = 2.28

where superscript j denotes the iteration number and iG

0∆Rψ = . This process

is repeated until convergence is achieved. ICFEP checks the convergence by

setting criteria for both the iterative nodal displacements and the residual loads.

These are checked against the incremental and accumulated nodal displacements

and global right hand side load vectors, respectively. The default convergence

criteria are set such that the scalar norm of the iterative nodal displacement

vector is less than 2% of both the incremental and the accumulated nodal

displacement norms and the norm of the residual load vector is less than 2% of

both the incremental and accumulated global right hand side load vector norms.

In the original Newton Raphson method, the stiffness matrix [ ]iGK is

recalculated and inverted for each iteration, based on the latest estimate of

stresses and strains obtained from the previous iteration. In the MNR method

54

however, the stiffness matrix is only calculated at the beginning of the increment,

i.e. for the first iteration, and is then kept constant throughout the increment for

all subsequent iterations. Although this technique usually requires more iterations

to converge, overall the MNR is computationally cheaper than the original

Newton Raphson method, as the assembly and the inversion of the stiffness

matrix is very time consuming. Furthermore, in ICFEP there is the flexibility to

calculate the stiffness matrix using either the elastic constitutive matrix [ ]D or

the elastoplastic matrix [ ]epD and to update the stiffness matrix for any number

of iterations within an increment. In addition, in order to reduce the number of

required iterations for convergence with the MNR method, ICFEP employs the

acceleration technique of Thomas (1984), in which the iterative displacements

( )ji

nG∆d , are increased prior to the calculation of the residual load vector, ψj.

A key issue in the MNR method is the accurate evaluation of the residual

load ψ. This is done by integrating the constitutive model along the incremental

strain paths and adding the obtained stress changes to the stresses determined at

the end of the previous iteration. Therefore, the accuracy of MNR essentially

depends on the precision of the stress point algorithm that performs the

integration of the constitutive model. For this the default option in ICFEP is a

substepping algorithm, which calculates directly the elastic proportion of stress

changes and only divides the elastoplastic proportion of the incremental strains

into a number of sub-steps. The constitutive equations are then integrated

numerically over each substep using a modified Euler integration scheme. The

size of each substep is controlled automatically to produce a required level of

accuracy (i.e the algorithm uses error control).

2.2.8 Consolidation theory

Real soil behaviour is often time related, as the pore fluid response often

depends on the soil permeability, the rate of loading and the hydraulic boundary

conditions. To model this behaviour, the pore fluid pressure must be incorporated

as a primary unknown, together with the displacement. Similar to displacement

55

variation (Equation 2.5) the pore fluid pressure variation across an element can

be expressed in terms of nodal values using the following formula:

[ ] nP ∆pN∆p = 2.29

where ∆p is the change in pore fluid pressure, n∆p is the change in nodal

pore fluid pressure and [ ]PN is the matrix of shape functions, similar to [ ]N . It

should be noted that if an incremental pore pressure degree of freedom is

assumed at each node of every consolidating element, [ ]PN is identical with the

displacement shape function matrix [ ]N . Consequently, for an 8-noded plane

element both the displacement and the pore pressure vary quadratically across

the element, whereas the strains and therefore the stresses vary linearly.

Although this inconsistency between the variation of stress and pore pressure is

theoretically acceptable, it is generally desired that the effective stresses and the

pore water pressures vary in the same manner. Hence for an 8 noded element the

pore water pressures should also vary linearly across the element. This can be

achieved by assigning pore pressure degrees of freedom only to corner nodes and

not to mid-side nodes of such an element.

To derive the governing equations for coupled consolidation analysis it is

necessary to combine the equations governing the deformation of soil due to

loading with the equations governing the pore fluid flow. Therefore, the first step

is to formulate the equations governing the deformation of soil, allowing the

solid and the fluid phases to deform independently. Similar to the uncoupled

formulation of Section 2.2.3, the principle of minimum potential energy needs to

be employed (Equation 2.14). Using the principle of effective stress (Equations

2.10 and 2.11), the incremental strain energy ∆WE, can be written as:

[ ] [ ]∫ +′=Vol

f

T

E dVol∆ε∆σ∆εD∆ε2

1∆W 2.30

The work done by external loads remains unchanged and is therefore still given

by Equation 2.16. Equilibrium is again found by minimizing the potential energy

of the body in the same manner as outlined in Section 2.2.3. Furthermore,

assembling the separate element equations for all the elements in the

56

computational domain, the global equilibrium equation in terms of effective

stresses is obtained:

[ ] [ ] GnGGnGG ∆R∆pL∆dK =+ 2.31

where

[ ] [ ] [ ] [ ][ ]

[ ] [ ] [ ] [ ]

[ ] [ ]

0111m

dSurf∆TNdVol∆FN∆R∆R

dVolNBmLL

dVolBDBKK

T

N

1iiSurf

T

iVol

TN

1i

EG

N

1iiVol

p

TN

1iiEG

N

1iiVol

TN

1iiEG

=

+

==

==

′==

∑ ∫∫∑

∑ ∫∑

∑ ∫∑

==

==

==

Obviously, Equation 2.31 cannot be solved as it includes two unknowns, the

nodal displacements nG∆d and the nodal pore pressures nGf∆p . Therefore,

another set of equations that governs the flow of the pore fluid is required to

solve the complete problem. This additional set of equations can be obtained

combining the equations of continuity and the generalised Darcy’s law. The

continuity equation is obtained considering the flow of pore fluid in and out of an

element of soil of unit dimensions and it is given by Equation 2.32:

t

ε∆Q

y

v

x

v vyx

∂∂

−=−∂

∂+

∂∂

2.32

where xv and yv are components of the seepage velocity of the pore fluid in the

coordinate directions, vε is the volumetric strain, t is the time and ∆Q denotes

any sources or/and sinks (the negative sign denotes outflow, i.e. sink). The above

equation assumes that the soil is fully saturated and that both the pore fluid and

the soil skeleton are incompressible. Furthermore, the generalised Darcy’s law

relates the seepage velocity to the pressure head h as follows:

[ ] hkv ∇−= 2.33

57

where v is the velocity vector with components xv and yv , [ ]k is the

permeability matrix of the soil and the hydraulic head h is defined as:

( )GyGx

f

iyixγ

ph ++= 2.34

TGyGxG iii = is the unit vector parallel, but in the opposite direction, to

gravity and fγ is the bulk unit weight of the pore fluid. In addition, employing the

principle of virtual work the continuity equation can be written as:

( ) ∆p∆QdVol∆pt

ε∆pv

Vol

vT =

∂∂

+∇∫ 2.35

Substituting Equations 2.33 and 2.34 into 2.35 and approximating t

εv

∂∂

as t

εv

∆∆

,

leads to:

[ ] ( ) ∆t∆p∆QdVol∆p∆εdt∆pipγ

1k

Vol

∆tt

t

vG

f

k

k

=

+∇

+∇−∫ ∫

+

2.36

Equation 2.36 can be written in finite element form as:

[ ] [ ] [ ] ( )∆t∆QndtpΦ∆dL G

∆tt

t

nGGnG

T

G

k

k

+=− ∫+

2.37

where

[ ] [ ] [ ] [ ][ ]

[ ] [ ] [ ] [ ]

[ ]T

ppp

N

1i i

G

Vol

TN

1iiEG

i

N

1i Vol f

TN

1iiEG

z

N

y

N

x

NE

dVolikEnn

dVolγ

EkEΦΦ

∂

∂

∂

∂

∂

∂=

==

==

∑ ∫∑

∑ ∫∑

==

==

The integral in Equation 2.37 can be approximated by:

58

( ) [ ]∆t∆pβtpdtpnGnGk

∆tt

t

nG

k

k

+=∫+

2.38

where β is an integration parameter introduced to indicate how the pore pressure

is assumed to vary during the increment and the integration limit tk refers to the

previous increment. Booker and Small (1975) suggest that the stability of the

marching process is ensured for 0.5≤β≤1.0. Throughout this thesis a value of

β=0.8 was employed. Utilising this time marching process, Equations 2.31 and

2.37 can be written in incremental form:

[ ] [ ][ ] [ ]

[ ] [ ] ( ) ( )[ ]

++=

− ∆ttpΦ∆Qn

∆R

∆p

∆d

Φ∆tβL

LK

nGkGG

G

nG

nG

G

T

G

GG 2.39

The coupled behaviour of the soil skeleton and pore fluid can be simulated by

solving the above system of simultaneous equations.

In coupled consolidation, in addition to the load or displacement

boundary conditions (see Section 2.2.5), it is necessary to prescribe hydraulic

boundary conditions along the mesh boundary. Therefore, either a nodal pore

fluid pressure or a nodal flow must be prescribed along the mesh boundary.

Nodal flows are included in the term ∆Q, whereas prescribed pore pressures

affect the global nodal pore water pressure vector nG∆p . Once all boundary

conditions have been specified, Equation 2.39 can be solved, using the

methodology described in Sections 2.2.6 and 2.2.7, to give the nodal

displacements and pore pressures.

2.3 Summary

This chapter detailed the basic steps that are required in the formulation

of the finite element method for static analysis. These are: element discretisation,

primary variable approximation, formulation of the element equations,

assemblage of the element equation to give the global equations, formulation of

the boundary conditions and solution of the global equations. Furthermore, when

the soil behaviour is described by nonlinear constitutive relationships, the

59

solution of the incremental global equations is not straightforward. Therefore the

modified Newton-Raphson method in conjunction with a substepping stress point

algorithm and a modified Euler integration scheme was used in the nonlinear

analyses presented in this thesis. Attention was also focused on the required

changes to the standard FE theory to model soil as a two phase material. Three

different types of possible soil response were discussed: the fully drained case in

which the pore fluid pressures remain constant, the fully undrained case and

finally the situation in between these two extremes that models the coupled

response of the pore fluid and the soil skeleton.

60

Chapter 3:

DYNAMIC FINITE ELEMENT FORMULATION

3.1 Introduction

In all the cases considered in the previous chapter, it has been assumed

that the variation of applied forces with time is sufficiently slow that inertia

forces can be neglected. However, if the loads are applied rapidly, with respect to

the natural frequencies of the system, inertia forces need to be taken into account.

For example, for a single degree of freedom (SDOF) system, inertia forces are

generally considered to become important when the frequency of loading is equal

to or greater than half the natural frequency of the system. Furthermore, a direct

consequence of inertia is the development of a damping force opposing the

motion. This force leads to dissipation of energy through various mechanisms

(e.g. friction, heat generation, plastic yielding) and it should be also included in

the finite element formulation.

Hardy (2003) discussed the fundamental aspects of dynamic finite

element theory and developed dynamic analysis capabilities within ICFEP. This

chapter repeats key issues of the dynamic finite element theory, with a particular

emphasis on the theory related to changes of the program undertaken by the

author. The first change concerns the implementation of a new time integration

scheme, the generalized-α algorithm of Chung & Hulbert (1993), which has to

date only been used in the field of structural dynamics. This algorithm was

further developed to deal with dynamic coupled consolidation problems.

3.2 Finite element formulation of the equation of motion

The finite element discretization in space as discussed in Section 2.2 also

applies in the case of dynamic analysis. The additional constituent of the

61

dynamic finite element analysis is the fact that inertia and damping forces should

also be considered when forming the equilibrium of a body (Equation 2.14). For

an undamped body dynamic equilibrium is obtained employing d’Alembert’s

principle. Hence the dynamic equilibrium of an element in incremental form is

expressed as:

∆L ∆I∆W∆E EEEE −+= 3.1

where E∆E is the incremental total potential energy and E∆I is the incremental

inertial energy. The incremental strain energy E∆W and the incremental work

done by the applied loads E∆L were defined in Equations 2.15 and 2.16

respectively. Assuming that the mass is constant, according to Newton’s second

law, the force applied to a body is equal to the rate of change of momentum.

Therefore the inertial force equals mass times acceleration and the incremental

inertial energy due to translational movement can be written as:

∫=Vol

E dVold∆ρ∆d∆I && 3.2

where ρ is the mass per unit volume and ∆d and d∆ && are the incremental

displacements and accelerations respectively. As mentioned earlier, in reality the

energy is dissipated during vibration and thus damping forces should also be

considered when investigating the dynamic equilibrium of a system. Hence, for a

damped vibration the dynamic equilibrium equation becomes:

∆L ∆D∆I∆W∆E EEEEE −++= 3.3

where E∆D is the incremental damping energy. The damping force is usually

assumed to be velocity dependent and thus the incremental damping energy can

be defined as:

∫=Vol

E dVold∆c∆d∆D & 3.4

where d∆& is the incremental velocity and c is a constant representing the damping characteristics of the material. This is a convenient but simplistic

62

approximation, as in reality it is not possible to describe mathematically the true

mechanism of damping employing just the constant c. The limitations of this

approximation, known as equivalent viscous damping, will be returned to later in

this chapter.

Furthermore, it is assumed that the velocity and acceleration variations

across an element are identical to that of the displacement. Hence, employing the

shape functions that were introduced in Section 2.2.3, the displacement, velocity

and acceleration can be expressed in terms of their nodal values, as follows:

[ ] [ ]

[ ] [ ]

[ ] [ ] nn

n

n

n

n

d∆Nv∆

u∆N

v∆

u∆d∆

d∆Nv∆

u∆N

v∆

u∆d∆

∆dN∆v

∆uN

∆v

∆u∆d

&&&&

&&

&&

&&&&

&&

&

&

&&

=

=

=

=

=

=

=

=

=

3.5

Writing the expressions of ∆W , ∆I , ∆L and ∆D in terms of their nodal values

and substituting them in Equation 3.3, the total incremental potential energy of

the body, expressed as the sum of potential energies of separate elements, is

given by:

( )

[ ] [ ][ ] [ ] [ ]

[ ] [ ] [ ]

[ ] i

N

1i

Surf

T

Vol T

n

T

n

T

n

T

i

T

n

dSurf∆TN

dVol

∆FNd∆NcN

d∆NρN∆dBDB2

1

∆d∆E ∑

∫

∫

=

−

−

++

= &

&&

3.6

To derive the governing dynamic finite element equilibrium equation, following

a similar procedure to that used for static analysis, the differential of the

incremental total potential energy E∆E with respect to the incremental nodal

displacements is set to zero. This leads to Equation 3.7 which represents the

incremental governing equation for the finite element method and which is

commonly known as the equation of motion.

63

[ ] ( ) [ ] ( ) [ ] ( ) ∑ ∑∑∑= ===

=++N

1i

N

1i

EiniE

N

1iiniE

N

1iiniE ∆R∆dKd∆Cd∆M &&& 3.7

where [ ] [ ] [ ]∫ ==Vol

T

E dVolNρNM element mass matrix;

[ ] [ ] [ ]∫ ==Vol

T

E dVolNcNC element damping matrix;

[ ] [ ] [ ][ ]∫ ==Vol

T

E dVolBDBK element stiffness matrix;

[ ] [ ] ∫ ∫ =+=Vol Surf

TT

E Surf∆TNdVol∆FN∆R Right hand side load vector.

The procedure to evaluate the stiffness matrix and the right hand side load vector

was described in Section 2.2.3. In a similar way, the consistent mass matrix of a

single element can be calculated in terms of the natural coordinates S and T as

shown in Equation 3.8.

[ ] [ ] [ ] dTdSJNρNtM

1

1

1

1

T

E ∫ ∫+

−

+

−

= 3.8

where t is the element thickness. The integral in Equation 3.8 is evaluated

numerically by employing the procedure described in Section 2.2.3. The term

consistent refers to the fact that the same interpolation functions are employed

for the calculation of the mass matrix as in the evaluation of the stiffness matrix.

The consistent mass matrix is fully populated and thus the mass is naturally

distributed throughout the element. On the other hand, quite commonly the mass

of an element is assumed to be lumped at its nodes, resulting in the diagonal

matrix of Equation 3.9.

[ ] [ ]IρME = 3.9

where ρ in the above equation is the mass density per unit length and [ ]I is a nxn identity matrix (i.e. n degrees of freedom). As it will be explained in Section

3.3.4, the use of a lumped mass matrix is very economical when it is combined

with an explicit integration scheme. Generally, lumping is based more on

64

convenience than theoretical arguments. Wood (1990), among others, compared

the properties of a lumped mass matrix against a consistent mass matrix and

concluded that the decision of lumping or not the mass matrix is heavily problem

dependent. Hardy (2003) notes that the relative merits of mass lumping appear to

be of less importance at present than during the early days of finite element

analysis and he therefore decided to implement a consistent mass matrix

formulation into ICFEP.

It was shown in Section 2.2.3, that beam elements have 2 translational

and 1 rotational degrees of freedom. Therefore, when considering the

incremental inertial energy of a beam element, the rotary inertia should also be

taken into account according to the Equation 3.10.

∫∫ +=Vol

rr

Vol

ttE dVold∆ρI∆ddVold∆ρ∆d∆I &&&& 3.10

where

Tr

T

r

T

t

T

t

θ00d∆

θ00∆d

0vud∆

0vu∆d

&&&&

&&&&&&

=

=

=

=

and u, v are the horizontal and the vertical displacement component respectively

and θ is the rotation. This leads to the element mass matrix of Equation 3.11.

[ ] [ ] [ ]rEtEE MMM += 3.11

where [ ] [ ] [ ]∫=length

t

T

ttE dlNρNM , [ ] [ ] [ ]∫=length

r

T

rrE dlNIρNM , I is the moment of

inertia and for a 3 noded beam element:

[ ]

[ ] [ ]321r

321

321

t

N00N00N00N

0N00N00N0

00N00N00NN

=

=

65

Generally, the inertia forces associated with node rotations are not significant.

Numerical tests by Stolarski and D’Costa (1997) suggest that the rotational

inertia only affects the high frequency response of non-slender beams.

Furthermore, the calculation of rotary inertia invokes the storage of an additional

variable, the second derivative of rotation ( )θ&& . It was decided to neglect the

rotary inertia contribution [ ]( )rEM when formulating the beam elements’ mass

matrix in ICFEP, as this does not significantly affect the rigorousness of the

formulation and it reduces the storage memory requirements. However, due to

this approximation the mass matrix is not fully consistent, even for second order

integration, with the stiffness matrix. The dynamic equilibrium formulation of

beam elements in ICFEP is validated in Chapter 4. It should be also noted that

interface (or joint) elements do not contribute to the global mass matrix, as they

have zero thickness and therefore zero mass.

As mentioned previously, the attenuation of energy of a vibrating body

occurs through different mechanisms. Generally two types of energy dissipation

exist in soil-structure-interaction problems: radiation and material damping.

Radiation damping is of geometrical origin due to spreading of energy over a

greater volume of material. Modelling the radiation damping usually involves the

incorporation of appropriate absorbing boundary conditions along the bottom and

side boundaries of a finite element mesh. This problem is addressed in detail in

Chapter 5. Material damping is a completely different mechanism and it can be

of hysteretic or viscous nature. Hysteretic damping is the dominant mechanism

and is caused by frictional loss and non-linearity of the stress strain relationship

of the material. Hysteretic damping depends on the strain level and the number of

vibration cycles, but is independent of the frequency of vibration. On the other

hand, viscous damping is caused by the viscosity of the fluid flow within the

pores of the soil matrix and is frequency dependent.

As mentioned earlier in this chapter, material damping is usually

modelled in a highly idealised way. An equivalent viscous damping is employed

to model both the hysteretic and the viscous parts of material damping. It is

therefore assumed that the damping matrix [ ]EC is a linear combination of the

stiffness [ ]EK and the mass [ ]EM matrices according to Equation 3.12.

66

[ ] [ ] [ ]EEE KBMAC += 3.12

where A is a mass proportional damping constant and B is a stiffness

proportional constant. This approximation is commonly known as Rayleigh

damping. For a single mode of a multiple degree of freedom system the

relationship between the damping ratio ξi and the constants A, B is given by

(Bathe, 1996):

2

Bω

2ω

Aξ i

i

i += 3.13

where ω is the angular frequency and the subscript i refers to the mode of

vibration under consideration. The damping ratio is a dimensionless measure of

damping and it expresses the damping of a system as a faction of the critical

damping (i.e. the damping that inhibits an oscillation completely). Figure 3.1

illustrates the variation of the critical damping ratio with the angular frequency

for mass proportionally damping, for stiffness proportionally damping and for

the sum of both components. Mass proportional damping is dominant in the low

frequency range while stiffness proportional damping dominates the high

frequency range. Since in reality damping is independent of frequency, the aim is

to evaluate the parameters A and B in such a way that the resulting damping is

reasonably constant for a desired frequency range. Woodward and Griffiths

(1996) suggest that the parameters A, B can be thus calculated by Equation 3.14:

21

t

21

t21

ωω

ξ2B

ωω

ξωω2A

+=

+=

3.14

where 1ω , 2ω are the two frequencies defining the frequency range over which

the damping is approximately constant and tξ is the target damping ratio. The

idea is to get the right “target” damping for the important frequencies of the

problem. According to Zerwer et al (2002) this can be achieved to a certain

extent by taking 1ω as the first natural frequency of the system and 2ω as the

highest natural frequency of the vibration modes with high contribution to the

67

response. Clearly, Rayleigh damping is only an approximate way to reproduce

material damping as it is not related to the strain level and is dependant on the

frequency of vibration. Theoretically, a rigorous non-linear elasto-plastic

constitutive model can capture hysteresis curves and can also model the energy

loss due to pore pressure generation (i.e. viscous effects). Hence Rayleigh

damping should be only used in linear analysis. However, when simple

elastoplastic constitutive models are employed, Rayleigh damping is widely used

to account for the lack of hysteretic damping (e.g. Smith, 1994, Woodward and

Griffiths, 1996).

Figure 3.1: Relationship between Rayleigh damping parameters and damping

ratio (after Zerwer et al, 2002)

3.2.1 Constitutive soil models

In general, two distinct methodologies are conventionally employed to

model non-linear cyclic soil stress-strain behaviour: equivalent linear and truly

non-linear. The equivalent linear method involves linear analyses in which the

soil shear stiffness and damping characteristics are iteratively adjusted until they

are compatible with the level of strain induced in the soil (Kramer, 1996).

Although this approach has the advantage of mathematical simplicity, it has

serious limitations as it is unable to predict the changes in stiffness that actually

occur due to increasing number of cycles of dynamic loading. Furthermore the

equivalent linear approach is unable to model plastic deformation and pore

pressure generation.

68

A reliable constitutive model should be able to capture features of the soil

behaviour when subjected to cyclic loading like: stiffness degradation, hysteretic

damping and plastic deformation during unloading. Various constitutive models

were used in the present study. A simple elastic perfectly plastic Mohr-Coulomb

model was employed in Chapter 4 to analyse a foundation subjected to various

earthquake loadings. In Chapter 7 a variant of the modified Cam Clay model

(Roscoe and Burland, 1968) was used to describe the plastic yielding behaviour

of the soil and the small strain stiffness model of Jardine et al (1986) was used to

describe the non-linear elastic pre-yield behaviour. A detailed description of the

above-mentioned constitutive models can be found in Potts and Zdravković

(1999). The major limitation of simple elasto-plastic models is their inability to

model hysteretic dissipation due to the unrealistic large extent of their yield

surface. Therefore, in Chapter 7 the analyses were repeated with a two-surface

kinematic hardening model (Grammatikopoulou, 2004) that allows non-linearity

and plasticity to develop within the conventionally defined yield surface.

3.2.2 Spatial discretization

As noted in Chapter 2, optimum mesh design requires having as few

elements as possible to reduce the computational cost and refining the mesh at

locations of stress concentration to give an accurate result. Away from areas of

stress concentration the mesh can become coarser to reduce the overall number

of degrees of freedom. While this is true for static analysis, in wave propagation

problems the spatial discretization of the mesh is closely related to the frequency

content of the excitation. Elements that are too large will filter waves of short

wavelengths. Kuhlemeyer and Lysmer (1973), using linear shape functions,

showed that for an accurate representation of wave transmission through a finite

element mesh the element side length, ∆l, must be smaller than approximately

one-tenth to one-eighth of the wavelength associated with the highest frequency

component of the input wave:

max

min

max

minminmin

f8

V

f10

V

8

λ

10

λ∆l ÷=÷≤ 3.15

69

where Vmin is the lowest wave velocity that is of interest in the simulation and

fmax is the highest frequency of the input wave. This frequency can be found by

performing a Fourier analysis of the input motion. The above-mentioned

condition agrees well with the results from a sensitivity analysis by Hardy

(2003). Kuhlemeyer and Lysmer (1973) also suggest that the condition of

Equation 3.15 is valid only when a consistent mass matrix is employed and they

note that this criterion should be stricter when a lumped mass matrix is used.

Furthermore, Bathe (1996) suggests that the length ∆l can be taken as the spacing

of the nodes and thus for an 8-noded solid element the condition of Equation

3.15 becomes:

max

min

max

minminmin

f4

V

f5

V

4

λ

5

λ∆l ÷=÷≤ 3.16

It should be highlighted that in nonlinear analysis the wave velocities represented

in the finite element model change during its response. Thus, the evaluation of ∆l

should take into account this wave velocity variation and the uncertainty

regarding the value of Vmin.

3.3 Direct integration method

In mathematical terms the governing dynamic finite element equation

(Equation 3.7) represents a system of second order differential equations. In

finite element analysis there are three distinct methodologies to solve this system

of equations: modal analysis, frequency domain analysis and direct integration.

The former approach transforms Equation 3.7 into a system of uncoupled

equations in modal coordinates. The response of each vibration mode can thus be

computed independently of the others and the modal responses can then be

superimposed to give the total response. Modal analysis is a very popular method

in structural dynamics, but it is not widely used in wave propagation problems as

it can properly model neither the spatial variation of material damping within the

soil mass nor the radiation damping (Chopra, 1995). On the other hand,

frequency domain analysis is widely used in problems of wave propagation as it

can deal with both material and radiation damping. This approach assumes that

70

any loading can be approximated by a Fourier series. Hence the total response is

a summation of the solution for each harmonic. Both classical modal analysis

and frequency domain analysis are based on the principle of superposition and

thus they are not applicable to nonlinear systems. In the present study only the

direct method is examined as it can be employed for the solution of nonlinear

systems. The fundamental idea of this method is to approximate the solutions of

the equation of motion with a set of algebraic equations which are evaluated in a

step-by-step manner. The step-by-step procedure involves discretising both the

excitation and the response into small time increments ∆t. This method assumes

that the equation of motion is then satisfied only at the discrete time intervals ∆t.

Furthermore the variation of displacement, velocity and acceleration within a

time increment is assumed. What distinguishes the various integration schemes is

the way that they approximate this variation of displacement, velocity and

acceleration within a time interval. While there are numerous time marching

schemes available, only some of the most popular are presented in this thesis.

The relative merits of each method are discussed, but more emphasis is put on

the generalized-α algorithm of Chung and Hulbert (1993) that was chosen to be

implemented into ICFEP.

3.3.1 Characteristics of integration schemes

While there is not yet a universally accepted “perfect” time integration

method, Hilber and Hughes (1978) gave a list of 6 characteristics that a marching

scheme should possess in order to be competitive and efficient:

1. Unconditional stability for linear problems

2. Only one set of implicit equations to be solved at each time step

3. Second order accuracy

4. Controllable algorithmic dissipation in the higher modes

5. Self-starting

6. No tendency for pathological overshooting

71

Stability

An integration scheme is said to be stable if the numerical solution, under

any initial conditions, does not grow without bound (Bathe, 1996). An algorithm

is unconditionally stable for linear problems if the convergence of the solution is

independent of the size of the time step ∆t. Otherwise the algorithm is

conditionally stable for values of ∆t less than a critical value ∆tcr. The value of

the critical time step is equal to a constant multiplying the smallest natural period

of the system and it depends also on the material damping of the system. For a

multiple degree of freedom system, like a finite element mesh, requiring stability

for all vibration modes imposes severe restrictions on the value of the critical

time step and can lead to high computational cost. Therefore in a finite element

analysis unconditionally stable schemes are generally preferred, as in that case

the size of the time step is determined only by the accuracy of the solution.

Furthermore, all integration schemes can be classified as either explicit or

implicit methods. The great advantage of explicit schemes is that the solution

does not involve the inversion of the stiffness matrix. However, Dahlquist (1963)

demonstrated that all explicit methods are conditionally stable with respect to the

size of the time step. On the other hand most implicit integration methods are

unconditionally stable, but the inversion of the stiffness matrix at each time step

makes them computationally expensive.

As mentioned earlier, in modal analysis the response of each vibration

mode is computed independently of the others and the modal responses are then

superimposed to give the total response. If all modes of a linear system are

integrated with the same time step ∆t, then modal superposition analysis is

completely equivalent to a direct integration analysis of the complete system

using the same time step ∆t and the same integration scheme (Bathe, 1996). To

investigate the stability characteristics of an integration scheme in the linear

regime, it is common practice to consider the modes of a system independently

with a common time step ∆t instead of considering the global Equation 3.7 (see

Appendix A). Furthermore the governing equation of one mode is equivalent to

the governing equation of a single degree–of–freedom (SDOF) model. Consider

the homogeneous equilibrium equation of a SDOF system:

72

0ukucum =++ &&& 3.17

where m is the mass, c is a constant representing the damping and k is the

stiffness of the SDOF system and u is a single degree of freedom in terms of a

displacement. Solving the above equation numerically will lead to the following

set of equations:

[ ]

=

+

+

+

k

k

k

1k

1k

1k

u

u

u

A

u

u

u

&&

&

&&

& 3.18

where A is the amplification matrix that determines algorithmic characteristics

like stability, accuracy and numerical dissipation. A marching scheme is said to

be stable if the amplification matrix is bounded:

||Aκ|| ≤ constant 3.19

where κ is a real number. The characteristic cubic equation of A is:

0AλAλA2λλI)det(A 32

2

1

3 =−+−=−− 3.20

where I denotes the identity matrix, λ is a solution of Equation 3.20, A1 is the

trace of A, A2 is the sum of the principal minors of A and A3 is the determinant

of A. Since Equation 3.20 is cubic, there can be 3 possible solutions (λ1, λ2, λ3)

which are also known as eigenvalues of A. The spectral radius ( )Aρ is then

defined as:

( ) 321 λ,λ,λmaxAρ = 3.21

An algorithm is said to be A-stable when the following conditions are fulfilled:

( ) 1Aρ ≤ 3.22

Eigenvalues of A of multiplicity greater than one are strictly less than one in

modulus.

73

Number of implicit systems to be solved

Hilber and Hughes (1978) suggest that the algorithm should not require the

solution of more than one implicit system, of the size of the mass and stiffness

matrices, at each time step. Although, algorithms that require two or more

implicit systems of the size of the mass and stiffness matrices to be solved at

each time step possess improved properties (e.g. Argyris et al. 1973), they

require at least twice the storage and computational effort of simpler methods.

Accuracy

After examining the stability of an algorithm, it is vital to investigate its

accuracy. In general, the accuracy depends on the size of the time step. The

smaller the time step, the more accurate is the solution. An integration scheme is

convergent if the numerical solution approaches the exact solution as ∆t tends to

zero. The magnitude of the numerical error is proportional to (∆t/T)ε, where ε

expresses the order of the accuracy. According to Hilber and Hughes (1978) the

second order accurate methods are immensely superior to the first order accurate

methods. Furthermore, Dahlquist (1963) theorem suggests that a third order

accurate unconditionally stable linear multistep method does not exist. Therefore,

a desirable characteristic of an algorithm is only second order accuracy.

It should be noted that there are two kinds of numerical errors controlling

the accuracy of an algorithm. Figure 3.2 illustrates the displacement response of

a SDOF oscillator subjected to an undamped free vibration normalised by the

initial displacement u0. The dashed curve is the numerical solution and the solid

one is the closed form solution. The divergence between the two curves indicates

the numerical error. The first type of error is the numerical dispersion which can

be expressed in terms of period elongation:

T)/T(T'PE −= 3.23

where T' is the natural period of the structure obtained by the numerical solution

and T is the actual period of the structure. The second type of error is the

numerical dissipation which can be expressed as either amplitude decay (AD) at

time tk defined by the following function:

74

)(tu

)T'(tu1AD

k

k +−= 3.24

or by the algorithmic damping ratio:

( )Ω'2

IRLnξ'

2

m

2

e +−= 3.25

where Re, Im are the real and the imaginary part respectively of the complex

eigenvalues λ1,2 (i.e. iIRλ me1,2 ±= ) and Ω' is the natural frequency of the

structure obtained by the numerical solution and which can be calculated as:

( )Q/RarctanT'

2πΩ' == 3.26

Time (t)

-1

0

1

Dis

pla

ce

me

nt

(u/u

0)

Closed form solution

Numerical solution

T/2

AD

T T'

PE

Figure 3.2: Illustration of period and amplitude error in numerical solution.

Algorithmic dissipation

The necessity for time integration algorithms to possess algorithmic

damping is widely recognized. Due to poor spatial discretization, the finite

element method cannot represent accurately high-frequency modes. Strang and

Fix (1973), among others, showed that modes corresponding to higher-

frequencies become more and more inaccurate. Thus, the role of the numerical

damping is to eliminate spurious high frequency oscillations. Specifically in

earthquake engineering, the highest modes of the system do not have to be

represented accurately anyway, since the seismic excitation can significantly

activate only the low frequency modes. Therefore, a desirable property of an

75

algorithm is the preferential numerical damping (“filtering”) of the inaccurate

high frequency modes and the preservation of the important low frequency

modes.

The spectral radius can also be used as a measure of algorithmic

dissipation. For ρ equal to 1, the dissipation is equal to zero, but as ρ decreases,

the algorithmic dissipation increases. A way to investigate the algorithmic

damping characteristics of an integration scheme is to plot the spectral radius

( )Aρ against the ratio ∆t/T (where T is the undamped natural period of a SDOF

and ∆t is the time step). Wood (1990) suggests that optimum dissipative

behaviour is attained when ρ stays close to unit level for as long as possible and

decreases to about 0.5-0.8 as ∆t/T tends to infinity.

Another way to filter the higher modes could be the use of viscous

damping. However, Hughes (1983) argues that the use of viscous damping

affects a middle band of frequencies, not the inaccurate higher frequency modes.

Figure 3.3 illustrates a plot of ρ versus ∆t/T for the constant average acceleration

(CAA) method (that is presented in detail in Section 3.3.4) for both the damped

(ξ=0.5) and the undamped case (ξ=0.0). Clearly the addition of viscous damping

does not affect the values of ρ at higher modes as ρ is still equal to 1 when the

ratio ∆t/T tends to infinity, whereas it seriously affects a middle band of

frequencies. Therefore, the only adequate way to damp out the spurious modes is

the use of controllable algorithmic damping.

0.001 0.01 0.1 1 10 100

∆t/T

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

ρ

CAA, (ξ=0.0)

CAA, (ξ=0.5)

Figure 3.3: CAA with and without viscous damping.

76

Self-starting

Integration schemes which are not self-starting require data from more than two

time steps to proceed the solution. In this case, the standard practice is to assume

the initial conditions. Thus, apart from the algorithm, a starting procedure should

be implemented and analysed. Obviously, this requires additional computational

effort and storage. Furthermore, the interaction of the fictitious values of the

starting procedure with the true solution is not straightforward. Therefore, self-

starting algorithms are generally preferred.

Overshooting

The term overshooting describes the tendency of an algorithm (for large

time steps) to exceed heavily the exact solution in the first few time steps, but

eventually to converge to the exact solution. This peculiar phenomenon was first

discovered by Goudreau and Taylor (1972) as a property of the Wilson θ-method

(that it is presented in detail in Section 3.3.6) and is not related to the stability

and accuracy characteristics of the algorithms discussed so far.

3.3.2 Houbolt method

Houbolt (1950) made one of the first attempts to develop an integration

scheme for the computer analysis of aircraft dynamics. This method employs the

following two backward difference formula for the acceleration and velocity at

time 1ktt += :

( ) ( ) ( ) ( ) ( )[ ]2k1kk1k1k tu2tu9tu18tu11∆t6

1tu −−++ −+−=& 3.27

( ) ( ) ( ) ( ) ( )[ ]2k1kk1k21k tutu4tu5tu2∆t

1tu −−++ −+−=&& 3.28

77

Substituting these variations of velocity and acceleration into the global1

equilibrium equation results in the following expression:

[ ] [ ] [ ] ( ) [ ] ( ) ( ) ( )

[ ] ( ) ( ) ( ) 1k2-k1-kk

2-k21-k2k21k2

Rtu∆t

1tu

∆t2

9tu

∆t

3C

tu∆t

1tu

∆t

4tu

∆t

5MtuKC

∆t6

11M

∆t

2

+

+

+

−−+

−−=

++

3.29

Although the method is unconditionally stable and second order accurate, it is

not very popular anymore, as it is not self-starting. Furthermore the Houblot

method does not allow parametric control of the amount of numerical dissipation

present.

3.3.3 Park method

Similarly to the Houbolt method, Park introduced a multistep algorithm

that is also second order accurate and unconditionally stable. Details of the

derivation can be found in the original publication (Park 1975) and will not be

repeated herein. Park’s algorithm is slightly more accurate than the Houbolt

method, but it also requires a special starting procedure and it does not allow

parametric control of the amount of numerical dissipation present.

3.3.4 Newmark method

The Newmark method (Newmark, 1959) is probably the most commonly

used family of algorithms for solving the system of Equations 3.7. It employs a

truncated form of Taylor’s expansions to approximate the displacement and the

velocity at time t+∆t, i.e. at 1ktt += . Hence using Taylor’s series and assuming

that the displacement, velocity and acceleration are known at the time ktt = , the

1 The subscripts G, n denoting global and nodal values respectively are omitted in all the

derivations of Section 3.3 for brevity.

78

displacement and velocity at time 1ktt += are given by Equations 3.30 and 3.31

respectively:

( ) ( ) ( ) ( ) ( ) K&&&&&& ++++=+6

∆ttu

2

∆ttu∆ttututu

3

k

2

kkk1k 3.30

( ) ( ) ( ) ( ) K&&&&&&& +++=+2

∆ttu∆ttututu

2

kkk1k 3.31

Newmark truncated these equations and expressed them in the following form:

( ) ( ) ( ) ( ) ( ) 2

1k

2

kkk1k ∆ttuα∆ttuα2

1∆ttututu ++ +

−++= &&&&& 3.32

( ) ( ) ( ) ( ) ( )∆ttuδ∆ttuδ1tutu 1kkk1k ++ +−+= &&&&&& 3.33

where the α and δ terms approximate the remaining terms in the Taylor series

expansions and they show how much of the third derivative of displacements at

the end of the time step enters into the relations for displacement and velocity.

The exact values of the remaining terms are not known, thus the selection of the

algorithmic parameters (α, δ) controls the stability and the accuracy of the

solution. Depending on their values, Newmark’s algorithm takes different forms.

Argyris and Mlejnek (1991) wrote Equations 3.32 and 3.33 in incremental form

to explain the role of the algorithmic parameters α and δ:

( ) ( ) 22

kk ∆tu∆α∆ttu2

1∆ttu∆u &&&&& ++= 3.34

( ) ∆tu∆δ∆ttuu∆ k&&&&& += 3.35

Figure 3.4, employing a dimensionless expression for the variable of

time∆t

ttτ k−= , illustrates an interpretation of the algorithmic parameters α and δ.

The second term of Equation 3.34 represents the triangular area ( )ktu2

1&& , while

the third term defines the incremental area A between the curve and the

triangular area ( )ktu2

1&& in Figure 3.4a. Furthermore, the first term of Equation

79

3.35 represents the shaded area in Figure 3.4b, whereas the second term

represents the additional area B due to the variation of acceleration over the time

step.

0 1 τ

(1−τ) u&& (τ)

A

0 1 τ

u&& (τ)

Bu&&

ktu&&

( )ktu2

1&&

ktu&&ktu&&

(a) (b)

Figure 3.4: Interpolation of acceleration and interpretation of the Newmark

parameters α and δ (after Argyris and Mlejnek 1991).

Approximating the areas A, B with zero and a triangular respectively

(α=0, δ=1/2), the algorithm collapses to the central difference method.

Considering Equations 3.32 and 3.33 (for α=0, δ=1/2) at times tk-1and tk+1 and

then rearranging them, the variations of velocity and acceleration for the central

difference method (CDM) can be obtained:

( ) ( ) ( )[ ]1k1kk tutu∆t2

1tu +− +=& 3.36

( ) ( ) ( ) ( )[ ]1kk1k2k tutu2tu∆t

1tu +− +−=&& 3.37

Substituting Equations 3.36 and 3.37 into the equilibrium equation and assuming

that the displacement is known at the time instances 1ktt −= and ktt = , the

solution in terms of displacement at time 1ktt += is given by equation 3.38.

[ ] [ ] ( ) ( ) [ ] [ ] ( )

[ ] [ ] ( ) 1-k2

k2kk1k2

tuC∆t2

1M

∆t

1

tuM∆t

2KtRtuC

∆t2

1M

∆t

1

−−

−−=

+ +

3.38

80

Obviously the CDM is explicit, as the solution of ( )1ktu + is based on the

equilibrium at time ktt = . The great advantage of this method is that when a

lumped mass matrix is employed and the damping is neglected, the system of

Equation 3.38 can be solved without inverting a matrix.

Furthermore, to obtain an implicit formulation of the Newmark method

for a displacement based FE program, Equations 3.32 and 3.33 need to be

rearranged and combined to give expressions for incremental velocity and

acceleration in terms of incremental displacements:

( ) ( )kk tu∆t2α

δ1tu

α

δ∆u

∆tα

δu∆ &&&&

−+−= 3.39

( ) ( )kk2tu

2α

1tu

∆tα

1∆u

∆tα

1u∆ &&&&& −−= 3.40

These relationships can then be substituted into the incremental equilibrium

equation and rearranged to give the following equation:

[ ] [ ] [ ] [ ] ( ) ( )

[ ] ( ) ( )

−++

++=

++

kk

kk2

tu1α2

δ∆ttu

α

δC

tuα2

1tu

∆tα

1M∆R∆uKC

∆tα

δM

∆tα

1

&&&

&&&

3.41

If it is assumed that the acceleration varies linearly over the time step (α=1/6 and

δ=1/2), the Newmark method collapses to the linear acceleration method.

Although this method is only conditionally stable, it is often used due to its

accuracy. The critical time step, for the conditionally stable members of

Newmark‘s family of algorithms, is given by Equation 3.42 (Newmark, 1959).

Tα41π

1∆t∆t cr

−=≤ 3.42

The unconditional stability of the Newmark method is guaranteed when:

81

( )2δ0.50.25α

0.5δ

+≥

≥ 3.43

The most widely used member of the Newmark family of algorithms is the

constant average acceleration method (α=1/4 and δ=1/2) (also known as the

trapezoidal rule). This method assumes a uniform value of acceleration during

the increment. Dahlquist (1963) proved that the constant average acceleration

method is the most accurate unconditionally stable scheme. The main

disadvantage of the method is that it does not possess numerical damping.

However, by selecting vales of δ greater than 0.5 numerical damping is

introduced into the algorithm. In the present study the constant average

acceleration method is denoted as NMK1 to distinguish it from the most popular

dissipative version of the Newmark method (α =0.3025, δ =0.6) which is denoted

as NMK2.

3.3.5 Quadratic acceleration method

Papastamatiou (1971) introduced the parabolic acceleration method

which is an explicit modification of Newmark’s algorithm. Tsatsanifos (1982)

revisited this method and studied its behaviour in terms of stability and accuracy.

This algorithm employs a higher order polynomial to represent the variation of

acceleration within an increment. Hence what distinguishes this method from the

explicit form of Newmark’s algorithm is the use of the third derivative of

displacement with time. The quadratic acceleration, like all the explicit methods,

is only conditionally stable. Hardy (2003) developed and implemented into

ICFEP the quadratic acceleration method (QAM) which is an implicit

modification of the parabolic acceleration method. To take into account the third

derivative of displacement with time (thrust), an additional parameter γ is added

to the original Newmark’s scheme. The quadratic acceleration method comprises

of the following equations:

( ) ( ) ( )k2

kk tu∆t6

γtu∆t

2α

δ1tu

α

δ∆u

∆tα

δu∆ &&&&&&& −

−+−= 3.44

82

( ) ( ) ( )kkk2tu∆tγtu

2α

1tu

∆tα

1∆u

∆tα

1u∆ &&&&&&&& −−−= 3.45

[ ] [ ] [ ] ( )u∆Ku∆CR∆Mu∆1

&&&&&&& −−= − 3.46

where ( ) ( ) ∆t

t∆Rt∆RR∆ k1k −

= +&

By setting γ = 1, δ = 1/3 and α = 1/12 an implicit form of the parabolic

acceleration scheme is obtained. As it will be shown in Section 3.3.12, the QAM

is only conditionally stable, but it achieves high accuracy.

3.3.6 Wilson θ-method

The basic assumption of this method (Wilson et al, 1973) is that the

acceleration varies linearly during the time interval kt to ∆tθtk + , where θ≥1

(Figure 3.5). Since θ≥1, the equilibrium is considered outside the original time

step and then the solution at the point of interest is found by backward

extrapolation. The parameter θ controls the stability and the accuracy of the

algorithm. The acceleration at any time in the interval ∆tθτ0 ≤≤ can then be

found employing Equation 3.47.

( ) ( ) ( ) ( ) −++=

kkktu∆tθtu

∆tθ

τtuτu &&&&&&&& 3.47

Integrating Equation 3.47 (assuming that the initial conditions are known,

( ) ( ) 00 u0u,u0u && == ) once and then twice gives the following variations of

velocity and displacement respectively:

( ) ( ) ( ) ( )[ ]kk

2

kk tu∆tθtu∆tθ2

τtuτtuτu &&&&&&&& −

+++= 3.48

( ) ( ) ( ) ( ) ( ) ( )[ ]kk

3

k

2

kk tu∆tθtu∆tθ6

τtu

2

τtuτtuτu &&&&&&& −+ +++= 3.49

At time τ=θ∆t the above expressions take the form:

83

( ) ( ) ( ) ( ) ( )[ ]kk

22

kkk tu2∆tθtu6

∆tθtu∆tθtu∆tθtu &&&&& ++++=+ 3.50

( ) ( ) ( )[ ]kkkk tu∆tθtu2

∆tθtu∆tθtu &&&&&& +

++=+ 3.51

tk

t + tk

θ∆t + tk

∆

τ

ktu&& θ∆tk

tu&&ktu&& ∆t

u&&

Figure 3.5: Linear acceleration assumption of the Wilson θ-method.

Equations 3.50 and 3.51 can then be substituted into the equilibrium equation to

find the solution to the problem at time ∆tθtk + . Since the equilibrium is

considered at some time in the future, the following linearly extrapolated load

vector should be used:

( ) ( ) ( ) ( )[ ]kkkk tR∆ttRθtRθ∆ttR −++=+ 3.52

It should be noted that the Wilson θ-method collapses to the linear

acceleration method for θ=1. Although the method is unconditionally stable for

values of θ≥1.37, is not widely used anymore mainly due to its characteristic to

overshoot.

3.3.7 Collocation method

Hilber and Hughes (1978) introduced the collocation method which

combines features of the Wilson θ-method and the Newmark method. Thus,

identically to the Wilson θ-algorithm, the acceleration varies linearly during the

time interval kt to ∆tθtk + according to Equation 3.47. The variation of

displacements and velocities is based on both the Newmark and Wilson methods:

84

( )

++

−++=+

θ∆ttuαtuα

2

1∆t)(θtu∆tθtuθ∆ttu kk

2

kkk &&&&& 3.53

( ) ( )[ ]

++−+=+ ∆tθtuδtuδ1∆tθtu∆tθtu kkkk

&&&&&& 3.54

Equations 3.53 and 3.54 can then be substituted into the equilibrium equation to

find the solution of the problem at time ∆tθtk + . Since the equation of motion is

satisfied at some time in the future, the linearly extrapolated load vector of the

Wilson θ-algorithm (Equation 3.52) should be used.

Obviously the collocation method for θ=1 reduces to the Newmark’s

scheme and for α=1/6 and δ=1/2, it collapses to the Wilson θ-method. An

optimum choice of the parameters α, δ and θ leads to an unconditionally stable

algorithm with satisfying accuracy characteristics. Hilber and Hughes (1978)

showed that second order accuracy is attained for δ=1/2. Furthermore

unconditional stability is achieved for:

1)(2θ4

1θ2α

1)(θ2

θ

1θ

3

2

−−

≥≥+

≥

3.55

3.3.8 HHT method

Hilber, Hughes and Taylor (1977) introduced a generalization of the

Newmark method in order to achieve controllable algorithmic dissipation of the

high frequency modes. A slightly modified version of the HHT scheme, which

was suggested by Hughes (1983), is examined in the present study. The method

employs Newmark equations for the displacement and velocity variations

(Equations 3.32 and 3.33 respectively) and introduces an additional parameter fα

into the equation of motion. The basic idea of the HHT method is to evaluate the

various terms of the equilibrium equation of motion at different points within a

time interval. Figure 3.6 shows that the inertia terms are evaluated at time

85

1ktt += of the considered interval ∆t, whereas all the other terms are evaluated at

some earlier time2

fα-1ktt += . Therefore, the equation of motion takes the form:

[ ] ( ) [ ] ( ) [ ] ( ) ( ) fα-1kfα-1kfα-1k1k tRtuKtuCtuM ++++ =++ &&& 3.56

where:

( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )kf1kffα1k

kf1kffα1k

kf1kffα1k

kf1kffα1k

tRαtRα1tR

tuαtuα1tu

tuαtuα1tu

tαtα1t

+−=

+−=

+−=

+−=

+−+

+−+

+−+

+−+

&&&

3.57

Clearly when 0αf = , the HHT scheme collapses to the Newmark method.

Substituting the above expressions into Equation 3.57, yields to Equation 3.58.

[ ] ( ) ( )[ ] ( ) [ ] ( ) ( )[ ] ( )

[ ] ( ) ( ) ( ) ( ) kf1kfkf

1kfkf1kf1k

tRαtRα-1tuKα

tuKα-1tuCαtuCα-1tuM

+=+

+++

+

+++ &&&& 3.58

It will be shown in Section 3.3.12 that the inclusion of a component of the terms

of the previous time step offers a selective filtering of the inaccurate high

frequency modes. Furthermore second order accuracy and unconditional stability

is achieved when:

( ) ( )

2

α21δ ,

4

α1α ,

3

1α0 f

2

ff

+=

+=≤≤ 3.59

Finally, Chung and Hulbert (1993) conveniently expressed the algorithmic

parameter fα as a function of the value of spectral radius at infinity ∞ρ :

2 For consistency with the presentation of the generalized-α method in Section 3.3.10, it is

assumed that 0αf ≥ , while in the original paper 0αf ≤ .

86

∞

∞

+=

ρ1

ρ-1αf 3.60

where [ ]1,1/2ρ ∈∞ .

tk tk+1tk+1- fα

∆t

stiffness anddamping terms

α ∆f t

inertia term


at different points within a time interval with the HHT algorithm.

3.3.9 WBZ method

Similar to the HHT method, the WBZ (Wood et al, 1981) employs the

Newmark equations for the displacement and velocity variations and it

introduces an additional parameter mα into the equation of dynamic equilibrium.

Again the basic idea is to evaluate the various terms of the equation of motion at

different points within a time interval. In this case, the inertia terms are evaluated

at time mα-1ktt += of the considered interval ∆t, while all the other terms are

evaluated at time 1ktt += . Hence the equation of motion takes the form:

[ ] ( ) [ ] ( ) [ ] ( ) ( ) 1k1k1kmα-1k tRtuKtuCtuM ++++ =++ &&& 3.61

where:

( )

( ) ( ) ( ) ( )km1kmα1k

km1kmα1k

tuαtuα1tu

tαtα1t

m

m

&&&&&& +−=

+−=

+−+

+−+

3.62

87

Obviously when 0αm = , WBZ collapses to the Newmark method. These

expressions can be substituted in Equation 3.61 to obtain Equation 3.63.

( )[ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) ( ) 1k1k1kkm1km tRtuKtuCtuMαtuMα1 ++++ =+++− &&&&& 3.63

As it will be shown in Section 3.3.12, the behaviour of the WBZ and the HHT

algorithms is very similar and it becomes almost identical for low values of

mα and fα . Second order accuracy and unconditional stability for the WBZ

algorithm is attained when:

( ) ( )

2

α21δ ,

4

α1α 0,α1- m

2

mm

−=

−=≤≤ 3.64

Chung and Hulbert (1993) also related the algorithmic parameter mα to the value

of spectral radius at infinity ∞ρ :

1ρ

1ραm +

−=

∞

∞ 3.65

where [ ]1,0ρ ∈∞ .

3.3.10 Generalized-α method

Chung and Hulbert (1993) introduced the generalized-α method (CH)

which combines features of the HHT and WBZ algorithms. Once more, the

fundamental idea of this method is the evaluation of the various terms of the

equation of motion at different points within the time step. The CH method also

employs Newmark’s equations for the displacement and velocity variations

(Equations 3.32 and 3.33 respectively), but it introduces two additional

parameters mα , fα into the equation of motion. Figure 3.7 shows that the inertia

terms are evaluated at time mα-1ktt += of the considered interval ∆t, whereas all

the other terms are evaluated at some earlier ( )mf αα ≥ time fα-1ktt += . Hence,

the equation of motion takes the form:

88

[ ] ( ) [ ] ( ) [ ] ( ) ( ) fffm α-1kα-1kα-1kα-1k tRtuKtuCtuM ++++ =++ &&& 3.66

where:

( )

( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )kf1kfα1k

km1kmα1k

kf1kfα1k

kf1kfα1k

km1kmmα1k

kf1kfα1k

tRαtRα1tR

tuαtuα1tu

tuαtuα1tu

tuαtuα1tu

tαtα1t

tαtα1t

f

m

f

f

f

+−=

+−=

+−=

+−=

+−=

+−=

+−+

+−+

+−+

+−+

+−+

+−+

&&&&&&

&&& 3.67

tk tk+1tk+1- fα tk+1- mα

∆t

st iffness anddamping terms

α ∆m t

α ∆f t

inertia term


at different points within a time interval with the CH algorithm.

As the name of the method suggests, it is a general scheme which includes the

HHT method ( )0αm = , the WBZ method ( )0αf = and the classic Newmark

method ( )0αα fm == . Substituting the above expressions into the equation of

dynamic equilibrium renders Equation 3.68.

( )[ ] ( ) [ ] ( ) ( )[ ] ( ) [ ] ( )

( )[ ] ( ) [ ] ( ) ( ) ( ) ( ) kf1kfkf1kf

kf1kfkm1km

tRαtRα-1tuKαtuKα-1

tuCαtuCα-1tuMαtuMα-1

+=++

+++

++

++ &&&&&& 3.68

The great advantage of the CH method is that for a desired user-controlled level

of high-frequency dissipation, it achieves minimum low-frequency impact

89

(Chung and Hulbert, 1993). The unconditional stability of the scheme is

guaranteed when:

( )4

αα21αand 0.5αα mf

fm

−+≥≤≤ 3.69

Furthermore, the CH method attains second order accuracy when:

fm αα2

1+−=δ 3.70

Finally the scheme achieves optimal high frequency dissipation with minimal

low frequency impact when the following three conditions hold:

( )2fmfm αα14

1α,

1ρ

ρα,

1ρ

12ρα +−=

+=

+−

=∞

∞

∞

∞ 3.71

where ∞ρ is the desirable value of spectral radius at infinity. A detailed stability

analysis of the CH method is included in Appendix A. Furthermore, to obtain an

incremental formulation of the CH method suitable for a displacement based FE

program, Equation 3.68 can be rearranged as follows:

( )[ ] ( )[ ] ( )[ ] ( )

( ) [ ] ( ) [ ] ( ) [ ] ( ) kkkk

fffm

tuKtuCtuMtR

∆Rα-1∆uKα-1u∆Cα-1u∆Mα-1

−−−+

=++

&&&

&&& 3.72

where the last four terms can be written as :

( ) [ ] ( ) [ ] ( ) [ ] ( ) ( ) ( ) 0∆RtRtRtuKtuCtuMtRktk

int

k

ext

kkkk ==−=−−− &&&

These terms express the out-of-balance force of the previous time step, as

( ) k

ext tR , ( ) k

int tR represent the external and internal forces respectively of the

previous increment. Furthermore, substituting Newmark’s recurrence expressions

for incremental velocity and acceleration (Equations 3.39 and 3.40 respectively)

into the above equation and rearranging to put all of the known terms on the right

hand side yields to Equation 3.73.

90

[ ] ( ) [ ] ( )[ ] ( )

( ) ( ) ( ) [ ]

( ) ( ) ( ) [ ] ktkkm

kkf

fff

2

m

∆R Mtuα2

1tu

∆tα

1α1

Ctu2α

δ-1∆ttu

α

δα-1

∆Rα-1∆uKα-1C∆tα

α-1δM

∆tα

α-1

+

+

−+

−

+

=

+

+

&&&

&&& 3.73

3.3.11 Other schemes

Zienkiewicz et al (1980b), based on a weighted process, derived the

general single step algorithm SPpj (where p denotes the order of the scheme and j

denotes the order of the differential equation to be solved). Katona and

Zienkiewicz (1985) using truncated Taylor series, developed the Generalized

Newmark method (GNpj) that shares very similar stability characteristics with

the SPpj method. These schemes employ higher order polynomials to maximize

accuracy and depending on the polynomial order, collapse to a number of

popular schemes (e.g. Newmark, HHT, WBZ, Wilson-θ). However, their

algorithmic parameters are not directly related to the value of spectral radius at

infinity and they loose their unconditional stability when the order of accuracy

exceeds two.

3.3.12 Comparative study of integration schemes

In order to choose the appropriate integration method, it is useful to

compare the accuracy and the numerical behaviour of some of the various

available integration schemes. For linear problems the most common procedure

is to perform a comparative study of the SDOF problem described by Equation

3.17. Such an analysis is often referred to as a spectral stability analysis.

Therefore this section compares the variation of spectral radius with the ratio

∆t/T and the accuracy in terms of algorithmic damping ratio (see Equation 3.25)

and period elongation (see Equation 3.23) of the various implicit schemes

presented so far.

91

0.01 0.1 1 10 100

∆t/T

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

ρ

QuadraticAcceleration

LinearAcceleration

NMK1

NMK2

Area of instability

Area of stability

Figure 3.8: Spectral radii for NMK1, NMK2, linear acceleration and quadratic

acceleration methods.

0 0.1 0.2 0.3 0.4

∆t/T

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

ξ'


NMK1, Linear Acceleration

NMK2

(a)

0 0.1 0.2 0.3 0.4

∆t/T

0

0.1

0.2

0.3

0.4

0.5

(T'-T)/T


Linear Acceleration

NMK2

(b)

NMK1

Figure 3.9: Algorithmic damping ratios (a) and period elongation (b) for NMK1,

NMK2, linear acceleration and quadratic acceleration methods.

Figure 3.8 presents the spectral radii of three members of the Newmark

method and of the quadratic acceleration method. The spectral radius variation

verifies the conditional stability of the linear acceleration and quadratic

acceleration methods. The normalised critical time step ( )/T∆tcr appears to be

stricter for the quadratic acceleration than for the linear acceleration method. As

mentioned previously, the NMK1 algorithm is unconditionally stable, but it

exhibits no numerical dissipation, since the spectral radius is always equal to one.

On the other hand, the NMK2 damps the high-frequency modes, but it also

affects the low frequency response. Furthermore, Figure 3.9a compares the

algorithmic damping ratios of the aforementioned schemes. Ideally, the

92

algorithmic damping ratio should be zero. If however it is not zero, the curve

should have a zero tangent at the origin and subsequently should turn smoothly.

Clearly the linear acceleration method and the NMK1 are the most accurate

schemes. The quadratic acceleration method appears to be superior to NMK2

algorithm as its curve has a zero tangent at the origin. In terms of period

elongation (Figure 3.9b), the quadratic acceleration method appears to be

superior to all the other examined schemes.

The spectral stability analysis presented so far shows that the

conditionally stable schemes (linear acceleration and quadratic acceleration

method) appear overall to be more accurate than the unconditionally stable

schemes. However, these two schemes will not be considered further in this

thesis due to their poor stability characteristics. Numerical tests by Hardy (2003)

showed the importance of this shortcoming in finite element analysis. Employing

the quadratic acceleration method Hardy (2003) found that in the absence of

material damping the numerical solution was unstable regardless of the size of

the time step.

Figure 3.10 illustrates the spectral stability analysis of Hilber and Hughes

(1978) for various unconditionally stable integration schemes. All examined

algorithms seem to have sufficient numerical damping in the high-frequency

range. The Houbolt and Park algorithms are known to asymptotically annihilate

the high-frequency modes (Fung, 2003) and thus they are too dissipative in the

medium and low frequency range. On the other hand, both the collocation and

the HHT (in Figure 3.10 this is denoted as the α method, where fαα −= )

methods allow parametric control of the amount of dissipation present. In

addition Hilber and Hughes (1978) studied the accuracy characteristics of the

above-mentioned schemes. Clearly, their results (Figures 3.11, 3.12) agree well

with the theorem of Dahlquist (1963) that the NMK1 (trapezoidal rule) is the

most accurate unconditionally stable scheme. Furthermore, among the other

schemes, the HHT method appears to be the most accurate method. Therefore,

taking into account both the dissipative and accuracy characteristics of the

methods considered so far, it can be concluded that overall the HHT is the most

appealing method.

93

Figure 3.10: Spectral radii for HHT (α-method), collocation, Houbolt and Park

methods (from Hilber and Hughes, 1978).

Figure 3.11: Algorithmic damping ratios for HHT (α-method), collocation,

Houbolt and Park methods (from Hilber and Hughes, 1978).

Figure 3.12: Period elongation for HHT (α-method), collocation, Houbolt and

Park methods (from Hilber and Hughes, 1978).

94

Moreover, Chung and Hulbert (1993) compared the behaviour of the CH scheme

with the other two α-methods (HHT, WBZ). This analysis is repeated herein and

for completeness the comparison also includes two popular members of the

Newmark family of algorithms (NMK1, NMK2). The algorithmic parameters of

the dissipative schemes are chosen such that for each algorithm the spectral

radius in the high-frequency limit is 0.818.

0.01 0.1 1 10 100

∆t/T

0.5

0.6

0.7

0.8

0.9

1

1.1

ρ

NMK1

NMK2

WBZ, αm=-0.1

HHT, αf=0.1

CH, αm=0.35, αf=0.45

Area of instability

Area of stability

Figure 3.13: Spectral radii for NMK1, NMK2, HHT, WBZ and CH methods.

The comparison of the spectral radii (Figure 3.13) shows that for a given value of

∞ρ , the CH curve has the smoothest transition from the low-frequency to the

high-frequency modes, whereas the NKM2 damps the most the low-frequency

modes. The WBZ scheme appears to behave only slightly worse than the HHT

method. In addition, Figure 3.14a compares the algorithms in terms of

algorithmic damping ratio. Clearly the CH method is more accurate than the

HHT and WBZ methods. The CH curve lies very close to that of NMK1 which

possess no dissipation. Regarding the period elongation plot, all algorithms

exhibit similar accuracy (Figure 3.14b).

95

0 0.1 0.2 0.3 0.4

∆t/T

0

0.1

0.2

0.3

0.4

0.5

(T'-T)/T

NMK2

(b)

WBZ, HHT

CH

NMK1

0 0.1 0.2 0.3 0.4

∆t/T

0

0.02

0.04

0.06

0.08

0.1

ξ'NMK2

(a)

WBZ

HHT

CHNMK1

Figure 3.14: Algorithmic damping ratios (a) and period elongation (b) for

NMK1, NMK2, HHT, WBZ and CH methods.

As mentioned earlier, in the case of unconditionally stable schemes the

size of the time step is determined only by the required accuracy of the solution.

A widely used practical rule is to choose the time step as T/10 (i.e., ∆t/T=0.1). It

should be noted that for this value of the normalized time step, the CH method

has zero algorithmic damping error and a period elongation error of 3.4%.

As mentioned previously, the great advantage of the CH method is that it

allows the user to control the amount of numerical dissipation at the high

frequency limit, without significantly affecting the lower modes. In this respect,

it is useful to investigate the behaviour of the algorithm for different values of ρ∞.

Thus, the spectral stability analysis of the CH algorithm was repeated for ρ∞

equal to 0.6, 0.42 and 0.0. The plot of spectral radius (Figure 3.15) shows that the

behaviour of the algorithm for ρ∞=0.0 is similar to that of Houbolt and Park

methods (Figure 3.10) and it leads to excessive dissipation in the low-frequency

range. Furthermore it is interesting to note that the CH algorithm even for low

values of ρ∞ (i.e. 0.42, 0.6) affects less the low-frequency modes than the NMK2

method (with ρ∞=0.818). The algorithmic damping ratio plot (Figure 3.16a)

shows the case of ρ∞=0.0 should be avoided as the error is high (5.5 % for

∆t/T=0.1). In all other cases, the CH performs satisfactorily as all the curves have

a zero tangent at the origin and subsequently a controlled turn upward. Apart

from the case of ρ∞=0, the period elongation error seems to be less sensitive to

the value of ρ∞ (Figure 3.16b).

96

0.01 0.1 1 10 100

∆t/T

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

ρ

NMK1

NMK2

CH, ρ∞=0.818

ρ∞=0.6

ρ∞=0.42 ρ∞=0.0

Figure 3.15: Spectral radii the CH (ρ∞=0.0, 0.42, 0.6, 0.818), NMK1 and NMK2

methods.

0 0.1 0.2 0.3 0.4

∆t/T

0

0.02

0.04

0.06

0.08

0.1

ξ'

(a)

NMK1CH, ρ∞=0.818

ρ∞=0.6

ρ∞=0.42

ρ∞=0.0

NMK2

0 0.1 0.2 0.3 0.4

∆t/T

0

0.1

0.2

0.3

0.4

0.5

(T'-T)/T

NMK2

(b)

CH, ρ∞=0.818

NMK1

ρ∞=0.0

ρ∞=0.6

ρ∞=0.42

Figure 3.16: Algorithmic damping ratios (a) and period elongation (b) for the CH

(ρ∞=0.0, 0.42, 0.6, 0.818), NMK1 and NMK2 methods.

In conclusion, the spectral analysis showed that the CH method is more

accurate and has better numerical dissipation characteristics than other

dissipative schemes. These advantageous characteristics and the flexibility that

such a general scheme offers, render the CH method the most attractive scheme

among those considered in this thesis. Therefore the method was further

developed to deal with dynamic coupled consolidation problems and was

implemented into ICFEP (Section 3.4.2).

97

3.3.13 The generalized-α method in dynamic nonlinear analysis

Dynamic nonlinear analysis employs very similar procedures to those

used in nonlinear static analysis (Section 2.2.7). The essential difference is that

both the out-of-balance force vector and the tangent stiffness matrix are modified

to include inertia and damping terms. In the case of the CH method, the

governing finite element equation for nonlinear problems can be written as3:

[ ] iiiR∆∆uK = 3.74

where [ ] [ ] ( ) [ ] ( )[ ]Kα-1C∆tα

α-1δM

∆tα

α-1K f

f

2

m +

+

= is the modified stiffness

matrix,

( ) ( ) ( ) ( ) [ ]

( ) ( ) ( ) [ ] ktkkf

kkmf

∆RCtu2α

δ-1∆ttu

α

δα-1

Mtuα2

1tu

∆tα

1α1∆Rα-1R∆

+

−

+

+

−+=

&&&

&&&

is the modified right hand side vector and the superscript i denotes the increment

number (i.e. time step). Similarly to the static case, the MNR method is

employed to iteratively solve the above equation. Hence, the modified out-of

balance force ψ is given by Equation 3.75.

[ ] ( ) 1-jjiiψ∆uK = 3.75

where the subscript j denotes the iteration number. As mentioned in Section 2.2.7

the MNR method is relatively insensitive to the increment size. While this is true

in conventional static analysis, Crisfield (1997) notes that in nonlinear dynamics

an error, which depends on the increment size, is introduced in the solution. This

error is associated with the time integration and can be controlled if an automatic

time step algorithm is employed. As the optimal time step size may change

3 All equations in this section are expressed for a system with a single degree of freedom, u.

98

during the computation, time step control algorithms automatically adjust the

time step to maximize accuracy. Hulbert and Jang (1995) and Chung et al (2003)

introduced strategies for automated adaptive selection of the time step in the CH

method. These approaches are not examined in the present study, but they offer

an obvious direction for research in the future.

As mentioned earlier the spectral stability theory is developed for linear

problems. It is widely recognised (e.g. Hughes, 1983, Argyris and Mlejnek,

1991) that the unconditional stability of an algorithm in the linear regime is a

necessary but not a sufficient condition for stable time integration in nonlinear

dynamics. Hughes (1976), for example, showed that the constant average

acceleration method (NMK1), which is regarded as unconditionally stable in

linear dynamics, can exhibit numerical instabilities in the nonlinear regime.

Erlicher et al (2002) list some of the various definitions of stability in nonlinear

dynamics. A widely used criterion for stability in nonlinear dynamics is that the

total energy of an unforced system within a time step should either remain

constant or reduce but not increase. The theoretical studies of Erlicher et al

(2002) showed the CH method is stable in an energy sense, but it can exhibit

overshoot in a nonlinear analysis.

It should be noted that both the spectral and the energy stability theories

are based on unforced systems and they do not examine the effect of the forcing

term on the performance of the integration scheme. Pegon (2001) proposed an

analysis in the frequency domain that examines the effect of the forcing term on

the accuracy properties of integration schemes under resonance conditions in the

linear regime. This study showed that the HHT method, in contrast with the

Newmark scheme, acts as a filter, as it does not amplify the unwanted high

frequency modes. Furthermore, it was concluded that the choice of the fα

parameter of the HHT algorithm should also take into account the spectrum of

the loading. Bonelli et al (2002) extended the study of Pegon (2001) to nonlinear

forced systems. Their results regarding the performance of the CH algorithm

under resonance conditions show that the algorithm can limit the resonance peak

of the high frequency response without significantly affecting the resonance peak

in the low frequency response. Furthermore, the performance of the algorithm is

99

critically controlled by the appropriate choice of the algorithmic parameters (αm,

αf). The studies of Pegon (2001) and Bonelli et al (2002) are restricted to

harmonic excitations, but they bring to light the need for further research on the

behaviour of commonly used integration schemes in transient analysis. Therefore

in Chapter 4 the CH algorithm is compared with the Newmark, the HHT and the

WBZ algorithms in a nonlinear boundary value problem of a deep foundation

subjected to various earthquake loadings. The aim is to investigate how the

earthquake response spectrum affects the accuracy and the numerical dissipation

characteristics of the above-mentioned algorithms in the nonlinear regime.

3.4 Dynamic consolidation theory

As discussed in Section 2.2.3, the assumptions of fully drained or

undrained soil behaviour are valid for a wide range of static geotechnical

applications. However, depending on the soil permeability, the rate of loading

and the hydraulic boundary conditions, it is often necessary to employ coupled

analysis to accurately model the two phase behaviour of the soil. In dynamic

problems the rate of loading is such that the assumption of fully drained

behaviour is often only valid for completely dry conditions. Furthermore, the

assumption of undrained behaviour is only a reasonable approximation for

relatively impervious materials, as it implies that no relative movement of the

pore fluid is allowed. Consequently, no inertia effects can affect the pore fluid

phase. On the other hand, in materials of low permeability the inertia effects

should be taken into account and therefore a fully coupled dynamic analysis is

required. Biot in a series of papers (Biot (1956a), Biot (1956b) and Biot (1962))

presented a general set of equations governing the behaviour of a saturated linear

elastic porous solid under dynamic loading. Zienkiewicz et al (1999)

distinguishes the following three approaches in the numerical formulation of

Biot’s theory:

(a) The “u-p-w” formulation which fully satisfies Biot’s theory and uses as

primary variables the solid phase displacement (u), the velocity of the fluid

relative to the solid component (w) and the pore fluid pressure (p).

100

(b) The “u-p” formulation which uses as primary variables the solid phase

displacement (u) and the pore fluid pressure (p). This method assumes that the

acceleration of the pore fluid relative to the soil matrix and the convective terms

of this acceleration are negligible. According to Zienkiewicz and Shiomi (1984)

the “u-p” approach is a reasonable approximation in the frequency range of

earthquake engineering problems or for frequencies lower than this range.

(c) The “u-U” formulation which employs as primary variables the solid phase

displacement (u) and the total displacement of the pore fluid (U). This approach

eliminates the pore fluid pressure (p) variable employing a “penalty” number to

approximate an incompressibility constraint on the fluid and the solid grains.

Furthermore it also ignores the convective terms of the acceleration of the pore

fluid relative to the solid.

While the “u-p-w” formulation is cumbersome and is rarely used, Smith (1994)

notes that of the other two types of fully coupled analysis (“u-p” and “u-U”)

there is no evidence to suggest which one is inherently superior. Hardy (2003)

implemented a “u-p” formulation of Biot’s theory in ICFEP and discretized the

coupled governing equation in time with the Newmark method. The original “u-

p” formulation of ICFEP is reviewed and a method to discretize the coupled

governing equation with the CH scheme is subsequently proposed in this section.

3.4.1 Dynamic finite element formulation for coupled problems

In a similar fashion to that described in Section 2.2.8, to derive the

governing equation for dynamic coupled consolidation analysis it is necessary to

combine the equations governing the deformation of soil due to loading with the

equations governing the pore fluid flow. Hence, the first step is to formulate the

equations governing the deformation of the soil allowing the solid and the fluid

phases to deform independently. Employing again d’Alembert principle the

equilibrium equation for a solid-fluid mixture is given by Equation 3.76.

∆L ∆D∆I∆W∆E EEEEE −++= 3.76

101

where ∆EE is the incremental total potential energy, ∆WE is the incremental

strain energy, ∆IE is the incremental inertial energy, ∆DE is the incremental

damping energy and ∆LE the incremental work done by the applied loads. As

mentioned earlier, the “u-p” formulation assumes that the acceleration of the

fluid relative to solid and the convective terms of this acceleration are negligible.

Therefore the incremental inertial energy and the incremental damping energy

are still given by Equations 3.2 and 3.4 respectively. The incremental strain

energy was defined by Equation 2.30 in terms of an effective stress and a pore

fluid component and the incremental work done by the applied loads is still given

by Equation 2.16. Assembling the contribution from each element in the

computational domain in the same manner as outlined in Section 2.2 and then

minimizing the incremental potential energy of the body, the global dynamic

incremental equilibrium equation for the solid-fluid mixture in terms of effective

stresses is obtained:

[ ] [ ] [ ] [ ] GnGGnGGnGGnGG ∆R∆pL∆dKd∆Cd∆M =+++ &&& 3.77

where

[ ] [ ] [ ] =

=∑ ∫

=i

N

1i Vol

T

G dVolNρNM global mass matrix;

[ ] [ ] [ ] =

=∑ ∫

= i

N

1i Vol

T

G dVolNcNC global damping matrix;

[ ] [ ] [ ][ ] =

=∑ ∫

=

N

1i iVol

T

G dVolBD'BK global stiffness matrix;

[ ] [ ] [ ] [ ] =

== ∑ ∫∑

==

N

1i iVol

p

TN

1iiEG dVolNBmLL global coupling matrix;

0111mT =

102

[ ] [ ] =

+

== ∑ ∫∫∑

==

N

1i iSurf

T

iVol

TN

1i

EG dSurf∆FNdVol∆FN∆R∆R Right hand

side load vector.

where d∆ ,d∆ ,∆d nGnGnG&&& and

nG∆p are the nodal displacement, velocity,

acceleration and pore pressure vectors respectively and ∆T,∆F are the body

forces and surface tractions respectively. As in the static case, the equilibrium

equation for the solid-fluid mixture has to be combined with the continuity

equation of the fluid phase. Considering the flow of pore fluid in and out of an

element of soil of unit dimensions, the equation of continuity for the pore fluid is

given by Equation 3.78.

t

ε∆Q

t

p

K

n

y

v

x

v v

f

yx

∂∂

−=−∂∂

+∂

∂+

∂∂

3.78

In Section 2.2.8 it was assumed that both pore fluid and the soil grains are

incompressible. Equation 3.78 also assumes that any volume change due to the

compressibility of the soil grains or due to thermal changes is negligible, but it

does include the term t

p

K

n

f ∂∂

that expresses the volume stored due to the

compressibility of the pore fluid. The pore fluid was assumed to be

incompressible relative to the soil skeleton for static analyses, but this

assumption may not be valid for dynamic problems, as the dilatational wave

velocity depends on the compressibility of the pore fluid (see Equation 4.12).

Therefore the pore fluid compressibility term was included in the dynamic

consolidation formulation. Furthermore, the motion of the pore fluid is assumed

to obey the Darcy fluid flow equation, which can be expressed as:

[ ]

+∇−= dg

1hkv && 3.79

where v is the velocity vector with components xv and yv , [ ]k is the

permeability matrix of the soil, h is the hydraulic head defined in Equation

2.34 and g is the acceleration due to gravity. The inherent assumption in the

103

above expression is that the acceleration of the pore fluid relative to soil skeleton

and the convective terms of this acceleration are negligible. Employing the

principle of virtual work the continuity equation can be written as:

( ) ∆p∆QdVol∆pt

ε∆p

t

p

K

n∆pv

Vol

v

f

T =

∂∂

+∂∂

+∇∫ 3.80

Substituting Equations 3.79 and 2.34 into 3.80 and approximating t

εv

∂∂

as ∆t

∆εv

and t

p

∂∂

as ∆t

∆pleads to:

[ ]

∆t∆p∆QdVol

∆p∆ε∆p∆pK

n

dt∆pdg

1ip

γ

1k

Vol

v

f

∆tt

t

G

f

k

k =

++

∇

++∇−

∫∫+

&&

3.81

Equation 3.81 can be written in finite element form as:

[ ] [ ] [ ]

[ ] [ ] ( )∆t∆Qn∆dL

∆pSdtdGdtpΦ

GnG

T

G

nGG

∆tt

t

nGG

∆tt

t

nGG

k

k

k

k

+=+

+−− ∫∫++

&&

3.82

where

[ ] [ ] [ ] [ ][ ]i

N

1i Vol

TN

1iiEG dVolEkN

g

1GG ∑ ∫∑

==

==

[ ] [ ] [ ] [ ][ ]i

N

1i Vol f

TN

1iiEG dVol

γ

EkEΦΦ ∑ ∫∑

==

==

[ ] [ ] [ ] [ ] ∑ ∫∑==

==

N

1i i

G

Vol

TN

1iiEG dVolikEnn

[ ] [ ] [ ] [ ]i

N

1i Vol

p

T

p

f

N

1iiEG dVolNN

K

nSS ∑ ∫∑

==

==

104

[ ]T

ppp

z

N

y

N

x

NE

∂

∂

∂

∂

∂

∂=

The integrals of Equation 3.82 can be approximated as:

( ) [ ]

( ) [ ]∆td∆βtddtd

∆t∆pβtpdtp

nGnGk

∆tt

t

nG

nGnGk

∆tt

t

nG

k

k

k

k

&&&&&& +=

+=

∫

∫

+

+

3.83

where β is the time marching parameter introduced in Section 2.2.8. Utilising this

time marching process, the final dynamic finite element continuity equation for

the pore fluid is obtained.

[ ] [ ]( ) [ ] [ ]

[ ] [ ] ( ) ( ) [ ] ( ) ( ) ( )∆t∆QtdGtpΦn

∆dLd∆G∆tβ∆pSΦ∆tβ

nGkGnGkGG

nG

T

GnGGnGGG

+++=

+−+−

&&

&&

3.84

Clearly the only new constituents of the above equation (i.e. those in addition to

those for a static analysis) are the matrices [ ]GS and [ ]GG which are related to

the compressibility of the pore fluid and to the inertia of the solid phase

respectively. Chan (1988) suggests that the influence of the inertia term in the

pore fluid equation is insignificant within the frequency range for which the “u-

p” approximation is valid. Therefore Zienkiewicz et al (1999) chose to neglect

this term, as it renders the system of equations non-symmetric. Later Chan

(1995) however included this term in computer program SWADYNE II, but only

in the right hand side load vector and then dealt with it iteratively. Due to the

recommendations made by Chan (1988), Hardy (2003) chose not to include this

term in ICFEP, but he suggested that its influence needs further investigation.

Therefore in the present study, both the compressibility of the fluid and the

inertia of the solid phase are taken into account when forming the continuity

equation for the pore fluid. Furthermore the effect of the inertia term on the

accuracy and the computational cost of the solution are examined in a validation

exercise in Chapter 4.

105

3.4.2 Implementation of the CH method for coupled problems

To solve the set of governing Equations 3.77 and 3.84 a time integration

method needs to be adopted. Taking into account the relative merits of the CH

scheme in uncoupled analysis (Section 3.3.12), it was chosen to extend this

algorithm to deal with coupled problems. Consequently, in a similar fashion to

that described in Section 3.3.10, the inertia terms are evaluated at time

mα-1ktt += and all the other terms are evaluated at some earlier ( )mf αα ≥ time

fα-1ktt += . Hence the dynamic equilibrium equation for the solid-fluid mixture

can be written as4:

[ ] ( ) [ ] ( ) [ ] ( )

[ ] ( ) ( ) ff

ffm

α-1nGnGα-1nG

nGα-1nGnGα-1nGnGα-1nG

tRtpL

tuKtuCtuM

++

+++

=+

++ &&&

3.85

where

( )

( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )kf1kfα1k

kf1kfα1k

km1kmα1k

kf1kfα1k

kf1kfα1k

km1kmα1k

kf1kfα1k

tpαtpα1tp

tRαtRα1tR

tuαtuα1tu

tuαtuα1tu

tuαtuα1tu

tαtα1t

tαtα1t

f

f

m

f

f

m

f

+−=

+−=

+−=

+−=

+−=

+−=

+−=

+−+

+−+

+−+

+−+

+−+

+−+

+−+

&&&&&&

&&& 3.86

Utilising the above expressions and rearranging Equation 3.85 into an

incremental form yields:

4 All equations in this section are expressed for a system with a single degree of freedom, u.

106

( )[ ] ( )[ ] ( )[ ] ( )[ ]

( ) ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) kGkGkGkGkGf

GfGfGfGm

tpLtuKtuCtuMtR∆Rα-1

∆pLα-1∆uKα-1u∆Cα-1u∆Mα-1

−−−−+

=+++

&&&

&&&

3.87

where the last five terms express the residual force kt

∆R from the previous

increment. Thus, when sufficient convergence is attained these terms cancel out.

Furthermore, as the continuity equation can only be expressed in an incremental

form, the CH method is applied directly to Equation 3.84. Hence multiplying the

inertia terms by (1-αm) and all the other terms by (1-αf) leads to:

( ) [ ] [ ]( ) ( ) [ ] ( )[ ]

( ) ktf

nG

T

GfnGGmnGGGf

∆F∆Fα1

∆uLα1u∆G∆tβα1∆pSΦ∆tβα1

+−=

−+−−+−− &&

3.88

where

[ ] [ ] ( ) [ ] ( ) ∆t∆QtuGtpΦn∆FnGkGnGkGG +++= && is the incremental right

hand side vector. It is assumed that the right hand side vector at fα-1ktt += can be

approximated as:

( ) ( ) ( ) ( )kf1kfα-1k tFαtFα1tFf

+−= ++ 3.89

The term kt

∆F expresses the out-of-balance flow from the previous increment

which ideally should be zero. Finally substituting in Equations 3.87 and 3.88 the

temporal recurrence relations of Newmark for velocity and acceleration

(Equations 3.39 and 3.40 respectively) yields the final coupled dynamic finite

element formulation.

[ ] ( )[ ]

( )[ ] ( ) [ ] ( ) [ ] [ ]( )

=

−−

−−−

−

G

G

nG

nG

GGfGmT

Gf

GfG

F∆

R∆

∆p

∆u

Φ∆tβSα1G∆tα

α1βLα1

Lα1K

3.90

where

107

[ ] [ ] ( ) [ ] ( )[ ]GfGf

G2

mG Kα1C

∆tα

δα1M

∆tα

α1K −+

−+

−=

( ) ( ) ( ) ( ) [ ]

( ) ( ) ( ) [ ] ktGnGknGkf

GnGknGkmGfG

∆RCtu∆tα2

δ1tu

α

δα1

Mtuα

1tu

∆tα

1α1∆Rα1R∆

+

−+−+

+−+−=

&&&

&&&

( ) [ ] [ ] ( ) ( ) [ ] ( ) ( )[ ]

( ) [ ] ( ) ( ) ktnGknGkGm

nGkGnGkGGfG

∆Ftuα2

1tu

∆tα

1G∆tβα1

tuGtpΦ∆Qn∆tα1F∆

+

+−−

+++−=

&&&

&&

3.5 Summary

This chapter detailed the extensions that are required to the static finite

element formulation to perform dynamic analyses. Hence a finite element

formulation of dynamic equilibrium was firstly presented. The various

constitutive procedures to model soil behaviour under cyclic loading and the

special spatial discretization requirement in wave propagation problems were

also briefly discussed. Attention was then focused on some of the most popular

time integration methods that are used to approximate the solution of the

dynamic equilibrium equation. These are: Houbolt, Park, Newmark, Quadratic

acceleration, Wilson-θ, collocation, HHT, WBZ and CH methods. A comparative

analytical study of these schemes in the linear regime showed the advantageous

properties of the CH algorithm. The implementation of the CH method in ICFEP

for both linear and nonlinear analyses was subsequently discussed. Furthermore

the consolidation theory presented in the previous chapter was extended for the

case of dynamic analyses. Finally, the CH method was extended to deal with

coupled problems and this new formulation was implemented into ICFEP.

108

Chapter 4:

NUMERICAL INVESTIGATION OF THE CH

METHOD

4.1 Introduction

Chapter 3 detailed key issues of the dynamic finite element theory, with

particular emphasis on the implementation of the generalised α-method (CH) into

ICFEP. The aim of this chapter is first to ensure that this integration scheme was

accurately implemented and then to compare its performance with other widely

used schemes.

The first part of this chapter presents a series of validation exercises. A

single degree of freedom problem, for which there is a known closed form

solution, was used to verify the uncoupled dynamic formulation of ICFEP for

both solid and beam elements. Furthermore, it was shown in Chapter 3 that the

CH method was extended to deal with coupled consolidation problems. Thus

another set of validation exercises was performed to verify the accuracy of

ICFEP’s coupled consolidation formulation. Hence, ICFEP’s results are initially

compared with the analytical solution of Zienkiewicz et al (1980a) for a

consolidating soil column subjected to harmonic loading. Subsequently,

numerical examples by Prevost (1982), Meroi et al (1995) and Kim et al (1993)

for both small and large deformation analysis, with constant and variable

permeability, were used to subject ICFEP’s dynamic coupled formulation to a

another series of validation tests.

The second part of this chapter compares the behaviour of the CH scheme

with more commonly used schemes (NMK1, NMK2, HHT and WBZ) in a

boundary value problem of a deep foundation subjected to various seismic

excitations.

109

4.2 Validation Exercises

4.2.1 Harmonically forced single degree of freedom system

Consider the damped single degree of freedom (SDOF) system of Figure

4.1, subjected to a simple harmonic loading of amplitude Po and loading

frequency Ωo. The loading is given by Equation 4.1.

( ) tsinΩPtP oo= 4.1

The closed form solution for the lateral displacement d(t) of this harmonically

forced SDOF for “at rest” initial conditions (zero initial displacement and

velocity) is given by Equation 4.2, (e.g. Sarma, 2000).

( ) ( ) ( )

−++

−= − φtΩsinθtωsine

ξ1

αM

k

Ptd oD

tωξ

2

o 4.2

where

( )[ ] 2

1

2222 αξ4α1M−

+−= is the dynamic magnification factor;

( )2α1

αξ2tan

−=ϕ ;

( )( )22

2

ξ21α

ξ-1ξ2tanθ

+−= ;

ω

Ωα o= ,

m

kω = is the natural frequency, ( )2D ξ1ωω −= is the damped

natural frequency and ξ is the damping ratio. The response of this system

(Equation 4.2) consists of two parts: the transient response, which is a damped

sinusoidal oscillation of frequency ωD and decays with time, and the steady-state

response, which occurs at the frequency of the applied loading, but it has a phase

shift φ with respect to the loading. The dynamic magnification factor M

110

expresses the amount by which the static displacement (Po/k) is magnified by the

harmonic loading. This is a function of the ratio of the loading frequency (Ωo) to

the natural frequency (ω) and the damping ratio ξ.

Figure 4.1: Single degree of freedom system

Solid elements were initially employed to model the SDOF problem in plane

strain. Figure 4.2 illustrates a sketch of the analysis arrangement. The free length

of the pendulum is 100m and was modelled with 300 elements (of dimensions

∆x=0.33m, ∆y=1.0m), whereas the pendulum’s mass was modelled with the top

3 elements (of dimensions ∆x=0.33m, ∆y=0.1m). Both horizontal and vertical

displacements were restricted along the base of the mesh and a harmonic point

load of amplitude Po=1kN and of frequency Ωo=10 (rad/s) was applied at point C

(Figure 4.2).

Figure 4.2: Sketch of the FE model (with solid elements) for the SDOF problem

Table 4.1 lists the assumed parameters in the FE element analysis and Table 4.2

lists the equivalent material properties of the pendulum. The top three elements

of the mesh, on which the mass is concentrated, were assigned a high value of

mass density, whereas the “free length” elements were assigned a very low value

111

of material density (see Table 4.1). The equivalent mass of the pendulum is given

by Equation 4.3.

mm Vρm = 4.3

where ρm and Vm are the material density and the volume respectively of the top

three elements representing the mass of the pendulum. Assuming that the FE

column behaves like a cantilever subjected to a point load on its free end, the

equivalent stiffness of the pendulum is given by Equation 4.4.

3

ff

L

IE3k = 4.4

where Ef and If are the Young’s modulus and the moment of inertia respectively

of the elements representing the free length of the pendulum.

Table 4.1: Single degree of freedom finite element analysis parameters

ρ

(Mg/m3)

E

(kPa)

I

(m4)

ν A B

“free length

elements” 10

-6 10

8 0.0833 0 0.589 0.002537

“mass

elements” 5.0

10

8 0.0833 0 0.589 0.002537

Table 4.2: Equivalent material properties of the pendulum

m

(Mg)

k

(kN/m2)

L

(m) ξ

ω

(rad/s)

ωD

(rad/s)

0.5 25

100 0.05 7.071 7.053

Furthermore, the response of the pendulum is assumed to be linear-elastic and

was modelled for both undamped and damped oscillations. In the case of the

damped analyses, Equation 3.14 was employed to calculate the Rayleigh

damping coefficients (A, B in Table 4.1), taking as ω1 the natural frequency of

the SDOF (ω1=ω) and as ω2 a high value (ω2=5ω1) to make sure that the resulting

112

damping is reasonably constant within the important frequency range. The values

of the Rayleigh damping coefficients correspond to an equivalent viscous

damping (ξ) of 5% for the pendulum.

0 2 4 6 8 10 12Time (s)

-0.16

-0.12

-0.08

-0.04

0

0.04

0.08

0.12

0.16

Dis

pla

ce

me

nt

(m)

(c) ∆t =To/100

0 2 4 6 8 10 12Time (s)

-0.16

-0.12

-0.08

-0.04

0

0.04

0.08

0.12

0.16

Dis

pla

ce

me

nt

(m)

Closed Form

NMK1

NMK2

CH

(a) ∆t =To/20

0 2 4 6 8 10 12Time (s)

-0.16

-0.12

-0.08

-0.04

0

0.04

0.08

0.12

0.16

(b) ∆t =To/50

Figure 4.3: SDOF undamped response modelled with solid elements

The aim of this validation exercise is to verify the implementation of the

CH algorithm and to examine how this algorithm compares with commonly used

members of Newmark’s family of algorithms. Therefore, the time integration

was conducted with three schemes: the CH, the NMK1 and NMK2. Figure 4.3

compares the undamped displacement time history of node C (Figure 4.2) with

the closed form solution for three time steps (∆t = To/20, To/50, To/100, where To

is the period of the harmonic loading). For all time steps, the CH and NMK1

responses are indistinguishable. For ∆t =To/20 the agreement with the closed

form solution is quite poor for NMK2, whereas the CH and NMK1 compare

reasonably well with the closed form solution. As one would expect, the

113

accuracy of all schemes improves as the size of the time step reduces. Hence for

∆t =To/100 all schemes show very good agreement with the closed form solution.

Furthermore it is interesting to note that in Figure 4.4 the introduction of

damping seems to improve the accuracy of the algorithms. When Rayleigh

damping is introduced (corresponding to ξ=5%), the displacement response of

node C for all schemes, even for ∆t= To/20, compares very well with the closed

form solution.

0 2 4 6 8 10 12Time (s)

-0.12

-0.08

-0.04

0

0.04

0.08

0.12

(b) ∆t =To/50

0 2 4 6 8 10 12Time (s)

-0.12

-0.08

-0.04

0

0.04

0.08

0.12

Dis

pla

ce

me

nt

(m)

Closed Form

NMK1

NMK2

CH

(a) ∆t =To/20

Figure 4.4: SDOF damped response modelled with solid elements (ξ=5%)

As noted in Chapter 3, it was chosen to neglect the rotary inertia

contribution when formulating the beam elements’ mass matrix in ICFEP. This is

a reasonable approximation as the inertia forces associated with node rotations

are generally not significant. Since the beam elements have a different

formulation, the SDOF problem analysis was repeated replacing the solid

elements with 3-noded beam elements. Instead of three columns of solid

elements, one column of beam elements was used (100 “free length elements” of

∆y=1.0m, and 1 “mass element” of ∆y=0.1m). The material properties were kept

the same and all degrees of freedom (horizontal, vertical displacements and

rotations) were restricted along the base of the mesh. The harmonic loading of

Equation 4.1 was applied on the middle node (D) of the “mass element”. Figure

4.5 compares the undamped displacement time history of node D with the closed

form solution for two time steps (∆t = To/20, To/50). Comparison of Figures 4.3

and 4.5 shows that the beam elements can model the undamped response of the

pendulum as well as the solid elements. In a similar fashion, the beam elements

114

predict very accurately the displacement response of node D when Rayleigh

damping is present (Figure 4.6).

0 2 4 6 8 10 12Time (s)

-0.16

-0.12

-0.08

-0.04

0

0.04

0.08

0.12

0.16

Dis

pla

ce

me

nt

(m)

Closed Form

NMK1

NMK2

CH

(a) ∆t =To/20

0 2 4 6 8 10 12Time (s)

-0.16

-0.12

-0.08

-0.04

0

0.04

0.08

0.12

0.16

(b) ∆t =To/50

Figure 4.5: SDOF undamped response modelled with beam elements

0 2 4 6 8 10 12Time (s)

-0.12

-0.08

-0.04

0

0.04

0.08

0.12

(b) ∆t =To/50

0 2 4 6 8 10 12Time (s)

-0.12

-0.08

-0.04

0

0.04

0.08

0.12

Dis

pla

ce

me

nt

(m)

Closed Form

NMK1

NMK2

CH

(a) ∆t =To/20

Figure 4.6: SDOF damped response modelled with beam elements (ξ=5%)

In conclusion, the results of the SDOF analyses verify the uncoupled

dynamic formulation of ICFEP for solid and beam elements. There is, however,

the option of considering the rotary inertia when formulating the beam elements’

mass matrix in the future and then investigating its importance to real

engineering problems. Furthermore, it was shown that the CH and NMK1

displacement responses are indistinguishable and that for ∆t= To/20 they damp

less the displacement response than NMK2. The theoretical analysis of Section

3.3.11 showed that the CH and NMK1 methods exhibit very similar algorithmic

damping ratio errors, whereas the NMK2 damps more the response. Hence the

115

SDOF problem results seem to agree qualitatively with the theoretical analysis of

the unforced vibration presented in the previous chapter.

4.2.2 Consolidating elastic soil layer subjected to cyclic loading

As noted in Chapter 3, Biot’s theory gives a general set of equations

governing the behaviour of a saturated linear elastic porous solid under dynamic

loading. Zienkiewicz et al (1980a) presented an analytical solution of Biot’s

equations for a soil medium of infinite lateral extent. Hardy (2003) used the “u-

p” form of this solution (see Section 3.4) to validate the dynamic coupled

formulation of ICFEP. The same benchmark problem is employed herein to

validate the implementation of the CH method for coupled analyses. Consider the

soil column of Figure 4.7 subjected to a surface harmonic pressure. Drainage is

restricted to the top of the layer, as all other boundaries are impervious.

Horizontal movement was restricted along the side boundaries and no movement

was allowed along the bottom of the layer. The symmetry conditions (i.e. no

lateral movement and no flow along the side boundaries) allow a soil layer of

infinite lateral extent to be idealised as a soil column. Zienkiewicz et al (1980a)

showed that neglecting the acceleration of the pore fluid relative to the soil

matrix (“u-p” approximation) and omitting the static gravity terms, the response

of the soil layer is governed by Equations 4.5 and 4.6.

uΠy

wκ

y

u22

2

2

2

−=∂∂

+∂∂

4.5

wΠ

iuΠβ

y

wκ

y

uκ

1

22

2

2

2

+−=∂∂

+∂∂

4.6

where u is the solid phase displacement and w is the velocity of the fluid relative

to the solid component. The dimensionless parameters κ, β, П1 and П2 are

defined as follows.

nKD

nKκ

f

f

+= 4.7

116

where Kf is the pore fluid compressibility, n is the porosity and D is the one-

dimensional constrained modulus given by Equation 4.8.

ν)2(1ν)(1

ν)(1ED

−+−

= 4.8

where E, ν are the Young’s modulus and the Poisson’s ratio respectively.

ρ

ρβ f= 4.9

where ρf, ρ are the densities of the fluid and the total mixture respectively.

2

o1

Tg

Tk

πβ

2Π

ˆ

= 4.10

where k is the soil permeability, g is the gravitational acceleration, To is the

period of excitation and T is the natural period of the layer defined as:

cV

H2T = 4.11

where H is the depth of the soil column and Vc is the compression wave velocity

in water given by Equation 4.12.

ρ

nKDV fC

+= 4.12

Finally the non-dimensional parameter П2 compares the period of the excitation

with the natural vibration period of the mixture.

2

o

2

2T

TπΠ

=

ˆ 4.13

117

H

p=0

u=w=dp/dy=0

qei tΩο

u=dp/dy=0

yx

Figure 4.7: Analysis arrangement for 1-D consolidation examples

Applying the boundary conditions shown in Figure 4.7, the simultaneous partial

differential equations (Equations 4.5 and 4.6) can be solved to give the response

of the soil layer in terms of u, w. Hence, the pore pressure response can then be

obtained from Equation 4.14.

∂∂

+∂∂

=y

w

y

u

n

Kp f 4.14

As noted earlier, the derivation of Zienkiewicz et al (1980a) omits the static

gravity terms from Biot’s equations. Therefore, Equation 4.14 gives the steady-

state response of the soil column in terms of excess pore pressure. Zienkiewicz et

al (1980a) also showed the range of values of Π1 and Π2 over which the “u-p”

approximation is valid (Figure 4.8). Hence, zone I includes very slow events in

which a static consolidation analysis is applicable, zone III represents high-

frequency events in which only the full “u-w-p” formulation is valid and zone II

represents the intermediate region in which the “u-p” approximation is valid. The

results of the ICFEP analysis were compared with the closed form solution for a

range of Π1 and Π2 values that the “u-p” approximation renders sufficient

accuracy (i.e. in zone II).

118

(III )(II )

(I)

UndrainedBehaviour Drained

Π2

102

10

1

10-1

10-2

10-3

10 -21 102

Π1

Figure 4.8 Zones of sufficient accuracy for various approximations (after

Zienkiewicz et al, 1980a)

The FE mesh consisted of one column of 200 4-noded elements (of dimensions

∆x=∆y=0.05m). A harmonic stress of amplitude q=100kPa and of frequency

Ωo=2π/To was applied normally to the free surface (Figure 4.7). Employing the

material properties listed in Table 4.3 the soil permeability and the period of the

loading are determined as functions of Π1 and Π2 according to Equations 4.15

and 4.16.

2

oΠ

π0.02T = (s) 4.15

π

ΠΠ1.0265x10k 215−= (m/s

2) 4.16

Table 4.3: Parameters for the FE analysis of the soil layer subjected to cyclic

load

ρ

(Mg/m3)

ρf

(Mg/m3)

E

(kPa) ν nK f

H

(m)

3.0 1.0

67500.0 0.25 2919000.0 10.0

To ensure that steady-state conditions had been reached, the excess pore

pressures were calculated after 20 cycles at the peak of the input wave. In all

analyses the time integration was performed with the CH scheme and the time

step was chosen equal to To/40. As noted in Chapter 3, Chan (1988) suggests that

119

the inertia of the solid phase in the fluid equilibrium equation (Equation 3.84) is

negligible within the range of frequencies that the “u-p” approximation is valid.

However, it was chosen for completeness to include this inertia term in the

present study. To highlight the role of this term, ICFEP’s results with (denoted as

CH+) and without (denoted as CH-) this inertia term are presented in this section.

Figures 4.9, 4.10 and 4.11 compare ICFEP analyses with the closed form

solution for various values of Π1 and Π2.

0 40 80 120 160 200

Pore Pressure (kPa)

1

0.8

0.6

0.4

0.2

0

No

rma

lise

d D

ep

th (

y/H

)

Closed Form

CH-

CH+

Π2=1.0

Π2=0.1

Figure 4.9: ICFEP results compared to closed form solution for Π1=0.1

In all cases, ICFEP results with the inertia term compare very well with the

closed form solution. Furthermore, ICFEP results without the inertia term depart

from the closed form solution for two pairs of Π1, Π2 values (Π1=Π2=1.0 and

Π1=10.0, Π2=0.1). These two combinations of Π1, Π2 lie on the limit up to which

the “u-p” approximation is valid (see Figure 4.8). Therefore, the decision of

Chan (1988) and Hardy (2003) to ignore this inertia term is largely justified.

Moreover, the above-mentioned numerical tests showed that the inertial term

does not increase the computational cost of the analysis. Hence, it was decided to

keep the inertia term in ICFEP’s formulation as it is theoretically more sound.

120

0 40 80 120

Pore Pressure (kPa)

1

0.8

0.6

0.4

0.2

0

No

rma

lise

d D

ep

th (

y/H

)

Closed Form

CH-

CH+

Π2=0.1Π2=1.0

Figure 4.10: ICFEP results compared to closed form solution for Π1=1.0

0 40 80 120 160

Pore Pressure (kPa)

1

0.8

0.6

0.4

0.2

0

No

rma

lise

d D

ep

th (

y/H

)

Closed Form

CH-

CH+

Π1=10.0, Π2=0.1

Figure 4.11: ICFEP results compared to closed form solution

4.2.3 Consolidating elastic soil layer subjected to a step load

A fundamental assumption in the previous validation exercise is that any

displacement of the mesh during the analysis is small compared to the

dimensions of the mesh. This assumption is commonly employed in FE analysis

and is referred to as the small displacement approximation. In earthquake

engineering related problems the intensity of the loading can be such that the

resulting displacements violate the small displacement assumption. It is therefore

useful to validate the dynamic coupled formation of ICFEP for large

displacement analyses. In this case an updated Lagrangian system is used to

redefine the mesh at the end of each increment according to the calculated

displacements. Bathe (1996), among others, presents in detail the large

displacement theory.

121

In this section two examples which compare the two approaches (i.e.

small and large displacement) are examined. The first problem refers to the work

of Prevost (1982). The same problem was later analysed by Meroi et al (1995).

ICFEP’s results are compared with the results of both studies. Consider the soil

column of Figure 4.7 subjected to a uniformly distributed step load at the free

surface. In accordance with Prevost (1982) the soil column was analysed in plane

strain with one column of 20 solid elements (of dimensions ∆x=∆y=0.5m) and

all boundary conditions were identical to those used in Section 4.2.2. The static

solution to this problem can be obtained “dynamically” utilising a large time

step. Hence, the time integration was conducted with the CH scheme, using a

variable time step: ∆tn=1.5 ∆tn-1, where n is the increment number and ∆t1=0.1.

Table 4.4 lists the assumed parameters in the FE element analysis. Five load

levels were considered (q=0.2E, 0.4E, 0.6E, 0.8E and 1.0E, where E is the

Young’s modulus) and in all cases the total load was applied in the first

increment.

Table 4.4: Parameters for the FE analysis of the soil layer subjected to a step load

ρ

(Mg/m3)

ρf

(Mg/m3)

E

(kPa) ν

k

(m/s)

H

(m)

2.0 1.0

10.06

0.0 0.01 10.0

Figure 4.12 is taken from Meroi et al (1995) and it plots the load level (q/E)

versus the maximum settlement (s) normalised by the column depth (H). It

should be noted that full consolidation and thus maximum settlement are reached

after three increments. Curves a and d correspond to the computational results

reported by Meroi et al (1995) for large and small displacement analysis

respectively, whereas curve e corresponds to the solution obtained by Prevost

(1982) for large displacements analysis. Curve c corresponds to the theoretical

solution, cited by Meroi et al (1995). ICFEP’s results are superimposed in Figure

4.12 and they are denoted as dots for large displacement and as triangles for

small displacement analysis. Clearly, the results of this geometrically nonlinear

analysis computed by both Meroi et al (1995) and ICFEP are in excellent

122

agreement with the theoretical solution. As one would expect, the higher the load

intensity, the more distinct is the difference between the small and the large

displacement approaches.

e

q

d

0

0.2

0.4

0.6

0.8

qq

c

a

s/H

s

Figure 4.12: Load level versus normalised settlement for an elastic consolidating

soil layer (from Meroi et al, 1995)

The second problem to be considered was initially analysed by Kim et al

(1993) and it was later repeated by Meroi et al (1995). The soil column of Figure

4.7 is again subjected to a uniformly distributed step load at the free surface and

all boundary conditions are identical to those used in Section 4.2.2. In this

example the load is applied incrementally over 10 increments up to 107 kPa. The

CH scheme was engaged for the time integration, using a variable time step

according to Table 4.5. The spatial discretization (Figure 4.13) and the assumed

parameters in the FE analysis (Table 4.6) are in accordance with Kim et al

(1993).

123

H=7m

Saturated

soil

Impervious

Insulated

2m

q

x

y2m

Figure 4.13: FE model for 1-D consolidation of Kim et al (1993)

Table 4.5: Variable time step for the FE analysis of a soil layer subjected to a

step load

Increment

Number

Time Step

∆t (sec)

1-10

11-19

20-28

29-37

38-55

56-75

0.01

0.1

1.0

10.0

50.0

500.0

Table 4.6: Material properties for the FE analysis of a soil layer subjected to a

step load

ρ

(Mg/m3)

ρf

(Mg/m3)

E

(kPa) ν

k

(m/s)

H

(m)

0.4 1.0

6.0E106

0.4 4.0E10-8 7.0

124

The variation of normalised settlement v/s (where s is the ultimate

settlement) of the free surface with dimensionless time is given in Figure 4.14.

The normalised time is defined as:

tcT vv = 4.17

where the consolidation factor cv is given by Equation 4.18.

( )

( )( )ν12ν1Hγ

ν-1Ekc

2

w

v +−= 4.18

In Figure 4.14, ICFEP’s results compare favourably with the results of Kim et al

(1993) both for small and large displacements analyses. Furthermore, the finite

element results (both of ICFEP and Kim et al, 1993) for small displacement

analyses agree very well with the analytical results of Terzaghi’s theory (as cited

in Kim et al, 1993).

-5 -4 -3 -2 -1 0 1

logTv

-1

-0.8

-0.6

-0.4

-0.2

0

v/s

Small

ICFEPKim et al (1993)Terzaghi

Large

Figure 4.14: Comparison of surface settlement history predictions of ICFEP with

Kim et al (1993)

Figure 4.15 plots the variation of normalised excess pore pressure p/q (where q is

the accumulated applied load) with dimensionless time for an integration point at

a distance y=0.2m from the free surface. ICFEP’s results agree well with Kim et

al’s results in the final stages of consolidation, but serious discrepancies can be

observed in the initial stages of the consolation process both for small and large

displacement analyses. It should be noted that for the initial stages of

consolidation, Kim et al predict the p/q ratio to be greater than one, which has no

125

physical meaning. On the other hand, ICFEP’s results underestimate the p/q ratio

for the initial stages of consolidation. As noted in Chapter 2, Gaussian integration

is employed to relate the pore pressures at the various Gauss points to the

corresponding nodal pore pressures. Therefore, the accuracy in terms of pore

pressures is expected to be more sensitive to the element size than it is in terms

of displacements.

-5 -4 -3 -2 -1 0 1

logTv

0

0.2

0.4

0.6

0.8

1

1.2

1.4

p/q

ICFEPKim et al (1993)Terzaghi

Small

Large

Figure 4.15: Comparison of pore pressure history predictions of ICFEP with Kim

et al (1993)

So far the analysis arrangement was kept consistent with the Kim et al’s

analysis. To investigate the poor agreement of ICFEP analyses with Terzaghi’s

solution in terms of pore pressures, a finer mesh was employed. In particular, 50

elements (instead of 5 in Figure 4.13) were used to model the top 2m of the

mesh. The results of the analyses with the finer mesh are shown in Figure 4.16

and they verify the sensitivity of the pore pressures to the spatial discretization.

Clearly, the small displacement curve agrees very well with the analytical

solution while the large displacement analysis seems to reach full consolidation

faster.

126

-5 -4 -3 -2 -1 0 1

logTv

0

0.2

0.4

0.6

0.8

1

1.2

1.4

p/q

ICFEP - smallICFEP - largeTerzaghi

Figure 4.16: Comparison of pore pressure history predictions of ICFEP with

Terzaghi’s solution using a finer mesh

Furthermore, Meroi et al (1995) addressed the same consolidation

problem for small and large displacement analyses. They also repeated the large

displacement analysis, assuming that the permeability is a linear function of the

void ratio, varying from an initial value to zero when porosity becomes zero. The

results of Meroi et al (1995) in terms of normalised settlement of the free surface

with dimensionless time are compared with ICFEP’s results in Figure 4.17. It

should be noted that Meroi et al employed an initial void ratio eo=1.0 that at high

load levels leads to negative void ratio. To avoid this problem, an initial void

ratio of eo=1.3 is used in the present study. ICFEP’s results compare very well

with those of Meroi et al for small displacement analyses. Furthermore, both

analyses with variable permeability (i.e. ICFEP’s and Meroi et al’s) show that

there is a significant increase in time to reach full consolidation. However, Meroi

et al’s analyses predict lower values of final settlement both for variable and

constant permeability than ICFEP’s analyses. Taking into account the excellent

agreement between ICFEP’s predictions and Kim et al’s predictions in terms of

surface settlement (see Figure 4.14), the above discrepancy can be presumably

attributed to an inconsistency of Meroi et al’s analysis.

127

-3 -2 -1 0 1 2

logTv

-1

-0.8

-0.6

-0.4

-0.2

0

v/s

Small

Large - variable k

Large - constant k

ICFEPMeroi et al (1995)

Figure 4.17: Comparison surface settlement history predictions of ICFEP with

Meroi et al (1995)

4.3 Performance of the CH method in a boundary value

problem

In Chapter 3 a comparative study of commonly used integration schemes

was presented that was useful to understand fundamental aspects of those

algorithms, but it was limited to linear elastic free vibration problems. As

mentioned in Section 3.3.12, Pegon (2001) proposed an analysis in the frequency

domain that examines the effect of the forcing term on the accuracy properties of

integration schemes under resonance conditions and Bonelli et al (2002)

extended this work to nonlinear forced systems. The studies of Pegon (2001) and

Bonelli et al (2002) showed that the frequency content of the loading should be

taken into account when selecting the algorithmic parameters. However their

conclusions are restricted to harmonic excitations and idealised systems. Hence,

the aim of this section is to investigate how the frequency content of the

earthquake excitation affects the accuracy and the numerical dissipation

characteristics of the CH and of other commonly used schemes (i.e. NMK1,

NMK2, HHT and WBZ) in a geotechnical application. For this purpose a deep

foundation was analysed for various earthquake loadings and for various soil

properties.

128

4.3.1 Description of the numerical model

A foundation 5 m deep and 1 m wide was analysed in plane strain, using

the finite element mesh shown in Figure 4.18. This model was employed by

Hardy (2003) to investigate the behaviour of deep foundations under seismic

conditions. The objective in the present study is to compare the performance of

different algorithms and not the thorough investigation of the seismic response of

deep foundations. Figure 4.18 also illustrates the boundary conditions employed

in the dynamic analyses. The vertical displacements were fixed on the bottom

boundary, since it was assumed that a very stiff soil layer exists at a depth of

20m. The mesh is 42 meters wide and on the lateral boundaries the displacements

are tied together in both directions (i.e. uB=uC and vB=vC in Figure 4.18)

(Zienkiewicz et al, 1988). Assuming that waves radiating away from the

foundation can be ignored, the tied degrees of freedom boundary condition

models accurately the free-field response at the lateral boundaries (see Sections

5.2.1, 7.7.4). Conversely, a numerical model with viscous dashpots along the

lateral boundaries would seriously underestimate the response. This issue is

discussed in detail in Chapter 7. Furthermore, interface elements were placed

along the two sides of the foundation, which allowed relative movement between

the foundation and the surrounding soil.

Tied degrees of freedom

uC

vCCBuB

5.0 m

1m

20 m

42 m

vB

Figure 4.18: Mesh and boundary conditions assumed in dynamic analyses

For simplicity dry conditions were assumed and a simple constitutive model was

used. Hence the soil and the interface elements were modelled as elastic perfectly

plastic Mohr – Coulomb materials and the foundation as linear elastic. It should

be noted that no material damping (i.e. Rayleigh damping) was used. The

129

implementation of the Mohr-Coulomb model in ICFEP allows the use of a non-

associated flow rule in which the angle of dilation can be different from the angle

of shearing resistance (see Potts and Zdravković, 1999). In all the analyses a zero

angle of dilation was assumed. The employed material properties are listed in

Table 4.7. The natural frequencies of a linear elastic soil column on a rigid base

are given by Equation 4.19 (Kramer, 1996):

( )H4

n21Vf Sn

+= n=0, 1, 2,…∞ 4.19

in which ν)(1ρ2

EVS +

= is the shear wave velocity of the soil and H is the

height of the soil column and n is the vibration mode. Assuming that the

presence of the foundation does not alter significantly the overall dynamic

behaviour of the finite element mesh, Equation 4.19 can be used to estimate the

initial fundamental frequency (for n=0) of the FE model. The height of the soil

column in this case is 20m and the shear wave velocity is 111.3 m/s resulting in a

fundamental frequency f0 of 1.391Hz. To investigate the role of the fundamental

frequency of the FE model some of the dynamic analyses of Section 4.3.4 were

repeated in Section 4.3.5 for various values of soil stiffness (see Table 4.8).

Table 4.7: Material properties for foundation analyses

Material

Properties Soil

Interface

Elements Concrete

E (MPa) 60

- 30.000

ν 0.25 - 0.2

γ (kN/m3) 19 1.0 24.0

φ΄(˚) 30 15 -

K5s, K

5n (MN/m

3) - 100 -

5 Ks, Kn denote the shear and normal stiffness of the interface elements respectively.

130

Initially, the static bearing capacity of the foundation was evaluated to be

3315 kN/m, by conducting a displacement controlled analysis as described by

Hardy (2003). Furthermore, prior to all dynamic analyses, a static working load,

corresponding to a factor of safety of 2.7 against the estimated static bearing

capacity was applied to the foundation over a series of load increments. A

common time step of ∆t=0.01sec was used in all dynamic analyses.

4.3.2 Input ground motion

Three acceleration time histories, recorded during the 1979 Montenegro

earthquake (ML=7.03), were considered. Specifically the foundation was

subjected to the east-west component of the Petrovac (PETO) recording, to the

north-south component of the Veliki (VELS) recording and to the east-west

component of the Titograd (TITO) recording. The above-mentioned filtered time

histories were obtained from the database of Ambraseys et al (2004). A brief

discussion on the filtering process of records is given in Section 7.7.2.

The acceleration time histories were applied incrementally to all nodes

along the bottom boundary of the FE model (Figure 4.18). Figure 4.19 illustrates

the three accelerograms and Figure 4.20 shows the corresponding elastic

acceleration response spectra for zero viscous damping. The spectral

accelerations of the PETO and VELS are higher than that of the TITO spectrum

with clear predominant periods of 0.5 sec (fo=2Hz) and 0.4 sec (fo=2.5Hz),

respectively. On the other hand, the spectrum of the TITO recording is

concentrated to the low period range, with small spectral values and a

predominant period of 0.065 sec (fo=15.38Hz).

131

(a) Veliki

0 20 40 60

Time (sec)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6 A

cc

ele

rati

on

(m

/se

c2

)

0 10 20 30 40

Time (sec)

-3

-2

-1

0

1

2

3

Ac

ce

lera

tio

n (

m/s

ec

2)

0 20 40 60

Time (sec)

-3

-2

-1

0

1

2

3

Ac

ce

lera

tio

n (

m/s

ec

2)

(b) Petrovac

(c) Titograd

Figure 4.19 Filtered accelerograms, obtained from Ambraseys et al (2004)

0 0.5 1 1.5 2 2.5

Period (sec)

0

5

10

15

20

25

30

35

40

45

50

55

Sp

ec

tra

l A

cc

ele

rati

on

(m

/se

c2

)

PETO

VELS

TITO

ξ=0%

Figure 4.20: Elastic acceleration response spectra

4.3.3 Parametric study of the CH algorithm

It was shown in Chapter 3 that the algorithmic parameters of the CH

method ( δα,αα fm ,, ) can be expressed as a function of the value of spectral

radius at infinity ∞ρ (see Equations 3.70 and 3.71). The great advantage of the

CH method is that by varying the value of spectral radius at infinity ∞ρ , the user

can control the amount of numerical dissipation at the high frequency limit,

132

without significantly affecting the lower modes. This feature was shown in

Chapter 3 by repeating the spectral stability analysis of the CH algorithm for

different values of ρ∞. However, it still remains to be shown whether the CH

scheme maintains this favourable property in nonlinear transient problems. It is

shown in the next section that from the three considered records (i.e. TITO

VELS and PETO) the performance of the algorithms is worse for the TITO

recording. Hence, to investigate the performance of the CH algorithm for the less

favourable case, the foundation was subjected to the TITO recording and its

dynamic response was compared for various levels of high frequency dissipation

(ρ∞ equal to 0.818, 0.6, 0.42 and 0.0) and for NMK1 (ρ∞=1.0) that possess no

dissipation. It should be noted that the low-frequency components of an

earthquake motion generally dominate the displacement response (Kramer,

1996). Thus, to assess the effect of numerical dissipation on the low frequency

response, displacement histories of the foundation were computed. Figure 4.21

shows the vertical displacement history of point A at the base of the foundation

(see Figure 4.18) for different values of spectral radius at infinity for the TITO

recording.

0.12

0.11

0.1

0.09

0.08

0.07

Se

ttle

me

nt

(m)

0 10 20 30 40 50 60

Time (sec)

ρ∞ = 0.0ρ∞ = 0.42

ρ∞ = 0.6

CH (ρ∞= 0.818), NMK1

initial settlementdue to working load

Figure 4.21: Settlement history of foundation base for various values of ρ∞ for

the TITO recording

All settlement histories start from an initial value (77mm), induced by the

applied working load, increase rapidly during the intense period of the

earthquake and then they stabilize. The curves for the CH algorithm with its

standard parameters ( ∞ρ =0.818) and the NMK1 are indistinguishable. Besides,

133

the curves for ∞ρ =0.42 and ∞ρ =0.6 have slightly different settlements during the

intense period of the earthquake and then they converge to a single value. The

final value of the settlement for ∞ρ =0.42 and ∞ρ =0.6 is only 1.4 % lower than

that for the CH with its standard parameters ( ∞ρ =0.818). For the case of ∞ρ =0.0

the response is damped more and the final settlement is 5.3% lower than the one

predicted by the CH with ∞ρ =0.818. The comparison of the settlement histories

indicates that the displacement response is not particularly sensitive to the value

of ∞ρ . However, both this numerical investigation and the theoretical analysis

presented in Section 3.3.12 suggest that the extreme value of ∞ρ =0.0 should be

avoided. Furthermore Figure 4.22 compares the Fourier amplitude spectra6 of the

horizontal acceleration time history of point A at the base of the foundation for

various levels of high frequency dissipation (ρ∞ equal to 0.818, 0.42 and 0.0) and

for NMK1. The Fourier amplitude spectrum of an accelerogram shows how the

amplitude of the motion is distributed with respect to frequency (Kramer, 1996).

Observing the frequency content of the response, it can be postulated whether

inaccurate high frequencies have been introduced into the solution. The Fourier

spectra of Figure 4.22 are very narrow banded with a dominant peak at a

frequency of 1.44 Hz. This value is very close to the estimated fundamental

frequency of 1.39 Hz (see Section 4.3.1). Furthermore a secondary peak can be

identified at a frequency of f1=4.13Hz that corresponds to the second natural

frequency of the system (i.e. n=1 in Equation 4.19). The Fourier spectrum of

NMK1 is dominated by spurious peaks at frequencies greater than 40Hz due to

lack of numerical damping. Some spurious frequencies can also be observed for

ρ∞=0.818, but are eliminated at higher levels of numerical dissipation (i.e.

ρ∞=0.42, 0.0). The comparison of the Fourier spectra highlights the need for

numerical dissipation of the high frequency modes which can not be adequately

calculated by the FE method. Although general conclusions cannot be drawn

from this brief parametric study, it seems that the CH scheme maintains its

6 All the Fourier amplitude spectra in this thesis were computed with the software SeismoSoft

(2004)

134

ability to allow parametric control of high frequency dissipation without

considerably affecting the low-frequency response in a boundary value problem.

0.1 1 10 100

Frequency (Hz)

0

4

8

12

16

20

Fo

uri

er

Am

plit

ud

e (

m/s

ec

)

(c) ρ∞ = 0.42

0.1 1 10 100

Frequency (Hz)

0

4

8

12

16

20

Fo

uri

er

Am

plit

ud

e (

m/s

ec

)

(d) ρ∞ = 0.0

0.1 1 10 100

Frequency (Hz)

0

4

8

12

16

20

Fo

uri

er

Am

plit

ud

e (

m/s

ec

)

(a) CH (ρ∞ = 0.818)

0.1 1 10 100

Frequency (Hz)

0

4

8

12

16

20

Fo

uri

er

Am

plit

ud

e (

m/s

ec

)

(b) NMK1

f1

f1

f1

f1

f0 f0

f0f0


at the foundation base

4.3.4 Analyses for various excitations

To allow the comparison of the CH scheme with other commonly used

schemes, the dynamic analyses of the foundation were also performed with the

NMK1, NMK2, HHT and WBZ schemes for the same value of spectral radius at

infinity (ρ∞ =0.818). Figure 4.23 presents the vertical displacement history of

point A at the base of the foundation for the TITO, VELS and PETO recordings.

Note that for all recordings the CH scheme gives identical results to that

computed with the NMK1. The curves for the HHT, WBZ and NMK1 are

indistinguishable for the VELS and PETO recording and only slightly deviate for

135

the TITO recoding. On the other hand the NMK2 always seems to damp out the

response significantly compared to the other algorithms. Analogous observations

can be made from plots of horizontal displacements and velocities that are not

presented for brevity. Note that in all cases the WBZ results were identical to the

ones predicted by the HHT. Thus, there is no need to include the WBZ results in

the future discussions.

Due to the different intensities of the recordings (Figure 4.19) the

resulting displacements are not directly comparable. Therefore, it was chosen to

normalize all vertical displacements with respect to the vertical displacements of

NMK1. This normalization is justified since NMK1 does not possess any

numerical damping. Hence, deviation from the displacement values of NMK1

can be used as a measure of how much dissipative algorithms damp the low

frequency modes of the solution. Besides, the effect of the frequency content of

the input motion on the performance of the algorithms cannot be assessed from

the simple displacement history plots of Figure 4.23. Hence, Figure 4.24a shows

a plot of percentage deviation of the NMK2 from the NMK1 in terms of vertical

displacements for the three recordings versus time. Clearly, as the predominant

frequency of the earthquake recording increases (predominant period decreases)

the NMK2 seems to damp the response more. Specifically, for the TITO

recording (fo=15.38Hz) the maximum deviation is 10.5 %, whereas for the PETO

recording (fo=2.0Hz), the maximum deviation is 7.2%. Figure 4.24b shows the

percentage deviation for the HHT algorithm. The HHT seems to follow the same

trend but with much lower values. So, for the TITO recording the maximum

deviation is 1.9% and for the PETO it is as low as 0.8%. On the other hand, the

CH seems to be insensitive to the frequency content of the input earthquake

(Figure 4.23). The deviation for the CH for all recordings was very low, less than

1%.

In addition, Figure 4.25 illustrates the percentage deviation of the CH for

different values of ρ∞ for the TITO recording. It is worth mentioning that even

for ρ∞=0.0 the CH has a smaller deviation than the NMK2 (Figure 4.24a).

Furthermore, the CH for ρ∞ =0.4 and ρ∞=0.6 performs similarly to the HHT

which has ρ∞ =0.818.

136

(c) PETO

(a) TITO (b) VELS

0 10 20 30 40

Time (sec)

0.2

0.16

0.12

0.08

Se

ttle

me

nt

(m)

NMK2

CH, NMK1, HHT, WBZ


0.12

0.11

0.1

0.09

0.08

0.07

Se

ttle

me

nt

(m)

0 10 20 30 40 50 60

Time (sec)

NMK2

HHT, WBZ

CH , NMK1


0.8

0.6

0.4

0.2

Se

ttle

me

nt

(m)

0 10 20 30 40 50 60

Time (sec)

NMK2

CH, NMK1, HHT, WBZ


Figure 4.23: Settlement history of foundation base for the TITO, VELS and

PETO recordings

-1

0

1

2

3

4

5

6

7

8

9

10

11

De

via

tio

n f

rom

NM

K1

(%

)

0 10 20 30 40 50 60

Time (sec)

TITO

PETO

(a) NMK2

VELS

-1

0

1

2

De

via

tio

n f

rom

NM

K1

(%

)

0 10 20 30 40 50 60

Time (sec)

TITO

PETO

(b) HHT

VELS

Figure 4.24: Percentage deviation from the NMK1 for the NMK2 (a) and the

HHT (b)

137

-3

-2

-1

0

1

2

3

4

5

6

De

via

tio

n f

rom

NM

K1

(%

)

0 10 20 30 40 50 60

Time (sec)

ρ∞ = 0.0

ρ∞ = 0.6

CH (ρ∞ = 0.818)

ρ∞ = 0.42

TITO

Figure 4.25: Percentage deviation from the NMK1 for various values of ρ∞

The displacement response of the foundation indicated the superior

accuracy characteristics of the CH compared to the other dissipative schemes

(NMK2, HHT and WBZ), as in all cases the curves of NMK1 and CH are

indistinguishable. As mentioned earlier the displacement response generally

reflects the low frequency response of the system. Hence, to compare the

behaviour of the integration schemes in the high frequency range, the

acceleration response needs to be examined. Considering the response for the

VELS record, Figure 4.26 plots the horizontal acceleration time history at point

A for NMK1, NMK2, CH and HHT. Spurious oscillations seem to dominate the

solution of NMK1. The cycles of the response are indistinguishable as large and

unrealistic numerical oscillations dominate the solution. Besides, the

performance of the CH and HHT is very similar. Their solution contains some

mild numerical oscillations, but it is generally satisfactorily. On the other hand

NMK2 is completely free from oscillations, but it seems to over-damp the

response. These observations are better illustrated by the Fourier amplitude

spectrum.

138

-9

-6

-3

0

3

6

9

Ac

ce

lera

tio

n (

m/s

ec

2)

0 10 20 30 40

Time (sec)

(a) NMK1

-4

-2

0

2

4

Ac

ce

lera

tio

n (

m/s

ec

2)

0 10 20 30 40

Time (sec)

(b) NMK2

-4

-2

0

2

4

Ac

ce

lera

tio

n (

m/s

ec

2)

0 10 20 30 40

Time (sec)

(c) CH

-4

-2

0

2

4

Ac

ce

lera

tio

n (

m/s

ec

2)

0 10 20 30 40

Time (sec)

(d) HHT

Figure 4.26: Horizontal acceleration time history of foundation base (for the

VELS record) for NMK1, NMK2, CH and HHT

Figure 4.27 plots the Fourier amplitude spectra of the acceleration time histories

of Figure 4.26. Three peaks can be immediately identified in all the Fourier

spectra: two peaks corresponding to the fundamental and the second natural

frequency (f0=1.44 Hz and f1=4.15 Hz respectively) of the soil layer and one

corresponding to the predominant frequency of the excitation (fo=1/To≈2.5Hz).

Similar to Figure 4.22b, the spectrum of NMK1 is dominated by spurious peaks

at frequencies greater than 15Hz. On the other hand the NMK2 eliminates the

spurious peaks, but it considerably damps the peak that corresponds to the

fundamental frequency of the soil layer.

139

f1

f1

f1

f1

0.1 1 10 100

Frequency (Hz)

0

4

8

12

16

20

Fo

uri

er

Am

plit

ud

e (

m/s

ec

)

(c) CH

0.1 1 10 100

Frequency (Hz)

0

4

8

12

16

20

Fo

uri

er

Am

plit

ud

e (

m/s

ec

)

(d) HHT

0.1 1 10 100

Frequency (Hz)

0

4

8

12

16

20

Fo

uri

er

Am

plit

ud

e (

m/s

ec

)

(a) NMK1

0.1 1 10 100

Frequency (Hz)

0

4

8

12

16

Fo

uri

er

Am

plit

ud

e (

m/s

ec

)

(b) NMK2

fo

fo

fo fo

f0f0

f0f0


at the foundation base (for the VELS record)

The predicted fundamental amplitude by NMK2 is 32% lower than the one

predicted by NMK1. In contrast to the CH, the HHT seems to damp the peaks

corresponding to the 3rd (f2=6.93Hz), 4

th (f3=9.55Hz), and 5

th (f4=12.0Hz) natural

frequencies. This however does not affect the overall accuracy of the response

which is clearly governed by the fundamental natural frequency. Unsurprisingly,

the fundamental amplitude computed with the CH and the HHT matches the one

predicted by the NMK1. Hence, Figures 4.26 and 4.27 demonstrate the

importance of numerical damping in FE analysis and that the α-schemes (i.e. CH,

HHT) efficiently filter the spurious frequencies, without seriously affecting the

accuracy of the solution.

140

4.3.5 Analyses for various soil properties

To investigate the effect of the numerical model’s natural frequencies on

the performance of integrations schemes, the dynamic analyses for the VELS

recording were repeated for various values of soil stiffness, as listed in Table 4.8.

Note that, apart from the soil’s Young’s modulus, all other parameters are the

same as before (see Section 4.3.1).

Table 4.8: Summary of analyses undertaken at different fundamental frequencies

Analysis Soil’s Young’s

modulus E (MPa)

Fundamental

frequency f0 (Hz)

1 37.53 1.11

2 60.0 1.39

3 121.6 1.98

4 190.0 2.5

5 760.0 4.95

Table 4.9 lists the maximum percentage deviation (PD) from NMK1 for the

settlement history at point A for NMK2, HHT and CH. Furthermore, the last

column of Table 4.9 gives the final settlement (S) at point A computed with the

NMK1. The fundamental frequency of the soil layer seems to play an important

role on the final value of the settlement. The soil layer with f0=2.5Hz is at

resonance with the excitation (which has a predominant frequency fo=2.5Hz) and

therefore the foundation settles as much as 127.0cm. Furthermore, the

performance of all algorithms is significantly affected by the resonance

condition. At resonance the NMK2 has the higher percentage deviation (28.4%),

but the α schemes also exhibit considerable deviation (17.3% for the HHT and

16.9% for CH). Excluding the analysis 4, the higher the fundamental frequency

of the layer, the higher is the percentage deviation of the integration schemes. It

is interesting to note that for f0=4.95Hz, the percentage deviations of the NMK2

141

and HHT are 40.9% and 9.5% respectively. However the settlement for this

frequency is only 5.6cm, thus the absolute error of the two schemes (i.e. NMK2

and HHT) is of minor practical importance. Excluding again the special case of

resonance, the CH scheme seems to be insensitive to the fundamental frequency

of the soil layer and its percentage deviation does not exceed the 3.5%. The

theoretical analysis of Chapter 3 showed that the accuracy of integration schemes

improves for low frequencies and for small time steps (see Figure 3.14a). Taking

into account that all the above-mentioned analyses were performed with the same

time step (∆t=0.01sec), the role of the fundamental frequency can be isolated.

The results of Table 4.9 are in qualitative agreement with the theoretical analysis.

Hence, for a given analysis (i.e. for given frequencies) to achieve the same level

of accuracy with different schemes one has to use a smaller time step when

employing the NMK2 (and to a certain extent when using the HHT) than when

using the CH algorithm. However for the special case of resonance condition,

one should carefully select a small time step even when using the CH scheme.

Table 4.9: Summary of results for various fundamental frequencies

NMK2 HHT CH NMK1 Fundamental

frequency f0

(Hz) PD

(%)

PD

(%)

PD

(%)

(%)

S

(cm)

(%) 1.11 6.0 0.5 0.0 22.6

1.39 8.3

8.3

0.5

0.5

0.2

0.1

18.9

0.1 1.98 13.5 3.0 1.5 47.1

2.5 28.4 17.3 16.9 127.0

4.95 40.9 9.5 3.5 5.6

4.3.6 Computational cost

While for the linear elastic problems of Section 4.2.1 all the algorithms

had the same computational cost, considerable differences were observed in the

elastoplastic analyses. Table 4.10 lists the run times of the dynamic analyses of

142

the foundation for the TITO recording for the 5 algorithms. The material

properties used in these simulations correspond to those of analysis 2 in Table

4.8. Besides, all the analyses were carried out on a 1.2GHz (64 bit) Sun-Blade

2000 workstation. Cleary, the CH is the most efficient scheme in terms of

computational cost, as it is 2.14 times quicker than the NMK1. Furthermore, the

HHT and the WBZ need 19% and 20% more run time respectively than the CH.

This difference in run time is due to the fact that the CH needs fewer iterations to

converge to the solution. It should be noted that the differences in Table 4.10 are

the minimum observed. For the higher intensity recordings (i.e. VELS and

PETO) the induced plasticity made the convergence much harder for NMK1 and

NMK2 than for the α-schemes. Thus, for these recordings the differences in

computational cost between Newmark’s schemes and the α-schemes were even

more pronounced than those listed in Table 4.10.

Table 4.10: Comparison of computational cost

Algorithms run time (min) Comparison with

CH

NMK1 1290 +115%

NMK2 798 +32%

CH 604 -

HHT 719 +19%

WBZ 728 +20%

4.4 Summary

The first part of this chapter detailed a series of validation exercises

which were used to verify the implementation of the CH algorithm into ICFEP.

The closed form solution of a SDOF system subjected to a harmonic oscillation

verified the uncoupled dynamic formulation of ICFEP for both solid and beam

elements. This example was also used to compare the behaviour of the CH

scheme with two commonly used variations of Nemark’s method (i.e. NMK1,

143

NMK2). It was demonstrated that for the same time step, the CH and the NMK1

methods achieve better agreement with the closed form solution than the NMK2

method. Furthermore, the analytical solution of Zienkiewicz et al (1980a) for the

steady state response of a consolidating soil column subjected to harmonic

loading, was used to verify the formulation of the CH algorithm for dynamic

coupled consolidation problems. It was also shown that the inclusion of the

inertia term in the pore fluid equation of continuity improves the accuracy of the

numerical solution for events that lie on the limit up to which the “u-p”

approximation provides sufficient accuracy (see Figure 4.8). Finally numerical

tests by Prevost (1982), Meroi et al (1995) and Kim et al (1993) were used to

validate ICFEP’s dynamic coupled consolidation formulation for both small and

large deformation analysis.

The second part of this chapter presents two-dimensional finite element

analyses of a deep foundation subjected to seismic excitations. In this study the

emphasis was placed on the behaviour of different integration schemes and not

on a thorough investigation of the seismic response of deep foundations. Hence,

a simple elastic perfectly plastic constitutive model was used. In the first set of

analyses, the foundation response to a seismic excitation was compared for

various levels of high frequency dissipation (i.e. ρ∞ equal to 1.0, 0.818, 0.6, 0.42

and 0.0). From the results of this parametric study it was shown that the CH

scheme maintains in elasto-plastic analyses its ability to filter the high frequency

modes without significantly affecting the low frequency response, at least for the

problems considered herein. Furthermore, the comparison of the Fourier

amplitude spectra of acceleration time histories for various levels of high

frequency dissipation highlighted the necessity for numerical dissipation of the

high frequency inaccurate modes in FE analyses. The second set of analyses

investigated the effect of the frequency content of the excitation on the behaviour

of five algorithms (CH, HHT, WBZ, NMK1 and NMK2). The CH algorithm was

found to be insensitive to the predominant frequency of the input motion and to

give similar results with the NMK1 scheme in terms of displacements. The

predominant frequency of the excitation affected more the performance of the

NMK2 than the HHT and WBZ algorithms. Furthermore the acceleration

response showed that spurious oscillations dominate the results of the NMK1,

144

whereas the α schemes (i.e. CH, HHT and WBZ) perform satisfactorily. The last

set of analyses investigated the effect of the numerical model’s natural

frequencies on the performance of integrations schemes. Hence the dynamic

analyses for one of the seismic excitations (VELS recording) were repeated for

various values of soil stiffness. The CH was found to be less sensitive than the

HHT and the NMK2 schemes to the fundamental frequency of the numerical

model. However the accuracy of all three schemes deteriorates in the case that

the fundamental frequency of the soil layer is equal to the predominant frequency

of the excitation (i.e. at resonance). Finally regarding the relative computational

costs, the CH was found to be the most efficient method, whereas the NMK1 was

the most expensive.

145

Chapter 5:

ABSORBING BOUNDARY CONDITIONS

5.1 Introduction

One of the major issues in dynamic analyses of soil-structure interaction

problems is to model accurately and economically the far-field medium. The

most common way is to restrict the theoretically infinite computational domain

to a finite one with artificial boundaries. The reduction of the solution domain

makes the computation feasible, but spurious reflections on the artificial

boundaries can seriously affect the accuracy of the results.

The first part of this chapter reviews some of the most popular boundary

conditions for solving wave propagation problems in unbounded domains.

The second and the third part of this chapter discuss in detail the two

boundary conditions that have been implemented into ICFEP: the standard

viscous boundary of Lysmer and Kuhlemeyer (1969) and the cone boundary of

Kellezi (1998, 2000). This includes a description of their physical and

mathematical formulation and also a description of their implementation into

ICFEP.

To ensure the absorbing boundaries were implemented accurately it is

necessary to conduct validation exercises. Therefore, the final section of this

chapter presents validation exercises for plane strain and axisymmetric analyses.

One set of the validation exercises concerns numerical plane strain examples

from the literature and the other involves an axisymmetric problem verified with

an analytical closed form solution. Once the accuracy of the implementation is

verified, further numerical examples are presented to investigate the

effectiveness of the newly implemented boundaries for the cases of soil layers

with vertically varying stiffness and with Rayleigh wave propagation. In all cases

the numerical tests compare results obtained from small meshes having

146

absorbing boundary conditions to those generated from extended meshes. The

extended meshes are made large enough to prevent reflections from the boundary

to the area of interest.

5.2 Literature review

5.2.1 Statement of the problem

Prior to the discussion of the various boundary conditions for solving

wave propagation problems in unbounded domains, it is necessary to introduce

some important facets of wave propagation in elastic infinite media. This brief

introduction of wave mechanics theory is based on Prakash (1981) and Kramer

(1996).

In an elastic half-space, two types of waves exist, body waves and surface

waves. Body waves fall in two categories: dilatational waves (P-waves) and

shear waves (S-waves). In the case of P-waves the particle motion is in the

direction of wave propagation. The dilatational waves are the faster body waves

and they are also known as primary, pressure, compression or longitudinal

waves. For consistency, only the term dilatational wave is invoked in this

chapter. On the other hand, the S-waves result in particle motion constrained in a

plane perpendicular to the direction of wave propagation. These waves are also

known as secondary or transverse waves, but again for consistency only the term

shear wave is employed herein. The direction of particle movement can be used

to resolve the S-waves into two components, SV-waves (vertical plane

movement) and SH-waves (horizontal plane movement). Therefore, performing

an analysis in plane strain conditions implies that only in-plane waves (P and

SV) can be considered.

Surface waves, as their name suggests, result from the interaction of body

waves with the free surface and they can be divided into two types: Rayleigh and

Love waves. Rayleigh waves can be considered as a combination of P and SV

waves. According to Snell’s law, the incidence of a P-wave on a free surface of a

homogeneous elastic half-space with an angle 1θ will result into two reflected

147

waves, one P and one SV (Figure 5.1). A special case occurs when 1θ reaches a

critical value such that the reflected P-wave is tangential to the surface, thus θ2

=π/2. The value of crθ can be calculated by the following equation:

P

Scr

V

Vθsin = 5.1

where VS, VP are respectively the shear and dilatational wave velocities. When

1θ = crθ a plane wave with constant amplitude will travel parallel to the free

surface. When 1θ > crθ an exponentially decaying wave is created. This kind of

wave, which is known as Rayleigh wave, propagates with a velocity VR < VS.

Free surface

z

Pinc

Pref

SVref

θ1

θ1

θ2

x

Figure 5.1: Incidence of a P-wave on a free surface

It should be noted that in a homogeneous half-space only body waves and

Rayleigh waves can exist. If, however, soil layering is present Love waves can

arise. Love waves typically develop in shallow surface soil layers overlying

layers of stiffer material properties. They basically consist of SH-waves that are

trapped by multiple reflections within the surface layer. Exactly like SH waves,

Love waves propagate in the out-of-plane direction and they have no vertical

component of particle motion.

Furthermore, it is interesting to examine the case of a body wave

impinging on a boundary between two different materials. Figure 5.2 illustrates a

soil deposit consisting of two soil layers of different material properties (e.g.

148

shear wave velocity, Vs2>Vs1), where the top layer is of finite thickness and the

bottom layer is of an infinite depth. It is for simplicity assumed that surface

waves cannot develop.

Ar

At A tMaterial 1Vs1

Transmitted waveA t

Reflected wave

A’ r

Transmitted wave

A’t

Material 2

Vs2

A t

8

Incident wave

A i

Figure 5.2: Shear wave vertically propagating through a layered soil deposit

When an incident shear wave (Ai) travelling vertically upwards arrives at the

interface of the two soil layers, part of the wave (At) will be transmitted to the

upper layer and part of the wave (Ar) will be reflected back to stiffer layer. The

partition of the wave depends on the elastic properties of the two media. The

wave reflected to the stiffer layer never returns to the interface between the

layers, as the stiffer layer is infinite. This loss of energy is termed radiation

damping. Furthermore, assuming zero material damping, the transmitted wave

will reach the free soil surface, where due to the zero stress condition, the entire

wave will be reflected back with an opposite sign. When this wave reaches again

the interface of the two layers, part of the energy is reflected back and part of the

energy is transmitted to the stiffer layer and thus it will be lost in the semi-

infinite domain of the bottom layer. This loss of energy represents again the

radiation damping. One of the big challenges of dynamic finite element analysis

is to model correctly this radiation damping.

Furthermore, the presence of a perturbation close to the soil surface (e.g.

any kind of geotechnical structure) poses an additional problem. As can be seen

149

in the sketch of Figure 5.3 multiple reflections and refractions of the incoming

wave take place due to the existence of the structure

Material 1

Vs1

Material 2

Vs2

Incident wave

Ai

8

Figure 5.3 : Multiple reflections and refraction of an incoming wave due to the

presence of a structure

In addition, in a shallow zone near to the surface, the occurrence of surface

waves should also be taken into account. In this case appropriate boundary

conditions, that can absorb both surface and body waves, should be applied at the

side boundaries.

To highlight the geometrical nature of the radiation damping, a vibrating

plate on a homogeneous halfspace is considered (Figure 5.4). The distance of the

boundary of the halfspace from the centre of the vibrating plate is r and the area

of the boundary surface corresponding to this distance is A(r). The vibration

generates a combination of body and surface waves which encounter an

increasingly larger volume of material as they travel outward. Considering the

far-field boundary surface (at r→∞), the radiation criterion states that for the

radiation of energy to occur, the displacement amplitude must thus decay at

infinity in inverse proportion to the square root of the surface area (A(r→∞)) at

infinity (Meek and Wolf, 1993). For body waves in three-dimensions, the surface

at infinity is a large hemisphere with area 2πr2 (where r→∞), whereas for

Rayleigh waves the surface is a flat cylinder with height approximately one

Rayleigh wave length λR.

150

Figure 5.4: Wave propagation at infinity (from Meek and Wolf, 1993)

Figure 5.5: Dynamic models of unbounded medium: Substructure method (a) and

Direct method (b) (from Kellezi, 2000)

To model numerically the dissipation of energy to the far-field medium, a

boundary called the interaction horizon (Wolf, 1996) is chosen up to which a

finite-element discretization is applied. The interaction horizon can either

coincide with the generalized structure-medium interface (substructure method,

Figure 5.5a) or it can be identical with an artificially chosen boundary ∞Γ (direct

method, Figure 5.5b).

The substructure technique employs the Boundary Element Method

(BEM) to satisfy the radiation condition at the soil-structure interface. The

151

analysis of the unbounded domain is carried out by discretized Green’s functions

along the interaction horizon and it gives a global solution in space and time. The

term global refers to the fact that the solution at a specific boundary degree of

freedom at a specific time depends on the response of all boundary degrees of

freedom at all previous instances (Wolf, 1996). Obviously, the global

formulation is computationally expensive, but the added expense is compensated

by the reduction in the total number of degrees of freedom needed to describe the

problem. It should be noted that the substructure method treats separately the

structure and the soil and uses the principle of superposition to couple the

responses. Therefore, it can be applied only for linear models and simple

geometries. Despite its accuracy, the substructure method is not widely used in

practice as it involves very sophisticated formulations. On the other hand, the

direct method employs FE discretization to model the structure and a region of

soil up to the artificial boundary. Various such artificial boundaries have been

developed to simulate the radiation condition. Kausel and Tassoulas (1981)

categorized them into three major groups:

Elementary boundaries. The most commonly used elementary boundaries are

conditions of either zero stress (Neumann condition) or zero displacement

(Dirichlet condition). These boundaries act as perfect reflectors. Therefore, they

cannot model the radiation of energy to infinity and as result trap the waves in

the mesh. To overcome this problem, the boundaries are usually placed far from

the area of interest and strong material damping is employed. Hence, the energy

is dissipated before the waves reflect back to the region of interest. This approach

leads to uneconomically large meshes and the incorporation of material damping

cannot satisfactorily model the radiation condition (Luco et al, 1974). However,

the elementary boundaries can be very efficient in cases where the radiation

damping is not important, like soft soil – stiff rock interfaces. Furthermore

Zienkiewicz et al (1988) introduced another kind of elementary boundary, using

tied degrees of freedom, that is employed on the lateral sides of the FE mesh.

This boundary condition constrains nodes of the same elevation on the two

lateral boundaries to deform identically. In the absence of a structure, this

approach can perfectly model the one-dimensional soil response. When a

structure is however included into the model, this method cannot absorb the

152

waves radiating away from the structure and thus it also results in wave-trapping

into the mesh.

Local boundaries. In this case, the radiation condition is satisfied approximately

at the artificial boundary, as the solution is local in space and time. When a

formulation is local (as opposed to a global formulation) in space and time, the

solution at a specific boundary degree of freedom depends on the response of

only adjacent boundary degrees of freedom at a specific time, or at most, during

a limited past period. Many of these local, transmitting, absorbing or non-

reflecting boundaries provide results of acceptable accuracy and are far less

computationally expensive than the more rigorous consistent boundaries.

Consistent (non local) boundaries. These boundaries are perfect absorbers and

they satisfy exactly the radiation condition. The majority of consistent

boundaries are frequency-dependent and thus in time domain analysis they are

restricted to steady state problems. Two distinct methodologies fall into this

category: the thin layer method and the coupled boundary element - finite

element method (BEM-FEM).

5.2.2 Local boundaries

The most widely used local boundary is the standard viscous boundary of

Lysmer and Kuhlemeyer (1969). This boundary was implemented in ICFEP and

is therefore comprehensively presented together with refinements made by more

recent studies, in Section 5.3. This section reviews other forms of local

boundaries.

Smith (1974) introduced the superposition boundary, which combines

Dirichlet and Neumann conditions to eliminate spurious reflections. For the

simple case of only one boundary interface, this method requires the

superposition of results from two boundary value problems. In the first problem

zero normal stresses and zero tangential displacements are imposed at the

boundary, whereas in the second problem tangential stresses and normal

displacements are set equal to zero at the boundary. In principle, dilatational and

shear waves are reflected with equal amplitude in both problems, but with

153

opposite signs. Hence superimposing the two solutions cancels the reflections. In

addition, Smith (1974) showed that this methodology can also deal with surface

waves. However, for more than one boundary interface (i.e. n interfaces) the

exact solution requires superposition of 2n independent boundary value problems.

For such cases, Smith (1974) indicates that some high order reflections of small

practical importance cannot be eliminated. Obviously, the numerical cost of the

method is high, since it requires computation of multiple problems. In addition,

to apply the principle of superposition in the time domain, it must be assumed

that the system behaves linearly. Cundall et al (1977) and Kunar and Marti

(1981) suggested an approach to overcome the high-order reflections by

performing the superposition of the independent solutions only in the vicinity of

the boundary and at intervals of few time steps. In particular, this method

consists of two independent overlapping narrow boundary zones, typically of

three to four elements wide, attached to the main mesh (Figure 5.6). A wave that

propagates from the main mesh will enter the two boundary zones

simultaneously. The boundary conditions in the two zones are such that the

waves resulting from reflection on the two artificial boundaries (A, B) have the

same magnitudes, but opposite sign. The superposition of the two solutions in the

boundary zones is performed before the reflected waves can reach the main

mesh, typically every three time steps. Kausel (1988), reviewing systematically

this method, suggests that the key idea of the modified Smith boundary is the

elimination of reflected waves as they occur. This modification prevents the

multiple reflections and the need for 2n solutions. However, Kausel points out

that the modified Smith boundary is not rigorously justified and that it often fails

to prevent the multiple reflections.

154

Figure 5.6: Illustration of the modified Smith boundary (after Wolf, 1988)

Moreover, Engquist and Majda (1977), for the scalar wave equation, and

Clayton and Engquist (1977), for the elastic wave equation, developed a

fundamental set of transmitting boundaries, which are called paraxial

boundaries. These boundaries are often called transparent as the key idea of the

method is to make the mesh boundary “transparent” to outward-moving waves.

To illustrate the basic concept of the paraxial boundary, an elastic half – space

subjected to an out-of-plane shear wave (SH), with inclination θ with respect to

the x-direction, is considered (Figure 5.7). The half-space (x ≤ 0) is limited by a

vertical boundary at x = 0. The governing wave equation is given by:

0utV

1

zx 2

2

S

2

2

2

2

=

∂∂

−∂∂

+∂∂

5.2

where VS is the shear wave velocity and the displacement u is a function of the

coordinates x and z.

A

B

155

Figure 5.7: Elastic half – space subjected to an out-of-plane shear wave (SH)

(after Kausel, 1988)

An out-of-plane perturbation, generated at x=0, results in two waves of equal

amplitude propagating towards two opposite directions (-x, +x). The aim of the

paraxial approach is to find a differential equation which allows waves moving in

one general direction (for Figure 5.7, this is the negative x-direction) and

neglects waves moving in the opposite direction. The solution of Equation 5.2 for

a harmonic wave of normal incidence (θ=0) has the form:

[ ]t)ωzkx(kiexpAu zx −+= 5.3

where A is the amplitude, kx, kz are the wave numbers (2π/wavelength) and ω is

the circular frequency of the wave. Substituting the expression 5.3 into Equation

5.2, results in:

0kVkVω 2

z

2

S

2

x

2

S

2 =−− 5.4

Factoring Equation 5.4 leads to:

156

0ω

kV1ωkV

ω

kV1ωkV

2

zS

xS

2

zS

xS =

−+

−− 5.5

For ω

zSkV<1 and considering only the second factor that corresponds to a wave

propagating in the negative x-direction, it leads to:

−−=

ω

zksV1

V

ωk

2

S

x 5.6

The square-root operator in the above equation can be expanded with a second

order approximation about small values of ω

kV zS :

0ω

kV

2

11

V

ωk

2

zS

S

x =

−−= 5.7

The paraxial boundary condition is given by the differential equation that

corresponds to Equation 5.7 and it therefore transmits only negative x-directed

waves:

0uz2

V

txV

t 2

22

S

2

S2

2

=

∂∂

−∂∂

∂+

∂∂

5.8

Engquist and Majda (1977) pointed out that a higher order paraxial

approximation based on Taylor series expansions leads to unstable schemes. The

use, however, of a 3rd order (or higher) approximation improves the accuracy of

the boundary. Clayton and Engquist (1980) suggested that good wave

transmission is obtained when the transparent boundary is employed in finite

difference computations. On the other hand, Cohen (1980) showed that the

paraxial boundary is unstable for values of Poisson’s ratio greater than 1/3.

Furthermore, Stacey (1988) introduced a more accurate paraxial approximation

without introducing higher order terms. Kausel (1992) showed that the Stacey

boundary is intrinsically stable, but he argues that its derivation is not

theoretically rigorous. A further shortcoming of the paraxial family of boundaries

157

is that it is not suitable for direct implementation in finite element programs.

Cohen and Jennings (1983) modified considerably the paraxial methodology to

make it suitable for finite element implementation. They showed analytically the

superiority of the paraxial boundary over the standard viscous boundary, but their

numerical results suggest that it performs only slightly better.

Liao and Wong (1984) introduced the extrapolation boundary which is

an explicit formulation for finite element applications. As the name of the

method suggests, the radiation condition on the boundary nodes is approximated

by extrapolating present and past data along a line normal to the boundary.

Kausel (1988) showed the close connection of the extrapolation boundary with

the paraxial family of boundaries. The two main disadvantages of the method are

the large storage requirement and the fact that the method fails if there are many

waves impinging at the boundary.

Underwood and Geers (1981) introduced the doubly asymptotic (DA)

transmitting boundary which absorbs completely waves propagating

perpendicularly to an artificial boundary at the low and high-frequency limits.

Specifically, at the high-frequency limit the infinite medium is modelled as an

array of viscous dashpots, whereas at the low-frequency limit it is modelled as an

array of springs. The boundary condition is derived by adding up the

contributions of the two extreme limits:

[ ] [ ] uCuKR ω0ω &∞→→ += 5.9

where R are the interaction forces along the boundary, u is the displacement,

u& is the velocity and [ ]0ωK → , [ ]∞→ωC are the stiffness and the damping matrices

respectively. The stiffness matrix [ ]0ωK → of the infinite medium is calculated

with a linear boundary element method. It is nonsymmetric as it couples all

degrees of freedom on the artificial boundary. The damping matrix [ ]∞→ωC is

calculated as follows:

[ ] [ ] [ ]mω CAρC =∞→ 5.10

158

where ρ is the mass density, [A] is a diagonal element-area matrix for the

medium and [ ]mC is a diagonal matrix of the propagation velocities of

dilatational and shear waves in the medium. The DA method is based on the idea

that the velocity vector is small relative to the displacement vector for low

frequencies, so the force is given by the static stiffness relationship. Conversely,

at high frequencies the opposite is true and thus the force is given by the

damping relationship. Clearly, the DA boundary is local in time, but the

calculation of the stiffness matrix makes it global in space. Neither the stiffness

nor the damping matrices are frequency dependent. Hence the DA boundary is

suitable for transient analyses in the time domain. The formulation of the method

dictates the material adjacent to the boundary to behave linearly, but it allows

nonlinear behaviour away from the boundary. The two main drawbacks of the

DA boundary are the lack of accuracy for the intermediate range of frequencies

and the fact that it achieves perfect absorption only for waves propagating

perpendicularly to the artificial boundary.

Higdon (1986, 1987) introduced the multi-directional (MD) boundary. Figure 5.8

illustrates an inclined out-of-plane shear wave at an artificial boundary. A

boundary condition for this out-of-plane wave is described by the following

equation:

0utxcosα

VS =

∂∂

+∂∂

5.11

where α is the assumed angle of incidence. A plane wave propagating at an angle

-α also satisfies the above equation.

159

α

Vα

VS

y

xO

Artificial boundary

Figure 5.8 : Inclined scalar wave at an artificial boundary with apparent velocity

in perpendicular direction (after Wolf and Song, 1996)

Higdon generalized this formulation for waves propagating at angles ± αi,

developing the multi-directional (MD) boundary which is formed as a product of

differential operators:

0utxcosα

Vm

1i i

s =

∂∂

+∂∂

∏=

5.12

where αi (i=1,2,…..m) are the predicted angles of incidence. This boundary

condition perfectly absorbs plane waves propagating outwardly at an angle αi and

to a large extent at other angles, as Higdon proved with numerical tests that the

amount of reflection is not very sensitive to the choice of αi. Thus, the fact that

the angle of incidence is unknown a priori is not a major restriction for the multi-

directional boundary, as Higdon (1991) suggests that in practice rough guesses of

αi are good enough. Theoretically, when the number of operators in Equation

5.12 goes to infinity, this boundary condition gives a global formulation.

However, in practice, only the product of two or three operators is taken and it is

applied at a specific location at a specific time. Thus, this boundary condition is

also local in space and time. It should be noted that the MD boundary reduces to

the paraxial approximation by setting m=2 and α1=α2=0. Clearly, the main

difference of the MD comparing with the paraxial method is that the predicted

angle of incidence α is not included in the formulation. Higdon (1991) expanded

160

his theory to elastic waves, by applying the following operator to each

component of the displacement vector:

0ut

βx

Vm

1i

i =

∂∂

+∂∂

∏=

5.13

where iβ is a dimensional constant and V is the assumed dominant wave

propagation velocity in each direction of the motion (either the P-wave velocity

PV or the S-wave velocity SV ). Generally, each component of the displacement

can experience both wave types. When the factor iβ takes values less than one,

the operator 5.13 mainly absorbs P-waves and partially S-waves. Conversely, for

values of iβ close to PV / SV , the operator mainly absorbs S-waves and partially

P-waves. Higdon (1991) found that the second order MD roughly attains the

same accuracy with the second order paraxial boundary. In addition, Higdon

showed that the MD boundaries of second or third order are stable for all values

of PV / SV , whereas the paraxial boundary is unstable for large values of PV / SV .

The above-mentioned formulation of the MD can be directly implemented only

in finite difference schemes. Kellezi (1998) introduced a modified formulation of

the MD boundary suitable for FE implementation. It was shown that the MD

boundary in FE schemes behaves slightly worse than the viscous boundary. Both

boundaries are inaccurate in the low-frequency limit. Therefore, Kellezi

concluded that the FE implementation of the MD is not attractive as it is

complicated, without gaining much in terms of accuracy.

Wolf and Song (1995) combined the advantages of the doubly asymptotic

and the multi-directional boundary and they derived the doubly asymptotic multi-

directional boundary (DAMD). The DAMD is derived for out-of-plane waves,

but it can be extended to plane waves. The boundary condition at the time station

n (t=n ∆t) is given by the following equation:

[ ] [ ] nbnb

ω

bnb

0ω

bb QuCuKR ++= ∞→→ & 5.14

where the subscript b refers to the artificial boundary. Comparing Equation 5.14

with the boundary condition of the simple DA approach, the only additional term

161

is nbQ , that represents the remaining interaction forces which are predicted

from the MD boundary. The differential operator of the MD boundary is now

formulated for forces instead of displacements:

0Qtxcosα

Vm

1i i

s =

∂∂

+∂∂

∏=

5.15

The DAMD boundary is local in time and global in space. However, Wolf and

Song (1995) also suggest a local in space formulation which employs an

approximate banded static-stiffness matrix instead of a stiffness matrix that

couples all the degrees of freedom on the artificial boundary. In all cases, the DA

part of the DAMD boundary is implemented implicitly. The DA contribution is

subtracted from the interaction forces and then the MD boundary is formulated

explicitly for the remaining interaction forces. Wolf and Song note that the

DAMD is rigorous for all frequencies and all pre-selected angles of incidence.

Figure 5.10 compares the results of the doubly asymptotic multi-directional

boundary for a semi-infinite rod on elastic foundation subjected to dynamic

loading (Figure 5.9), to the results of other methodologies for the same problem.

Obviously, the accuracy of the second order (m=2) DAMD boundary is greater

than the other boundaries. However, an obvious disadvantage of this boundary is

the explicit formulation of the MD contribution which can raise time step

limitations.

R

u0

∆l Artificial Boundary

Figure 5.9 : FE mesh up to the artificial boundary of a semi-infinite rod on elastic

foundation (after Wolf and Song, 1996)

162

Figure 5.10: Comparison of various boundaries for a semi-infinite rod on elastic

foundation (after Wolf and Song, 1996)

A frequency dependent approach based on Kelvin elements was

introduced by Novak et al (1978) for plane strain problems and by Novak and

Mitwally (1988) for axisymmetric problems. In both cases their derivation is

based on waves propagating away from an infinitely long harmonically vibrating

cylinder. The physical interpretation of this boundary is a series of springs and

dashpots in parallel configuration. The constants of the springs and the dashpots

for the axisymmetric case can be evaluated using the following expression:

[ ]β)ν,,(αSiβ)ν,,(αSr

Gk o2o1

o

*

r += 5.16

where *

rk is the complex stiffness, G is the shear modulus of the soil, S1 and S2

are dimensionless parameters from closed form solutions, ν is the Poisson’s ratio,

β is the material damping ratio, i is the imaginary unit, αo is the dimensionless

frequency (= roω/VS where ω is the angular frequency of excitation) and ro is the

distance from the centre of the vibrating body to the boundary node. The real and

163

imaginary parts of Equation 5.16 represent the stiffness and the damping

coefficients respectively. Novak and Mitwally (1988) note that the presence of

the spring term gives to this boundary a distinct advantage over the standard

viscous boundary. They particularly found it very advantageous in the study of

pile driving. The use, however, of this frequency dependent approach in the

study of transient excitations, like earthquake, requires some simplistic

approximations. Maheshwari et al (2004) studied the three dimensional response

of pile groups to seismic excitation employing the boundary of Novak and

Mitwally at the sides of the FE mesh and applying an acceleration time history at

the bottom of the mesh. They used the predominant frequency of excitation to

calculate the constant of the springs and the dashpots. This is clearly a crude

approximation that implies that the transmitting boundary is not accurate for

other frequencies. In addition, Maheshwari et al (2004) calculated the ro as the

distance in plan from the centre of the foundation to the transmitting boundary

node. They claim that this approximately represents the corresponding radial

distance of the cylindrical model of Novak and Mitwally (1988). This is another

simplistic approximation, as the derivation of the boundary of Novak and

Mitwally (1988) is based on the idea that the excitation is applied at the centre of

the cylinder. Therefore this boundary can only be accurately used when the

excitation is directly applied on the structure.

Deeks and Randolph (1994) derived approximate frequency independent

boundaries for shear and dilation waves in axisymmetric problems. Similarly to

the approach of Novak and Mitwally (1988), they considered the propagation of

cylindrical waves. The physical interpretation of the shear wave boundary is a

spring with a distributed spring constant of G/2ro and a dashpot with a distributed

damping constant of ρVS (where ρ is the material density) in parallel

configuration. The dilatational wave boundary consists of the same spring, a

dashpot with a distributed damping constant of ρVS and a lumped mass with a

distributed mass constant of 2ρro. The numerical tests of Deeks and Randolph

(1994) showed that the accuracy of this boundary is better than either the viscous

boundary or DA boundary.

164

Naimi et al (2001), following the ideas of Sarma (1990) and Sarma and

Mahabadi (1995), developed transmitting boundary conditions for two inclined

mutually perpendicular planes. The two planes divide the halfspace to a finite

domain ABC and to a semi-infinite region outside the boundaries (Figure 5.11).

The derivation of this method is based on two dimensional elasto-dynamic

equations for propagation of P and SV body waves in plane strain conditions. It

is assumed that any structure is included in a region ABC and that it is allowed to

behave non-linearly. However, for the region close to the boundary the material

is assumed to be linear. The essential assumptions of the method are that the

boundary AB is normal to the wavefront of the incoming waves and therefore

this method does not consider the propagation of surface waves. The travelling

(from the far-field towards the inner region) P and/or SV wave(s) arrive at the

boundary AB and then part of these waves is reflected at the boundary and the

other part is refracted into the interior region. Subsequently, the refracted waves

in the interior region are reflected at both the free surface and the assumed

structure and they finally travel back towards the boundaries AB and BC. These

latter waves refracted into the far-field through the boundaries AB and BC are

assumed to be P and S body waves in directions normal and parallel to the

boundaries. These directions can be considered as components of the real waves.

The boundary condition for example for the plane AB is defined by the following

two equations:

in21PS v2y

'vA

x

uAV

x

'vVv

11′=

′∂

∂+

′∂

′∂−

′∂∂

−′ &&&&&

&& 5.17

in21PS u2y

'vA

x

uAV

x

uVu

11′=

′∂

∂+

′∂

′∂−

′∂

′∂−′ &&

&&&&& 5.18

where A1, A2 ,A3 and A4 are defined as:

( ) ( )2

P1

2

S1

2

S

42

S1

2

S32

P1

2

P22

P1

2

S

2

P1

2

S

2

P

1

1

1

111

11

Vρ

VρρVA,

Vρ

ρVA,

Vρ

ρVA,

Vρ

2VVρ2VVρA

−===

−−−=

u′ , v′ are the displacement in the x′ and y′ directions respectively, in'u&& , in'v&& are

the accelerations caused by incoming waves in the x′ and y′ directions

165

respectively, the suffix 1 refers to the semi-infinite medium whereas there is no

suffix for the region ABC and ρ is the material density. It should be noted that

the accelerations of the incoming waves in'u&& , in'v&& are equal to 0.5 g'u&& , 0.5 g'v&&

where g'u&& , g'v&& are the free field accelerations (i.e on a rock outcrop).

Figure 5.11 : Two inclined mutually perpendicular boundaries AB and AC (from

Naimi et al, 2001)

Naimi et al (2001) also developed three different sets of equations, of similar

form to Equations 5.17, 5.18, for the plane BC, depending on the inclination of

the boundary AB to the horizontal (for an angle less than, equal to, or greater

than 45°). Furthermore, the suggested implementation of this method concerns

coupling of FE discretization for the core region with an explicit finite difference

approximation for the boundary region. Naimi et al (2001) conducted a series of

numerical tests for different angles of incidence of a harmonic P wave. The

results of both transient and steady state analyses indicated a good performance

of this method. The method needs to be further tested for incident shear waves.

A completely different approach, the modelling of the unbounded

medium using infinite elements, was first proposed by Bettess and Zienkiewicz

(1977). In this case there is no attempt to truncate the unbounded domain and, as

the name of the method suggests, the whole domain is modelled using elements

of infinite extent. Bettess (1992) notes that the theory of infinite elements for

static problems involves either the use of decay functions in the infinite direction,

166

which multiply the parent element shape function (decay function infinite

elements), or the use of some completely new shape functions in the infinite

direction (mapped infinite elements). In dynamic problems for both types of

infinite elements an extra exponential term is added. For the case of decay

function infinite elements, the shape function Ni(T,S) can be written as follows:

exp(iks)(T)fS)(T,PS)(T,N ii = 5.19

where P denotes the shape functions of the parent finite elements, T, S are the

local coordinates (T is assumed to be in the radial direction, extending to

infinity), (T)f is a decay function that is assumed to depend only on T and s is a

coordinate that is directly related to T, defined as follows:

∂∂

+

∂∂

=∂∂

22

T

y

T

x

T

s 5.20

where x, y are the global coordinates. For the case of the mapped infinite

elements, the shape function Ni(T,S) is given by the following equation:

exp(ikr)S)(T,PS)(T,N ii′= 5.21

where P’ denotes the shape function of the parent element (see Bettess, 1992), T,

S are the local coordinates (-1≤S≤1 and -1≤T≤1) and r is the global radial

coordinate. The exponential terms in Equations 5.20 and 5.21 represent the wave

propagation towards infinity, but they obviously depend on the frequency of the

wave. Therefore, in time-domain, the analysis is restricted to the steady state

case. Kim and Yun (2000, 2003) however employed conventional FE to model

the near-field medium and frequency-dependent infinite elements to model the

far-field for transient analysis. In this case the exponential terms of the shape

functions are given by approximate expressions for waves propagating in

isotropic layered elastic media. The resulting mass and stiffness matrices of each

infinite element are then expressed in terms of the exciting frequency and

solution in the time domain is obtained using inverse Fourier transforms.

Astley (1983) introduced wave envelop finite elements, which are

essentially modifications of infinite elements. The novel characteristic of the

167

wave envelop finite elements is that the complex conjugates of the shape

functions are used as weighting functions. Figure 5.12 shows a typical wave

envelop model. The computational model is based on a subdivision of the

external area R into a near-field conventional FE mesh (Rin) matched to a single

layer of infinite wave envelope (WE) elements in Rout.

Figure 5.12: Typical wave envelope model (from Astley, 1994)

The initial formulation of the WE method was solely applicable to the frequency

domain. Astley (1996) and Astley et al (1998) also introduced a transient

formulation of the WE method by applying an inverse Fourier transformation to

a time-harmonic WE model.

The use of Fourier transforms in general restricts the solution to linear

elastic systems, where the method of superposition is valid. This is not however a

major restriction of the method, since the near-field area, which is discretized by

conventional FE, can behave nonlinearly.

The perfectly matched layer (PML) method was first introduced in the

field of electromagnetic waves (Bérenger, 1994), but it is rapidly becoming

popular in problems of elastodynamic wave propagation. The main idea of the

method is to surround the truncated computational domain with a highly

168

absorbing boundary layer. This absorbing layer is called “perfectly matched”

because it does not allow any reflection from the truncated domain – PML

interface. Basu and Chopra (2003a) presented a PML formulation for out-of-

plane and plane-strain motion of visco-elastic media in the frequency domain. A

wave of unit amplitude caused by out-of-plane motion in the truncated two-

dimensional isotropic elastic domain ΩBD of Figure 5.13 is of the form:

+−= t)ωp

V

ω(iexpt),u(

s

xx 5.22

where x denotes the vector of coordinates and p is a unit vector denoting the

propagation direction. When this wave leaves the truncated domain ΩBD, it is

attenuated in the PML domain (ΩPM) and is then reflected back from the rigid

boundary towards the domain ΩBD. To attenuate the wave in the PML area, the

wave solution in the domain ΩPM is of the form:

+−

−= ∑ t)ωpx

V

ω(iexpp)(xFexpt)u(x,

si

iii 5.23

The term

−∑

i

iii p)(xFexp represents an attenuation function. Basu and

Chopra (2003a) suggest that the choice of the attenuation function and the depth

of the layer control the amplitude of the reflected wave and that this amplitude

can be infinitesimally small for non-tangential incident waves. Furthermore,

Basu and Chopra (2004) transformed the PML formulation in the time-domain

applying a special attenuation function and using inverse Fourier transforms.

They also implemented the method in a FE program and conducted several

numerical tests to examine its performance. Basu and Chopra showed that the

PML method attains more accurate results than the standard viscous boundary at

the expense of 50%-75% higher computational cost. They note that the main

disadvantage of the time domain formulation of PML is its inadequacy to

attenuate quickly fading waves. Furthermore, to the author’s knowledge a

formulation of PML has yet to be introduced to deal with problems in which the

source of excitation is located outside the PML domain (ΩPM) (e.g. earthquake

excitation). To cope with this limitation, Basu and Chopra (2003b) suggest the

169

use of the effective seismic input method of Bielak and Christiano (1984). The

effective seismic input method is an early formulation of the domain reduction

method that is examined in detail in the next chapter of this thesis.

Figure 5.13: A PML adjacent to a truncated domain (from Basu and Chopra,

2004)

5.2.3 Consistent boundaries

Lysmer (1970) and Lysmer and Waas (1972) introduced the first

consistent boundaries that couple all the boundary points and constitute perfect

absorbers for any kind of wave and any kind of angle of incidence. Their

approach is known as the thin-layer method (TLM) and it deals with the radiation

condition at the side boundaries of layered strata overlaying rigid rock subjected

to out-of-plane motion. The basic concept of the method is that a natural soil-

layer is discretized over the depth into thin sub-layers and that an interpolation

function is used for the variation of displacement in the direction of layering. It is

a semi-analytical approach, as the FE solution is used in the direction of layering,

whereas closed-form solutions are employed for the remaining directions. The

170

method was subsequently extended for in-plane motion in layered media and for

a three-dimensional model exhibiting cylindrical geometry. Although Kausel

(1994) presented a time-domain approach of the method, TLM is

computationally cheaper and more flexible when it is formulated in the

frequency-domain. The two main shortcomings of the TLM are that it only

provides a means for the analysis of soil deposits of finite depth and that it is

restricted to linear systems. A thorough review of the method and various

examples of its application are given by Kausel (2000).

5.3 Standard viscous boundary

5.3.1 Theory

Lysmer and Kuhlemeyer (1969) introduced the standard viscous

boundary which absorbs waves impinging normally to the interaction horizon

(Figure 5.5b). The fundamental idea of this method is the application of a

traction condition at a free artificial boundary which dictates any reflected

stresses to be zero:

0t

s)(t,uVραs)(t,σ P =

∂∂

+ 5.24

0t

s)(t,vVρbs)(t,τ S =

∂∂

+ 5.25

where s)σ(t, , s)τ(t, are the normal and shear stresses on the boundary, s)u(t, ,

s)v(t, are the normal and tangential displacements, s denotes the coordinate on a

convex artificial boundary and α , b are dimensionless parameters. The standard

viscous boundary can be described by two series of dashpots oriented normal and

tangential to the boundary of the FE mesh. The analytical study of Lysmer and

Kuhlemeyer (1969) suggests that the performance of the boundary is optimised

for α =b=1.0. The numerical investigations of Cohen (1980) showed that the

viscous boundary is not very sensitive to the viscosity coefficients α , b.

171

It should be noted that perfect absorption can only be achieved for

perpendicularly impinging waves. Therefore, the method is exact only for one-

dimensional propagation of body waves. For two-dimensional and three-

dimensional cases, perfect absorption is achieved for angles of incidence greater

than 30° (when the angle is measured from the direction parallel to the

boundary).

x x

z z

dz

A

G

ρ

v

u

dz

Q

dzz

QQ

∂∂

+

2

2

t

vdzA

∂∂

ρ

z=r

CH

(a) (c)

(b)

Figure 5.14: Semi-infinite rod model

Wolf (1988) considered the problem of a semi-infinite prismatic homogeneous

and elastic rod to show that the standard viscous boundary is based on one-

dimensional wave theory. Figure 5.14a illustrates a shear wave propagating in a

rod with area A, shear modulus G and mass density ρ. The equilibrium of an

infinitesimal element (Figure 5.14b) is given by:

0t

vdzAρdz

z

Q2

2

=∂∂

−∂∂

5.26

where Q represents the shear force and v the transverse displacement. The force-

displacement relationship can be expressed as:

z

vAGQ

∂∂

= 5.27

172

Considering that G = ρ 2

SV , the equation of motion is obtained by substituting

Equation 5.27 into Equation 5.26:

0t

v

V

1

z

v2

2

2

S

2

2

=∂∂

−∂∂

5.28

The general solution of the equation of motion is of the form:

t)Vg(zt)Vf(zt)v(z, SS ++−= 5.29

where t)Vf(z S− represents a wave of arbitrary shape travelling in the positive z

direction and t)Vg(z S+ represents a wave with the same characteristics, but

travelling in the opposite direction. Considering only the wave that encounters

the artificial boundary at z=r, the g wave can be ignored. The strain, the stress

and the particle velocity at any coordinate z are given by the following formulas:

t)V(zfz

vt)γ(z, S−′=

∂∂

= 5.30

t)V(zfGγGt)τ(z, S−′== 5.31

t)V(zfVt

vSS −′−=

∂∂

5.32

where f ′ denotes the derivative of f with respect to the argument. The radiation

condition at z=r allows the wave t)Vf(zt)v(z, S−= to pass through the artificial

boundary without modification. Therefore combining the Equations 5.31 and

5.32 for z=r, one gets the boundary condition of Equation 5.25 for α =b=1.0.

Furthermore, multiplying both terms of Equation 5.25 by the surface A, the

equation of equilibrium at the artificial boundary between the shear force and the

force of the viscous dashpot is derived:

0t)(r,t

vCQ H =

∂∂

+ 5.33

173

where SH VρAC = represents the viscosity of the dashpot. Hence, the shear force

of the damper can replace the part of the rod that extends to infinity (Figure

5.14c). Similarly, the boundary condition of Equation 5.24 can be derived

considering the propagation of a dilatational wave that causes axial deformation

along the prismatic rod and that the Young’s modulus E is equal to ρ 2

PV . In this

case the part of the rod that extends up to infinity can be replaced by a

longitudinal dashpot. The viscosity of the dashpot is then defined as,

PV VρAC = .

In addition, White et al (1977) introduced a modification of the standard

viscous boundary, the unified viscous boundary which can deal with anisotropic

materials. The only novelty of the unified viscous boundary is that the

dimensionless parametersα , b are evaluated as:

2S)(315π

8b

)2S2S(515π

8α 2

+=

−+=

5.34

where S is the ratio of VP over VS. Numerical tests of Cohen (1980) and Kellezi

(1998) showed that for isotropic materials the boundary of White et al (1977) is

slightly worse than the standard viscous boundary.

The standard viscous boundary is probably the most commonly used

scheme, as it gives acceptable accuracy for low computational cost. The great

advantage of this approach is that the absorption characteristics are independent

of frequency and thus the viscous boundary is suitable for both harmonic and

non-harmonic waves. The performance of the boundary improves significantly

the farther it is placed away from the source of excitation or the area of interest

(i.e. structure) of the model. Wolf (1988) suggests that generally at large

distances from the vibrating source (or structure) body waves propagate one-

dimensionally in the direction of the normal to the artificial boundary. Hence in

large computational domains waves of extreme angle of incidence, that cannot be

absorbed by the viscous boundary, are less likely to develop. Therefore, one

should find a balance between accuracy and an economically acceptable mesh

174

size. Furthermore, it is well recognized that the viscous boundary is more

accurate for high frequency excitations. In the low-frequency range, as it is

demonstrated in Section 5.5, the viscous boundary leads to permanent

displacement even in elastic systems, especially at the points of the mesh

adjacent to the artificial boundary. According to Wolf (1988) the directionality of

the waves increases for high frequencies and therefore the wave propagation is

closer to the one-dimensional case.

Moreover, a drawback of the viscous boundary that is commonly

encountered in the literature (Lysmer and Kuhlemeyer, 1969; Urich and

Kuhlemeyer, 1973; Valliappan and Favaloro, 1977) is its inability to absorb

Rayleigh waves. To tackle this shortcoming, Lysmer and Kuhlemeyer (1969)

also introduced a frequency-dependent modification of the viscous boundary that

can completely absorb Rayleigh waves. Obviously, the application of this

boundary is limited to steady state problems. It should be noted that some more

recent studies (Cohen 1980, Wolf 1988, Kellezi 1998) show that the standard

viscous boundary can, to a certain extent, absorb Rayleigh waves. The ability of

the viscous boundary to absorb Rayleigh waves is addressed in detail in Section

5.5.3.

5.3.2 Implementation

The standard viscous boundary is implemented into ICFEP for two-

dimensional plane strain and axisymmetric analyses. The discretised equilibrium

equation is of the form:

[ ] ( ) [ ] ( ) [ ] ( ) ∑∑∑∑====

=++N

1i

E

N

1i

iniE

N

1i

iniE

N

1i

iniE ∆R∆uKu∆Cu∆M &&& 5.35

All the terms in the above equation are defined in Section 3.2. To implement the

standard viscous boundary, only the element damping matrix has to be modified.

The new element damping matrix is formulated as:

[ ] [ ] [ ]BEE CCC += 5.36

175

where [ ]BC is the contribution from the viscous boundary to the element

damping matrix. When the viscous stresses are applied continuously along the

boundary of the mesh, the dashpots have first to be converted to equivalent nodal

ones, before they can be assembled into the global damping matrix. In this case

the contribution of a single element side takes the form:

[ ] [ ] [ ][ ] SrfdNCNCSrf

C

T

B ∫= 5.37

where [ ]N contains the interpolation functions on the element side, the

evaluation of the constitutive viscous damping matrix [ ]CC is based on the

material properties of the elements adjacent to the boundary, according to the

following formula:

[ ]

=

P

S

CbV0

0αVρC 5.38

and Srf is the element side over which the dashpot acts. In nonlinear analysis the

modified Newton-Raphson method is employed to solve the finite element

equations (Chapter 2) and consequently the constitutive damping matrix is

updated every increment. The surface integral of Equation 5.37 must be first

transformed into one dimensional form in the natural coordinate system (see

Chapter 2):

[ ] [ ] [ ][ ] TdJBCBtC

1

1-

C

T

B ∫= 5.39

where t is the element thickness, J is the determinant of the Jacobian matrix

and [ ]B contains the derivatives of the interpolation functions on the element

side. For the 4-noded isoparametric element in Figure 5.15, assuming that the

dashpots are applied along the right hand side of the element, the shape functions

are linear and they are given by the following expressions:

176

T)(12

1N

T)(12

1N

0NN

3

2

41

+=

−=

==

5.40

The Jacobian determinant for each point on the element side is given by:

2

1

22

dT

dy

dT

dxJ

+

= 5.41

where x, y are the global coordinates.

Figure 5.15: 4 noded isoparametric element

5.4 Cone Boundary

5.4.1 Theory

In vibration analysis of foundations it is common practice to employ

simple physical models (e.g. lumped-parameter models, cone models) to

represent the soil medium. The so called “strength of material” theories are based

on physical assumptions regarding the deformation behaviour of the soil. Wolf

(1994) gives a thorough review of these methods, whereas Wolf and Deeks

(2004) present the “state of the art” on the cone models. Kellezi (1998, 2000)

suggested that the cone model can be employed in the FE analysis as a local

transmitting boundary. As the one-dimensional rod model can be considered as

177

the physical interpretation of the viscous boundary, the cone boundary can be

represented by a one-dimensional conical rod model.

dz

Q

dzz

QQ

∂

∂+

2

2

t

vdzA(z)

∂∂

ρ

x

z

dz

G

ρ

v

uA(z)

A(z+dz)

α

x

z

CHKH

z = r

(a) (c)

(b)

Figure 5.16: Semi infinite conical rod model

When a load is applied at the free surface of a half-space this leads to

stresses acting on an area that increases with depth. This cannot be properly

modelled with the semi-infinite rod model of Figure 5.14a. A better

approximation is a semi-infinite rod with variable cross section A(z) (Figure

5.16a). Considering the equilibrium of an infinitesimal element (Figure 5.16b)

for shear wave propagation:

0t

vdzA(z)ρdz

z

Q2

2

=∂∂

−∂∂

5.42

where the area A(z) can be expressed as:

22 zα)(tanπA(z) = 5.43

and the force-displacement relationship is specified as:

z

vA(z)GQ

∂∂

= 5.44

Substituting Equations 5.43 and 5.44 into 5.42, the equation of motion is

obtained:

178

0t

v

V

1

z

v

z

2

z

v2

2

2

S

2

2

=∂∂

−∂∂

+∂∂

5.45

The general solution of the equation of motion for waves propagating with a

spherical wavefront (e.g. body waves) is of the form:

z

t)Vg(z

z

t)Vf(zt)v(z, SS +

+−

= 5.46

Comparing Equation 5.46 with the general solution of the prismatic rod

(Equation 5.29), the only difference is that the amplitude decreases inversely to

the distance travelled. Consider again only the wave that encounters the artificial

boundary at z=r, the g wave can be ignored. The strain, the stress and the particle

velocity at any coordinate z are given by the following formulas:

t)V(zfz

1t)V(zf

z

1

z

vt)γ(z, SS2

−′+−−=∂∂

= 5.47

−′+−−== t)V(zfz

1t)V(zf

z

1GγGt)τ(z, SS2

5.48

t)V(zfz

V

t

vS

S −′−=∂∂

5.49

Combining the Equations 5.48 and 5.49 for z=r, one gets the boundary condition

for shear wave propagation:

t

t)(r,vVρt)v(r,

r

Gt)(r,τ S ∂

∂−−= 5.50

Likewise, considering the propagation of a wave that causes axial deformation

along the rod, a boundary condition for dilatational wave propagation can be

derived:

t

t)(r,uVρt)(r,u

r

Et)(r, P ∂

∂−−=σ 5.51

179

where E is the Young’s modulus. Furthermore, multiplying both terms of

Equations 5.50 and 5.51 by the surface area A, the Equations of equilibrium at

the artificial boundary are obtained:

0t)(r,t

vCt)(r,vKQ HH =

∂∂

++ 5.52

0t)(r,t

uCt)(r,uKN VV =

∂∂

++ 5.53

where Q, N are respectively the shear and the axial force and HK , VK , HC VC

are the frequency-independent stiffness and viscosity coefficients which are

defined as:

PV

SH

2

PV

2

SH

VρA(r)C

VρA(r)C

r

Vρ(r)AK

r

Vρ(r)AK

=

=

=

=

5.54

Looking closer to equilibrium Equations 5.52 and 5.53 it becomes evident that

the part of the conical rod that extends up to infinity can be replaced by a

mechanical system containing a spring and a dashpot. Figure 5.16c illustrates this

mechanical system for the case of one-dimensional shear wave propagation. It

should be noted that the viscosity coefficients HC , VC are identical to the ones

of the standard viscous boundary. Consequently, the cone boundary has the same

absorbing characteristics as the viscous boundary. Overall, in FE analysis, the

cone boundary can be described by two series of dashpots and springs oriented

normal and tangential to the boundary of the mesh. The greater advantage of the

cone boundary over the standard viscous boundary is that it approximates the

stiffness of the unbounded soil domain. Thus, it eliminates the rigid body

movement that occurs for low frequencies with the viscous boundary. However,

a drawback of the cone boundary is that the stiffness coefficients HK , VK

180

depend on the distance r of the boundary from the source of excitation.

Therefore, its use is restricted to problems with surface excitations (e.g. dynamic

pile loading, moving vehicles) where the distance of the boundary from the

source is known.

It has already been mentioned in Section 5.2.1 that in three-dimensional

space the wavefront of body waves at infinity is a large hemisphere with an area

2πr2(r→∞), whereas for Rayleigh waves it is a flat cylinder with a height

approximately of one Rayleigh wave length λR. Equation 5.46 is the general

solution of the equation of motion (Equation 5.45) for waves propagating with a

spherical wavefront (spherical waves) and thus it is not adequate for describing

the propagation of Rayleigh waves. Kellezi (1998, 2000) suggests a modification

of the previously derived cone boundary conditions (Equations 5.50 and 5.51) to

deal with Rayleigh waves. While it is not possible to find an exact expression for

waves travelling with a cylindrical wavefront (cylindrical waves), Whitham

(1974) showed that they can be closely approximated by:

z

Vt)g(z

z

Vt)f(zt)v(z,

++

−= 5.55

where V denotes the velocity of propagation. It should be noted that cylindrical

waves attenuate more slowly than spherical waves, as the wave amplitude

decreases at a rate of z

1 . Considering again only the wave that encounters the

artificial boundary at z=r, the g wave can be ignored. The strain, the stress and

the particle velocity at any coordinate z due to the shear component of a Rayleigh

wave that propagates with a velocity VR are given by the following formulas:

t)V(zfz

1t)V(zf

z2

1

z

vt)γ(z, RR

3−′+−−=

∂∂

= 5.56

−′+−−== t)V(zf

z

1t)V(zf

z2

1VργVρt)τ(z, RR

3

2

R

2

R 5.57

t)V(zfz

V

t

vR

R −′−=∂∂

5.58

181

where the Rayleigh wave velocity (VR) is slightly smaller than the shear wave

velocity and it can be approximated as (Achenbach, 1973):

ν)(1

ν)1.14(0.862VV SR +

+= 5.59

where ν is the Poisson’s ratio. Combining Equations 5.57 and 5.58 for z=r, one

obtains the boundary condition for the shear component of a Rayleigh wave:

t

t)(r,vVρt)v(r,

r2

Vρt)(r,τ R

2

R

∂∂

−−= 5.60

Similarly, considering the dilatational component of a Rayleigh wave, one

obtains the longitudinal boundary condition:

t

t)(r,uVSρt)(r,u

r

VSρt)(r,σ R

2

R

2

∂∂

−−= 5.61

where S is, as previously defined, the ratio of VP over VS. Multiplying both terms

of equations with the area A(z) and formulating the equilibrium equations at the

artificial boundary at z=r, the stiffness and viscosity coefficients can be obtained:

RV

RH

2

R

2

V

2

RH

VSρA(r)C

VρA(r)C

r2

VSρ(r)AK

r2

Vρ(r)AK

=

=

=

=

5.62

In the case of Rayleigh waves, the modification to the cone boundary involves

both the use of the Rayleigh wave velocity (VR), instead of the shear (VS) or

dilatational (VP) wave velocity, and halving the stiffness of the spring terms.

Kellezi (1998, 2000) suggests that the boundary conditions for Rayleigh waves

should be employed along the lateral boundaries of the FE mesh up to a depth of

one Rayleigh wave length (λR), whereas the boundary conditions for body waves

should be used for the rest of the mesh. Obviously for transient excitations an

182

approximation is needed, thus the calculation of the wave length λR is based on

the predominant frequency of the pulse.

Thus far the derived cone boundary conditions can deal with body and

Rayleigh waves for three-dimensional and axisymmetric conditions. An

adjustment is however needed for two-dimensional plane strain analyses. In this

case body waves propagate along a cylindrical wavefront. Therefore in plane

strain analysis Kellezi (1998, 2000) suggests the use of Equations 5.60 and 5.61

for body waves, simply by using VS instead of VR. Furthermore in plane strain

conditions Rayleigh waves propagate along an infinitely long rectangular surface

with height equal to λR. Hence it is suggested that the standard viscous boundary

with viscosity coefficients HC , VC given by Equation 5.62 can be employed up to

depth of λR to absorb Rayleigh waves in plane strain problems.

5.4.2 Implementation

The cone boundary is implemented into ICFEP for two-dimensional

axisymmetric and plane strain analyses. It was shown in the previous section that

the cone boundary consists of both dashpots and springs. Although the

implementation of the viscous part is essentially the same as that for the standard

viscous boundary (Section 5.3.1), it is repeated in this section for completeness.

In addition to modifications to the element damping matrix, the element stiffness

matrix of Equation 5.35 needs to be modified. The new matrices are formulated

as:

[ ] [ ] [ ]

[ ] [ ] [ ]BEE

BEE

KKK

CCC

+=

+= 5.63

where [ ]BC , [ ]BK are the contributions from the cone boundary to the element

damping and stiffness matrices respectively. Both the dashpots and the springs

are applied continuously along the boundary of the mesh and therefore they have

first to be converted to equivalent nodal ones before they can be assembled to the

global matrices. The contribution of a single element side takes the form:

183

[ ] [ ] [ ][ ]

[ ] [ ] [ ][ ] SrfdNKNK

SrfdNCNC

Srf

C

T

B

Srf

C

T

B

∫

∫

=

=

5.64

where the constitutive matrices [ ]BC , [ ]BK are defined in Table 5.1, depending

on the type of the analysis and the type of waves. As it was already mentioned in

the previous section, Kellezi (1998, 2000) suggests that the formulae for

Rayleigh wave absorption should be employed along the lateral boundaries of the

FE mesh up to a depth of one wavelength λR. Consequently, the formulas for

body waves are employed for the remaining length of the lateral boundaries and

the bottom boundary of the mesh. It should be also noted that the derivation of

the cone boundary in Section 5.4.1 is based on curved boundary geometries.

Figure 5.17 illustrates a homogeneous plane strain model with a rectangular

boundary. A source of vibration P is located on the free surface and the cone

boundary is employed along the bottom and the lateral boundaries. The wave

direction vector is denoted as r and n represents a vector normal to cone

boundary. According to the cone boundary derivation the vectors r, n coincide.

Theoretically, at any boundary node (e.g. node B) a cone can be assigned with its

axis perpendicular to the plane that is tangential to the boundary surface.

Although this is true for a curved boundary (see the dashed line), for a

rectangular boundary the two vectors (r, n) coincide only for surface nodes (i.e.

node A). To take into account non-coincidence between r, n the interaction

factor η is introduced in the stiffness term (Table 5.1). This factor is a function

of the scalar product of the vectors n and r and it is equal to one for a curved

boundary. According to Kellezi (1998) the interaction factor is very much

dependent on the location of the boundary from the excitation source and on the

mesh boundary size. Furthermore, Kellezi suggests that η is determined

experimentally by carrying out numerical tests and comparing them with

analytical solutions.

184

n

r

Source of vibrationP

B

Cone boundary

A

r, n

Figure 5.17: Application of the cone boundary on a homogeneous model with

rectangular boundary

Table 5.1: Summary of constitutive damping and stiffness matrices

Axisymmetric Analysis Plane Strain Analysis

Rayleigh Waves

[ ]

=

R

R

CV0

0SVρC

[ ]

=

2

R

2

R

2

CV0

0VS

rη2

ρK

[ ]

=

R

R

CV0

0SVρC

[ ] 0KC =

Body Waves

[ ]

=

S

P

CV0

0VρC

[ ]

=

2

S

2

P

CV0

0V

rη

ρK

[ ]

=

S

P

CV0

0VρC

[ ]

=

2

S

2

P

CV0

0V

rη2

ρK

Finally, the surface integrals of Equation 5.64 must be first transformed into one

dimensional form in the natural coordinate system:

185

[ ] [ ] [ ][ ]

[ ] [ ] [ ][ ] TdJNKNtK

TdJNCNtC

1

1-

C

T

B

1

1-

C

T

B

∫

∫

=

=

5.65

5.5 Verification and validation of absorbing boundary

conditions

5.5.1 Plane strain analysis

The numerical examples of Kellezi (1998, 2000) for plane strain analysis

were repeated to check the implementation of the transmitting boundaries.

Kellezi (1998, 2000) used three FE models, which are illustrated in Figure 5.18.

In all three models, along the left hand side boundary, the horizontal movement

was restricted for the case of vertical loading and the vertical movement was

restricted for the case of the horizontal loading. Furthermore, the soil surface is

always modelled as a stress-free boundary. The viscous and the cone boundaries

were applied along the bottom and the right hand boundary of the M10x10 and

M15x15 models, which have 100 and 225 elements respectively. To take into

account the interaction factor η of the cone boundary (Section 5.4.2), the

stiffness of the springs varies linearly along each boundary. The minimum and

the maximum values of the stiffness are based on the maximum and the

minimum radial distance respectively, of the boundary nodes from the source of

excitation. The M35x35 model is the extended mesh with 1225 elements and was

used as the reference solution. The analyses of the small models were also

repeated with Dirichlet (rigid) boundary conditions. Kellezi (1998, 2000)

considers also Neumann (free) boundary conditions for the small models,

however these boundary conditions lead to unacceptable high rigid body

movement of the mesh and thus they were not considered herein.

186

Figure 5.18: FE models for the extended and small meshes (from Kellezi, 2000)

A transient vertical line load of Delta type time function was chosen as

the input excitation (applied in the top left hand corner of the mesh, Figure 5.19).

By changing the duration of the pulse, different frequencies are generated. Three

pulses were considered with unit amplitude and different frequency content. The

objective of Kellezi (1998, 2000) was to check the performance of the

transmitting boundaries for a frequency range that is relevant to soil-structure

interaction problems. Regarding the time integration method, Kellezi (1998,

2000) uses the Wilson θ -method. The Wilson algorithm is an unconditionally

stable scheme for values of θ greater than 1.38. Kellezi (1998, 2000) employed a

θ =1.4 that introduces numerical damping into the scheme. Since the Wilson

method is not available in ICFEP, the generalized α-method is used instead.

Furthermore, in all analyses full Gaussian integration (3rd order) was employed.

187

Figure 5.19 : Delta function type loads and their Fourier transforms (from

Kellezi, 2000)

In the first investigation example, a Delta pulse with Tp=0.4s was applied.

The soil was assumed to be homogeneous and linear elastic with VS=224m/s,

ρ=2000 kg/m3 and ν=0.25. The finite element model comprised 8-noded

rectangular elements. The size of the element side is ∆x=∆z=λS/9=10m, where λS

is the wavelength of the SV-waves and it corresponds to the predominant period

(Tp=0.4s). In all analyses the wavelength calculation λS was based on the

predominant period Tp (λS = VS x Tp) and the time step was chosen equal to

Tp/20. Kellezi (1998, 2000) suggests that the time load function should be

discretized at least to 20 values. Along the lateral boundary of the mesh, the cone

boundary formulation consists only of dashpots for a depth equal to 80m. This

value is slightly lower than the Rayleigh wavelength (λR=0.92λS=82.2m).

In Figure 5.20 the displacement time histories for two surface points (E,

B, see Figure 5.18), as calculated by Kellezi (1998, 2000), are presented. The

same responses, as computed with ICFEP, are shown in Figure 5.21. The

comparison of these responses is perfect for the viscous boundary. Regarding the

cone boundary, some minor discrepancies can be noticed in the graph of

horizontal displacements.

Apart from verifying the implementation of the absorbing boundaries, it

is also necessary to validate their performance. In the plots of Figure 5.21 the

reflections from the rigid boundary are evident, whereas the transmitting

boundaries seem to absorb the outgoing waves fairly well. The reflections from

the rigid boundary show that the investigation time is long enough for a wave to

188

be reflected at the boundary of the M10x10 model and then to return to the

monitoring points (E, B). Therefore, the dimensions of the M10x10 model are

adequate to test the ability of the viscous and the cone boundaries to absorb

outgoing waves. Furthermore, for the case of the viscous boundary, the FE mesh

shows a rigid body movement in the vertical direction. This can be attributed to

the lack of stiffness of this boundary.

Figure 5.20: Comparison of the response at surface points (E, B) for vertical

excitation, Tp=0.4sec (from Kellezi, 2000)

189

Horizontal displacement (m)Point E

0 0.4 0.8 1.2 1.6

Time (s)

-4E-006

-2E-006

0

2E-006

4E-006

-5E-006

-3E-006

-1E-006

1E-006

3E-006

5E-006

0 0.4 0.8 1.2 1.6

Time (s)

-4E-006

-2E-006

0

2E-006

4E-006

-5E-006

-3E-006

-1E-006

1E-006

3E-006

5E-006

Horizontal displacement (m)Point B

0 0.4 0.8 1.2 1.6

Time (s)

-1E-005

-5E-006

0

5E-006

1E-005

1.5E-005

Vertical displacement (m)Point B

0 0.4 0.8 1.2 1.6

Time (s)

-1E-005

-5E-006

0

5E-006

1E-005

1.5E-005

M35x35 Extended

M10x10 Rigid BC

M10x10 Viscous BC

M10x10 Cone BC

Vertical displacement (m)Point E

Figure 5.21: Comparison of the response at surface points for vertical excitation

and Tp=0.4sec (ICFEP results)

Although the meshes were too coarse to determine the stresses accurately,

it is useful to compare the stress response from the analyses with absorbing

boundaries with that from the extended mesh. Two monitoring points close to the

free surface were selected: integration point F (x=15.0m, z=5.0m) and integration

point G (x=25.0m, z=5.0m). In Figure 5.22 horizontal and vertical stresses for

these two points, as computed with ICFEP, are presented. The results of the

extended mesh (M35x35 model) are used as a reference solution. Some mild

oscillations arise in the response of both absorbing boundaries, whereas the

reflections from the rigid boundary are again significant. In all cases, the cone

boundary seems to give slightly more accurate results than the viscous boundary.

190

0 0.4 0.8 1.2 1.6

Time (s)

-0.04

-0.02

0

0.02

0.04

Horizontal stress (kPa)Point G

0 0.4 0.8 1.2 1.6

Time (s)

-0.04

-0.02

0

0.02

0.04M35x35 Extended

M10x10 Rigid BC

M10x10 Viscous BC

M10x10 Cone BC

Horizontal stress (kPa)Point F

0 0.4 0.8 1.2 1.6

Time (s)

-0.008

-0.004

0

0.004

0.008

0.012

Vertical stress (kPa)Point F

0 0.4 0.8 1.2 1.6

Time (s)

-0.004

-0.002

0

0.002

0.004

0.006

Vertical stress (kPa)Point G


Tp=0.4sec (ICFEP results)

In the second investigation example, a Delta pulse with Tp=0.2s was

applied vertically on the free surface of all the three models (M10x10, M15x15

and M35x35). The size of the element side was ∆x=∆z=λS/4.5=10m. It should be

noted that the mesh discretization is very coarse. Hardy (2003) suggests as a rule

of thumb that the mesh should have at least 10 elements per wavelength to model

accurately the wave propagation. However, in order to be consistent with the

numerical example of Kellezi (1998, 2000) a coarse mesh was used. Time

integration was performed with ∆t=Tp/20=0.01s. Along the right hand side of the

mesh, the cone boundary consists only of dashpots for a depth of 40m

(λR=41.1m). The response is investigated at the nodes C and D which are located

inside the domain (Figure 5.18).

191

In Figure 5.23 the vertical displacement time histories of nodes C, D for

all three meshes, as calculated by Kellezi (1998, 2000), are presented. The same

responses, as computed by ICFEP, are given in Figure 5.24. Clearly, the results

of ICFEP compare very well with the results of Kellezi (1998, 2000) and show

the accuracy of the implementation.


excitation, Tp=0.2s (from Kellezi, 2000)

192

0 0.4 0.8 1.2 1.6

Time (s)

-5E-006

-2.5E-006

0

2.5E-006

5E-006

7.5E-006

1E-005M35x35 Extended

M10x10 Rigid BC

M10x10 Viscous BC

M10x10 Cone BC

Vertical displacement (m)Point C

0 0.4 0.8 1.2 1.6

Time (s)

-5E-006

-2.5E-006

0

2.5E-006

5E-006

7.5E-006

1E-005

M35x35 Extended

M15x15 Rigid BC

M15x15 Viscous BC

M15x15 Cone BC

Vertical displacement (m)Point C

0 0.4 0.8 1.2 1.6

Time (s)

-5E-006

-2.5E-006

0

2.5E-006

5E-006

7.5E-006

1E-005

Vertical displacement (m)Point D

0 0.4 0.8 1.2 1.6

Time (s)

-5E-006

-2.5E-006

0

2.5E-006

5E-006

7.5E-006

1E-005

Vertical displacement (m)Point D


excitation, Tp=0.2s (ICFEP results)

The behaviour of the cone boundary is very similar to the extended mesh

behaviour for both the small (M10x10) and the medium mesh (M15x15). The

viscous boundary shows again a rigid body movement especially for the small

mesh. Clearly, the reliability of the transmitting boundaries depends on the size

of the model. This is to be expected, as Castellani (1974) and Wolf (1988)

emphasized the dependence of the behaviour of the standard viscous boundary

on the distance from the source and the frequency content of the excitation. For

practical applications, Kellezi (1998) suggests that the boundary should not be

placed closer than (1.2-1.5)λS from the excitation source.

Furthermore, in Figure 5.25 the stresses recorded at the integration points

F (x=15.0m, z=5.0m) and G (x=25.0m, z=5.0m), as computed with ICFEP, are

193

presented. Both boundary conditions produce nearly the same response, which is

in a good agreement with the extended mesh results.

0 0.4 0.8 1.2 1.6

Time (s)

-0.04

-0.02

0

0.02

0.04M35x35 Extended

M10x10 Rigid BC

M10x10 Viscous BC

M10x10 Cone BC

Horizontal stress (kPa)Point F

0 0.4 0.8 1.2 1.6

Time (s)

-0.008

-0.004

0

0.004

0.008

0.012

Vertical stress (kPa)Point F

0 0.4 0.8 1.2 1.6

Time (s)

-0.008

-0.004

0

0.004

0.008

0.012

Vertical stress (kPa)Point G

0 0.4 0.8 1.2 1.6

Time (s)

-0.04

-0.02

0

0.02

0.04

Horizontal stress (kPa)Point G


Tp=0.2sec (ICFEP results)

The last example of Kellezi (1998, 2000) concerns a Delta pulse with

Tp=0.1s which was applied as a horizontal line load, at the top left hand corner of

the free surface of two models (M10x10 and M35x35). Along the left hand side

boundary any movement in the vertical direction was constrained. The soil was

again assumed to be homogeneous and linear elastic but with different material

properties, VS=200m/s, ρ=1800kg/m3 and ν=0.4. Since the wavelength of SV-

waves was only 20m, the mesh had to be denser. Thus, the size of the elements

was taken ∆x=∆z = λS/5=4m. The investigation time was taken as only 0.5sec to

avoid any reflection from the extended mesh. Time integration was performed

194

with ∆t=Tp/20=0.01s. Along the right hand side lateral boundary of the mesh, the

cone boundary consists only of dashpots for a depth equal to 20m (λR=18.83m).

For this example the Dirichlet boundary conditions were not considered, but the

behaviour of the unified viscous boundary of White et al (1977) (see Section

5.3.1) was examined instead.

Similar to the previous examples, in Figure 5.26 the horizontal

displacement time histories of the surface nodes A, E and B for the two meshes,

as calculated by Kellezi (1998, 2000), are presented. The same responses, as

computed by ICFEP, are given in Figure 5.27. Once more a comparison of the

two figures shows very good agreement for the viscous and the unified boundary,

but some minor discrepancies can be observed between the responses using the

cone boundary.


excitation and Tp = 0.1 sec. (from Kellezi, 2000)

195

0 0.1 0.2 0.3 0.4 0.5

Time (s)

-1E-005

-5E-006

0

5E-006

1E-005

1.5E-005

2E-005

M35x35 Extended

M10x10 Viscous BC

M10x10 Unified BC

M10x10 Cone BC

Horizontal displacement (m)Point A

0 0.1 0.2 0.3 0.4 0.5

Time (s)

-5E-006

0

5E-006

1E-005

1.5E-005

Horizontal displacement (m)Point E

0 0.1 0.2 0.3 0.4 0.5

Time (s)

-5E-006

0

5E-006

1E-005

-2.5E-006

2.5E-006

7.5E-006

Horizontal displacement (m)Point B


excitation and Tp = 0.1 sec. (ICFEP results)

It is interesting to note that the unified boundary is no better than the

viscous boundary and often gives less accurate results. The cone boundary

exhibits the same ability to absorb waves as the viscous boundary. This is not

surprising, since the absorption of waves is controlled by the dashpot coefficients

which are the same for both boundaries. The greater advantage of the cone

boundary is that it approximates the stiffness of the unbounded soil domain.

Thus, as noted above, it reduces the rigid body movement that occurs with the

viscous boundary.

In conclusion, the results of the plane strain analyses verify the

implementation of the transmitting boundaries. Comparing ICFEP results with

the results of Kellezi (1998, 2000) some minor discrepancies occurred for the

196

cone boundary. The cone boundary formulation for one part of the lateral

boundary consists only of dashpots (Section 5.4.2). Kellezi (1998, 2000) employs

these dashpots at a depth approximately equal to Rλ without determining the

exact depth value. Furthermore, it is stated in Kellezi (1998) that the interaction

factor is taken equal to 5.13.1 −=η , but its exact value is not specified. Hence,

the small differences between the ICFEP and the Kellezi (1998, 2000) results can

be attributed to the above-mentioned approximations.

5.5.2 Axisymmetric analysis

The accuracy of the standard viscous and the cone boundary for the case of

axisymmetric analysis was tested with an analytical solution for the problem of a

spherical cavity subjected to an internal blast. The analytical solution was

derived by Blake (1952) and it assumes that the material is elastic and isotropic.

The cavity was subjected to an impulsive pressure function P(t) that jumps from

zero to p0 at t=0 and then decays exponentially with time. So the pressure

function can be defined by:

0) (t 0P(t)

0)(tt)αexp(pP(t) 00

<=

≥−= 5.66

where 0α is a time constant. The axisymmetric nature of the problem eliminates

the SV-waves. Thus, the blast causes only radial displacement and no

circumferential one. The radial displacement is given as a function of both time

and radial distance from the source:

( )[ ]

( )[ ]

−−−+

−−−+−−

−++

−−−+−−

−+−=

−

−

−

0

0ξ1

0ξ

P

0

0

0ξ1

0ξ

P

ξ

0

P

0

2

oξ

2

0

0

0ξ1

0ξ02

0ξ

2

0

2

0r

ω

ααtanτωsinΛτ)αexp(

V

ω

ω

ααtanτωcosΛτ)αexp(

V

ατ)αexp(

V

α

ααωrρ

αp

ω

ααtanτωcosΛτ)αexp(τ)αexp(

ααωrρ

αpu

5.67

where α = radius of the cavity,

197

( )( )( )ν12ν1ρ

ν1EVP +−

−= = dilatational wave velocity

E = Young’s modulus, ν = Poison’s ratio, ρ = mass density,

ν)21(2

ν)(1K

−−

= ,

r = radial coordinate,

Kα2

Vα Pξ = = radiation damping constant,

PV

αrtτ

−−= ,

14KKα2

Vω P0 −= = natural frequency and

2

12

0

0ξ

ω

αα1Λ

−+=

The mesh that was used to analyze the problem is shown in Figure 5.28. The top

half of the cavity was modelled by using axial symmetry, while the bottom half

was included by utilizing the horizontal plane of symmetry. Therefore, only one

quarter of the problem had to be modelled. In order to investigate the effect of

the mesh size on the performance of the boundaries, two meshes were analysed,

one with D=10m and one with D=15m. Transmitting boundaries were applied

along the top and the right hand side of the mesh. Since there is not a free surface

in the model, Rayleigh waves cannot develop. Therefore, in the cone boundary

formulation only dashpots and springs for body waves were employed.

Furthermore, due to the axisymmetric nature of the solution a mesh with curved

boundaries would be more appropriate to model this problem. However, a

rectangular mesh was employed in order to subject the cone boundary to more

severe test conditions. Similar to the previous plane strain examples, the stiffness

of the springs varies linearly along each boundary, based on the maximum and

the minimum radial distance of the boundary nodes from the source of excitation.

Initially, the response to a Heaviside unit step function was considered, by setting

p0 = 1 and 0α = 0 in Equation 5.67. The impulse load was applied over one

198

increment of duration 0.002 seconds and then maintained for the rest of the

analysis. Time integration was performed with the generalized-α method

(ρ∞=0.8). The parameters chosen for the analysis are as follows:

E = 10.0MPa, ρ=1900 kg/m3

α = 2.0 m, ν = 0.25,

r = current radius, p0 = 1.0 kPa

y

x

P(t)

D

Figure 5.28: FE model for the cavity problem

In Figure 5.29 ICFEP results in terms of the displacements normal to the

cavity for nodal points A (r =3.4m) and B (r =9.1m) located at the bottom

boundary of the mesh, are compared against the closed form solution for the

small mesh (i.e. D=10m). The investigation time was long enough for the P-wave

to be reflected twice at the boundary. The finite element response shows some

mild oscillations. Presumably, these oscillations are due to the mesh

discretization not accurately modelling the response of the higher modes and

they are not associated with the transmitting boundaries. Therefore the numerical

damping was increased, by changing the parameters of the integration scheme

(ρ∞=0.4). The response for higher numerical damping is plotted in Figure 5.30.

199

The finite element response is now free from numerical oscillations. Thus, this

amount of numerical damping was used for all the following analyses of the

cavity.

0 0.2 0.4 0.6

Time (s)

0

5E-006

1E-005

1.5E-005

2E-005

2.5E-005

(b) Point B, r=9.1 m

0 0.2 0.4 0.6

Time (s)

0

2E-005

4E-005

6E-005

8E-005

No

rma

l D

isp

lac

em

en

t (m

)

Closed Form

M10x10 Viscous BC

M10x10 Cone BC

(a) Point A, r=3.4 m

Figure 5.29: Displacement normal to the cavity (ρ∞ =0.8, D=10m)

0 0.2 0.4 0.6

Time (s)

0

2E-005

4E-005

6E-005

8E-005

No

rma

l D

isp

lac

em

en

t (m

)

Closed Form

M10x10 Viscous BC

M10x10 Cone BC


0 0.2 0.4 0.6

Time (s)

0

5E-006

1E-005

1.5E-005

2E-005

2.5E-005



Regarding the performance of the transmitting boundaries, the cone boundary

compares quite well with the closed form solution for both nodal points. As

noted for the plane strain analyses, the viscous boundary absorbs the P-waves

similarly to the cone boundary. Furthermore, the inaccuracy of the standard

viscous boundary is mainly due to a rigid body movement. It is interesting to

note that the nodal point B has 2 -3 times higher rigid body movement than the

near field point A. On the other hand, the accuracy of cone boundary appears to

be only slightly worse for the far field node. It is also evident that the waves

200

impinging on both boundaries are not completely absorbed. Due to the simplicity

of the problem it is easy to identify the arrival time of the P-wave at the point A,

after it is first reflected at the boundary of the mesh:

t =S / Vp ≈ (14.6m) / (79.47m/s) ≈ 0.18s

where S (S=2xD-r-α ) is approximately the distance travelled by the P-wave.

Theoretically, the response should depend only on the radial distance from

source and not on the nodal coordinates (i.e. x, y in Figure 5.30). Since the

stiffness of the cone boundary was assumed to vary linearly along each

boundary, there is a need to check whether the cone boundary model satisfies this

theoretical requirement. Therefore, the response was monitored at point C

(x=2.69m, y=2.09m), which is located inside the domain at a radial distance r

=3.4m. Nodal points A, C are located at the same radial distance from the source,

but they have different nodal coordinates. Figure 5.31 compares the

displacements normal to the cavity for nodes A, C, as computed with the cone

boundary, with the closed form solution. Clearly, the response at points A, C is

very similar and it compares well with the closed form solution. The portion of

the P-wave, which was not absorbed from the cone boundary, arrives slightly

later at point C than at point A. This was to be expected as, due to the rectangular

mesh shape, point C is located at a larger radial distance from the absorbing

boundary than point A. Note that in Figure 5.31 a larger scale had to be

employed (comparing with the two previous figures) to identify differences

between the responses at the two nodal points (A,C).

201

0 0.2 0.4 0.6

Time (s)

4E-005

5E-005

6E-005

7E-005

No

rma

l D

isp

lac

em

en

t (m

)

Closed Form

Point A

Point C

Figure 5.31: Displacement normal to the cavity at points A, C (ρ∞ =0.4, D=10m)

Radial displacements from the larger mesh (D=15m) are plotted in Figure

5.32 for nodes (A, B), but the investigation time has been increased to 1sec.

Hence, the P-wave can be reflected two times at the boundary. In order to

highlight the necessity of the transmitting boundary nodes, the response for node

A is computed also with Dirichlet boundary conditions. Again the first arrival

time of the P-wave at t ≈ 0.31s can be identified at point A.

0 0.2 0.4 0.6 0.8 1

Time (s)

0

5E-006

1E-005

1.5E-005

2E-005

2.5E-005


0 0.2 0.4 0.6 0.8 1

Time (s)

0

2E-005

4E-005

6E-005

8E-005

No

rma

l D

isp

lac

em

en

t (m

)

Closed Form

M15x15 Viscous BC

M15x15 Cone BC

M15x15 Rigid BC



The FE results with transmitting boundaries compare almost perfectly with the

closed form solution for the near field node. The rigid body movement of the

viscous boundary has vanished. However, at the far field node some small rigid

body movement can be detected. As for the plane strain analyses, a comparison

202

of Figures 5.30 and 5.32 shows that the reliability of the transmitting boundaries

depends on the size of the model. In the cavity problem, it is more difficult to

determine the boundary location, since the wavelength of the excitation is not

known. Therefore, the distance to the boundary from the excitation has to be

determined experimentally, by carrying out analyses with different meshes.

Apart from the Heaviside unit step function, the three pressure functions

illustrated in Figure 5.33 were also considered. By varying the value of the time

constant 0α , the duration of loading changes. The shorter the time of action, the

higher is the frequency content of the motion.

0 0.2 0.4 0.6 0.8 1

Time (sec)

0

0.2

0.4

0.6

0.8

1

Pre

ss

ure

(k

Pa

) α0=1

α0=5

α0=50

Figure 5.33: Exponential decay functions

Figures 5.34-5.36 compare predicted displacements normal to the cavity at the

same two monitoring points (A and B) for the different pressure functions with

the closed form solution. In all three cases the large model (D=15m) was

employed. The predicted displacements at the near field node are almost identical

to the closed form solution, irrespective of the applied pressure function. On the

contrary, the predicted displacements at the far field node B highlight the effect

of the frequency content of the excitation on the performance of the transmitting

boundaries. Hence, for higher frequencies ( 0α =50) both boundaries perform very

well. In fact, the cone boundary shows no improvement compared to the viscous

203

boundary. Regarding the lower frequency plots ( 0α =1 and 5), the cone boundary

appears to be more accurate than the viscous boundary.

Overall the results of the axisymmetric analyses verify the

implementation of the transmitting boundaries. The ability of both boundaries to

absorb reflected waves was shown in all cases. The use of the cone boundary is

preferred to the viscous boundary for low frequencies.

0 0.2 0.4 0.6 0.8 1

Time (s)

0

2E-005

4E-005

6E-005

8E-005

No

rma

l D

isp

lac

em

en

t (m

)

Closed Form

M15x15 Viscous BC

M15x15 Cone BC


0 0.2 0.4 0.6 0.8 1

Time (s)

-1E-005

-5E-006

0

5E-006

1E-005

1.5E-005

2E-005

2.5E-005


Figure 5.34: Displacement normal to the cavity for 1α0 =

0 0.2 0.4 0.6 0.8 1

Time (s)

0

2E-005

4E-005

6E-005

8E-005

No

rma

l D

isp

lac

em

en

t (m

)

Closed Form

M15x15 Viscous BC

M15x15 Cone BC


0 0.2 0.4 0.6 0.8 1

Time (s)

-1E-005

-5E-006

0

5E-006

1E-005

1.5E-005

2E-005

2.5E-005



204

0 0.2 0.4 0.6 0.8 1

Time (s)

-1E-005

0

1E-005

2E-005

3E-005

4E-005

No

rma

l D

isp

lac

em

en

t (m

)

Closed Form

M15x15 Viscous BC

M15x15 Cone BC


0 0.2 0.4 0.6 0.8 1

Time (s)

-8E-006

-4E-006

0

4E-006

8E-006

1.2E-005



5.5.3 Rayleigh wave absorption

In the previous two sections, the investigation of the transmitting

boundaries was focused on the absorption of body waves. Apart from body

waves, as noted in Section 5.2.1, Rayleigh waves also play a very important role

in near-surface soil structure interaction problems. In order to simulate accurately

the far field response, numerical models should be able to absorb Rayleigh waves

at the lateral sides of the truncated domain. It was widely believed that the

standard viscous boundary cannot absorb Rayleigh waves as well as body waves.

However, numerical tests by Cohen and Jennings (1983) and Kellezi (1998)

showed that the viscous boundary can absorb Rayleigh waves to an acceptable

degree. Further numerical tests were carried out herein, to investigate the ability

of the standard viscous and the cone boundaries to absorb Rayleigh waves for

different frequencies and Poisson’s ratios.

As noted earlier, Rayleigh waves can be considered as a combination of P

and SV waves and they thus consist of both a horizontal and a vertical

component. A Rayleigh wave can be defined in terms of displacement as:

x)kt(ωcosz)(kgwx)kt(ωsinz)f(ku

−=−=

5.68

where u , w are the horizontal and the vertical displacement respectively,

z)(kg z),(kf are functions that vary with Poisson’s ratio, k is the wave number,

205

z is the depth from the surface, ω is the circular frequency and finally x is the

horizontal distance from the wave front. From Equation 5.68 it is evident that the

horizontal and vertical displacements are out of phase by 90°. Figure 5.37

illustrates the variation of displacement amplitudes gf, with depth for various

values of Poisson’s ratio. The horizontal axis is normalized by the

values (0)g(0),f at the surface and the vertical axis is normalized by the

wavelength Rλ . Both displacement components decay with depth. Generally,

Rayleigh waves influence a surface layer of a depth z= 1-1.5 Rλ .

-0.5 0 0.5 1

Normalized Amplitudes of Displacements

2.5

2

1.5

1

0.5

0

No

rma

lize

d D

ep

th (

z/λ

R)

Horizontal Component

Vertical Component

ν=0.25

ν=0.33

ν=0.4

ν=0.25

ν=0.33

ν=0.4

Figure 5.37: Horizontal and vertical amplitudes of Rayleigh waves

To investigate the ability of the transmitting boundaries to absorb

Rayleigh waves, the two plane-strain models shown in Figure 5.38 were

compared. In the first model (small mesh), the displacements in both directions

were fixed along the base of the mesh, and on the right hand side lateral

boundary transmitting boundaries were applied. In the second model (extended

mesh), the displacements were fixed in both directions along the base and the

right hand side lateral boundary. In all analyses the time step was chosen equal to

T/20 and the generalized-α method was employed for the time integration.

206

Absorbing Boundary

184 elements

Rayleigh WaveInput

Ax

z

184 elements

Rayleigh WaveInput

A

184 elements18 elements

x

z

Figure 5.38: FE models subjected to Rayleigh wave excitation

The first numerical test investigates the effectiveness of the small model

to simulate Rayleigh wave propagation for 4 different periods (To = 0.1s, 0.4s,

1.0s, and 2.0s). The load was applied by prescribing continuously the horizontal

and vertical displacements along the left hand side of the mesh, according to the

Rayleigh wave solution for v =0.25:

[ ]

[ ] t)(ωcosz)k0.3933(exp1.4679z)k0.8475(exp0.8475Av

t)(ωsinz)k0.3933(exp0.5773z)k0.8475(expAu

−+−−=

−−−= 5.69

where A is the amplitude taken equal to 1 and k (k = ω / VR) is the wave number.

The soil was assumed to be homogeneous and linear elastic with SV =100m/s,

ρ=1800 kg/m3 and ν=0.25. The size of the element side was determined by the

analysis with the smallest wavelength (To=0.1s) as ∆x=∆z= Rλ /9=1m.

Depending on the period of the input (To = 0.1s, 0.4s, 1.0s, 2.0s), the cone

boundary consists only of dashpots, along the right hand side of the mesh, up to

depths of 9, 37, 92 and 184m respectively. For the remaining part of the

boundary, the spring stiffness varies linearly according to the variation of the

radial distance from the excitation source (as explained in Sections 5.5.1 and

5.5.2).

It was chosen to monitor the displacements at the surface node A (Figure

5.38), as the Rayleigh waves develop only close to the free surface. Furthermore,

the stresses were recorded at the closest integration point Q (x=9.9m, z=0.11m)

207

to node A. Figures 5.39-5.42 compare the predicted displacements and stresses of

the three models (viscous boundary, cone boundary, extended mesh) for the 4

periods (To= 0.1s, 0.4s, 1.0s, 2.0s). Furthermore, Figure 5.39 also includes the

predicted response from an analysis with Dirichlet boundary conditions. The

width of the extended mesh is 184m. Therefore reflected waves have not arrived

at the monitoring point during the investigation time. On the other hand, the

width of the small mesh is only 18m, so the reflected waves, if they are not

absorbed, can reach the artificial boundary at least twice during the investigation

time.

0 0.2 0.4 0.6 0.8 1

Time (s)

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

Ho

riz

on

tal

Str

es

s (

kP

a)

0 0.2 0.4 0.6 0.8 1

Time (s)

-1

0

1

Ho

riz

on

tal

Dis

pla

ce

me

nt

(m)

Viscous BC

Cone BC

Extended Mesh

Rigid BC

0 0.2 0.4 0.6 0.8 1

Time (s)

-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

Ve

rtic

al

Str

es

s (

kP

a)

To=0.1s

0 0.2 0.4 0.6 0.8 1

Time (s)

-1

-0.5

0

0.5

1

Ve

rtic

al

Dis

pla

ce

me

nt

(m)

Figure 5.39: Response for Rayleigh wave loading of To=0.1s

208

0 0.4 0.8 1.2 1.6 2

Time (s)

-0.8

-0.4

0

0.4

0.8

Ho

riz

on

tal

Dis

pla

ce

me

nt

(m)

Viscous BC

Cone BC

Extended Mesh

To=0.4s

0 0.4 0.8 1.2 1.6 2

Time (s)

-0.8

-0.4

0

0.4

0.8

Ve

rtic

al

Dis

pla

ce

me

nt

(m)

0 0.4 0.8 1.2 1.6 2

Time (s)

-1200

-800

-400

0

400

800

1200

Ve

rtic

al

Str

es

s (

kP

a)

0 0.4 0.8 1.2 1.6 2

Time (s)

-600

-400

-200

0

200

400

600

Ho

riz

on

tal

Str

es

s (

kP

a)

Figure 5.40: Response for Rayleigh wave loading of To=0.4s.

For To=0.1s, the extended mesh results compare very well with the absorbing

boundary results, both in terms of displacement and stresses, whereas the rigid

boundary response exhibits substantial amplitude and period elongation errors.

For To=0.4s, the agreement is only slightly worse in terms of displacements and

vertical stresses. Surprisingly, the analyses with absorbing boundaries tend to

overestimate the results. Regarding the horizontal stresses, both the viscous and

the cone boundary alter severely the period of motion. The results of Figures

5.39 and 5.40 show that the transmitting boundaries can prevent the reflection of

Rayleigh waves for low periods. In contrast to above results, for To=1.0s, the use

of absorbing boundaries results in a slightly damped displacement response. Both

absorbing boundaries modify considerably the period of the horizontal stresses

and moderately the period of the vertical stresses. For To=2.0s, the amplitude

decay error is smaller in both the horizontal and the vertical displacements.

However, a moderate period elongation error is introduced in the horizontal

209

displacements for the analysis with absorbing boundaries. The period error in the

horizontal stresses is once again substantial.

0 1 2 3

Time (s)

-0.8

-0.4

0

0.4

0.8

Ho

riz

on

tal

Dis

pla

ce

me

nt

(m)

Viscous BC

Cone BC

Extended MeshTo=1.0s

0 1 2 3

Time (s)

-600

-400

-200

0

200

400

600

Ho

riz

on

tal

Str

es

s (

kP

a)

0 1 2 3

Time (s)

-1200

-800

-400

0

400

800

1200V

ert

ica

l S

tre

ss

(k

Pa

)

0 1 2 3

Time (s)

-0.8

-0.4

0

0.4

0.8

Ve

rtic

al

Dis

pla

ce

me

nt

(m)

Figure 5.41: Response for Rayleigh wave loading of To=1.0s.

A comparison of Figures 5.39-5.42 shows that the period of the input

wave, with absorbing boundaries, does not considerably affect the effectiveness

of the absorbing boundaries in terms of displacements. On the other hand, the

small models fail to predict accurately the stress response for all periods greater

than To=0.1s, due to substantial period elongation errors. It is important to note

that in all cases the viscous boundary appears to behave identically to the cone

boundary. This was to be expected, as for plane strain analysis up to a depth of

approximately Rλ the cone boundary consists only of dashpots (Table 5.1).

Furthermore the dashpots of the two boundaries have very similar viscosity

values, since VR is only slightly lower than VS.

210

0 1 2 3

Time (s)

-0.8

-0.4

0

0.4

0.8

Ve

rtic

al

Dis

pla

ce

me

nt

(m)

0 1 2 3

Time (s)

-500

-250

0

250

500

Ho

riz

on

tal

Str

es

s (

kP

a)

0 1 2 3

Time (s)

-0.8

-0.4

0

0.4

0.8

Ho

riz

on

tal

Dis

pla

ce

me

nt

(m)

Viscous BC

Cone BC

Extended Mesh

0 1 2 3

Time (s)

-400

-300

-200

-100

0

100

200

300

400

Ve

rtic

al

Str

es

s (

kP

a)

To=2.0s

Figure 5.42: Response for Rayleigh wave loading of To=2.0s

The second numerical test investigates the effectiveness of the small

models to simulate Rayleigh wave propagation for 3 different Poisson’s ratios

(ν= 0.25, 0.33 and 0.4). The period of the wave was taken equal to To=0.4s. The

Rayleigh wavelength does not significantly vary with the Poisson’s ratio.

Therefore, the depth up to which the cone boundary consists only of dashpots,

along the right hand side boundary, was kept the same for all the analyses, equal

to 37m. Depending on the value of the Poisson’s ratio, the prescribed

displacement time history along the left hand side of the mesh is defined by a

different Rayleigh wave solution. The Rayleigh wave solution for ν=0.25 was

defined previously by Equation 5.69. For ν= 0.33, 0.4 the Rayleigh wave

solutions are given by the following two expressions respectively:

211

[ ]


t)(ωsinz)k0.3624(exp05656z)k0.8829(expAu

−+−−=

−−−= 5.70

[ ]


t)(ωsinz)k0.3372(exp0.5590z)k0.9232(expAu

−+−−=

−−−= 5.71

Figures 5.43 and 5.44 plot the predicted displacements and stresses of the

three models (viscous boundary, cone boundary, extended mesh) for ν= 0.33, 0.4

respectively.

0 0.4 0.8 1.2 1.6 2

Time (s)

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

Ho

riz

on

tal

Dis

pla

ce

me

nt

(m)

Viscous BC

Cone BC

Extended Meshν=0.33

0 0.4 0.8 1.2 1.6 2

Time (s)

-0.8

-0.4

0

0.4

0.8

Ve

rtic

al

Dis

pla

ce

me

nt

(m)

0 0.4 0.8 1.2 1.6 2

Time (s)

-600

-400

-200

0

200

400

600

Ve

rtic

al

Str

es

s (

kP

a)

0 0.4 0.8 1.2 1.6 2

Time (s)

-600

-400

-200

0

200

400

600

Ho

riz

on

tal

Str

es

s (

kP

a)

Figure 5.43: Response for Rayleigh wave loading of To=0.4s and ν= 0.33

212

0 0.4 0.8 1.2 1.6 2

Time (s)

-0.8

-0.4

0

0.4

0.8

Ho

riz

on

tal

Dis

pla

ce

me

nt

(m)

Viscous BC

Cone BC

Extended Meshν=0.4

0 0.4 0.8 1.2 1.6 2

Time (s)

-0.8

-0.4

0

0.4

0.8

Ve

rtic

al

Dis

pla

ce

me

nt

(m)

0 0.4 0.8 1.2 1.6 2

Time (s)

-1200

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

1200

Ve

rtic

al

Str

es

s (

kP

a)

0 0.4 0.8 1.2 1.6 2

Time (s)

-1500

-1200

-900

-600

-300

0

300

600

900

1200

1500

Ho

riz

on

tal

Str

es

s (

kP

a)

Figure 5.44: Response for Rayleigh wave loading of To=0.4s and ν= 0.4

Comparing Figures 5.40, 5.43 and 5.44, it becomes clear that the value of

Poisson ratio does not significantly affect the performance of the absorbing

boundaries in terms of displacements. On the other hand, the ability of the

transmitting models to predict the correct stress response improves as the Poisson

ratio increases from to 0.25 to 0.33. For ν=0.4 the agreement between the small

models and the extended mesh is slightly worse than for ν=0.33, but it is still

acceptable.

As noted earlier, it was widely believed that the viscous boundary is

ineffective in the case of Rayleigh waves. On the other hand, the numerical tests

of Cohen and Jennings (1983) showed that the standard viscous boundary

absorbs Rayleigh waves in the same manner as it does for body waves. The

numerical tests presented herein showed that both the viscous and the cone

boundary can absorb Rayleigh waves to a certain extent. Models with both

213

boundaries predicted reasonably the displacement response for all wave periods

and Poisson’s ratios. However with respect to the stress response, the errors were

tolerable only for small periods or for values of Poisson’s ratio of 0.33 and 0.4.

The errors associated with the standard viscous boundary are of the same

magnitude as those introduced by the cone boundary.

5.5.4 Soil layer with vertically varying stiffness

All the validation examples presented so far concern homogeneous soil

profiles. However natural soil deposits rarely have a uniform distribution of shear

modulus. Usually, the soil stiffness increases with depth as a result of changes in

effective confining pressure. As one would expect, wave propagation in a

vertically heterogeneous profile is far more complicated than in a homogeneous

one.

As noted in Section 5.2.1, Love waves develop only when soil layering is

present. A vertically inhomogeneous half-space can be considered as a layered

medium composed of a series of infinitesimally small layers. In this case Vrettos

(2000) suggests that generalized SH surface waves develop. Generalized SH

waves are essentially very similar to Love waves and they also consist of shear

waves propagating in the out-of-plane direction. The generalized SH surface

waves, in contrast to the simple SH waves, are dispersive. Kramer (1996) defines

dispersion as a phenomenon in which waves of different frequency propagate at

different velocities. For example, the application of any of the delta pulses of

Figure 5.19 in a vertically inhomogeneous half-space, will result in SH surface

waves of various propagation velocities. However, each dashpot and each spring

are designed for only one set of wave velocities.

On the other hand, generalized SV/P waves develop in the plane direction

of a vertically inhomogeneous half-space. Vrettos (1990, 2000) considers

Rayleigh waves as a special case of generalized SV/P waves for a homogeneous

halfspace. It was shown in Figure 5.37 that the depth to which Rayleigh waves

cause considerable displacements increases with increasing wavelength. For a

vertically increasing velocity halfspace though, this means that Rayleigh waves

with longer wavelengths will propagate faster. According to Vrettos (1990, 2000)

214

these dispersive Rayleigh waves should be called generalized SV/P waves, as for

high frequency ranges they exhibit very distinctive characteristics from the

simple Rayleigh waves of a homogeneous halfspace.

The dispersive nature of surface SH and SV/P waves complicates

considerably the wave field of a vertically heterogeneous half-space. Since in

reality soil stiffness varies with depth, it is vital to investigate the performance of

the transmitting boundaries for such a scenario.

To check the behaviour of the transmitting boundaries the plane strain

model M10x10 of Figure 5.18 was employed. The in-plane nature of the problem

does not allow generalized SH waves to develop. As a reference solution an

extended mesh M55x55 with 3025 elements was used. In the vertical direction

the soil stiffness varied with depth according to the following expression:

(kPa)z1009150000Ev ×+= 5.72

In the horizontal direction the Young’s modulus was constant, equal to 150000.0

kPa. Furthermore, the soil was assumed to be linear elastic with ρ=1800 kg/m3

and ν=0.25. In the first investigation example, a Delta pulse of Tp=0.2s (Figure

5.19) was applied as a vertical point load on the free surface. The time step was

chosen equal to Tp/20. The cone boundary consists only of dashpots, along the

right hand side boundary of the mesh, for a depth of 40m. This depth value is

slightly greater than the Rayleigh wavelength (λR=38.7m) that corresponds to the

Young’s modulus value in the middle of the soil layer. For the remaining part of

the lateral boundary, the stiffness of the springs varies linearly according to the

variations of both the Young’s modulus and the radial distance from the source

of excitation (r). Along the bottom boundary the spring stiffness is based on the

maximum value of the Young’s modulus and it varies linearly according to

variation of the radial distance form the source (as explained in Sections 5.5.1

and 5.5.2). In all cases the viscosity values of the dashpots were calculated based

on the material properties of the elements adjacent to the boundary.

In Figure 5.45 the displacement time histories of nodes B, D of the four

analyses (extended mesh, rigid boundary, viscous boundary and cone boundary)

215

are presented. The inhomogeneity of the soil layer does not seem to significantly

affect the performance of the absorbing boundaries. Similar to the previous

numerical examples, the displacement response of the cone boundary is very

comparable to the one of the extended mesh. The viscous boundary shows again

a substantial rigid body movement in the vertical direction and a minor one in the

horizontal direction. Furthermore, the stress time histories were monitored in two

points close to the free surface: integration point P (x=8.5m, z=7.5m) and

integration point S (x=85.0m, z=5.0m). The plot of the stress response in Figure

5.46, demonstrates that both transmitting boundaries can prevent reflections of

stress waves.

0 0.4 0.8 1.2 1.6

Time (s)

-4E-006

-2E-006

0

2E-006

4E-006

Horizontal displacement (m)

0 0.4 0.8 1.2 1.6

Time (s)

-5E-006

-2.5E-006

0

2.5E-006

5E-006

Vertical displacement (m)

0 0.4 0.8 1.2 1.6

Time (s)

-4E-006

-2E-006

0

2E-006

4E-006


0 0.4 0.8 1.2 1.6

Time (s)

-5E-006

-2.5E-006

0

2.5E-006

5E-006


M55x55 Extended

M10x10 Rigid BC

M10x10 Viscous BC

M10x10 Cone BC

Point D Point D

Point BPoint B

Figure 5.45: Comparison of the displacement response for vertical excitation,

Tp=0.2sec

216

0 0.4 0.8 1.2 1.6

Time (s)

-0.02

0

0.02

Horizontal Stress (kPa)

0 0.4 0.8 1.2 1.6

Time (s)

-0.02

0

0.02

Vertical Stress (kPa)

M55x55 Extended

M10x10 Rigid BC

M10x10 Viscous BC

M10x10 Cone BC

0 0.4 0.8 1.2 1.6

Time (s)

-0.02

-0.01

0

0.01

0.02


0 0.4 0.8 1.2 1.6

Time (s)

-0.02

0

0.02


Point PPoint P

Point S Point S

Figure 5.46: Comparison of the stress response for vertical excitation, Tp=0.2sec

In the second investigation example the same Delta pulse of Tp=0.2s was

applied as in the last numerical model, but in the horizontal direction. The mesh

discretization was kept the same as in the previous computation, but the nodes at

the axis of symmetry were constrained in the vertical direction. The displacement

response of nodes B and D is given in Figure 5.47. Interestingly, the cone

boundary exhibits for first time a considerable rigid body movement. This

movement, although smaller than the one predicted by the viscous boundary, is

quite significant in the horizontal direction. The cone boundary formulation for

the lateral boundary consists only of dashpots for a depth approximately equal

to Rλ . However, the rigid body movement is an indication of lack of stiffness at

the lateral boundary. Therefore, the cone boundary was modified such that

springs extend up to the free surface. The computations with the modified cone

boundary compare quite well with the extended mesh results in Figure 5.47.

217

Finally, the stress time histories of integration points P, S are also given in Figure

5.48. All transmitting boundaries predict the stress response quite accurately.

The last two numerical tests showed that the viscous and the cone

boundary can absorb reflected waves in a vertically inhomogeneous half-space.

Models with both boundaries predicted reasonably the displacement and the

stress response for vertical excitation. In the case of horizontal excitation, the

cone boundary showed a rigid body movement that can be attributed to lack of

stiffness. The modified cone boundary has springs along the whole length of the

lateral boundary. In this way, the rigid body movement is minimized.

0 0.4 0.8 1.2 1.6

Time (s)

-4E-006

-2E-006

0

2E-006

4E-006


0 0.4 0.8 1.2 1.6

Time (s)

-5E-006

-2.5E-006

0

2.5E-006

5E-006


0 0.4 0.8 1.2 1.6

Time (s)

-4E-006

-2E-006

0

2E-006

4E-006


0 0.4 0.8 1.2 1.6

Time (s)

-5E-006

-2.5E-006

0

2.5E-006

5E-006


M55x55 Extended

M10x10 Rigid BC

M10x10 Viscous BC

M10x10 Cone BC

M10x10 Modified Cone BC

Point D Point D

Point BPoint B

Figure 5.47: Comparison of the displacement response for horizontal excitation,

Tp=0.2sec

218

0 0.4 0.8 1.2 1.6

Time (s)

-0.03

-0.02

-0.01

0

0.01

0.02

0.03


M55x55 Extended

M10x10 Rigid BC

M10x10 Viscous BC

M10x10 Cone BC

M10x10 Modified Cone BC

0 0.4 0.8 1.2 1.6

Time (s)

-0.03

-0.02

-0.01

0

0.01

0.02

0.03


0 0.4 0.8 1.2 1.6

Time (s)

-0.008

-0.004

0

0.004

0.008


0 0.4 0.8 1.2 1.6

Time (s)

-0.03

-0.02

-0.01

0

0.01

0.02

0.03


Point P Point P

Point S Point S

Figure 5.48: Comparison of the stress response for horizontal excitation,

Tp=0.2sec

5.5.5 Nonlinear waves

The derivation of both transmitting boundaries is based on linear elastic

wave propagation theory and therefore the numerical examples presented herein

concern linear elastic media. However, the absorbing boundaries can be applied

to nonlinear models, in cases where the region adjacent to the boundaries

remains linear elastic. This assumption is realistic for most problems of soil-

structure interaction, as the nonlinearities are mainly confined to the vicinity of

the structure.

On the other hand, it is suggested by Cohen and Jennings (1983) that in

some problems the nonlinearity of the wave motion at the far field cannot be

disregarded. In these cases slow-moving nonlinear waves have to be absorbed by

219

the transmitting boundaries. This can be achieved to a certain extent, as in ICFEP

the dashpot coefficients are updated at every increment. Furthermore, the

numerical examples presented herein, and in particular the one with a soil layer

with vertically varying stiffness, suggest that the transmitting boundaries are

relatively insensitive to dashpot and spring coefficients. Thus, it can be

postulated that the boundaries might absorb waves travelling at different speeds.

Unfortunately this hypothesis cannot be verified with a closed form solution,

since these solutions are restricted to the linear case. Another way to check the

above-mentioned hypothesis would be to undertake numerical tests that compare

results obtained from a small mesh having absorbing boundary conditions to

those generated from an extended mesh with rigid boundary conditions. In this

case, however, the area of nonlinear behaviour should be limited well within the

dimensions of the small mesh. If the area of nonlinear behaviour exceeds the

dimensions of the small mesh, the response of the extended mesh may include

waves resulting from reflections (due to stiffness degradation) in the part of the

medium that is not modelled by the small mesh. In this case the extended mesh

response is not an adequate reference solution.

5.6 Conclusions

Some of the most important boundary conditions for solving wave

propagation problems in unbounded domains were reviewed in this chapter. The

emphasis was put on the local boundaries that can be used in time-domain

formulations. The literature review showed that the standard viscous boundary of

Lysmer and Kuhlemeyer (1969) and the cone boundary of Kellezi (1998, 2000)

give acceptable accuracy for low computational cost. Therefore both boundaries

were implemented into ICFEP for two-dimensional plane strain and

axisymmetric analyses. The implementation of transmitting boundary conditions

for three-dimensional and Fourier series analyses was not considered in this

thesis, but does offer an obvious direction for research in the future.

The numerical examples of Kellezi (1998, 2000) and the closed form

solution of Blake (1952) verified the implementation of the standard viscous and

220

the cone boundary into ICFEP for plane strain and axisymmetric analysis

respectively. Furthermore, the validation exercises highlighted important features

of the transmitting boundaries. It was shown that the reliability of the

transmitting boundaries depends on the size of the model. The findings of this

chapter agree with the general suggestion of Kellezi (1998) that the absorbing

boundary should not be placed closer than (1.2-1.5) λS from the excitation

source.

It was also observed that the ability of both boundaries to absorb reflected

waves is very similar. This is not surprising since they have the same dashpot

coefficients. The greater advantage of the cone boundary is that it approximates

the stiffness of the unbounded soil domain. Thus, it eliminates the rigid body

movement that can occur for low frequencies with the viscous boundary.

In addition, the ability of the transmitting boundaries to absorb Rayleigh

waves was investigated. Models with both boundaries predicted reasonably the

displacement response for all wave periods and Poisson’s ratios. However with

respect to the stress response, the errors were tolerable only for small periods or

for values of Poisson’s ratio greater than 0.25. Regarding the Rayleigh wave

absorption, the cone boundary did not appear to be more accurate than the

standard viscous boundary.

Finally the performance of the boundaries was examined for the case of

plane strain analysis of a soil layer with vertically varying stiffness. The

dispersive nature of generalized SV/P waves complicates considerably the wave

field of a vertically heterogeneous half-space. However, both boundaries

predicted reasonably well the displacement and the stress response. This implies

that the boundaries might absorb waves travelling at different speeds. Thus, it

can be postulated that even slow-moving nonlinear waves can be absorbed by the

transmitting boundaries.

221

Chapter 6:

DOMAIN REDUCTION METHOD

6.1 Introduction

Many problems dealing with the dynamic response of soil-structure

interaction systems involve enormous computation domains. The domain

reduction method (DMR) is a two-step procedure that aims at reducing the

domain that has to be modelled numerically by a change of governing variables.

The seismic excitation is directly introduced into the computational domain and

an artificial boundary is needed only to absorb the scattered energy of the system.

The DRM is implemented into ICFEP based on the derivation of Bielak et al

(2003). The method was further extended and developed to deal with dynamic

coupled consolidation problems.

Prior to the final derivation of Bielak et al (2003) several basic

formulations of the DRM were presented in the literature. The first part of this

chapter reviews the evolution of the method over the last years. The theory and

the assumptions that form the basis of the DRM are also presented and their

limitations are discussed.

The second part of this chapter illustrates the development of the method

to deal with dynamic coupled consolidation problems and its implementation into

ICFEP.

The final part validates the implementation of the method and explores

the use of the DRM in conjunction with the cone boundary. The results using the

cone boundary are compared with those using the viscous boundary and with

those using an extended mesh.

222

6.2 Theoretical background to the method

6.2.1 Literature Review

The DRM has its basis in the work of Herrera and Bielak (1977) that

transformed the problem of seismic response of soil-structure interaction systems

to an equivalent continuum diffraction problem. They assumed that the seismic

ground motion is known in the absence of the structure in the elastic halfspace

SL RR U of Figure 6.1a. The boundary ϑRNL separates the sub-region RS, that

will eventually be occupied by a structure, from the surrounding region RL. The

seismic excitation is expressed in terms of free-field tractions (σo) and

displacements (uo) at the interface ϑRNL. In Figure 6.1b the region RN represents

the area occupied by a structure that can behave nonlinearly. The problem is to

determine the total response (ut, σ

t) in the regions RL and RN for a given free field

excitation (σo , u

o) applied at the interface RNL. This problem is solved as one of

diffraction by applying the continuity condition for tractions and displacements

at the interface ϑRNL and the stress free condition at the surface ϑRF. Although

this approach is not directly applicable to discretized domains, it is the first

attempt to introduce the source of excitation into the domain of computation.

ϑRNL

RS

RL

ϑRS ϑRLϑRL

ϑRNL

RL

ϑRF

ϑRF ϑRF

RN

Figure 6.1: (a) Model of soil in natural state and (b) model of soil-structure

system (after Bielak and Christiano 1984)

Bielak and Christiano (1984) developed two equivalent techniques to

solve the diffraction problem of Herrera and Bielak (1977) using the finite

element method. Their first technique employs a direct method of analysis, in

which the response of the structure and the soil medium is determined

simultaneously. The second technique uses a substructure method in which the

223

structure and the soil are analysed separately. In both techniques the seismic

excitation is expressed in terms of effective forces and it is applied at the soil-

structure interface ϑRNL. The effective forces are determined from the free-field

tractions (σo) and displacements (u

o) at the boundary ϑRNL in the unaltered soil

medium (Figure 6.1a). Cremonini et al (1988) implemented the above-mentioned

direct formulation and applied it to a simple two-dimensional soil-structure

system.

The method of Herrera and Bielak (1977) and Bielak and Christiano

(1984) was developed to solve building-soil-foundation interaction problems.

Loukakis and Bielak (1994) further modified the method to model the ground

motion of 2D sedimentary valleys in a half-space due to incident plane SV

waves. In their formulation the term “structure” refers generally to a region RN

with localized geological or structural features. The essential modification of the

latter authors is that the free-field displacements are stored on a one-element

thick band of elements adjacent to the interface between the soil and the

“structure”. Furthermore, there is no need to evaluate the free-field tractions,

since the calculation of the effective forces is based only on the free-field

displacements.

Bielak (2005) reviewing the evolution of the DRM, points out that

although the implemented equation of Loukakis and Bielak (1994) is correct, the

derivation of their formulation is deficient. The shortcoming of the derivation of

both Loukakis and Bielak (1994) and Bielak and Christiano (1984) is that they

introduced certain unknown forces that must be applied at the exterior boundary

to produce the free-field displacements within the unaltered soil medium. To

overcome this deficiency, Bielak et al (2003) suggested a two-step procedure. In

the first step the seismic excitation is included into the unaltered computational

domain through equivalent body forces. This allows the calculation and storage

of the free-field displacements and thus of the effective forces on a one-element

thick band of elements. In the second step the calculated effective forces are

applied on the “structure” of interest. In a companion paper Yoshimura et al

(2003) verified the DRM by comparing the FE results with those obtained by the

theoretical Green’s function method. Therefore, Bielak (2005) argues that the

224

latest derivation is the first rigorous presentation of the DRM. Yoshimura et al

(2003) showed also the applicability of the DRM in large scale three dimensional

domains containing the causative fault and strong geological and topographical

irregularities (e.g. sedimentary basins).

The formulation of the DRM presented in the following section is based

on the work of Bielak et al (2003). However there are 3 modifications herein:

(a) The addition of viscous damping terms in the equation of motion.

(b) The introduction of the algorithmic parameters αm, αf into the equation of

motion to perform the time integration with the Generalised-α scheme.

(c) The equations are expressed in incremental form.

6.2.2 Formulation of the Domain Reduction Method7

Figure 6.2a illustrates a large scale problem that includes the source of

the dynamic loading (e.g. fault slip) and an area with geotechnical structures or

localised geological features. The aim of finite element analysis is to examine the

response of geotechnical structures or localised geological features due to

dynamic loading. In practise the source of dynamic loading can be very far from

the area of interest. Thus, finite element modelling of such a problem can be

extremely expensive computationally. The DRM is a two-step procedure that

aims at reducing the domain that has to be modelled numerically by transferring

the excitation closer to the region of interest. A new equivalent excitation is to be

applied at the fictitious boundary Г of Figure 6.2a. The interface Г divides the

domain to the internal region Ω that contains the area of interest and the external

region Ω+ that includes the far field and the source of excitation. The outer

7 In this chapter to keep the notation consistent with the one of Bielak et al (2003) ∆u

represents the vector of incremental displacements previously denoted as ∆d , while P∆

expresses the incremental right hand side vector previously denoted as ∆R .

225

boundary of the truncated semi-infinite domain is denoted as Г+ and the dynamic

loading is expressed by the incremental nodal forces ∆Pe. The incremental nodal

displacements for the internal region Ω, the fictitious boundary Г and the

external region Ω+ are denoted as ∆ui, ∆ub and ∆ue respectively.

∆Pe

∆ub

∆u i

Fault∆ueΩ+

Ω

Γ+

Γ

∆Pe

∆ub0

∆u i

0

Fault∆ue

0

Ω+

Ω0

Γ+

Γ

(a) (b)

Figure 6.2: (a) Initial complete model (b) background model

(after Bielak et al 2003)

The equation of motion for the internal region Ω can be written as:

−=

−+

−+

−

b

f

b

i

Ω

bb

Ω

bi

Ω

ib

Ω

ii

f

b

i

Ω

bb

Ω

bi

Ω

ib

Ω

ii

f

b

i

Ω

bb

Ω

bi

Ω

ib

Ω

ii

m

∆P

0α1

∆u

∆u

KK

KKα1

u∆

u∆

CC

CCα1

u∆

u∆

MM

MMα(1

)()(

)()&

&

&&

&&

6.1

and the one for the external region Ω+ as :

−=

−+

−+

−

++

++

++

++

++

++

e

b

f

e

b

Ω

ee

Ω

eb

Ω

be

Ω

bbf

e

b

Ω

ee

Ω

eb

Ω

be

Ω

bbf

e

b

Ω

ee

Ω

eb

Ω

be

Ω

bbm

∆P

∆P-α1

∆u

∆u

KK

KKα1

u∆

u∆

CC

CCα1

u∆

u∆

MM

MMα(1

)()(

)()&

&

&&

&&

6.2

In the preceding equations, the matrices M, C and K denote mass, damping and

stiffness matrices and the subscripts i, e and b refer to nodes in the interior

domain, in the exterior domain and on the boundary Г respectively. The

superscripts Ω and Ω+ refer to the domain over which the matrices are defined,

αm, αf are algorithmic parameters and ∆Pb are the nodal forces transmitted by Ω+

to Ω.

226

The conventional governing equation for the total domain is obtained by

adding Equations 6.1 and 6.2:

−=

+−+

+−+

+−

++

++

++

++

++

++

e

f

e

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

f

e

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

f

e

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

m

∆P

0

0

)α(1

∆u

∆u

∆u

KK0

KKKK

0KK

)α(1

u∆

u∆

u∆

CC0

CCCC

0CC

)α(1

u∆

u∆

u∆

MM0

MMMM

0MM

)α(1

&

&

&

&&

&&

&&

6.3

An auxiliary model, called the background model, is employed to transfer

the dynamic excitation from the source (e.g. fault) to the fictitious boundary Г.

This model, illustrated in Figure 6.2b, comprises of the same external region, but

the internal area Ω of the actual model has been replaced by the simplified region

Ω0. Thus, the background model represents the free-field of the original model,

since any geotechnical structures or localised geological features with possibly

short wavelengths have been eliminated. The superscript 0 in Figure 6.2b refers

to free-field response. The equation of motion for the external region Ω+ of the

background model can be written as:

−=

−+

−+

−

++

++

++

++

++

++

e

0

b

f0

e

0

b

Ω

ee

Ω

eb

Ω

be

Ω

bbf

0

e

0

b

Ω

ee

Ω

eb

Ω

be

Ω

bbf0

e

0

b

Ω

ee

Ω

eb

Ω

be

Ω

bbm

∆P

∆P-)α(1

∆u

∆u

KK

KK)α(1

u∆

u∆

CC

CC)α(1

u∆

u∆

MM

MM)α(1

&

&

&&

&&

6.4

Given that there is no change in the external area, the mass, damping and

stiffness matrices in Equation 6.4 are the same as in Equation 6.2. Using equation

6.4 of the background model, the incremental forces ∆Pe can be expressed in

terms of the incremental free-field displacements, velocities and accelerations:

0

e

Ω

ee

0

b

Ω

eb

0

e

Ω

ee

0

b

Ω

eb

0

e

Ω

ee

0

b

Ω

eb

f

me ∆uK∆uKu∆Cu∆C)u∆Mu∆(M

α1

α1∆P

++++++

+++++

−−

= &&&&&&

6.5

227

Substituting Equation 6.5 into 6.3, the right hand side is expressed in terms of the

free-field response. However it includes the terms 0

e

Ω

ee u∆M &&+

, 0

e

Ω

ee u∆C &+

and

0

e

Ω

ee ∆uK+

that require the free field response to be stored throughout the domain

Ω+. Linear elastic soil behaviour is assumed in the external area Ω

+ and the total

response of the original model can be expressed as the sum of the free field

response and the relative response with respect to the background model:

e

0

ee

e

0

ee

e

0

ee

wuu

wuu

wuu

&&&&&&

&&&

∆+∆=∆

∆+∆=∆

∆+∆=∆

6.6

where ew∆ , ew&∆ and ew&&∆ are the relative incremental displacements, velocities

and accelerations respectively. Substituting Equations 6.5 and 6.6 into 6.3 and

rearranging to put all the free-field terms on the right hand side gives the

following expression:

++

−=

+−+

+−+

+−

+++

+++

++

++

++

++

++

++

0

b

Ω

ebf

0

b

Ω

ebf

0

b

Ω

ebm

0

e

Ω

bef

0

e

Ω

bef

0

e

Ω

bem

e

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

f

e

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

f

e

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

m

∆u)Kα-(1u∆)Cα-(1 u∆)Mα-(1

∆u)Kα-(1u∆)Cα-(1 - u∆)Mα-(1-

0

∆w

∆u

∆u

KK0

KKKK

0KK

)α(1

w∆

u∆

u∆

CC0

CCCC

0CC

)α(1

w∆

u∆

u∆

MM0

MMMM

0MM

)α(1

&&&

&&&

&

&

&

&&

&&

&&

6.7

In the right hand side of Equation 6.7 the dynamic excitation is now expressed by

the incremental effective forces ∆Peff :

++

−=

=+++

+++

0

b

Ω

ebf

0

b

Ω

ebf

0

b

Ω

ebm

0

e

Ω

bef

0

e

Ω

bef

0

e

Ω

bem

eff

ei

eff

b

eff

i

eff

∆u)Kα-(1u∆)Cα-(1 u∆)Mα-(1

∆u)Kα-(1u∆)Cα-(1 - u∆)Mα-(1-

0

∆P

∆P

∆P

∆P

&&&

&&& 6.8

228

Interestingly, the terms 0

e

Ω

ee u∆M &&+

, 0

e

Ω

ee u∆C &+

and 0

e

Ω

ee ∆uK+

that involve the response

of the whole domain Ω+ have cancelled out. The incremental effective forces

∆Peff depend only on the free-field displacements, velocities and accelerations of

a single layer of elements in Ω+ adjacent to the fictitious boundary Г. So with the

change of variables and the use of the background model, the dynamic excitation

is no longer expressed in terms of incremental nodal forces at the source ∆Pe, but

is “brought” close to the area of interest. It is important to emphasize that the

effective forces ∆Peff do not depend on the material properties of the internal

region. Therefore, any material in regions Ω0, Ω can be described by a nonlinear

constitutive model.

Figure 6.3 summarizes the two steps of the DRM. In step I (Figure 6.3a)

the simplified background model is analysed that includes the source of

excitation, but not the area of interest (that contains geotechnical structures or

localised geological features). The aim of the step I analysis is to calculate and

store the incremental displacements, velocities and accelerations of a single layer

of elements within the boundaries Гe and Г. Storing this free field response, one

has all the necessary information to calculate the effective forces ∆Peff which are

required for step II. As discussed in Chapter 3, the coarseness of the mesh is

dictated by the modelling of the shortest wavelength of the propagating wave.

Since structures or geological features of short wavelengths are eliminated from

the background model, the computation cost of the step I analysis is very small

compared to the cost of analysing the complete domain (Figure 6.2a). The second

step is performed on a reduced domain (Figure 6.3b) that comprises of the area

of interest Ω and of a small external region Ω+. Although the domains Ω

+ and

Ω+ have identical material properties, Bielak et al (2003) used distinct notation

for the external area of step II, just to highlight the reduction in size. The

effective nodal forces ∆Peff, from the incremental displacements, velocities and

accelerations computed in step I, are applied to the model of step II at the

elements located within the boundaries Гe and Г. The perturbation in the external

area Ω + is only outgoing and corresponds to the deviation of the area of interest

from the background model. Suitable absorbing boundary conditions should be

applied along the boundary Γ + to absorb any spurious reflections. It has been

229

shown in a previous chapter that the performance of the absorbing boundaries

significantly depends on the size of the mesh. Therefore, the dimensions of the

external area Ω + are determined only by the reliability of the absorbing

boundary conditions. The numerical examples of Yoshimura et al (2003) showed

that the ground motion in the external area Ω + is generally small compared to the

motion in the area Ω+ of the free-field model. Hence the absorbing boundaries

perform better when incorporated in the DRM, as they are required to absorb less

energy.

(a) (b)

∆Pe

∆u i

0

Fault

∆ue

0

Ω+

Ω0

Γ+

Γ

∆w e

Γ+

Ω∆Pe

eff

Γe

Ω+

∆P b

eff

Γ∆ub

∆u b

0Γe

∆u i

Figure 6.3: Summary of the two steps of DRM (after Bielak et al 2003)

Yoshimura et al (2003) showed the efficiency of the DRM when

analysing large scale 3D problems which include the causative fault. The DRM

can also be a useful tool when the modelling of the fault is not considered in the

analysis. Generally, in seismic soil-structure-interaction problems the excitation

is applied as an acceleration time history at the bottom of the mesh. The bottom

boundary coincides with the level of the bedrock and can be located at a great

depth. When analysing such a problem with DRM, the background model is

extended up to the bedrock, but the step II model can be significantly smaller.

Additional savings in the computational cost can be made by using a 1-D FE

column (extending up to the bedrock) as a background model in the step I

analysis to calculate the free-field response. Besides, in conventional analysis,

absorbing boundaries cannot be employed at the bottom of the mesh together

with the excitation. The great advantage of the DRM is that the excitation is

directly introduced into the computational domain leaving more flexibility in the

choice of appropriate boundary conditions.

Besides, Yoshimura et al (2003) underline that the use of the DRM is

beneficial in many practical cases in which parametric studies have to be

230

undertaken for the area of interest. In these cases the step I analysis is performed

just once and only the analysis of the reduced domain has to be repeated for the

various sets of parameters.

However, it should be noted that an obvious shortcoming of the method is

the storage cost associated with saving the free-field response in a layer of

elements of the background model in step I.

6.3 Formulation of the DRM for dynamic coupled

consolidation analysis

The formulation of the DRM presented so far has been restricted to deal

with either fully drained or undrained soil behaviour. In dynamic problems the

fully drained behaviour is rarely the case, while undrained behaviour is a valid

assumption for very quick loading of impermeable soils. When considering the

intermediate case, the pore fluid response is coupled to the response of the solid

phase. This results in both displacement and pore fluid pressure degrees of

freedom at element nodes. It was shown in Chapter 3 that the overall equilibrium

of the soil-fluid mixture is described by a system of two simultaneous equations

(see Equation 3.90).

∆Pe

∆u i

Fault∆ueΩ+

Γ+

Γ

∆Pe

∆pb

0

Fault∆ue

0

Ω+

Γ+

Γ

(a) (b)

∆pi

∆p b∆ub

∆pe

∆u i

0∆p i

0

∆ub

0

∆p e

0

Ω Ω0

Figure 6.4: (a) Initial complete model (b) background model for coupled

consolidation problems

The aim of this section is to illustrate the development of the DRM to

deal with dynamic coupled consolidation problems. Figure 6.4 illustrates the

semi infinite half-space described in the previous section. The additional

constituents are the incremental pore fluid pressures ∆pi, ∆pb and ∆pe which refer

to the internal (Ω), boundary (Г) and external (Ω+) area respectively.

231

The equation of motion for the internal area Ω can be written as:

−=

−+

−+

−

b

f

b

i

Ω

bb

Ω

bi

Ω

ib

Ω

ii

f

b

i

Ω

bb

Ω

bi

Ω

ib

Ω

ii

f

b

i

Ω

bb

Ω

bi

Ω

ib

Ω

ii

m

∆P

0)α(1

∆p

∆p

LL

LL)α(1

∆u

∆u

KK

KK)α(1

u∆

u∆

MM

MM)α(1

&&

&&

6.9

where L is the matrix coupling the solid and fluid phases. The inclusion of

damping was considered in the previous section but it is ignored in the present

formulation for brevity. The dynamic consolidation equation for the internal area

Ω can be written as:

=

+

+

−

+

− ∫∫

++

b

i

b

i

T

Ω

bb

Ω

bi

Ω

ib

Ω

ii

b

i

Ω

bb

Ω

bi

Ω

ib

Ω

ii

∆tt

t b

i

Ω

bb

Ω

bi

Ω

ib

Ω

ii

∆tt

t b

i

b

i

Ω

bb

Ω

bi

Ω

ib

Ω

ii

∆Q

∆Q∆t

∆u

∆u

LL

LL

∆p

∆p

SS

SSdt

u

u

GG

GGdt

n

n

p

p

ΦΦ

ΦΦ

&&

&&

6.10

All the terms of Equation 6.10 are defined in Section 3.4.1. The two integrals of

Equation 6.10 can be written as:

[ ]

[ ]∆tu∆βudtu

∆t∆pβpdtp

t

∆tt

t

t

∆tt

t

&&&&&& +=

+=

∫

∫

+

+

6.11

where β is an integration parameter to indicate how the pore pressure and the

acceleration vary during the increment and the subscript t refers the previous

increment. Substituting 6.11 into 6.10 and introducing the algorithmic parameters

mα , fα , one obtains the dynamic consolidation equation for the internal area Ω

in incremental form:

232

+

+

+

−=

−+

−−

+

−−

tb

i

Ω

bb

Ω

bi

Ω

ib

Ω

ii

tb

i

Ω

bb

Ω

bi

Ω

ib

Ω

ii

b

i

b

i

f

b

i

T

Ω

bb

Ω

bi

Ω

ib

Ω

ii

f

b

i

Ω

bb

Ω

bi

Ω

ib

Ω

ii

m

b

i

Ω

bb

Ω

bi

Ω

ib

Ω

ii

Ω

bb

Ω

bi

Ω

ib

Ω

ii

f

u

u

GG

GG

p

p

ΦΦ

ΦΦ

n

n

∆Q

∆Q∆t)α(1

∆u

∆u

LL

LL)α(1

u∆

u∆

GG

GGβ∆t)α(1

∆p

∆p

SS

SS

ΦΦ

ΦΦβ∆t)α(1

&&

&&

&&

&&

6.12

Similarly, the equation of motion and the dynamic consolidation equation in

incremental form for the external area Ω+ are given by:

−=

−+

−+

−

++

++

++

++

++

++

e

b

f

e

b

Ω

ee

Ω

eb

Ω

be

Ω

bbf

e

b

Ω

ee

Ω

eb

Ω

be

Ω

bbf

e

b

Ω

ee

Ω

eb

Ω

be

Ω

bbm

∆P

∆P-)α(1

∆p

∆p

LL

LL)α(1

∆u

∆u

KK

KK)α(1

u∆

u∆

MM

MM)α(1

&&

&&

6.13

+

+

+

−=

−+

−−

+

−−

++

++

++

++

++

++

++

++

++

++

++

++

+

te

b

Ω

ee

Ω

eb

Ω

be

Ω

bb

te

b

Ω

ee

Ω

eb

Ω

be

Ω

bb

e

b

e

b

f

e

b

T

Ω

ee

Ω

eb

Ω

be

Ω

bbf

e

b

Ω

ee

Ω

eb

Ω

be

Ω

bbm

e

b

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

ee

Ω

eb

Ω

be

Ω

bbf

u

u

GG

GG

p

p

ΦΦ

ΦΦ

n

n

∆Q

∆Q-∆t)α(1

∆u

∆u

LL

LL)α(1

u∆

u∆

GG

GGβ∆t)α(1

∆p

∆p

SS

SS

ΦΦ

ΦΦβ∆t)α(1

&&

&&

&&

&&

6.14

The equation of motion for the total domain is obtained by adding Equations 6.9

and 6.13, while the dynamic consolidation equation for the total domain is

obtained by adding Equations 6.12 and 6.14 :

233

−=

+−+

+−+

+−

++

++

++

++

++

++

e

f

e

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

f

e

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

f

e

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

m

∆P

0

0

)α(1

∆p

∆p

∆p

LL0

LLLL

0LL

)α(1

∆u

∆u

∆u

KK0

KKKK

0KK

)α(1

u∆

u∆

u∆

MM0

MMMM

0MM

)α(1

&&

&&

&&

6.15

++

++

++

−=

+−+

+−−

++

+−−

++

+++

++

+++

++

+++

++

+++

++

+++

++

+++

+

te

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

te

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

e

bb

i

e

i

f

e

b

i

T

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

f

e

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

m

e

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

f

u

u

u

GG0

GGGG

0GG

p

p

p

ΦΦ0

ΦΦΦΦ

0ΦΦ

n

nn

n

∆Q

0

∆Q

∆t)α(1

∆u

∆u

∆u

LL0

LLLL

0LL

)α(1

u∆

u∆

u∆

GG0

GGGG

0GG

β∆t)α(1

∆p

∆p

∆p

SS0

SSSS

0SS

ΦΦ0

ΦΦΦΦ

0ΦΦ

β∆t)α(1

&&

&&

&&

&&

&&

&&

6.16

In the same way as the original derivation in the previous section, the

background model of Figure 6.4b is employed to transfer the dynamic excitation

from the source (e.g. fault) to the fictitious boundary Г. To describe the free field

response, apart from incremental displacements ( 0

e∆u ), the incremental pore

pressures ( 0

e∆p ) are needed. The equation of motion for the external region Ω+ of

the background model can be written as:

234

−=

−+

−+

−

++

++

++

++

++

++

e

0

b

f0

e

0

b

Ω

ee

Ω

eb

Ω

be

Ω

bbf

0

e

0

b

Ω

ee

Ω

eb

Ω

be

Ω

bbf0

e

0

b

Ω

ee

Ω

eb

Ω

be

Ω

bbm

∆P

∆P-)α(1

∆p

∆p

LL

LL)α(1

∆u

∆u

KK

KK)α(1

u∆

u∆

MM

MM)α(1

&&

&&

6.17

The dynamic consolidation equation for the external region Ω+ of the background

model is given by:

+

+

+

−=

−+

−−

+

−−

++

++

++

++

++

++

++

++

++

++

++

++

+

t

0

e

0

b

Ω

ee

Ω

eb

Ω

be

Ω

bb

t

0

e

0

b

Ω

ee

Ω

eb

Ω

be

Ω

bb

e

b

e

b

f

0

e

0

b

T

Ω

ee

Ω

eb

Ω

be

Ω

bbf

0

e

0

b

Ω

ee

Ω

eb

Ω

be

Ω

bbm0

e

0

b

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

ee

Ω

eb

Ω

be

Ω

bbf

u

u

GG

GG

p

p

ΦΦ

ΦΦ

n

n

∆Q

∆Q-∆t)α(1

∆u

∆u

LL

LL)α(1

u∆

u∆

GG

GGβ∆t)α(1

∆p

∆p

SS

SS

ΦΦ

ΦΦβ∆t)α(1

&&

&&

&&

&&

6.18

Using Equation 6.17 of the background model the incremental forces ∆Pe can be

expressed in terms of the incremental free-field displacements, accelerations and

pore pressures:

0

e

Ω

ee

0

b

Ω

eb

0

e

Ω

ee

0

b

Ω

eb

0

e

Ω

ee

0

b

Ω

eb

f

me ∆pL∆pL∆uK∆uK)u∆Mu∆(M

α1

α1∆P

++++++

+++++

−−

= &&&&

6.19

Likewise, using Equation 6.18 of the background model, the incremental

discharges ∆Qe can be expressed in terms of both incremental and accumulated

free-field pore pressures and accelerations:

235

)uGu(G∆pΦpΦn-)∆uL∆u(L∆t

1

)u∆Gu∆(Gβ)α(1

)α(1)∆pS∆p(S

∆t

1)∆pΦ∆p(Φβ∆Q

0

teee

0

tbeb

0

teee

0

tbebe

0

eee

0

beb

0

eee

0

beb

f

m0

eee

0

beb

0

eee

0

bebe

&&&&

&&&&

++++++

++++++

+−−−++

+−−

−+++−=

6.20

Assuming linear elastic soil behaviour in the external area Ω+, the total response

of the original model can be expressed as the sum of the free field response and

the relative response with respect to the background model:

te

0

tete

e

0

ee

te

0

tete

e

0

ee

e

0

ee

ppp

p∆∆p∆p

wuu

w∆u∆u∆

∆w∆u∆u

+=

+=

+=

+=

+=

&&&&&&

&&&&&&

6.21

where the pore pressure terms denoted with represent the relative response.

Equations 6.22 and 6.23 are the desired governing equations for the reduced

model obtained by substituting Equations 6.19 and 6.21 into 6.15 and Equations

6.20 and 6.21 into 6.16 and rearranging to put all free-field terms on the right

hand side. The right hand side of the final set of equations expresses the

incremental effective forces. Evidently, the effective forces depend only on the

free-field displacements, accelerations and pore pressures of a single layer of

elements in Ω+ adjacent to the fictitious boundary Г of the background model.

The essential difference from the original formulation of the DRM is the need to

store in that layer of elements the incremental free-field pore pressures and the

accumulated free-field accelerations and pore pressures of the previous time step.

The introduction of pore pressure as an additional degree of freedom and the

need to satisfy a set of two simultaneous equations render dynamic analyses of

coupled consolidation problems to be computationally expensive. It is therefore

desirable to reduce as much as possible the domain that has to be modelled

numerically employing the DRM methodology. The implementation of the new

formulation of the DRM is assessed in the following section.

236

++

−=

+−+

+−+

+−

+++

+++

++

++

++

++

++

++

0

b

Ω

ebf

0

b

Ω

ebf

0

b

Ω

ebm

0

e

Ω

bef

0

e

Ω

bef

0

e

Ω

bem

e

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

f

e

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

f

e

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

m

∆p)Lα-(1∆u)Kα-(1 u∆)Mα-(1

∆p)Lα-(1 -∆u)Kα-(1 u∆)Mα-(1-

0

p∆

∆p

∆p

LL0

LLLL

0LL

)α(1

∆w

∆u

∆u

KK0

KKKK

0KK

)α(1

w∆

u∆

u∆

MM0

MMMM

0MM

)α(1

&&

&&

&&

&&

&&

6.22

+++−−

−+

−−+−−

+++−+

++

+

++

+

−=

+−+

+−−

++

+−−

++++++

+++++++

++

+++

++

+++

++

+++

++

+++

++

+++

++

+++

0

bbe

0

beb

0

tbeb

0

beb

f

m0

tbeb

0

beb

0

eeb

0

ebe

0

tebe

0

ebe

f

m0

tebe

0

ebebf

te

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

te

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

b

ii

f

e

b

i

T

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

f

e

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

m

e

b

i

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

Ω

ee

Ω

eb

Ω

be

Ω

bb

Ω

bb

Ω

bi

Ω

ib

Ω

ii

f

∆uL∆pSuG∆t)u∆(G)α(1

)α(1∆tβ)pΦ∆pΦ(β∆t-

∆uL∆pSuG∆t)u∆(G)α(1

)α(1∆tβ)pΦ∆pΦβ(n∆t

0

)α(1

w

u

u

GG0

GGGG

0GG

p

p

p

ΦΦ0

ΦΦΦΦ

0ΦΦ

0

n

n

0

0

∆Q

∆t)α(1

∆w

∆u

∆u

LL0

LLLL

0LL

)α(1

w∆

u∆

u∆

GG0

GGGG

0GG

β∆t)α(1

p∆

∆p

∆p

SS0

SSSS

0SS

ΦΦ0

ΦΦΦΦ

0ΦΦ

β∆t)α(1

&&&&

&&&&

&&

&&

&&

&&

&&

&&

6.23

237

6.4 Verification and validation of the DRM

An essential feature of the DRM is that the ground motion in the external

region Ω+ of the reduced model is only outgoing and corresponds to the

deviation of the local structures from the background model. A way to test the

DRM is to take the internal area Ω0 of the background model to be identical with

the internal area Ω of the reduced model. Since there is no deviation from the

background model, zero response should be calculated in the external Ω+ area of

the reduced model. Furthermore, the computed responses in steps I and II should

be identical for the internal areas Ω0, Ω. These features of the DRM were used to

check the implementation of the method in ICFEP for both linear and nonlinear

problems. For this purpose two dimensional dynamic coupled consolidation

analyses of a cut and cover tunnel were undertaken.

6.4.1 Verification of the DRM formulation for dynamic coupled consolidation

linear analysis

Figure 6.5 illustrates the plane strain arrangement of the background (a)

and the reduced model (b) employed for the verification. Typically, when

utilising the DRM the background model represents the free-field response and it

does not contain any structures. However, the purpose of this example is to test

the implementation of the DRM and thus both models (a,b) comprise of the same

internal area. The tunnel is 30m wide and 14m deep and its top slab is 1m below

the ground level. The soil is assumed to be fully saturated and the water table is

at the ground level. To test the ability of the program to deal with external areas

Ω+, Ω

+ of different sizes, the area Ω

+ of the background model is considerably

larger. The background model consists of 4362 8-noded elements, whereas the

reduced model consists of 3302 8-noded elements. In the first numerical test the

soil is modelled as linear elastic material with the following material properties:

E = Young’s modulus = 20.6505×104 kPa

v = Poisson’s ratio = 0.25

K0 = earth pressure coefficient at rest = 1.0

γ = unit weight = 19 kN/m3

238

k = permeability = 0.01m/s

Kf = water compressibility = 18.05×105 kPa

The wall of the tunnel is modelled with 3-noded beam elements and it is assumed

to behave in a linear elastic manner. The material properties chosen for the beam

elements were as follows:

E = Young’s modulus = 30×106 kPa

v = Poisson’s ratio = 0.2

γ = unit weight = 24 kN/m3

t = wall thickness = 1m

Prior to the dynamic analysis, a static analysis was undertaken to model

the construction sequence. During the static analysis vertical displacements were

restricted along the outer boundaries +Γ , +Γ of the background and the reduced

model respectively, while horizontal displacements were restricted along the

bottom mesh boundary in both models. Initially, the side walls were constructed

as wished in place and the excavation was then performed in ten stages. During

the excavation (i.e. of the elements originally occupying the tunnel), the walls

were supported by restricting their horizontal movement. Subsequently, the

bottom and the top slabs were constructed, the horizontal support of the wall was

removed (prescribed displacements were released) and the area above the top

slab was backfilled with soil. During the static analysis the soil is assumed to

behave in a drained manner. The modelling of the construction sequence tests the

ability of the program to correctly employ the DRM in a latter stage of the

analysis.

239

60m

120m

252m

126m

A

B

ΓeΓ

Ω0

Ω+

Γ +

z

x

(a)

ΝΒ

120m

172m

86m

60m

A

B

D

C

Γe

Ω

Ω+

Γ +

(b)

Γ

ΝΒ

Figure 6.5: Background model (a) and reduced model (b) of the verification

example

During the step I of the dynamic analysis, vertical displacements were

restricted along the boundary +Γ and an acceleration time history was applied in

the horizontal direction along the bottom part of this boundary. In the reduced

model, vertical displacements were restricted along the boundary +Γ and

horizontal displacements were restricted along its bottom part. Since the

perturbation in the external area +Ω is expected to be zero, the choice of the

boundary conditions on the outer boundary +Γ should not affect the solution.

240

However, it was felt that the use of absorbing boundaries could suppress any

nonzero values of the response, whereas the restraint of displacements would

amplify them. The generalized–α scheme with its standard parameters (δ=0.6,

α=0.3025, αf=0.45 αm=0.35) was used for these analyses to give an

unconditionally stable time scheme with some added numerical damping. The

north-south component of the Veliki acceleration time history, recorded during

the 1979 Montenegro earthquake, was the input motion for the step I analysis.

The background model was subjected to the first 20 seconds of the filtered

recording (Figure 6.6) with a time step of 0.01 sec. No material damping was

considered.

0 4 8 12 16 20

Time (sec)

-3

-2

-1

0

1

2

3

Ac

ce

lera

tio

n (

m/s

ec

2)

Figure 6.6: Filtered Montenegro 1979 earthquake record

Figure 6.7 shows snapshots of the deformed mesh of both the background

and the reduced model at t=4.5sec. The response due to the static analysis has

been subtracted in all cases and the time is measured from the onset of the

dynamic excitation. In the reduced model, displacements in the interior region

are total, whereas displacements in the exterior region are relative to those

corresponding to the background model. Since the internal area of the

background model is identical to the one of the reduced model the relative

response is expected to be zero. Indeed, the perturbation in Figure 6.7b is

restricted in the internal region only, while in the case of background model the

whole mesh is deformed (Figure 6.7a).

241

Furthermore, displacements and accelerations were monitored at node A

(x=0.0m, z=-20.0m) of the internal area and at node B (x=0.0m, z=-66.0m) of the

external area (Figure 6.5). Pore pressures were recorded at the closest integration

points to nodes A, B namely points E (x=0.7887, z=-19.12) and F (x=0.7887, z=-

65.12) respectively.

(a)

(b)

Figure 6.7: Deformed meshes of the background model (a) and the reduced

model (b)

242

0 10 20

Time (s)

-0.02

-0.01

0

0.01

0.02


Background model

Reduced model

0 10 20

Time (s)

-0.02

-0.01

0

0.01

0.02


Point A(a) Point B(b)

Figure 6.8: Comparison of horizontal displacements of nodes A, B for linear

analyses

0 10 20

Time (s)

-4

-2

0

2

4

Horizontal acceleration (m/sec2)

Background model

Reduced model

0 10 20

Time (s)

-4

-2

0

2

4



Figure 6.9: Comparison of horizontal accelerations of nodes A, B for linear

analyses

Figures 6.8-6.10 compare the horizontal displacement, horizontal

acceleration time histories of nodes A and B and the pore pressure time histories

of integration points E, F of the background model with the ones of the reduced

model. Clearly the curves of the background and the reduced model are

indistinguishable for node A of the internal area. Furthermore, the response is

barely visible at node B and integration of the reduced model, whereas

significant values of displacement and acceleration are recorded at node B of the

background model. In a similar way, almost zero values of pore pressure are

computed at integration point F of the reduced model, while some pore pressure

variation is recorded at integration point F of the background model. The above-

243

mentioned observations verify the implementation of the DRM in ICFEP for

linear coupled consolidation problems.

0 10 20

Time (s)

-8

-4

0

4

8

Pore pressure (kPa)

Background model

Reduced model

0 10 20

Time (s)

-0.4

-0.2

0

0.2

0.4

Pore pressure (kPa)

IntegrationPoint E

(a) IntegrationPoint F

(b)

Figure 6.10: Comparison of pore pressures of integration points E, F for linear

analyses

6.4.2 Verification of the DRM formulation for dynamic coupled consolidation

nonlinear analysis

As noted in Section 6.2.2, the formulation of the DRM is such that linear

elastic soil behaviour is dictated in the external areas Ω+, Ω

+. Conversely,

materials in regions Ω0, Ω can be described by nonlinear constitutive models. To

verify this, the numerical test of the previous section was repeated assigning a

nonlinear constitutive law to the area which is enclosed by the interface NB (see

Figure 6.5) in both the background and the reduced model. Hence, the same

arrangement for the background and reduced models was used as before, except

that the small strain stiffness constitutive model of Jardine et al (1986) was

introduced to describe the nonlinear soil behaviour. The small-strain stiffness

model is discussed in more detail in Section 7.7.5 (see Equations 7.20-7.23),

while the assumed parameters for this model are listed in Table 6.1.

244

Table 6.1: Parameters used in the small strain stiffness model

G1 G2 G3

(%) α γ

Ed(min)

(%)

Ed(max)

(%)

Gmin

(MPa)

705.0 605.0 1.5x10-5 1.105 0.82 0.001 1.0 23.0

K1 K2 K3

(%) δ µ

εv(min)

(%)

εv(min)

(%)

Kmin

(MPa)

400.0 400.0 6.0x10-4 0.999 0.74 0.001 1.0 33.0

0 10 20

Time (s)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06


Background model

Reduced model

0 10 20

Time (s)

-0.04

-0.02

0

0.02

0.04



Figure 6.11: Comparison of horizontal displacements of nodes A, B for nonlinear

analyses

In a similar fashion to the linear analyses of the previous section, Figures

6.11-6.13 compare the horizontal displacement, horizontal acceleration time

histories of nodes A and B and the pore pressure time histories of integration

points E, F of the background model with the ones of the reduced model. In all

cases the response due to the static analysis has been subtracted and the time is

measured from the onset of the dynamic excitation. Once more the curves of the

background and the reduced model are indistinguishable for node A of the

internal area. It is interesting to note that the nonlinear response, in terms of

displacement and acceleration, appears amplified with respect to the

corresponding linear one (see Figures 6.8 and 6.9). This is not surprising, as

245

during the nonlinear analysis the soil stiffness reduces in the area enclosed by the

interface NB, while no damping is introduced.

0 10 20

Time (s)

-15

-10

-5

0

5

10

15


Background model

Reduced model

0 10 20

Time (s)

-8

-4

0

4

8



Figure 6.12: Comparison of horizontal accelerations of nodes A, B for nonlinear

analyses

Moreover, the displacement response in Figure 6.11b is barely visible at node B

of the reduced model, whereas significant values are recorded at node B of the

background model. It should be noted that in the reduced model some negligible

values of relative (with respect to the background model) acceleration and pore

pressure were recorded at node B and integration point F respectively. This

indicates that the iterative process of the nonlinear analysis introduces some

small error in the DRM. However, the response, in terms of acceleration and pore

pressure, recorded in the external area of the reduced model is much smaller than

the corresponding one of the background model. Thus, it can be concluded that

in the case of nonlinear analysis the introduced error in the DRM is insignificant.

Hence the fact that the steps I and II of the DRM give identical results in

the internal area and that negligible response is computed in the external area of

the reduced model, provides a useful numerical check both in the linear and

nonlinear regime.

246

0 10 20

Time (s)

-40

-20

0

20

Pore pressure (kPa)

Background model

Reduced model

0 10 20

Time (s)

-40

-20

0

20

40

Pore pressure (kPa)

IntegrationPoint E

(a) IntegrationPoint F

(b)

Figure 6.13: Comparison of pore pressures of integration points E, F for

nonlinear analyses

6.5 Performance of absorbing boundary conditions in the

DRM

6.5.1 Application of the cone boundary in the step II model of the DRM

It was shown in Section 5.4.1 that the mechanical equivalent of the cone

boundary is a system consisting of a spring and a dashpot, whereas the viscous

boundary is described only by a system of dashpots. Thanks to the spring term,

the cone boundary approximates the stiffness of the unbounded soil domain and

it eliminates the permanent movement that occurs with the viscous boundary at

low frequencies. The limitation of the cone boundary is that the spring stiffness

term is inversely proportional to the distance (r) of the boundary from the source

of excitation. This decrease of stiffness with distance is of geometrical origin and

is consistent with radiation damping. Body waves travelling outward with a

hemispherical wavefront encounter an increasingly larger volume of material.

Consequently, the wave amplitude decreases at a rate of 1/r due to spreading of

the energy over a greater volume of material.

The cone boundary is mainly employed in problems with surface

excitations (e.g. dynamic pile loading, moving vehicles) where the distance of a

boundary from the source is known. In seismic soil-structure interaction

247

problems the distance from the seismic source (fault) is difficult to be accurately

determined. Furthermore, even in cases that the location of the fault is known,

modelling of the fault is rarely undertaken because it results in excessively large

computational domains. The seismic excitation is typically applied as an

acceleration time history along the bottom mesh boundary. Thus, no absorbing

boundary condition can be specified at the bottom boundary together with the

excitation. In addition the cone boundary cannot be used at the lateral boundaries

of the mesh since the concept of geometrical spreading towards infinity does not

apply in this case.

The aim of this section is to investigate the possibility of employing the

cone boundary in the reduced model of the DRM. It was noted in Section 6.2.2

that the absorbing boundaries perform better when they are incorporated in the

DRM, since they are required to absorb less energy. Waves reaching the outer

boundary +Γ are only due to the deviation of the area of interest from the

background model. In cases where the only additional element of the reduced

domain is a structure (e.g. tunnel, retaining wall), the perturbation in +Ω is only

due to waves reflecting from this structure. Therefore this structure can be

considered as the “excitation source” for the external area +Ω . The idea is to

calculate the stiffness terms of the cone boundary based on the distance of the

structure from the boundary. Since the structure is not a point source and has

finite dimensions, the theoretical value of the distance r for each boundary node

needs to be approximated. Figure 6.14 illustrates a step II model of the DRM

containing a structure ABCD. If the structure is considered as the “excitation

source” for the area +Ω , the r of each boundary node can be approximated as the

distance to the closest point of the structure. For example, along the boundary

A1A2 r is the distance from the point A, along the boundary A2B1 r is constant

equal to AA2 and along the boundary B1B3 r is the distance from the point B. In

a similar way, the distance r can be calculated for the rest of the boundary nodes

of +Γ . The numerical results of Chapter 5 and the findings of Kellezi (1998)

agree that the cone boundary is relatively insensitive to the dashpot and spring

coefficients. Therefore, one would expect that the preceding approximation of r

248

is sufficient. The performance of the cone boundary, when employed in the step

II model of DRM, is examined in the following section.

A

B

D

C

A1

A2

B1

B3 C1 C2

C3

D1

D2

B2

Γe

Γ

Γ+ Ω+

Ω

Figure 6.14: Step II model of the DRM containing a structure ABCD.

6.5.2 Numerical results and discussions

A 1-D model of a soil column 4m wide and 612m deep was considered

for the background analysis. The column was assigned the linear material

properties of the previous example (Section 6.4.1) and consists of 408 (2x204) 8-

noded elements. Vertical displacements were restricted along the bottom and the

side boundaries. The background analysis was repeated for three acceleration

sinusoidal pulses of periods To =1sec, 2sec and 4sec. The excitations with

amplitude of 1m/sec2 were applied in the horizontal direction along the bottom

boundary. The investigation time is only 6sec and the 1-D extended mesh is

taken long enough to prevent reflections from the boundary to the area of

interest. In all analyses the time step was chosen equal to To/20 and the

generalized–α scheme with standard parameters (δ=0.6, α=0.3025, αf = 0.45

αm=0.35) was employed for the time integration.

To illustrate the applicability of the cone boundary to the DRM, the

reduced model of Figure 6.5b was used for the step II calculations. Prior to the

dynamic analyses, the construction sequence was simulated to establish the

249

initial stress state as described in Section 6.4.1. In all analyses the soil was

assumed to behave in a drained manner. The cone boundary was applied along

the boundary +Γ and the stiffness of the springs was calculated according to the

procedure described in the previous section. Furthermore all step II analyses

were repeated with the viscous boundary.

To verify the applicability of the cone boundary, the step II analyses were

also repeated with an extended mesh 933m wide and 466m deep. This model is

taken big enough to prevent reflections from the boundary to the area of interest.

Along the boundary +Γ of the extended mesh displacements were restricted in

both directions. The verification model consists of 29522 8-noded elements and

has the same element dimensions as the reduced model of Figure 6.5b.

During step I analyses the effective forces were calculated at various

depths of the 1-D model. These forces were subsequently applied to the

corresponding nodes of the step II models which are located between the

boundaries eΓ and Г. It should also be highlighted that this verification example

subjects the absorbing boundaries to severe test conditions. It was shown in

Chapter 5 that the performance of both the cone and the viscous boundaries is

more accurate for high frequencies. Therefore the selected low frequency

excitation pulses challenge the limits of their capabilities. Furthermore, Kellezi

(1998) suggests that the absorbing boundary should not be placed closer than

(1.2-1.5)λS from the excitation source. Considering the suggestion of Kellezi

(1998), it becomes clear that the absorbing boundaries have been placed very

close (0.2 λS -0.7 λS) to the tunnel which in this case is the assumed “source”.

The response was monitored at the surface node C (50.0, 0.0) and at node

D (60.0, -60.0) which is located inside the field (Figure 6.5b). It should be noted

that these nodes lie in the internal area Ω, and they therefore record the total

response (free-field response plus reflections from the structure). Figures 6.15-

6.17 compare the predicted displacements of the three models (cone boundary,

viscous boundary and extended mesh) for pulses of 3 periods (To = 1.0s, 2.0s and

4.0s).

250

Since the loading is applied only in the horizontal direction, the

horizontal response is dominant. However, due to multiple reflections on the

tunnel, vertical displacements are also recorded at both nodes. Regarding the

horizontal displacements, the results of both absorbing boundaries (cone,

viscous) compare perfectly with the ones of the extended mesh irrespective of the

period of the loading.

0 1 2 3 4 5 6

Time (s)

0

0.1

0.2

0.3

0.4


0 1 2 3 4 5 6

Time (s)

-0.02

-0.01

0

0.01

0.02


0 1 2 3 4 5 6

Time (s)

0

0.1

0.2

0.3

0.4


Extended Mesh

Viscous BC

Cone BC

0 1 2 3 4 5 6

Time (s)

-0.008

-0.004

0

0.004

0.008


Point DTo=1s

(c)Point DTo=1s

(d)

Point CTo=1s

(a) Point CTo=1s

(b)


of To =1.0s.

251

0 1 2 3 4 5 6

Time (s)

0

0.4

0.8

1.2

1.6


0 1 2 3 4 5 6

Time (s)

-0.014

-0.007

0

0.007

0.014


0 1 2 3 4 5 6

Time (s)

0

0.4

0.8

1.2

1.6


Extended Mesh

Viscous BC

Cone BC

0 1 2 3 4 5 6

Time (s)

-0.014

-0.007

0

0.007

0.014


Point DTo=2s

(c)Point DTo=2s

(d)

Point CTo=2s

(a) Point CTo=2s

(b)


of To =2.0s.

On the other hand, observing the vertical response it becomes clear that

the accuracy of both absorbing boundaries deteriorates as the period of the

loading increases. Considering that this numerical test is quite challenging for

both absorbing boundaries, it can be said that their performance is unexpectedly

good. This can be attributed to their application in the external area of the DRM

model, where they are required to absorb less energy. Comparing the viscous

boundary with the cone boundary, it is clear that the cone boundary performs

better for all periods.

252

0 1 2 3 4 5 6

Time (s)

0

2

4

6


0 1 2 3 4 5 6

Time (s)

-0.016

-0.008

0

0.008

0.016


0 1 2 3 4 5 6

Time (s)

0

2

4

6


Extended Mesh

Viscous BC

Cone BC

0 1 2 3 4 5 6

Time (s)

-0.016

-0.008

0

0.008

0.016


Point DTo=4s

(c)Point DTo=4s

(d)

Point CTo=4s

(a) Point CTo=4s

(b)


of To=4.0s.

Figure 6.18 shows the vertical acceleration response at nodes C, D for

pulses of 2 periods (To = 2.0s and 4.0s). Both absorbing boundaries seem to give

more accurate results in terms of accelerations than in terms of displacements.

This is not surprising, as the acceleration response is dominated by the higher

frequencies of the system.

Furthermore, Figure 6.19 shows the displacement response at the node G

(82.0, -82.0), which is located very close to the outer boundary +Γ , for pulses of

2 periods (To = 2.0s and 4.0s). The response recorded at node G is purely due to

reflections from the structure. Hence, the horizontal displacements are much

smaller than the ones recorded in nodes C, D, whereas the vertical displacements

are of the same order of magnitude. The errors associated with both absorbing

boundaries are significant in the plots of horizontal displacements. However, the

253

cone boundary predicts more accurately the vertical displacements than the

viscous boundary. The comparison of the cone boundary with the extended mesh

showed that the cone boundary can be used in the reduced model of the DRM. It

was also observed that the ability of both absorbing boundaries to absorb

reflected waves is very similar, although the cone boundary seems to give

slightly more accurate results.

0 1 2 3 4 5 6

Time (s)

-0.06

-0.03

0

0.03

0.06

Vertical acceleration (m/s2)

Point CTo=4s

(c)

Point CTo=2s

(a) Point DTo=2s

(b)

0 1 2 3 4 5 6

Time (s)

-0.16

-0.08

0

0.08

0.16


Extended Mesh

Viscous BC

Cone BC

0 1 2 3 4 5 6

Time (s)

-0.2

-0.1

0

0.1

0.2


0 1 2 3 4 5 6

Time (s)

-0.06

-0.03

0

0.03

0.06


Point DTo=4s

(d)

Figure 6.18: Comparison of the acceleration response at nodes C, D for pulses of

To=2.0, 4.0s.

254

0 1 2 3 4 5 6

Time (s)

-0.008

-0.004

0

0.004

0.008


Point GTo=4s(c)

Point GTo=2s

(a) Point GTo=2s

(b)

0 1 2 3 4 5 6

Time (s)

-0.008

-0.004

0

0.004

0.008


Extended Mesh

Viscous BC

Cone BC

0 1 2 3 4 5 6

Time (s)

-0.014

-0.007

0

0.007

0.014


0 1 2 3 4 5 6

Time (s)

-0.016

-0.008

0

0.008

0.016


Point GTo=4s

(d)

Figure 6.19: Comparison of the displacement response at node G for pulses of

To=2.0s ,4.0s.

In the preceding example the excitation was a simple pulse and the

investigation time was limited to avoid reflections from the Dirichlet boundaries

of the extended model. In order to compare the performance of the two absorbing

boundaries in a more realistic scenario, the previous analysis was repeated with

an earthquake excitation. The UNAM acceleration time history, recorded during

the 1985 Mexico earthquake, was the input motion for the step I analysis. The 1-

D background model was subjected to 60 seconds of the filtered recording

(Figure 6.20a) with a time step of 0.01 sec. This acceleration time history was

specifically selected for its low frequency content, as demonstrated in the

acceleration response spectrum of Figure 6.20b.

255

Figure 6.21 shows the displacement response recorded at nodes C, D for

both absorbing boundaries. As observed with the sinusoidal excitation results,

both boundaries give identical results in terms of horizontal displacements.

Regarding the vertical displacements the viscous boundary seems to

underestimate to some extent the response. Furthermore, the two absorbing

boundaries predict identical acceleration time histories, which are not included

herein for brevity. As the system was subjected to a particularly low frequency

excitation, one would expect considerable differences in the predicted responses

of the two boundaries. This is not the case, probably due to the improved

performance of the viscous boundary when is used in the external area of the

DRM model.

(a) (b)

0 20 40 60

Time (sec)

-1

0

1

Acceleration (m/sec2)

0 1 2 3 4

Period (sec)

0

2

4

6

8

Response Acceleration (m/sec2)

Figure 6.20: Filtered acceleration time history (a) and response acceleration

spectrum (b) of the 1985 Mexico earthquake.

256

0 10 20 30 40 50 60

Time (s)

-0.8

-0.4

0

0.4

0.8


0 10 20 30 40 50 60

Time (s)

-0.02

-0.01

0

0.01

0.02


0 10 20 30 40 50 60

Time (s)

-0.2

-0.1

0

0.1

0.2


Viscous BC

Cone BC

0 10 20 30 40 50 60

Time (s)

-0.015

-0.01

-0.005

0

0.005

0.01

0.015


Point D(c) Point D(d)

Point C(a) Point C(b)

Figure 6.21: Comparison of the displacement response of nodes C, D.

6.6 Summary

The DRM presented in this chapter is a two step procedure that aims at

reducing the domain that has to be modelled numerically. The DRM is

implemented into ICFEP based on the derivation of Bielak et al (2003) and is

further developed to deal with dynamic consolidation problems.

In step I of the DRM a simplified background model (Figure 6.3a) is

analysed that includes the source of excitation, but not the area of interest (that

contains geotechnical structures or localised geological features). The aim of the

step I analysis is to calculate and store the incremental free field response of a

single layer of finite elements within the boundaries Гe and Г. The second step is

performed on a reduced domain (Figure 6.3b) that comprises of the area of

257

interest Ω and of a small external region Ω+. The nodal effective forces ∆P

eff,

calculated from the results of step I, are applied to the model of step II at the

elements located within the boundaries Гe and Г. In the case of coupled

consolidation analysis the effective forces ∆Peff include additional terms derived

from the free field incremental pore pressures. The perturbation in the external

area Ω + is only outgoing and corresponds to the deviation of the area of interest

from the background model.

The development of the DRM for coupled consolidation problems and its

implementation was verified numerically both for linear and nonlinear analyses.

For the numerical test, the internal area Ω0 of the background model was

identical to the internal area Ω of the reduced model. In the step II analysis since

there is no deviation from the background model, zero response was calculated in

the external Ω+ area of the reduced model. Furthermore, the computed responses

in steps I and II were found to be identical for the internal areas Ω0 and Ω.

The great advantage of the DRM is that the excitation is directly

introduced into the computational domain, leaving more flexibility in the choice

of appropriate boundary conditions. Hence, a methodology was suggested which

employs the cone boundary on the external boundary +Γ of the reduced domain.

A cut and cover tunnel was analysed with both the cone and the viscous

boundary. To verify the applicability of the cone boundary, the step II analyses

were repeated with an extended mesh. The cone boundary was found to be

slightly superior to the viscous boundary. Both boundaries were subjected to a

quite challenging numerical test and they both performed very well. This agrees

well with the conclusion of Yoshimura et al (2003) that the absorbing boundaries


energy.

258

Chapter 7:

CASE STUDY ON SEISMIC TUNNEL

RESPONSE

7.1 Introduction

Until recently, it was widely believed that underground structures are not

particularly vulnerable to earthquakes. However, this perception changed after

the severe damage and even collapse of a number of underground structures that

occurred during recent earthquakes (e.g. the 1995 Kobe, Japan earthquake, the

1999 Chi-Chi, Taiwan earthquake and the 1999 Duzce, Turkey earthquake).

The present study considers the case of the Bolu highway twin tunnels

that experienced a wide range of damage during the 1999 Duzce earthquake in

Turkey. The Bolu tunnels establish a well-documented case, as there is

information available regarding the ground conditions, the design of the tunnels,

the ground motion and the earthquake induced damage. It should be noted that

most of this data was made available by Dr. C.O. Menkiti of the Geotechnical

Consulting Group (GCG) who was involved in the design of the tunnels.

The focus in the present study is placed on a part of the tunnels that was

still under construction when the earthquake struck and that suffered extensive

damage. In particular, the aim of this chapter is to use dynamic finite element

analyses to investigate the seismic tunnel response of two sections and to

compare the results with post-earthquake field observations.

The first part of this chapter describes the case study. This includes an

overview of the Bolu tunnels project, a description of the ground conditions and

a summary of construction issues for the analysed sections. The seismicity of the

Bolu area is also briefly discussed, while more emphasis is placed on the

description of the 1999 Duzce earthquake. Furthermore, post-earthquake field

259

observations of the damage are presented and linked to a general discussion

regarding the seismic hazards associated with underground structures.

In the second part of the chapter a thorough discussion on the adopted

numerical model is given. Theoretical issues presented in previous chapters

regarding spatial discretization, absorbing boundary conditions, time integration

and constitutive modelling are investigated and applied in the present case study.

For this purpose linear and nonlinear FE analyses are compared with linear and

equivalent linear analyses undertaken with the site response software EERA

(Bardet et al 2000).

The third part of the chapter presents the results of the dynamic and quasi

static FE analyses. The results are compared qualitatively and quantitatively with

simplified analytical methods and with post-earthquake damage observations.

7.2 Project description

This section presents a brief description of the Bolu tunnels project and is

based on the papers by Menkiti et al (2001a, 2001b). Particular emphasis has

been placed on the tunnels section that is relevant to the present case study.

7.2.1 Background

The Bolu tunnels project is part of the Trans European Motorway (TEM),

in the Gumusova-Gerede branch that links Ankara to Istanbul via a series of

viaducts, tunnels and embankments. The 23.7 km long part (Stretch 2 in Figure

7.1) of the Gumusova-Gerede motorway, which crosses the Bolu Mountain, is of

particular interest as it is constructed in complex ground conditions. It comprises

of several major structures including 5 viaducts, 3.3km long twin tunnels and 10

bridges.

The construction of the 3.4km long twin tunnels, the so called Bolu

tunnels, started at the Asarsuyu (west portal) in 1993 and at the Elmalik (east

portal) in 1994. Figure 7.2 illustrates a longitudinal section of the left tunnel. The

twin tunnels cross-sections range from 133m2 to 260m

2 and they are separated by

260

a 50m wide rock/soil pillar. The maximum overburden cover is 250m with most

of the cross-sections under a cover of 100-150m. The pre-tunnelling ground

water levels, are also indicated in Figure 7.2 and vary from 45% to 85% of the

overburden cover. The ground conditions along the tunnels alignment are highly

variable, consisting of extremely tectonised and sheared mudstones, siltstones

and limestones, with thick layers of stiff highly plastic fault gouge clay.

Figure 7.1: Layout of a 27.3 km part of the Gumusova-Gerede motorway (from

Menkiti et al, 2001b)

Figure 7.2: Longitudinal section of the left tunnel (from Menkiti et al, 2001b)

7.2.2 Construction details

The initial design of the tunnels, based on the standard Austrian rock

classification system, adopted the New Austrian Tunnelling Method (NATM). In

this design option a flexible shotcrete temporary lining was initially employed to

support the immediate loads allowing some controllable deformation. The final

lining of cast in-situ concrete was subsequently installed to complete the tunnel.

The NATM system proved to be inadequate for the zones involving tunnels

through the gouge clay (see Figure 7.2), as large uncontrolled deformations of

261

the temporary lining were observed. As a consequence of this, in 1997 a section

of the tunnel on the Elmalik side partially collapsed. This incident led to a

thorough design review, which (in 1998/1999) included a more detailed site

investigation and geotechnical characterisation of the ground conditions. Due to

the highly variable ground conditions, the project was divided into various

“design areas”. The design solution that is relevant to the present study is the one

developed for the worst ground conditions, namely for the thick zones of fault

gouge clay (see Figure 7.3). For such ground conditions two pilot tunnels were

first constructed in the bench area and back-filled with reinforced concrete. It

should be noted that the bench pilot tunnels themselves are substantial structures

with an external diameter of 5m. These two bench concrete beams provided a

stiff foundation for the top heading. In addition, the primary shotcrete support

(40cm thick) of the main tunnel was enlarged with an additional 60-80cm cast-

in-situ concrete layer (intermediate lining) which provided a stronger support

close to the face before the inner lining was cast. When the Duzce earthquake

struck the Bolu region, on 12/11/1999, only 2/3 of the project had been

completed. As will be discussed in Section 7.4, the uncompleted sections of the

tunnels suffered most of the damage.

Clearanceprofile

Top heading

Bench

Invert

Invert concrete

Bench pilot tunnelwith infill concrete

Inner concrete lining

Intermediateconcrete lining

Shotcrete primary lining

0 5m

Scale

Figure 7.3: Design solution for the thick zones of fault gouge clay (after Menkiti

et al 2001b)

262

7.3 The 1999 Duzce earthquake

The dominant tectonic feature in Turkey is the North Anatolian Fault

Zone (NAFZ). The NAFZ is a right lateral strike-slip fault, generally running

east-west, with a total length of about 1500km. It extends from the Karliova

triple junction in eastern Turkey to the mainland of Greece, branching into a

series of sub-parallel fault systems at the Marmara Region. In 1999, Turkey

suffered two major earthquakes on the NAFZ. First in August, the Kocaeli

earthquake struck with a moment magnitude of Mw=7.4 and a bilateral rupture of

at least 140km long, extending from Gölcük to Melen Lake. Three months later

(12/11/1999), a second earthquake, known as the Duce earthquake, struck with a

moment magnitude of Mw=7.2. The surface rupture associated with the second

event also propagated bilaterally in an east-west direction, but was significantly

smaller (30km) (Sucuoğlu, 2002).

Table 7.1: Summary of ground motion records from Duzce and Bolu stations

(from Menkiti et al, 2001a)

Station PGA

8

(m/sec2)

PGV8

(cm/sec)

Distance to

surface

rupture (km)

Distance to

subsurface

rupture (km)

Bolu 0.81g 65

18.3 6.0

Duzce 0.51g

80

6.8 6.8

The Bolu tunnels did not suffer any major damage during the first event.

Conversely, due to the close proximity of the tunnels to the Duzce rupture,

extensive damage in various sections of the tunnels was observed after the

second event (see Section 7.4). The west portals of the Bolu tunnels are located

within 3km from the east tip of the Duzce rupture and within 20km from the

8 PGA, PGV denote the peak ground acceleration and velocity respectively and refer to east-west

components of the records.

263

earthquake’s epicenter. Ground motion records from the November event close

to the project site and to the causative fault are available from the Duzce and the

Bolu strong motion stations. Menkiti et al (2001a) summarized the peak motion

characteristics of the two stations in Table 7.1.

(a) E-W, PGA=0.81g

0 20 40 60

Time (sec)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Ac

ce

lera

tio

n (

g)

0 20 40 60

Time (sec)

-0.8

-0.4

0

0.4

0.8

Ac

ce

lera

tio

n (

g)

0 20 40 60

Time (sec)

-0.8

-0.4

0

0.4

0.8

Ac

ce

lera

tio

n (

g)

(b) N-S, PGA =0.74g

(c) Vertical, PGA =0.2g

Figure 7.4: Unmodified strong motion records of the Bolu Station (from

Ambraseys et al, 2004)

Due to the proximity of the project to the fault rupture, the ground motion

at the tunnels was presumably influenced by near fault effects. Typically,

accelerograms of near-field ground motions exhibit pulse flings and are

frequently affected by the direction of the rupture propagation (rupture directivity

effects). Although the Bolu station motion is located at a distance of 18.3km

from the fault rupture, it has some features which are characteristic of near-field

motions. In particular, Akkar and Gülkan (2002) identified a strong pulse fling in

the E-W accelerogram (see Figure 7.4a). Furthermore, Sucuoğlu (2002) suggests

that the short duration of the strong motion in Duzce station (25.9 sec) compared

to Bolu (55.9sec) indicates that the Bolu station was in the forward directivity of

the ruptured segment of the fault. In the absence of any ground motion record at

the tunnels area, taking into account the bilateral mechanism of the rupture and

264

the relative positions of the stations with respect to the project and the rupture

(see Figure 7.5), it was decided that the ground motion of the Bolu station is the

most representative for the case study.

Figure 7.5: The surface rupture of the November 1999 Düzce earthquake and

active faults around Bolu (from Akyüz et al, 2002)

Table 7.2 summarizes the ground conditions at the Bolu station. Menkiti et al

(2005, personal communication) visited the Bolu station in 2000 and reported

that the Bolu seismometer is founded on an isolated concrete pillar at a

foundation depth of 2m. Therefore it can be postulated that the recorded ground

motion reflects the response of the stronger second layer (Vs=580.0m/s).

However, one can not completely rule out the possibility of the motion having

been affected by site effects (e.g. basin amplification).

Table 7.2: Shear wave velocity profile at the Bolu station (Menkiti 2005,

Personal communication)

Depth (m) Unit Vs (m/s)

0-2.2 Soil 111.0

2.2 -6.6 Soil

580.0

Below 6.6 Sandstone 1178.0

265

Since the causative fault generally runs in an east-west direction, the E-W

and N-S accelerograms in Figure 7.4 represent the fault parallel and normal

components of the motion respectively. Furthermore, as one would expect for a

lateral strike-slip fault, the vertical component of the motion is significantly

smaller.

7.4 Post-earthquake field observations

Due to the Duzce earthquake the Bolu tunnels experienced a wide range

of damage severity, depending on the ground conditions, the construction method

and the construction phase. Menkiti et al (2001a) and O’Rourke et al (2001)

provide a detailed description of the tunnels performance during the earthquake.

It is reported that the completed sections of the tunnels performed remarkably

well given the severity of the strong motion. Slight to moderate damage was

observed in the 16.5m diameter tunnels excavated in the metasediments,

although they were only supported by the primary shotcrete lining (30cm thick).

On the other hand, 360m of the right tunnel and 160m of the left tunnel

experienced collapses at the Elmalik side. These tunnels, with an external

diameter of 16.8m, were supported by thicker shotcrete (45cm-75cm), but were

excavated in the zone of poor fault gouge cay. It is remarkable to note that a 8m

diameter sinkhole was observed on the ground surface, although the overburden

is more than 50 m at this location.

Furthermore, collapses occurred over a length of 30m in both bench pilot

tunnels (BPTs) of the Asarsuyu left tunnel (see Figure 7.7). The failure of the 5m

external diameter BPTs was limited to the zone of poor fault gouge clay. This

part of the project sets an interesting example for back-analysis and it is thus the

focus of the present study. When the Duzce earthquake struck, the BPTs of the

left tunnel had not yet been back-filled with concrete and were only supported by

30cm thick shotcrete and HEB 100 steel ribs set at 1.1m longitudinal spacing. On

the other hand, the BPTs of the right tunnel had been back-filled with concrete

and thus did not suffer any major damage. Consequently only the BPTs of the

266

left tunnel are considered herein. Figure 7.6 shows a picture of the collapsed left

bench pilot tunnel (LBPT) after it was re-excavated and back-filled with foam

concrete. The excessive deformation of the cross-section involved crushing of

the shotcrete and buckling of the steel ribs at shoulder and knee locations and

invert uplift of 0.5m-1.0m. It is also reported that the buckled steel ribs indicated

shortening of 0.3-0.4m. Furthermore, Figure 7.7 shows a plan view of Asarsuyu

left tunnel progress when the earthquake struck. The bench pilot tunnels have a

center-to-center separation of 19.0 m, with the left tunnel leading the right one.

The post-earthquake investigations showed that the damage was limited to the

zone in which the two tunnels overlap within the zone of the fault gouge clay (as

indicated in Figure 7.7). Generally the damage was found more pronounced in

the LBPT. It is also interesting to note that the leading portion of the LBPT in the

same material (i.e. fault gouge clay) did not collapse. Various explanations can

be postulated. One possible reason could be the interaction of the tunnels and

consequent “wave trapping” in the soil pillar. Furthermore, as the fault gouge

clay layer is steeply inclined, the stratigraphy varies along the axis of the tunnels.

The difference in stratigraphy could have also affected the tunnel response. To

investigate these two postulations, two cross-sections AB and CD (see Figure

7.7) were analysed in Sections 7.7.8 and 7.7.12.

Figure 7.6: The collapsed LBPT after it had been re-excavated and back filled

with foam concrete (Menkiti 2005, personal communication)

267

62+810 62+820 62+830 62+840 62+850 62+860 62+870 62+880 62+890

LBPT

RBPT

shotcrete 30cm shotcrete 50cm shotcrete 30cm

shotcrete 30cm shotcrete 50cm shotcrete 30cm

Area of failure No severe damage in this area

Ch: 62+835 Ch: 62+865Completed Section

Design option without BPTs

Metasediments Fault Gouge ClayGray sandstone with Marlfragments

Quarzitic rock

A

B

C

D

Figure 7.7: Plan view of the Asarsuyu left tunnel

7.5 Ground conditions

As noted in Section 7.2.2, the design reassessment in 1998/1999 included

a detailed site investigation and geotechnical characterisation of the ground

conditions. An exploratory pilot tunnel was driven from each portal which

allowed a detailed characterisation of the ground conditions ahead of the drive.

Furthermore, the ground investigation included sub-surface boreholes drilled

from the pilot tunnels and surface boreholes. Hence the relevant geotechnical

units for cross-sections AB, CD were identified, as illustrated in Figures 7.8 and

7.9 respectively, and the water table was established at a depth of 62m below the

ground surface.

83.0m

58.0m

37.0m

12.0m

5.0m

water table

Calcareous Sandstone

Metasediments

Fault Gouge Clay

Sandstone, Marl

Quarzitic Rock (bedrock)

Fault Breccia

and Fault Gouge Clay

19.0m

14.0m

RBPT LBPT

Figure 7.8 Ground profile at chainage 62+850 (cross-section AB)

268

83.0m

44.0m

37.0m

14.0m

5.0m

water table


Metasediments

Fault Gouge Clay

Sandstone, Marl

Quarzitic Rock (bedrock)

Fault Breccia


28.0m

LBPT

Figure 7.9: Ground profile at chainage 62+870 (cross-section CD)

Table 7.3: Geotechnical description and index properties

Unit Consistency PI (%) CP

9 (%) &

Mineralogy

Calcareous

sandstone

Brown coloured slightly to

highly weathered/ fractured.

? ?

Fault breccia

and fault

gouge clay

heavily slicken-sided, highly

plastic, stiff to hard fault gouge

55 30-60

Metasediments Gravel, cobble and boulder

sized shear bodies in soil matrix.

10-15 5-25; illite

(58%),

smectite

(23%) Fault gouge

clay

Red to brown coloured, heavily

slicken-sided, highly plastic,

stiff to hard fault gouge

40-60 20-50,

smectite

Sandstone,

siltstone with

marl fragments

Gray sandstone with green,

weathered, medium strong to

weak marl fragments.

15 0-20

Bedrock Strong to very strong, faulted-

fractured quarzitic rock.

? ?

Moreover, based on information obtained by Menkiti (2005, personal

communication), Table 7.3 summarizes a description of the various geotechnical

9 Clay percentage by weight.

269

units and their index properties. The strength properties (the angle of shearing

resistance ϕ΄, the cohesion c΄ and the undrained strength Su) and the estimated

maximum shear modulus (Gmax) values are listed in Table 7.4.

The strength properties of the two clay layers and the metasediments are

based on laboratory shear strength tests, as reported by Menkiti et al (2001a),

while the calcareous sandstone and the sandstone overlaying the bedrock were

assumed to have the same drained strength properties as the metasediments.

Moreover, the estimated maximum shear modulus (Gmax) values of the two clay

layers and metasediments are based on pressuremeter tests, as reported by

Menkiti et al (2001a), while the Gmax values of the two sandstones are based on

the values published by O’Rourke et al (2001).

Table 7.4: Estimated strength and stiffness parameters

ϕ΄ c΄ (kPa) Unit

peak residual peak residual

Su

(kPa)

Gmax

(MPa)

Calcareous

sandstone 25˚-30˚ 20˚-25˚ 50 25 700 1000

Fault breccia

and fault gouge

clay

13˚-16˚ 9˚-12˚ 100 50 1000 750

Metasediments 25˚-30˚ 20˚-25˚ 50 25 1350 1500

Fault gouge clay 18˚-24˚ 6˚-12˚ 100 50 1000 850

Sandstone,

siltstone with

marl fragments

25˚-30˚ 20˚-25˚ 50 25 1500 2500

7.6 Earthquake effects on tunnels

Before looking into the details of the case study, it is useful first to

discuss the seismic hazards associated with underground structures and then to

assess which of these hazards are applicable to the case of the Bolu tunnels. It is

270

widely accepted (e.g. Hashash et al, 2001) that damage to underground structures

can be attributed to the following factors:

(a) Liquefaction of the soil adjacent to the structure. In this phenomenon, the

severe reduction of the soil’s strength leads to an increase of the loads acting on

the lining and to excessive deformations. Soils susceptible to liquefaction are

fully saturated loose to medium density sands, silts and gravels. Clearly (see

Figures 7.8 and 7.9) the soil units of the Bolu stratigraphy are not susceptible to

liquefaction and therefore this hazard is not relevant to the case study.

(b) Slope instability. Landslides intersecting the tunnel could lead to

concentrated shearing displacements and collapse of the cross section. This

hazard is usually more critical for tunnel portals and for cross sections at shallow

depth (St John and Zahrah, 1987). The cross sections considered in this study are

located at great depth. Furthermore, the post-earthquake field observations of

Menkiti et al (2001b) regarding the tunnels’ performance do not indicate signs of

slope instability. Therefore it can be postulated that land-sliding is not a relevant

hazard for this case study.

(c) Fault rupture. Displacements in the form of lateral movement, heave or

subsidence along a fault that crosses the alignment of the tunnel can be very

damaging. The causative fault of the Duzce earthquake did not cross the Bolu

tunnels, although its surface rupture was within 3km from the west portals

(O’Rourke et al, 2001). Therefore the hazard related to fault rupture

displacements is not applicable for this case study.

(d) Ground shaking. Damage of underground structures can be most commonly

attributed to ground shaking. Ground shaking was identified as the most relevant

hazard for the Bolu tunnels. Ground deformation due to propagation of seismic

waves in the soil may induce large dynamic loads on the tunnel lining. These

dynamic loads are superimposed on the existing static loads in the tunnel lining

and can lead to damage or collapse. St John and Zahrah (1987) listed the main

factors controlling the shaking damage:

- shape, dimensions and depth of the structure

271

- soil properties

- tunnel lining properties

- severity of the ground shaking

Furthermore, Owen and Scoll (1981) suggest that the response of circular

tunnels to seismic shaking can be described by the following types of

deformation: axial compression or extension (Figure 7.10a), longitudinal bending

(Figure 7.10b), and ovaling (Figure 7.11). Axial deformation and longitudinal

bending are produced by seismic waves propagating in planes parallel to the

tunnel’s axis. The components of these seismic waves which produce particle

motion parallel and perpendicular to the tunnel’s axis are responsible for the

axial deformation and the longitudinal bending respectively. On the other hand,

the ovaling deformation is mainly caused by shear waves propagating in planes

perpendicular to the tunnel axis. In this case, Owen and Scoll (1981) identified

two critical modes of lining failure:

- Compressive failure: the compressive capacity of the lining is

exceeded due to compressive dynamic stresses added to existing

static stresses.

- Tensile failure of the lining: tensile dynamic stresses are subtracted

from the existing compressive static stresses resulting overall in

tensile stresses.

Tunnel

Tension Compression

Tunnel

Negative

curvature

Positivecurvature

(a) Compression- extension (b) Longitudinal bending

Figure 7.10: Axial (a) and bending (b) deformation along the tunnel axis (after

Owen and Scoll, 1981)

272

Shear wave front

Tunnel before wave motion

Tunnel duringwave motion

Figure 7.11: Ovaling deformation of a circular tunnel’s cross section (after Owen

and Scoll, 1981)

Clearly, in order to describe all three modes of deformation (i.e. axial,

longitudinal bending and ovaling) a three-dimensional finite element model is

required. However, Penzien (2000), among others, suggests that the most critical

deformation of a circular tunnel is the ovaling of the cross-section. Furthermore,

the post-earthquake field observations in the area where the BPTs collapsed

agree with the ovaling form of deformation, as the damage was concentrated at

shoulder and knee locations of the lining (see Figure 7.6). Therefore a simplified

2D plane strain finite element model can be used to capture the most important

aspects of the seismic tunnel response.

In addition, a number of simplified methods have been developed to

quantify the seismic ovaling effect on circular tunnels. The so called “free-field

deformation” approach, ignores any soil-structure interaction effects and it

provides a first estimate of the deformation of the structure. Depending on the

relative stiffness of the tunnel lining with respect to the surrounding ground, this

method can be conservative in some cases and non-conservative in others. This

approach is extensively presented by Hashash et al (2001) and is not considered

herein. On the other hand, there are several analytical solutions which consider

the soil-structure-interaction (SSI) effects in a quasi-static way, ignoring though

any inertial interaction effects. Generally, dynamic SSI effects are important for

cases in which the dimensions of the cross-section are comparable with the

dominant wavelengths of the ground motion, for shallow burial depths and in

cases of stiff structures in soft soil. The dimensions and the burial depth of the

BPTs are such that the dynamic SSI effects are not expected to have played a

273

significant role in the collapse of the tunnels. Two analytical methods (Wang,

1993 and Penzien, 2000) were applied in the case of the Bolu tunnels and their

results are compared against the results of the FE analyses in Section 7.7.11.

These methods assume that the soil is an infinite, elastic, homogeneous and

isotropic medium and that the lining is an elastic thin walled tunnel under plane

strain conditions. Figure 7.12 illustrates the circumferential forces and moments

in a circular tunnel caused induced by waves propagating perpendicular to the

tunnel axis.

The compressibility ratio (C) and the flexibility ratio (F), as defined by

Hoeg (1968), are employed to quantify the relative stiffness between the tunnel’s

lining and the surrounding medium:

( )

( )( )mml

2

lm

ν21ν1tE

rν1EC

−+−

= 7.1

( )( )ml

32

lm

ν1IE6

rν1EF

+−

= 7.2

where:

Em, El are the Young’s moduli of the medium and lining respectively;

vm, vl are the Poisson’s ratio of the medium and lining respectively;

r, t, I are the radius, the thickness and the moment of inertia (per unit width)

respectively of the lining.

The compressibility ratio expresses the extensional stiffness, while the

flexibility ratio is a parameter of high significance as it represents the resistance

to ovaling. Values of the flexibility ratio F greater than 1 suggest that the tunnel

has lower stiffness than the surrounding medium. Hence, values of F→∞ imply

that the tunnel undergoes identical deformations to that of an unlined tunnel

without resisting the ovaling deformation.

274

Figure 7.12: Forces and moments induced by seismic waves (from Power et al

1996)

The analytical solutions of Wang (1993) and Penzien (2000) express the

maximum thrust (Tmax) and the maximum bending moment (Mmax) of the lining

as a function of the maximum free-field shear strain (γmax) at the level of the

tunnels. The inherent assumption of this approach is that that the tunnel is

subjected to a seismically induced uniform stress-strain field of intensity γmax.

Furthermore both methods consider separately the cases of full-slip and non-slip

conditions along the interface between the ground and the lining. The full-slip

condition assumes that no tangential shear force is developed along the ground-

lining interface. Hashash et al (2001) suggest that in reality for most tunnels the

interface condition is between these two limits (i.e. no-slip and full-slip).

However they recommend the use of the non-slip assumption as the full-slip

condition can lead to underestimation of the maximum thrust. Equations 7.3 and

7.4 give Wang’s expressions for Mmax and Tmax for full-slip conditions:

( ) max

m

m1max γr

ν1

EK

6

1T

+±= 7.3

( ) max

2

m

m1max γr

ν1

EK

6

1M

+±= 7.4

where: ( )

( )m

m1

6ν-5F2

ν112K

+−

= .

275

In the case of non-slip conditions, Wang’s method still employs Equation 7.4 to

calculate the maximum bending moment, but the maximum thrust is now

calculated by Equation 7.5.

( ) max

m

m2max γr

ν12

EKT

+±= 7.5

where ( )( ) ( )

( ) ( )[ ] m

2

mmmm

2

mm

2

ν86ν6ν82

5CCν21ν23F

2ν212

1C1ν21F

1K

−+

+−+−+−

+−−−−+=

Furthermore, Penzien (2000) introduced the lining-soil racking ratio R to

estimate the distortion of the tunnel:

f

l

∆d

∆d=R 7.6

where ∆dl is the lining diametric deflection and ∆df is the free-field diametric

deflection. Equations 7.7 and 7.8 give Penzien’s expressions for Mmax and Tmax

for non-slip conditions:

)ν(1r

∆dRIE3T

2

l

3

flmax −

±= 7.7

)ν(1r2

∆dRIE3M

2

l

2

flmax −

±= 7.8

where the lining-soil racking ratio (R) is defined as:

( )( )1α

ν14R

m

+

−= 7.9

where ( )( )2lm

3

ml

ν1Gr

ν43IE3α

−−

= and Gm is the shear modulus of the surrounding

medium. For the case of full-slip conditions the racking ratio under normal

loading Rn is defined as:

276

( )( )1α

ν14R

n

mn

+

−= 7.10

where ( )

( )2lm

3

mln

ν1Gr2

ν65IE3α

−−

= . Furthermore, Equations 7.11 and 7.12 give Penzien’s

expressions for Mmax and Tmax for full-slip conditions:

)ν(1r2

∆dRIE3T

2

l

3

f

n

lmax −

±= 7.11

)ν(1r2

∆dRIE3M

2

l

2

f

n

lmax −

±= 7.12

7.7 Finite element analyses

7.7.1 Spatial discretization

Plane strain analyses of the Bolu bench pilot tunnels (BPTs) were

undertaken for the cross sections AB and CD. Figure 7.13 illustrates the finite

element mesh used in the analyses of the cross-section AB, which consists of

5574 8-noded solid elements and 62 3-noded beam elements. While the depth of

the mesh was dictated by the stratigraphy at chainage 62+850 (see Figure 7.8),

the width of the mesh was decided to be 219m, based on numerical tests which

are separately discussed in Section 7.7.4.

Furthermore, as noted in Chapter 3, to accurately represent the wave

transmission through a finite element mesh, the element side length (∆l) must be

smaller than approximately one-tenth to one-eighth of the wavelength associated

with the highest frequency component of the input wave (see Section 3.2.2).

Hence, to specify the mesh discretization, the lowest shear wave velocity that is

of interest in the simulation and the highest frequency of the input wave need to

be first determined.

277

20D=100m 20D=100m19m

195m


Metasediments

Fault Gouge Clay

Sandstone, Marl

Fault Breccia


x

z


the tunnels

Table 7.5: Summary of estimated minimum shear wave velocity and resulting

maximum element side length

Unit Layer number Vsmin (m/s) ∆lmax(m)

Calcareous

sandstone 1 490.0 4.0

Fault breccia and

fault gouge clay 2 390.0 3.2

Metasediments 3 770.0 6.5

Fault gouge clay 4 420.0 3.5

Sandstone,

siltstone with

marl fragments 5 940.0 7.8

Considering the Fourier amplitude spectrum of the E-W component of the Bolu

acceleration time history (Figure 7.15a), it was decided that there was no need to

accurately model frequencies greater than 15Hz. In addition, since in nonlinear

problems the soil stiffness changes during the analysis, an estimate of the

minimum shear wave velocity for each layer, was obtained by undertaking

278

equivalent linear analyses with the software EERA (Bardet et al 2000). These

analyses are discussed in detail in Section 7.7.6. Table 7.5 presents the estimated

Vsmin and the resulting maximum element side length for each layer for

fmax=15Hz and ∆lmax=λmin/8.

7.7.2 Input ground motion

As noted in Section 7.6 the most critical deformation of a circular tunnel

is the ovaling of the transverse cross-section caused by shear waves propagating

in planes perpendicular to the tunnel’s axis. The alignment of the Bolu tunnels is

approximately perpendicular to the fault rupture. Therefore, the component (E-

W, see Figure 7.4a) of the ground motion parallel to the fault rupture is the one

responsible for the shear deformation of the tunnels' transverse cross-section.

The Bolu strong motion recording device is a digital apparatus with

sufficient memory to preserve the full history of the ground shaking. While

digital accelerographs have many advantages with respect to analog devices, the

need to apply a filtering process to the record, especially in the low frequency

range, is not entirely eliminated (Boore and Bommer, 2005). Employing the

software SeismoSoft (2004) a fourth order band-pass Butterworth filter was used

to remove the extreme low and high frequency components of the record. Figure

7.14 illustrates both the raw and filtered acceleration, velocity and displacement

time histories.

Generally the high-frequency noise of digital accelerographs is not

significant. This is also true for the Bolu record, as the Fourier amplitude values

of the uncorrected record in the high frequency limit (e.g. greater than 10Hz) are

almost zero (Figure 7.15a). Therefore, the choice of the maximum cut-off

frequency does not significantly affect the accuracy of the process and it was

taken equal to 15Hz. On the other hand, the noise in the low frequency range is

important both for analog and digital accelerograms as it results in unphysical

velocities and displacements. The unfiltered displacement history (Figure 7.14c)

shows an unrealistic drift from the zero displacement axis. Boore and Bommer

(2005) note however that nonzero values of final displacement should not be

always attributed to low-frequency noise. Permanent displacements are also

279

associated with plastic deformation of near surface materials or with co-seismic

slip on the fault. Since, no offset of the Bolu accelerometer was reported, the

drift in the displacement history can be attributed to low-frequency noise.

Therefore, a low-frequency cut-off of 0.1Hz was applied to the raw record. This

is the lowest possible cut-off frequency that ensures a realistic displacement

response and it was determined by a trial and error procedure.

0 20 40 60

Time (sec)

-0.3

-0.2

-0.1

0

0.1

Dis

pla

ce

me

nt

(m)

0 20 40 60

Time (sec)

-8

-4

0

4

8

Ac

ce

lera

tio

n (

m/s

ec

2)

raw record

filtered record

0 20 40 60

Time (sec)

-0.8

-0.4

0

0.4

0.8

Ve

loc

ity

(m

/se

c)

(a) (b)

(c)

Figure 7.14: Acceleration (a), velocity (b) and displacement (c) time histories of

the E-W component of the Bolu record

0.01 0.1 1 10

Frequency (Hz)

0

1

2

3

4

Fo

uri

er

Am

plit

ud

e (

m/s

ec

)

raw record

filtered record

0 1 2 3 4

Period (sec)

0

4

8

12

16

Sp

ec

tra

l A

cc

ele

rati

on

(m

/se

c2)

(a) (b)

ξ=5%

Figure 7.15: Fourier amplitude spectrum (a) and elastic acceleration response

spectrum (b) of the E-W component of the Bolu record

280

Furthermore, as shown in Figures 7.8 and 7.9, the bedrock is located at a

considerable depth from the ground surface (193m and 185m for chainages

62+850 and 62+870 respectively). Since there is no bedrock strong motion

record in the vicinity of the tunnels, the surface accelerogram was scaled to 70%

to account for strong motion attenuation with depth. The reduction of

acceleration with depth is evident in numerous records from vertical arrays. It is

attributed both to the free surface effect, which approximately doubles the

incoming ground motion, and to the impedance contrast of near-surface

materials. However due to the complexity of the problem, a reliable method has

not yet been developed to assess the reduction of ground motion with depth. The

scaling factor (i.e. 0.7) adopted in this study is in agreement with the

recommendations of the Federal Highway Agency (FHWA, 2000) for depths of

more than 30m and is an upper bound for data collected from down-hole arrays

(e.g. Archuleta et al, 2000). In any case there is a degree of uncertainty in this

approach which cannot be avoided.

0 10 20 30 40

Time (sec)

-4

-2

0

2

4

6

Ac

ce

lera

tio

n (

m/s

ec

2)

PGA=5.61m/sec2 at t=5.83sec

Figure 7.16: Scaled and truncated accelerogram used in the FE analyses

Furthermore, there is no need to use the full duration of the strong

motion, as the important shaking is limited to the time interval of 5sec-40sec.

Figure 7.16 illustrates the processed and scaled acceleration time history that was

employed in all the analyses. The peak value of the input acceleration time

history is 0.57g (5.61m/sec2) and it occurs approximately 5.8sec after the onset

of the excitation. The response spectrum of the record (Figure 7.15b) indicates

that the accelerogram is particular strong in the period range of 0.12sec to 1.0sec.

281

7.7.3 Construction sequence

When the earthquake struck, considerable static stresses were acting on

the tunnel linings due to the overburden pressure and the construction process.

Hence, prior to all 2D dynamic analyses presented in this chapter, a static

analysis was undertaken to establish the initial stresses acting on the lining.

During the static analysis displacements were restricted in both directions along

the bottom mesh boundary and vertical displacements were restricted along the

side boundaries.

As noted in Section 7.4 when the earthquake struck the BPTs were under

construction and they were therefore only supported by a 30cm thick shotcrete

preliminary lining with HEB 100 steel ribs sets at 1.1m longitudinal spacing.

While in a 3Diamensional model it is sensible to model the steel ribs, in plane

strain analyses the moment of inertia contribution from the steel ribs is very

small compared to that provided by the shotcrete. Therefore the steel ribs were

ignored in all the analyses. It should also be noted that at the time of the

earthquake, the shotcrete had not yet developed its full operational strength.

Menkiti (2005, personal communication), based on in-situ measured shotcrete

strength development curves, estimated the strength and stiffness properties of

the tunnel linings at the instant of the earthquake at chainage 62+850 (Table 7.6).

Table 7.6: Strength and stiffness properties of the BPTs at the time of earthquake

at chainage 62+850

LBPT

(shotcrete15 days old )

RBPT

(shotcrete 7 days old )

Cube Strength

(fcu, MPa)

Young’s

Modulus (GPa)

Cube Strength

(fcu, MPa)

Young’s

Modulus (GPa)

40 28 30 21

The lining was modelled with beam elements and for all the analyses it was

assumed to behave in a linear elastic manner. The beam elements were generated

within the mesh at the beginning of the analysis (i.e. in increment 0 which

282

corresponds to the mesh generation stage). ICFEP has a special facility that

allows initial excavation of elements, which are to be constructed in a latter stage

of the analysis, without the application of any loads. Thus the beam elements

were excavated as soon as they were generated (i.e. in increment 0). The tunnel

construction was then modelled using the convergence-confinement method

which is described in detail by Potts and Zdravkovic (2001). Starting from a

green-field profile, the excavation of the tunnels causes stress relief in the

ground. To model this excavation process, equivalent nodal forces along the

tunnel boundary, which represent the stresses exerted by the excavated soil, are

calculated and are then removed over several increments of the analysis. During

this process the elements representing the excavated soil are non active. These

forces are assumed to vary linearly with the number of increments over which

the excavation is to take place. The excavation of the BPTs was performed in ten

increments and the linings were constructed prior to the completion of

excavation. In particular the LBPT lining was constructed at 50% of stress

relaxation (i.e. increment 5), whereas the RBPT lining was constructed at 60% of

stress relaxation (i.e. increment 6). For both tunnels an initial Young’s modulus

of 5GPa was assigned which was increased to 28GPa and to 21GPa for the LBPT

and RBPT linings (see Table 7.6) respectively after the completion of excavation

(i.e. increment 11).

Table 7.7: Geometrical and material properties of tunnel linings

t

(m)

I

(m4/m)

El

(GPa) νl

ρ

(Mg/m3)

C F

LBPT 0.3 0.00225

28.0 0.2 2.45 1.21 67.46

RBPT 0.3

0.00225

21.0 0.2 2.45 1.62 89.95

All the geometrical and material properties of the BPTs linings are summarized

in Table 7.7. It should be noted that the flexibility ratios (F) of both BPTs in

Table 7.7 suggest that the gouge clay is much stiffer than the tunnel linings.

283

7.7.4 Discussion on the boundary conditions and mesh width

As discussed earlier, the FE mesh models the ground stratigraphy down

to the interface of the sandstone with the quartzic rock (see Figures 7.8 and 7.9)

which is a very stiff formation. The impedance contrast of the bedrock with the

overlying sandstone is sufficient to assume that this interface acts as a rigid

boundary. Therefore, the acceleration time history of Figure 7.16 was applied

incrementally in the horizontal direction to all nodes along the bottom boundary

of the FE model (i.e. along the bedrock-sandstone interface), while the

corresponding vertical displacements were restricted.

Furthermore, the width of the mesh and the lateral boundary conditions

should be such that free-field conditions (i.e. one-dimensional soil response)

occur near to the lateral boundaries of the mesh. Initially, the side boundaries

were placed at a distance of 20 tunnel diameters (D=5.0m) from the centreline of

the tunnels and the standard viscous boundary condition was employed along

them. A series of drained linear elastic analyses were undertaken to check the

adequacy of this model. Prior to all 2D dynamic analyses a static analysis was

carried out as described in the previous section, while the assumed material

properties are listed in Table 7.8. Rayleigh damping coefficients (A, B in Table

7.8) were employed corresponding to an equivalent viscous damping (ξ) of 5%

for layers 1, 3 and 5 and 6% for layers 2 and 4. The objective in this set of

analyses is to compare the response at various distances from the axis of

symmetry of the 2D FE model, illustrated in Figure 7.13, with the one-

dimensional response. Both the FE method and the EERA approach were used to

compute the free-field response.

The 1D FE model is 1m wide and comprises 212 (4x53, ∆x=0.25m) 8-

noded solid elements. The spatial discretization in the vertical direction is the

same as that of the 2D mesh at a horizontal distance of x=80.0m from the axis of

symmetry of the model (see Figure 7.17). The boundary conditions along the

bottom boundary of the mesh are the ones previously described for the 2D

model, while the vertical movement was constrained along the lateral boundaries.

284

z

x

Figure 7.17: FE mesh with boundary conditions

Table 7.8: Material properties used in elastic analyses

Unit Gmax

(MPa)

ρ

(Mg/m3)

ν A B

Calcareous

sandstone 1000 2.04 0.3 0.4712 3.98E-3

Fault breccia and

fault gouge clay 750 2.04 0.3 0.518 5.97E-3

Metasediments 1500 2.04 0.3 0.4712 3.98E-3

Fault gouge clay 850 2.04 0.3 0.518 5.97E-3

Sandstone,

siltstone with marl

fragments

2500 2.04 0.3 0.4712 3.98E-3

Moreover, the free-field response was also computed with the one-

dimensional site response software EERA, developed by Bardet et al (2000).

EERA is based on the same principles as the widely used program SHAKE

(Schnabel et al, 1972). It calculates the response in a layered visco-elastic soil-

rock system of infinite horizontal extent subjected to vertically propagating shear

waves. The visco-plastic constitutive relationship used in EERA is formulated in

a uniaxial stress-strain space and therefore only the strain component associated

285

with pure shear is computed. Although the analysis is executed in the frequency

domain, the response is expressed in the time domain by using an inverse Fast

Fourier Transform. This methodology relies on the principle of superposition and

thus it is restricted to linear systems. However, nonlinear behaviour can be

approximated using an iterative procedure. This type of analysis, which is

usually referred to as equivalent-linear analysis, is addressed in Section 7.7.5.

For simple linear elastic analysis the only required parameters are the thickness

of the soil layers, the shear modulus (G), the mass density (ρ) and the damping

ratio (ξ) for each layer.

In all the following FE analyses, the time integration was performed with

the CH method and with a time step ∆t=0.01sec. A discussion on the selection of

the time step is included in Section 7.7.6. The results from the 2D analyses will

refer to a distance x from the axis of symmetry of the FE model (see Figure

7.13), the response due to the static analysis has been subtracted in all cases and

the time is measured from the onset of the dynamic excitation. Figure 7.18a

shows the shear strain history at a depth z=157.9 (i.e. approximately at the

tunnels’ crown level) computed with the 2D model (at x=70.0m), 1D model and

EERA. While the curves for the 1D model and EERA are indistinguishable, the

2D response has fewer cycles and it is seriously damped. The excellent match

between the 1D model and the EERA analysis verifies the adopted element

discretization in the vertical direction and poses serious questions regarding the

inability of the 2D analysis to reproduce the free-field response. Besides, Figure

7.18b plots the locus of maximum shear strain with depth from the 2D model (at

x=70.0m and 90.0m), the 1D model and the EERA analysis. It is again evident

that the 1D FE analysis agrees very well with the solution of EERA, whereas the

2D FE model seriously underestimates the response. It is interesting also to note

that the 2D response seems to be even lower at a larger distance from the axis of

symmetry (i.e. for x=90.0m). This suggests that the 2D model is sensitive to the

width of the mesh.

286

0 10 20 30 40

Time (sec)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Sh

ea

r S

tra

in (

%)

(on

ly d

ue

to

e/q

)

ICFEP 2D viscous, x=70.0m

ICFEP 1D

EERA

(a)

0 0.1 0.2 0.3 0.4

Maximum Shear Strain (%) (only due to e/q)

200

160

120

80

40

0

De

pth

(m

)



ICFEP 1D

EERA

(b)

depth=157.9m


computed with the 2D model (with viscous boundary conditions), 1D model and

EERA

Brown et al (2001) and Maheshwari et al (2004) studied the 3D response of pile

groups to seismic excitation employing the boundary of Novak and Mitwally

(1988) at the sides of a 3D FE mesh and applying an acceleration time history at

the bottom of the mesh (see Figure 7.19). As discussed in Section 5.2.2, the

Novak and Mitwally (1988) boundary is based on Kelvin elements and it

comprises both springs and dashpots. The elastic analyses of Brown et al (2001)

and Maheshwari et al (2004) showed that the 3D free-field response is

considerably lower than that obtained using a 1D model (Figure 7.20).

Figure 7.19: Cross-section view of the 3D FE mesh and the boundary conditions

used in the analyses of Brown et al (2001) and Maheshwari et al (2004)

287

Figure 7.20: Free-field acceleration time histories obtained by 3D and 1D models

(after Brown et al 2001)

To explain this descrepancy, they suggest that the total damping in the 3D

analysis is higher than that in the 1D analysis, as the 3D model allows wave

propagation and thus energy dissipation in all directions. This implies that the 2D

response in Figure 7.18 is lower than the 1–D response due to energy dissipation

in the vertical direction. This is not a satisfying explanation, as it is well

established that the free-field response of a layered soil-rigid rock system of

infinite horizontal extent subjected to vertically propagating shear waves is one-

dimensional. The inability of the 2D model to reproduce the 1D free-field

response at the side boundaries of the mesh can be attributed to the poor

performance of the viscous boundary. It was extensively discussed in Chapter 5

that the viscous boundary method is exact for perpendicularly impinging waves.

Furthermore, for the 2D and 3D cases, perfect absorption is achieved for angles

of incidence greater than 30° (when the angle is measured from the direction

parallel to the boundary). Besides at large distances from the excitation source

the waves propagate one-dimensionally in approximately the direction of the

normal to the artificial boundary. Consequently, the performance of the boundary

improves significantly the farther it is placed away from the source of excitation.

However, in the 2D model of Figure 7.17, the dashpots were placed very close to

the seismic excitation, especially at the bottom corners of the mesh, and the shear

waves propagate in a direction parallel to the viscous boundary.

Since the viscous boundary failed to reproduce the free-field response,

the 2D analysis was repeated using the tied degrees of freedom (TDOF)

boundary condition along the sides of the mesh. This boundary condition

288

constrains nodes of the same elevation on the two side boundaries to deform

identically. Although this method can perfectly model the one-dimensional soil

response, it cannot absorb the waves radiating away from the tunnels and thus it

can result in wave-trapping into the mesh. In Figure 7.21a, the shear strain

history of the 2D model with the TDOF boundary condition (z=157.9m,

x=70.0m) compares very well with those of the 1D model and EERA. In

addition, Figure 7.21b plots the maximum shear strain profile of the 2D model

(at x=0.0m, 13.0m, 50.0m, 70.0m and 90.0m), the 1D model and the EERA

analysis. It is interesting to note that even at a distance of x=50.0m the free-field

profile is recovered in the 2D model, while the presence of the tunnels is evident

only in the maximum shear strain profile at the axis of symmetry (i.e. x=0.0) and

at a distance of x=13.0m. This observation indicates that a mesh of smaller width

could be used. Therefore the 2D analysis was repeated placing the lateral

boundaries at a distance x=75m (i.e. x=15D), resulting in a model with 5032

solid elements. The small mesh yielded identical response to that obtained by the

large model, both in the far-field area and in the vicinity of the tunnels. Although

the small model is accurate and saves computational time, it was decided to use

the large mesh in the following analyses in order to further examine the effect of

the mesh width in nonlinear analyses.

0 10 20 30 40

Time (sec)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Sh

ea

r S

tra

in (

%)

(on

ly d

ue

to

e/q

)

ICFEP 2D TDOF, x=70.0m

ICFEP 1D

EERA

(a)

0 0.1 0.2 0.3 0.4


200

160

120

80

40

0

De

pth

(m

)






ICFEP 1D

EERA

(b)

depth=157.9m


computed with the 2D model (with tied degrees of freedom boundary

conditions), 1D model and EERA

289

It is postulated, that the reason that the FE model of Figure 7.17 failed to

represent the free-field response, is that the dashpots were placed too close to the

seismic excitation. To check the validity of this hypothesis, the 2D analysis was

repeated with the Domain Reduction Method (DRM) in conjunction with viscous

dashpots, as illustrated in Figure 7.22.

z

x87.5m 87.5m

Ω

Ω+

Γe

Γ

Figure 7.22: FE mesh with boundary conditions used in the step II DRM analysis

In the DRM, which was extensively presented in the previous chapter, the

dynamic part of the analysis is performed in two steps. In step I the 1D FE model

was used to calculate the free-field response, in terms of effective forces at

various depths. In the step II analysis, the excitation (i.e. the effective nodal

forces calculated in step I) is introduced at the corresponding nodes in a zone of

elements (located between the boundaries eΓ and Г in Figure 7.22) within the FE

mesh. Note that in the step II model, any movement was restricted along the

bottom boundary of the mesh while viscous dashpots were placed along the

lateral boundaries. The interface Г divides the domain into the internal region Ω

in which the absolute response is calculated and to the external region Ω+ in

which the relative response, with respect to the free-field response, is computed.

Therefore, the perturbation in area Ω+ is only outgoing and corresponds to the

deviation of the 2D model from the 1D one. Hence, the dynamic interaction of

the tunnels with the surrounding ground conditions can be in a qualitative way

assessed, by examining the response in the area Ω+. Furthermore, in the DRM

model the dashpots were placed far away from the excitation (i.e. from the

290

tunnels), so they are expected to perform better. In Figure 7.21a, the shear strain

history of the DRM model (z=157.9m, x=70.0m) compares very well with those

of the 1D FE model and EERA. The viscous dashpots that were used in this

analysis did not affect the accuracy of the response. Furthermore, Figure 7.21b

plots the maximum shear strain profile of the 2D model (at x=70.0m and

x=90.0m), the 1D FE model and the EERA analysis. The maximum shear strain

profile of the DRM model at x=70.0m coincides with the 1D profiles up to a

depth of 183.0m (i.e. in the internal region Ω), whereas the profile at x=90.0m

which entirely lies in the external area Ω+ is much lower than the 1D profiles.

This suggests that the tunnels do not significantly interact with the surrounding

ground. This is to be expected as the diameter of the tunnels (5m) is much lower

than the dominant wavelengths in the gouge clay (77.5m to 645m for the period

range of 0.12sec to 1.0sec in Figure 7.15b), the burial depth is very large

(157.5m) and the flexibility ratios are high (see Table 7.7) Therefore, the simple

TDOF boundary condition can be safely used in the following analyses, as the

waves radiating away from the tunnels are negligible. So, to avoid the

inconvenience when using the DRM of saving every increment of the step I

model, the TDOF boundary condition was employed in all the subsequent

analyses.

0 10 20 30 40

Time (sec)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Sh

ea

r S

tra

in (

%)

(on

ly d

ue

to

e/q

)

ICFEP 2D DRM & viscous BC, x=70.0m

ICFEP 1D

EERA

(a)

0 0.1 0.2 0.3 0.4


200

160

120

80

40

0

De

pth

(m

)



ICFEP 1D

EERA

(b)

depth=157.9m


computed with the 2D model (with DRM and viscous boundary conditions), 1D

model and EERA

291

7.7.5 Constitutive models used in the analyses

Two constitutive models, a simple elasto-plastic and a two-surface

kinematic hardening model were considered in the FE analyses. Furthermore, 1D

equivalent linear analyses were undertaken with the site response software

EERA.

Simple elasto-plastic analyses

For the simple elasto-plastic analyses, a variant of the modified Cam Clay

(MCC) model (Roscoe and Burland, 1968) was used to describe the plastic

yielding behaviour of the soil and the small strain stiffness model of Jardine et al

(1986) was used to describe the elastic pre-yield behaviour.

In the MCC model any volume change along the virgin consolidation line

is elasto-plastic, while volume changes along swelling lines are assumed to be

elastic. Hence, the yield surface in the J (deviatoric stress), p΄ (mean effective

stress), v (specific volume) space plots above each swelling line, as illustrated in

Figure 7.24. The projection of this yield surface in the J-p΄ space, as

implemented in ICFEP, is an ellipse that takes the form:

( )

−

′′

−

′= 1

p

p

θgp

JF o

2

7.13

where the p΄, J are defined in Equations 7.14 and 7.15 respectively, θ is Lode’s

angle defined in Equation 7.16 and op′ is the value of p΄ at the intersection of the

current swelling line with the virgin consolidation line (see Figure 7.24).

( )321 σσσ3

1p ′+′+′=′ 7.14

( ) ( ) ( )213

2

32

2

21 σσσσσσ6

1J ′−′+′−′+′−′= 7.15

( )( )

−

′−′′−′

= − 1σσ

σσ2

3

1tanθ

31

321 7.16

292

where 1σ′ , 2σ′ and 3σ′ are the principal effective stresses.

lnp ’ p’ o

J

v

Swelling line

Virgin consolidationline

Yield surface

Figure 7.24: Yield surface (from Potts and Zdravković, 1999)

The original MCC model is formulated in the triaxial stress space (i.e. 1σ′ - 3σ′ ,

p΄). However, the implementation of the MCC model in a FE program requires

the extension of the model to the general stress space, making some assumption

on the shape of the yield and plastic potential surfaces in the deviatoric plane.

The function g(θ) in Equation 7.13 defines the shape of the yield surface in the

deviatoric plane. In the conventional model associated plasticity is assumed and

consequently the plastic potential function is also given by Equation 7.13. While

several shapes of the yield (g(θ)) and the plastic potential surfaces (gpp(θ)) are

available in ICFEP, in all the analyses of this chapter a Mohr Coulomb hexagon

and a circle have been adopted for the g(θ) and gpp(θ) surfaces respectively:

( )ϕ′+

ϕ′=

sinsinθ3

1cosθ

sinθg 7.17

( ) ϕ′= sinθgpp 7.18

where φ΄ is the angle of shearing resistance.

Furthermore, an isotropic hardening/softening rule is used to define the change in

the size of the yield surface, relating the parameter op′ to the plastic volumetric

strain, p

vε as follows:

293

κλ

vdε

p

pd p

v

o

o

−=

′

′ 7.19

where λ, κ are the slopes of the virgin consolidation and swelling lines in the v-ln

p΄ space respectively. Hence, the MCC model in the form used in the present

study requires the three consolidation parameters (λ, κ and the specific volume at

unit pressure v1), one drained strength parameter (φ΄) and one elastic parameter

(the maximum shear modulus G). The values of these parameters for the

different layers are given in Appendix B.

Furthermore a small strain stiffness model was used in combination with

the MCC model to describe the elastic pre-yield behaviour. Jardine et al (1986),

based on the results of high quality test data obtained with local measurements of

strain, developed empirical trigonometric expressions which represent the elastic

soil behaviour reasonably well. Equations 7.20 and 7.21 give the expressions, as

implemented into ICFEP, that describe the variation of the secant shear modulus,

G, and the bulk modulus, K, with the mean effective stress, p΄, and strain level in

the nonlinear elastic region.

+=

′

γ

3

d10

21sec

G3

Elogαcos

3

G

3

G

p

G 7.20

+=

′

µ

3

vol

1021sec

K

εlogδcos

3

K

3

K

p

K 7.21

where the deviatoric strain invariant, Ed, and the volumetric strain invariant, εv

are given by:

( ) ( ) ( )( )232

2

31

2

21d εεεεεε6

12E −+−+−= 7.22

321v εεεε ++= 7.23

where ε1, ε2 and ε3 are the principal strains. G1, G2, G3, α, γ, K1, K2, K2, δ and µ

are constants which can be obtained from a fit to laboratory or field test data.

Due to the trigonometric nature of Equations 7.20 and 7.21, minimum (Ed(min),

294

εv(min)) and maximum (Ed(max), εv(max)) strain limits need to be specified, below

and above which the secant shear and bulk moduli vary only with mean effective

stress. In addition, minimum values for the secant shear and bulk moduli, Gmin

and Kmin, are also specified. The parameters used in the small-strain stiffness

model for the different geological units are given in Appendix B.

Analyses with a kinematic hardening model

As mentioned earlier a two-surface kinematic hardening model (M2-

SKH) was also employed in the FE analyses. The M2-SKH model of

Grammatikopoulou (2004) is an improved version of the kinematic hardening

model of Al-Tabbaa and Wood (1989). The former model has a modified

hardening rule that achieves a smoother transition from elastic to elasto-plastic

behaviour than the latter model. The model is extensively presented by

Grammatikopoulou (2004) and only a brief overview is included herein. The M2-

SKH model is an extension of the MCC model, as it introduces a small kinematic

yield surface (denoted as “bubble” in Figure 7.25) within the MCC bounding

surface. The behaviour within the small kinematic yield surface (KYS) is elastic,

while it becomes elasto-plastic when the stress state engages the KYS. Plasticity

is introduced by both the movement (kinematic hardening) and the change of size

(isotropic hardening) of the KYS. The bounding surface is represented by the

elliptical MCC yield surface given by Equation 7.13. Furthermore, Equation 7.24

describes the inner kinematic surface, which has always the same shape as the

bounding surface, but is scaled to a smaller size.

( )( ) 4

pR

θg

J-Jp-pF

2

o2

b

2

a2

ab

′−

+′′= 7.24

where Rb is the ratio of the size of the KYS to that of the bounding surface and

g(θ) is defined in Equation 7.17. It should be noted that when the KYS is in

contact with the bounding surface the M2-SKH model predicts the same

behaviour as the MCC model. Hence, the M2-SKH model collapses to the MCC

model when monotonic loading is applied to normally consolidated material.

295

(p , Ja’ a)

ab

po’ p’

Jg( )θ

Bounding surface

“Bubble”

Figure 7.25: Two-surface kinematic hardening model (after Potts and

Zdravković, 1999)

The two-surface kinematic hardening (M2-SKH) model requires in total 7

parameters. Five of them have their origin in the MCC model (λ, κ, v1, φ΄ and G).

The remaining two are the ratio of the two surfaces (Rb) and the parameter α

which is related to the hardening rule (for details see Grammatikopoulou, 2004).

The adopted values of these parameters are given in Appendix B.

Equivalent linear analyses

As mentioned in Section 7.7.4 the software EERA (like any “SHAKE”

type software) employs the equivalent linear method to approximate nonlinear

behaviour. The equivalent linear method, introduced by Seed and Idriss (1969),

is the most widely used approach for site response analysis. This method

employs for each layer laboratory-derived variations of shear modulus (G) and

damping ratio (ξ) with shear strain, as illustrated in Figure 7.26. The shear

modulus and damping curves adopted in the presented study are included in

Appendix B. In each layer, some initial estimates are made for the shear modulus

and the damping ratio (denoted as G (1) and ξ

(1) in Figure 7.26). These initial

estimates are then used to compute the ground response in each layer. A

percentage (denoted as γeff) of the maximum shear strain recorded in each layer

(γeff is usually taken (0.5-0.7)γmax) is then used to determine new values for the

shear modulus and damping ratio (i.e. G (2) and ξ

(2)). These new values are

subsequently employed to repeat the computation. The whole process is repeated

several times, until differences between the computed shear modulus and

damping ratio values in two successive iterations fall below some predetermined

value in all layers. It should be noted that the strain compatible soil properties

(i.e. G (i), ξ

(i), where i is the iteration number) change in a step wise fashion, but

do not follow the G-γ and ξ-γ relationships implicity. Furthermore, the method

296

uses equivalent viscous damping to mimic the hysteretic behaviour of soil and is

unable to predict plastic deformation or pore pressure generation.

Shear modulus, G

G(1)

(2)G

(2)

(3)

G(3)

γ eff

(1)

Shear strain (log scale)

Damping ratio, ξ

ξ (1)

(2)ξ(2)

(3)ξ (3)

γ eff(1)

Shear stra in (log scale)

(1)

(1)(a) (b)

Figure 7.26: Iteration of shear modulus (a) and damping ratio (b) with shear

strain in equivalent linear analysis.

7.7.6 1D nonlinear dynamic analyses

Due to the complexity of the 2D nonlinear model it is helpful to perform

some preliminary tests using the 1D column model to examine the behaviour of

the constitutive models and to verify the adequacy of the time discretization.

Hence, undrained FE 1D analyses were undertaken with the MCC model in

combination with the small strain stiffness model of Jardine et al (1986) (denoted

as MCCJ in all future discussions) and with the M2-SKH model. Furthermore,

the FE analyses are compared with equivalent linear analyses undertaken with

the software EERA. Since the equivalent linear approach is well-established and

widely used, the EERA analyses provide a useful reference solution. However, it

should be clarified that the comparison with the equivalent linear method does

not serve any validation purposes, as this method has several limitations and it is

based on different assumptions to that of the FE models (see Sections 7.7.5 and

3.2.1). Besides, as mentioned in Section 7.7.1, the equivalent linear analyses

were used to specify the mesh discretization of the 2D model. The final shear

moduli (i.e. the G values of the last iteration) of the EERA analyses were used to

estimate the minimum wave length for each layer.

The arrangement of the 1D model was taken the same as before (see

Section 7.7.4). The material parameters of all constitutive models are

297

summarized in Appendix B and the time step was taken 0.01sec. Figure 7.27

plots the locus of maximum shear strain with depth computed with the MCCJ,

the M2-SKH models and EERA. While the M2-SKH model and EERA analysis

predict similar strain values in most layers, the MCC model predicts much higher

values, especially in the clay layers.

0 0.4 0.8 1.2

Maximum Shear Strain (%)

200

160

120

80

40

0

De

pth

(m

)

ICFEP M2-SKH

ICFEP MCCJ

EERA

Figure 7.27: Maximum shear strain profile computed with the MCCJ, the M2-

SKH models and EERA

Figure 7.28 plots strain time histories at depths of z=84.25m (i.e. in layer

2) and of z=157.5m (i.e. in layer 4, at the level of the tunnels’ crown) for all 3

models. For both layers the MCCJ model predicts unrealistic behaviour, as the

intense period of the motion cannot be distinguished and the response appears to

be undamped. Figure 7.29 compares the predicted strain history at the depth of

z=157.5m by a simple linear elastic analysis without any Rayleigh damping (i.e.

A=B=0 in Table 7.8), with that predicted by the MCCJ model. The two strain

histories appear to be quite similar, as the introduced plasticity in the MCCJ

analysis does not provide adequate damping in the response.

298

0 10 20 30 40

Time (sec)

-1

-0.5

0

0.5

1

1.5

Sh

ea

r S

tra

in (

%)

ICFEP M2-SKH

ICFEP MCCJ

EERA

0 10 20 30 40

Time (sec)

-0.8

-0.4

0

0.4

0.8

Sh

ea

r S

tra

in (

%)

(b) depth=157.5m (layer 4)(a) depth=84.25m (layer 2)

Figure 7.28: Representative strain time histories for the two clays layers (i.e.

layer 2 (a) and layer 4 (b))

0 10 20 30 40

Time (sec)

-0.8

-0.4

0

0.4

0.8

Sh

ea

r S

tra

in (

%)

ICFEP Linear elastic without damping

ICFEP MCCJ

Figure 7.29: Comparison of strain time histories at a depth of z=157.5m

The yield surface of the MCCJ model is unrealistically large and the

model cannot develop hysteretic dissipation. This limitation of the MCCJ model

is better illustrated in Figure 7.30, which presents the shear strain time history,

the shear stress-strain curve and the p΄-J stress path of an integration point at a

depth of z=157.5m computed with the MCCJ model for the first 11.38sec of the

earthquake. Starting from point A, the stress state oscillates inside the yield

surface (i.e. zero plastic strains) following a nonlinear elastic stress-strain curve

until the time instance t=9.81sec (i.e. point B). At t=9.81sec (i.e. point B) the

stress state reaches for the first time the yield surface but it only stays there for 1

increment (i.e. time step) and due to unloading moves back in the elastic region.

At t=10.56 (point C) the stress state reaches again the yield surface, moves along

299

the yield surface for a few increments, but at t=10.6 (point D) due to unloading it

is obliged to return again to the elastic region.

(a)

0 4 8 12

Time (sec)

-0.4

-0.2

0

0.2

0.4

0.6

Sh

ea

r S

tra

in (

%)

A

D

E

C

(b)

-0.4 -0.2 0 0.2 0.4

Shear Strain (%)

-1200

-800

-400

0

400

800

1200

Sh

ea

r S

tre

ss

(k

Pa

)

A

C D

E

B

B

0 2000 4000 6000

p'

0

400

800

1200

J (

kP

a)

A

B,C

D

E

p'o

(c)


stress path (c) of an integration point at a depth z=157.5m computed with MCCJ

model for the first 11.38sec of the earthquake.

Furthermore, Figure 7.31 shows the shear stress-strain curve and the p΄-J stress

path of the same integration point for the whole duration of the earthquake.

Clearly the above-mentioned process is repeated several times, resulting in a

more or less nonlinear elastic behaviour rather than in a nonlinear elasto-plastic

behaviour. It should be also noted that the yield surface changes size during the

earthquake, but this change is so small that it is impossible to be distinguished in

Figure 7.30c.

300

0 2000 4000 6000

p'

0

400

800

1200

J (

kP

a)

p'o

(a) (b)

-0.8 -0.4 0 0.4

Shear Strain (%)

-1200

-800

-400

0

400

800

1200

Sh

ea

r S

tre

ss

(k

Pa

)


point at a depth of z=157.5m computed with the MCCJ model for the whole

duration of the earthquake.

In Figure 7.28, on the other hand, the M2-SKH predicts a much lower

response than the EERA analysis for the thin clay layer (i.e. layer 2).

Furthermore, the M2-SKH model and the EERA analysis predict similar

behaviour in the thick clay layer (i.e. layer 4) for the first few seconds of the

earthquake, but as plasticity is introduced in the M2-SKH analysis the two time

histories depart Figure 7.32 compares the relative horizontal displacement (with

respect to the rigid base of the mesh) and acceleration time histories at a depth of

z=163.5m (layer 4) computed with the M2-SKH model and with EERA. The

displacement histories predicted by the two analyses compare quite well until

approximately t=5.6sec which is just before the peak of the input ground motion

(t=5.83, in Figure 7.16). As the intensity of the shaking increases the permanent

displacements predicted by the M2-SKH model cannot be modelled by the

equivalent linear method. The acceleration time histories predicted by the two

approaches are in general agreement.

301

0 10 20 30 40

Time (sec)

-0.08

-0.04

0

0.04

0.08

Re

lati

ve

ho

riz

on

tal

dis

pla

ce

me

nt

(m)

ICFEP M2-SKH

EERA

0 10 20 30 40

Time (sec)

-0.4

-0.2

0

0.2

0.4

0.6

Ho

riz

on

tal

ac

ce

lera

tio

n (

g)

(b)(a)

Figure 7.32: Relative horizontal displacement (a) and horizontal acceleration (b)

time histories at a depth of z=163.5m

Moreover, the ability of the M2-SKH model to predict hysteretic

behaviour is illustrated in Figure 7.33, which presents the shear strain time

history, the shear stress-strain curve and the p΄-J stress path of an integration

point at a depth of z=157.5m (layer 4) computed with the M2-SKH model for the

first 6.48sec of the earthquake. The stress state starts from a point inside the KYS

(point A), oscillates for a while within the kinematic surface along a linear stress-

strain path and at t=3.68sec reaches the extremity of the KYS (point B) for the

first time. From that point onwards, plasticity is introduced in the analysis, as the

KYS changes size (only slightly though) and most importantly as it is dragged

around within the bounding surface. From point B onwards the behaviour is

highly nonlinear and the stress reversals (e.g. at point C) produce a significant

hysteresis loop (Figure 7.33b). The area of this loop represents the dissipated

energy during the corresponding cycle. Figure 7.34, shows the p΄-J stress path

and the shear stress-strain curve of the same integration point for the whole

duration of the earthquake. As the intensity of the excitation reduces, the

hysteresis loops become very narrow resulting in lower damping.

302

(a)

0 2 4 6 8

Time (sec)

-0.2

-0.1

0

0.1

0.2

Sh

ea

r S

tra

in (

%)

A

C

D

B

(b)

-0.2 -0.1 0 0.1 0.2

Shear Strain (%)

-400

-200

0

200

400

Sh

ea

r S

tre

ss

(k

Pa

)

A

B

C

D

0 2000 4000 6000

p'

0

400

800

1200

J (

kP

a)

A

B

C

D

p'o

(c)

C, D

B


stress path (c) of an integration point at a depth z=157.5m computed with the

M2-SKH model for the first 6.48sec of the earthquake.

(b)

-0.2 -0.1 0 0.1 0.2

Shear Strain (%)

-400

-200

0

200

400

Sh

ea

r S

tre

ss

(k

Pa

)

A

B

C

D

0 2000 4000 6000

p'

0

400

800

1200

J (

kP

a)

A

B

C

D

p'o

(a)

C, D

B


point at a depth of z=157.5m computed with the M2-SKH model for the whole

duration of the earthquake.

303

Similar observations can be made by comparing representative strain time

histories in the remaining rock layers obtained by all three models (Figure 7.35).

The MCCJ model predicts unrealistically large oscillations in all layers, while the

M2-SKH model and EERA analyses compare reasonably well for the fist few

seconds of the excitation, but then diverge as the intensity of the shaking

increases.

0 10 20 30 40

Time (sec)

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Sh

ea

r S

tra

in (

%)

0 10 20 30 40

Time (sec)

-0.2

-0.1

0

0.1

0.2

Sh

ea

r S

tra

in (

%)

0 10 20 30 40

Time (sec)

-0.15

-0.1

-0.05

0

0.05

0.1

Sh

ea

r S

tra

in (

%)

ICFEP M2-SKH

ICFEP MCCJ

EERA(a) depth=70.31m (layer 1) (b) depth=114.1m (layer 3)

(c) depth=186.9m (layer 5)

Figure 7.35: Representative strain time histories for the rock layers (i.e. layer 1

(a), layer 3 (b) and layer 5 (c))

As mentioned earlier, the numerical tests of the 1D column were also

used to verify the adequacy of the time discretization. Hence the analysis with

the M2-SKH model was repeated halving the time step (i.e. ∆t=0.005sec). Figure

7.36 shows the displacement and acceleration time histories at a depth of

z=163.5m for ∆t=0.005 and ∆t=0.01sec. The displacement response curves of the

analyses with ∆t=0.005 and ∆t=0.01sec are indistinguishable, while some minor

304

differences can be observed in the acceleration plot. Similar checks were

performed for all the layers, which are not included herein for brevity. Since the

response for ∆t=0.005 is almost identical to the response for ∆t=0.01sec, it may

be assumed that the ∆t=0.01sec is sufficient small to ensure accurate results.

0 10 20 30 40

Time (sec)

-0.4

-0.2

0

0.2

0.4

0.6

Ho

riz

on

tal

ac

ce

lera

tio

n (

g)

(b)(a)∆t=0.01sec

∆t=0.005sec

0 10 20 30 40

Time (sec)

-0.12

-0.08

-0.04

0

0.04

0.08

Ho

riz

on

tal

dis

pla

ce

me

nt

(m)

Figure 7.36: Horizontal displacement (a) and acceleration time histories (b) at a

depth of z=163.5m for ∆t=0.005sec and ∆t=0.01sec

In conclusion, the numerical tests with the 1D model showed that the

inability of the MCCJ model to mimic hysteretic behaviour leads to unrealistic

predictions, while the M2-SKH model appropriately captures features of the soil

behaviour when subjected to cyclic loading, such as hysteretic damping and

plastic deformation during unloading. Therefore, most of the 2D investigations

presented in the rest of this chapter were carried out with the M2-SKH model.

However, in order to evaluate how much the employed constitutive model can

affect the predicted tunnel response, the MCCJ model was used in the static and

dynamic analyses of Section 7.7.9.

7.7.7 2D nonlinear static analyses at chainage 62+850

As discussed in Section 7.7.3, prior to any dynamic analysis a static

analysis had to be performed to establish the stresses that were acting on the

tunnels’ lining prior to the earthquake. Hence a plane strain static analysis, as

described in Section 7.7.3, was undertaken, assuming undrained conditions for

the clay layers and drained conditions for the rock layers. As noted earlier, the

305

parameters used in the M2-SKH model for the different geological units are

given in Appendix B.

26.0m

7.0m

Deformation scale: 0.0284m

LBPTRBPT

Figure 7.37: Mesh configuration around the tunnels at the end of the static

analysis

Figure 7.37 shows an enlarged view of the original and final mesh (i.e. at

the last increment of the static analysis) configurations around the tunnels. The

static stresses acting on the tunnels’ lining cause an elliptical deformed shape,

which is slightly more pronounced in the RBPT. The amount by which the

tunnels deformed is summarized in Table 7.9.

Table 7.9 :Summary of the diametral movements and strains after the static

analysis

LBPT RBPT Diametral

Convergence (mm) (%) (mm) (%)

Horizontal 40.72 0.81 51.86 1.03

Vertical 28.59 0.57 37.88 0.76

Measurements from monitoring the exploratory pilot tunnel in a flyschoid

clay (not at the sections considered herein) reported by Menkiti et al (2001b)

indicate a horizontal convergence of 15mm-25mm, which is lower than the FE

predictions of Table 7.9. However, it is also reported that the exploratory pilot

tunnel experienced much larger movements in the fault gouge clay which in

some cases led to failure. Furthermore, measurements from a completed section

306

of the left tunnel (main tunnel) in the gouge clay show a horizontal diametral

convergence of the BPT concrete beams of 0.9% (Menkiti et al, 2001b).

Therefore, overall the FE results are in agreement with the observed behaviour of

the tunnels. The FE analysis also indicates that the RBPT, which was constructed

at 60% stress relaxation and is more flexible than the LBPT (see Table 7.6),

experienced larger deformations. Figure 7.38 shows the accumulated thrust

(compression positive), bending moment and hoop stress distribution at the beam

elements around the tunnels’ lining. The maximum hoop stress distribution of

outer fibre reflects the combined effect of the compressive thrust and bending

moment and it was calculated as follows:

I

yM

A

TσH += 7.25

where y is the distance from the neutral axis to the extreme fibre of the lining

cross-section and A is the area per unit width of the lining cross-section.

The thrust distribution is more or less uniform around the tunnel linings,

while the bending moment values are quite low and show a fluctuation around

the lining. Furthermore, the thrust and hoop stress distribution indicate that the

stiffer tunnel (i.e. LBPT) attracted higher loads than the RBPT. Menkiti et al

(2001b), based on the performance of the exploratory tunnel, estimated the

immediate ground loads as being 40-65% of the overburden, which corresponds

to hoop stresses of 7450-12120kPa on the tunnels’ lining. The predicted hoop

stresses for the RBPT lie within this range, while the ones for the LBPT are

marginally above the upper limit of this range.

307

0 60 120 180 240 300 360

Angle around tunnel lining, θ

10000

11000

12000

13000

14000

Ho

op

Str

es

s (

kP

a/m

)

0 60 120 180 240 300 360


-20

-10

0

10

20

Be

nd

ing

Mo

me

nt

(kN

m/m

)

0 60 120 180 240 300 360


2800

3200

3600

4000

Th

rus

t (k

N/m

)

LBPT

RBPT

(a)

θ

(b)

(c)


distribution around the tunnels’ lining at the end of the static analysis

Furthermore Figure 7.39 presents contours of the pore water pressure

distribution in the vicinity of the tunnels at the end of the static analysis. The FE

results show that the excavation process causes the generation of pore water

suctions. The contours of this tensile pressure are concentrically aligned around

the tunnels and they gradually decay with distance, so that a compressive pore

pressure is recovered at a distance from the tunnel linings approximately equal to

D/2 (i.e. D is the tunnel diameter). In a similar fashion, Figure 7.40 shows

contours of plastic shear strain distribution. Clearly, a plastic zone is formed

which approximately extends up to a distance of 1.2D and 1.0 D from the RBPT

and LBPT’s linings respectively.

308

CONTOUR LEVELSCOMPRESSION POSITIVE

875.0 kPa

426.0 kPa

-21.0 kPa

-471.0 kPa

-919.0 kPa

A

B

C

D

E

33.12m

18.7m RBPT

B

C

D

B

B

B

LBPTCC

B

B

B

B

BB

B BB B

B

C

B

D

E

EE

B

Figure 7.39: Contours of pore pressure distribution around the tunnels at the end

of the static analysis.

CONTOUR LEVELSCOMPRESSION POSITIVE

1.5%

1.17%

0.83%

0.5%

0.16%

A

B

C

D

E

-0.17%

-0.51%

-0.84%

-1.12%

-1.15%

F

G

H

I

J

33.12m

18.7m

J

ABCD

E

F

G

HIJ

F

GH

I

E

DC

BA

F

GHI

E

D CB

E

D

CB

GHI

F

RBPT LBPT

Figure 7.40: Contours of plastic shear strain around the tunnels at the end of the

static analysis.

309

7.7.8 2D nonlinear dynamic analyses at chainage 62+850

Once the static stresses acting on the tunnel linings were established, a

dynamic analysis, as described in previous sections, was undertaken assuming

that all materials behave in an undrained manner. Figure 7.41 compares the

maximum shear strain profiles (caused only by the dynamic excitation) at various

distances x from the axis of symmetry of the 2D FE model (i.e. x=0.0m, 13.0m,

50.0m, 70.0m and 90.0m) with the response of the corresponding 1D FE model.

The free-field response is recovered at distances greater than x=50.0m, as the

maximum shear strain profiles at distances x=50.0m, 70.0m and 90.0m agree

well with the 1D results. It is interesting to note that this agreement is slightly

worse than the one observed for the linear elastic case (see Figure 7.21b).

Furthermore, the response at the level of the tunnels (the centre of the tunnels is

at z=160.0m) is significantly de-amplified with respect to the free-field response

at a distance x=13.0m, while it is amplified in the pillar (i.e. x=0.0m).

0 0.1 0.2 0.3 0.4


200

160

120

80

40

0

De

pth

(m

)

2D M2-SKH, x=70.0m

2D M2-SKH, x=90.0m

2D M2-SKH, x=50.0m

2D M2-SKH, x=13.0m

2D M2-SKH, x=0.0m

1D M2-SKH

(b)

Figure 7.41: Maximum shear strain profile computed with the M2-SKH model

for 1D and 2D analyses

As discussed in Section 7.6, analytical studies suggest that circular

tunnels, subjected to shear waves propagating in planes perpendicular to the

tunnel axis, undergo an ovaling deformation. This form of deformation was

verified by the FE analysis. Figure 7.42 illustrates an enlarged view of the

deformed mesh shortly after the peak of the excitation (i.e. at t=8.0sec). The

310

ovaling deformation is evident in both BPTs and it implies a stress concentration

at the shoulder and knee locations of the lining.

26.0m

7.0m

Deformation scale: 0.08m

Figure 7.42: Enlarged view of the deformed mesh at t=8.0sec

CONTOUR LEVELS

A

B

C

D

E

(a) t=5.0sec

CONTOUR LEVELS

A

B

C

D

E

(b) t=6.0sec

CONTOUR LEVELS

A

B

C

D

E

(c) t=7.0sec

CONTOUR LEVELS

A

B

C

D

E

(d) t=8.0sec

DD

D

C

D

D

D

C

C

BA

A

B

C

D

CD

D

C

C

D D

C

AA

B

BB

B

CD

B

B

CD

B

CD

B CC CD

B

B

B

B

C B CD

CCD

AB

CD C

B

A

A

B

C

C

C

BA

A

C

B

C

ABC

A

B CA B B

A

AB

C

A

A

BC

BC

A

A

BC

Figure 7.43: Snapshots (at t=5.0, 6.0, 7.0 and 8.0sec) of deviatoric stress (J)

contours in the vicinity of the tunnels (for the area indicated in Figure 7.42)

311

Figure 7.43 illustrates snapshots of contours of deviatoric stress (J) in the

vicinity of the tunnels (i.e. for the area indicated in Figure 7.42) at various

instances before and after the peak of the earthquake (i.e. at t=5.0, 6.0, 7.0 and

8.0 sec). Initially (i.e. at t=5.0) the stress contours have an almost vertical

configuration, later they gradually concentrate around the shoulder and knee

locations of the linings. Interestingly, shear planes at 45˚ seem to form in the

pillar at t=8.0sec.

0 10 20 30 40

Time (sec)

-800

-400

0

400

800

Po

re P

res

su

re (

kP

a/m

)

LBPT

RBPT

0 10 20 30 40

Time (sec)

-0.6

-0.4

-0.2

0

0.2

0.4

Sh

ea

r S

tra

in (

%)

LBPT

RBPT

(a) (b)


integration points adjacent to the crowns of the BPTs

Figure 7.44 presents the pore water pressure and shear strain time

histories recorded at two integration points E (x=9.1m, z=157.4m) and F (x=-

9.9m, z=157.4m) adjacent to the crowns of the LBPT and the RBPT respectively.

As discussed in the previous section, the excavation process caused the

generation of pore water suction around the tunnel linings. During the first

seconds of the earthquake, the tensile pore fluid pressure is maintained around

both tunnels, but approximately at the peak of the input excitation (see Figure

7.16) an abrupt jump is observed in Figure 7.44a, which results in compressive

pore pressure. Subsequently, the compressive pore pressure continues to build up

for a few more seconds (approximately until t=10.0sec) and then stabilizes. It

should be noted that for both tunnels these stabilised values are lower than the

green-field hydrostatic pore pressure at the crown level (i.e. 936.0kPa).

Furthermore, in a similar fashion to the shear strain history of the 1D mesh

computed with the M2-SKH model (see Figure 7.28b), the intense period of the

shaking generates significant permanent strains. The maximum shear strain

312

adjacent to the crown is 0.52% and 0.46% for the LBPT and the RBPT

respectively. These values are more than two times larger than the maximum

free-field shear strain at the same level (i.e. at z=157.4m) which is 0.19% (see

Figure 7.27).

Figure 7.45 shows the accumulated thrust (compression positive),

bending moment and hoop stress distribution, due to the combined effects of

static and dynamic loading, in the beam elements around the BPTs’ lining at

t=10.0sec. In all three plots the distribution is highly non-uniform around the

tunnel linings and the maxima of the thrust, bending moment and hoop stress

occur at shoulder and knee locations (i.e. θ=137˚ and 317˚ respectively). This is

in agreement with the post-earthquake field observations at the collapsed section

of the LBPT, which showed crushing of shotcrete and buckling of the steel ribs

at shoulder and knee locations of the lining (see Figure 7.6). The hoop stresses at

θ=137˚, 317˚ are approximately three times larger than the corresponding static

stresses in Figure 7.38, while in other locations the stresses are on average two

times larger. The thrust and bending moment time histories at θ=137˚ of both

BPTs are presented in Figure 7.46. In both tunnels, the axial forces start from an

initial value, induced by the static loading, and during the most intense period of

shaking they significantly increase. In a similar fashion to the pore pressure time

histories (see Figure 7.44), when the shaking intensity reduces the loads stabilise.

While the thrust developed in the RBPT is initially lower than that in the LBPT,

during the intense period of the shaking the thrust curves of the two BPTs

become indistinguishable. While the bending moment variations start from a

very small initial value, they significantly increase during the intense period of

the earthquake and finally stabilize to a relatively large value. It should be noted

that the maximum and stabilised values of bending moment in the RBPT are

considerably lower than those observed in the LBPT. Overall, the dynamic

analysis results indicate that the LBPT attracted higher loads than the RBPT.

This is in agreement with post-earthquake field observations suggesting that the

LBPT experienced more severe damage than the RBPT.

313

0 60 120 180 240 300 360


20000

24000

28000

32000

36000

40000

44000

Ho

op

Str

es

s (

kP

a/m

)

0 60 120 180 240 300 360


-300

-200

-100

0

100

200

Be

nd

ing

Mo

me

nt

(kN

m/m

)

0 60 120 180 240 300 360


6000

6400

6800

7200

7600

8000

Th

rus

t (k

N/m

)

LBPT

RBPT

(a)

θ

(b)

(c)


distribution around the tunnels’ lining at t=10.0sec

0 10 20 30 40

Time (sec)

2000

3000

4000

5000

6000

7000

8000

Th

rus

t (k

N/m

)

LBPT

RBPT

0 10 20 30 40

Time (sec)

-300

-200

-100

0

100

Be

nd

ing

mo

me

nt

(kN

m/m

)

θ=137°(a) (b)

Figure 7.46: Thrust (a) and bending moment (b) time histories at θ=137˚ for both

BPTs

314

Table 7.10 summarizes the values of maximum hoop stress recorded at

shoulder and knee locations (i.e. at θ=137˚, 317˚) of the lining due to static and

dynamic loading. The predicted maximum total hoop stresses exceed the strength

of the shotcrete in both tunnels, which is 40MPa and 30MPa for the LBPT and

RBPT respectively (see Table 7.6), and they thus match favourably with the

observed failure. However, it should be noted that the beam elements were

assumed to behave as a linear elastic material. Therefore the present FE analysis

cannot actually model the cracking of the lining and thus the predicted loads

might overestimate to some extent the loads that were actually acting on it.


lining computed with the M2-SKH model

Maximum Hoop Stress ( Hσ ) (MPa)

Point Static Earthquake Total

LBPT, θ=137˚ 12.1 29.2 41.3

LBPT, θ=317˚ 12.5 29.0 41.5

RBPT, θ=137˚ 10.5 26.4 36.9

RBPT, θ=317˚ 10.5 29.6 40.1

7.7.9 2D static and dynamic analyses with the MCCJ model

In order to evaluate how much the choice of constitutive model can affect

the computed tunnel response, the static analyses of Section 7.7.7 and the

dynamic analyses of Section 7.7.8 were repeated with the MCCJ model. The

assumed parameters for the different layers are given in Appendix B.

Figure 7.47 shows the accumulated thrust (compression positive),

bending moment and hoop stress distribution at the beam elements around the

tunnels’ lining at the last increment of the static analysis. In a similar fashion to

the M2-SKH static analysis results, the bending moment values are quite low and

show a significant fluctuation around the lining. On the other hand, for both

tunnels the predicted thrust values by the MCCJ model are lower than those

315

predicted by the kinematic hardening model. Consequently, for both tunnels the

hoop stresses computed with the MCCJ model are lower than those of Figure

7.38c and lie well within the estimated range by Menkiti et al (2001b) (i.e. 7450-

12120kPa). Hence, the static analysis’s results of the MCCJ model are, to some

extent, in better agreement with the estimates of Menkiti et al (2001b) than those

of the M2-SKH model. However, overall the static behaviour predicted by both

models is fairly similar, as the observed differences in hoop stresses are not

significant. Therefore, it can be concluded, that the MCCJ and M2-SKH dynamic

analyses start from a similar static configuration.

0 60 120 180 240 300 360


8000

9000

10000

11000

12000

13000

Ho

op

Str

es

s (

kP

a/m

)

0 60 120 180 240 300 360


-8

-4

0

4

8

12

Be

nd

ing

Mo

me

nt

(kN

m/m

)

0 60 120 180 240 300 360


2400

2600

2800

3000

3200

3400

3600

Th

rus

t (k

N/m

)

LBPT

RBPT

(a)

θ

(b)

(c)


distribution around the tunnels’ lining at the end of the static analysis computed

with MCCJ model

Figure 7.48 shows the pore water pressure and shear strain time histories

recorded at two integration points E (x=9.1m, z=157.4m) and F (x=-9.9m,

316

z=157.4m) adjacent to the crowns of the LBPT and the RBPT respectively. In a

similar fashion to the M2-SKH analysis’s results, the pore pressure time history

in both BPTs starts from negative values caused by the excavation process. In

both tunnels during the first seconds of the earthquake, the tensile pore pressures

are maintained, but around the peak of the input excitation the pore pressure

gradually increases. Unlike the M2-SKH analysis (see Figure 7.44a) that at

t≈10sec the pore pressure reaches a plateau, the pore pressure response predicted

by the MCCJ model is dominated by unrealistically large oscillations from

t≈10sec onwards. It should be noted that these oscillations are more pronounced

in the RBPT. Besides, the predicted strain time history adjacent to the RBPT

crown appears to be undamped and it seems to be essentially elastic. On the other

hand, the MCCJ model predicts significant plastic shear strains at the integration

point adjacent to the LBPT crown. However, due to the unrealistically large yield

surface of the MCCJ model, plasticity is introduced from t≈10.0sec onwards

which is late with respect to the peak of the excitation. Therefore the computed

shear strain values at the RBPT crown, although smaller than those computed at

the LBPT crown, are still much larger than those predicted by the M2-SKH

model (see Figure 7.44b).

0 10 20 30 40

Time (sec)

-4000

-3000

-2000

-1000

0

1000

Po

re P

res

su

re (

kP

a/m

)

LBPT

RBPT

0 10 20 30 40

Time (sec)

-2

-1

0

1

2

3

4

Sh

ea

r S

tra

in (

%)

LBPT

RBPT(a) (b)


integration points adjacent to the crowns of the BPTs computed with the MCCJ

model

Figure 7.49 presents the thrust and bending moment time histories at

θ=137˚ for both BPTs. Although the MCCJ model predicts some permanent

317

loads, the behaviour seems to be effectively elastic. Notably, the oscillations in

the bending moment variation are more pronounced in the LBPT.

0 10 20 30 40

Time (sec)

2000

3000

4000

5000

6000

7000

8000

9000

Th

rus

t (k

N/m

)

LBPT

RBPT

0 10 20 30 40

Time (sec)

-600

-400

-200

0

200

400

Be

nd

ing

mo

me

nt

(kN

m/m

)

θ=137°(a) (b)

Figure 7.49: Thrust (a) and bending moment (b) time histories at θ=137˚ of both

BPTs computed with the MCCJ model

Finally, Table 7.11 summarizes the values of maximum hoop stress

recorded at shoulder and knee locations (i.e. at θ=137˚, 317˚) of the lining due to

static and dynamic loading. For both tunnels the predicted maximum total hoop

stress values by the MCCJ model are approximately 42%-82% larger than those

computed by the M2-SKH model (see Table 7.10) and they are unrealistically

high. In conclusion, although the MCCJ model predicted very well the static

response of the BPTs, its inability to mimic hysteretic behaviour leads to a

substantial overestimation of the seismic loads acting on the tunnel linings.


lining computed with the MCCJ model


Point Static Earthquake Total

LBPT, θ=137˚ 10.9 52.7 63.6

LBPT, θ=317˚ 11.3 63.8 75.1

RBPT, θ=137˚ 9.9 57.3 67.2

RBPT, θ=317˚ 9.7 47.2 56.9

318

7.7.10 Quasi-static analyses

Due to the complexity and the high computational cost of dynamic FE

analyses, it is often preferred to employ simplified quasi-static methods to

investigate dynamic phenomena. Although quasi-static methods cannot properly

model the changes in soil stiffness and strength that take place during an

earthquake, they often give a reasonable estimate of the seismic loads. Therefore,

it is interesting to examine how the results of the dynamic analysis presented in

Section 7.7.8 compare with those obtained by a quasi-static method.

uff

usus

level of tunnels centre

Figure 7.50: Schematic representation of FE mesh configuration in quasi-static

analysis

Usually, the quasi-static analysis approximates the earthquake induced

inertia forces as a constant horizontal body force applied throughout the mesh. In

the present study however, a different approach was followed. Initially a

conventional static analysis, as described in Sections 7.7.3 and 7.7.7, was

undertaken to establish the static loads acting on the tunnels. Once the

construction sequence was modelled, the mesh was subjected to simple shear

conditions, as shown schematically in Figure 7.50. During the quasi-static

analysis the vertical displacements were restricted along all mesh boundaries,

while the horizontal displacements were restricted along the bottom boundary.

Furthermore, a uniform displacement us and a triangular displacement

distribution, as illustrated in Figure 7.50, were applied over 200 increments along

the top and the lateral boundaries of the mesh respectively. The displacement us

was calculated as follows:

Hγu maxs = =0.0019x195.0m=0.3705m

319

where H is the depth of the mesh and γmax is the maximum free-field shear strain

at the level of the tunnels calculated by the 1D analysis of the M2-SKH model in

Section 7.7.6.

0 60 120 180 240 300 360


10000

20000

30000

40000

50000

Ho

op

Str

es

s (

kP

a/m

)

0 60 120 180 240 300 360


-400

-200

0

200

400

Be

nd

ing

Mo

me

nt

(kN

m/m

)

0 60 120 180 240 300 360


4000

5000

6000

7000

8000

Th

rus

t (k

N/m

)

LBPT

RBPT

(a)

θ

(b)

(c)


distribution around the tunnels’ lining at the end of the quasi-static analysis

Figure 7.51 illustrates the maximum (i.e. calculated at the last increment)

accumulated thrust, bending moment and hoop stress distribution around the

tunnel linings computed with the M2-SKH model. In a similar fashion to the

results of the corresponding dynamic analysis (see Figure 7.45) the load

distribution is highly non-uniform around the tunnel linings and the maxima of

the thrust, bending moment and hoop stress variations occur at shoulder and knee

locations. Comparison of Figures 7.45 and 7.51, shows that the quasi-static

analysis predicts lower values of thrust than the dynamic analysis. Conversely,

the quasi-static analysis predicts much higher bending moments. The predicted

320

hoop stress variation by the two analyses, which combines the effect of the axial

force and the bending moment, is fairly similar.

While it is difficult to draw general conclusions from this set of analyses,

it seems that the quasi-static analysis’s results in terms of hoop stresses compare

reasonably well with those obtained by the corresponding dynamic analysis.

7.7.11 Comparison with analytical solutions

As discussed earlier, a number of simplified methods have been

developed to quantify the seismic ovaling effect on circular tunnels. In this

section, two of these approaches, the methods of Wang (1993) and Penzien

(2000) (see Section 7.6), are employed to calculate the seismic response of the

BPTs at chainage 62+850. The results of these simplified methodologies, in

terms of maximum hoop stress, are then compared with those obtained by the FE

method in Section 7.7.8 and with post-earthquake field observations.

Table 7.12: Analytical methods parameters

Parameter Soil (layer 4) LBPT RBPT

Em (kPa) 2.21x106 - -

νm 0.3 - -

El (kPa) - 28.0 x106 21.0 x10

6

νl - 0.2 0.2

t (m) - 0.3 0.3

I (m4/m) - 0.00225 0.00225

r (m) - 5.0 5.0

As noted in Section 7.6, the methods of Wang (1993) and Penzien (2000),

assuming either full-slip or non-slip conditions along the interface between the

ground and the lining, express the maximum thrust (Tmax) and the maximum

bending moment (Mmax) of the tunnel lining as a function of the maximum free-

field shear strain (γmax) at the level of the tunnel and properties of the soil and the

lining. The assumed parameters are listed in Table 7.12, while the γmax at the

321

level of the tunnels was taken from the 1D analysis with the M2-SKH model

equal to 0.19% (see Figure 7.27). Furthermore, Tables 7.13 and 7.14 summarize

the analytical results for the LBPT and RBPT respectively.

The Penzien approach seems not to be sensitive to the assumed condition

along the interface between the ground and the lining and in all cases predicts

much lower hoop stress values than those predicted by the Wang method.

Assuming that the maximum hoop stress acting on the tunnels lining due to static

loading was on average 10MPa (see Section 7.7.7), the total hoop stress based on

the Penzien method is then 24.15MPa and 20.7MPa for the LBPT and the RBPT

respectively. These values are much lower than the estimated strength of

shotcrete at the time of the earthquake (40MPa and 30MPa for the LBPT and

RBPT respectively). Consequently, as failure was observed in the field, it can be

concluded that the Penzien (2000) methodology underestimates the maximum

hoop stress developed due to the earthquake in the BPTs.

Table 7.13: Summary of analytical results for the LBPT

Wang (1993) Penzien (2000)

LBPT

Full Slip No Slip Full Slip No Slip

Tmax (kN/m) 81.8 3959.2 81.9 163.2

Mmax (kNm/m) 204.6 204.6 204.6 204.0

Hσ max(MPa) 13.9 26.8 13.9 14.15

Table 7.14: Summary of analytical results for the RBPT

Wang (1993) Penzien (2000)

RBPT

Full Slip No Slip Full Slip No Slip

Tmax (kN/m) 61.7 3742.7 61.7 123.2

Mmax (kNm/m) 154.4 154.4 154.4 154.0

Hσ max(MPa) 10.5 22.8 10.5 10.7

322

On the other hand, the Wang method gives much higher values of

maximum thrust for the no-slip assumption than for the full-slip assumption. As

discussed in Section 7.6, the full-slip condition is a reasonable approximation in

cases of tunnels in soft soils, while for tunnels in stiff soils (i.e. like the BPTs) it

leads to underestimation of the maximum thrust. It should be noted that the FE

analyses presented herein are more consistent with the no-slip assumption, at

least until the shear strength of the soil is mobilised at the tunnel-soil boundary.

For both BPTs the predicted seismic hoop stresses by the Wang method under

no-slip assumption compare reasonably well to those predicted by the FE

analysis in Table 7.10 (i.e. compare earthquake values).

Furthermore, assuming again that the static hoop stress was on average

10MPa, the total hoop stress acting on the lining based on the Wang method for

the no-slip assumption is then 36.8MPa and 32.8MPa for the LBPT and the

RBPT respectively. Thus, the Wang (1993) method, for the no-slip assumption,

predicts hoop stresses that match quite well with the post-earthquake field

observations.

7.7.12 2D nonlinear analyses at chainage 62+870

As discussed in Section 7.4, the Duzce earthquake caused striking

damage to the BPTs in the area that the two tunnels overlapped, but the leading

portion of the LBPT in the same material (i.e. fault gouge clay) did not suffer

extensive damage (see Figure 7.7). Two possible explanations were identified:

- During the seismic event the BPTs presumably interacted, as the

pillar between the tunnels is small. Thus, wave trapping in the pillar

possibly caused amplification of the ground motion at the area

where the BPTs overlap.

- The different stratigraphy of the cross section CD (i.e. at chainage

62+870) resulted in lower seismic loads at the LBPT compared to

those acting at the LBPT at the cross section AB (i.e. chainage

62+850).

323

To investigate these postulations, two sets of analyses were undertaken.

In the fist set of analyses, denoted in future discussions as 1BPT-AB, the

analyses of the cross-section AB (presented in Sections 7.7.7 and 7.7.8 and

denoted in future discussions as 2BPTs-AB) were simply repeated without the

RBPT. The purpose of this is to investigate whether the two tunnels interacted

during the seismic event by comparing the 1BPT-AB dynamic analysis results

with those previously obtained by the dynamic analysis 2BPTs-AB.

The second set of analyses, denoted in future discussions as 1BPT-CD,

concerns static and dynamic analyses of the stratigraphy of cross-section CD (see

Figure 7.9). The purpose of this set of analyses is to examine whether the

different stratigraphy resulted in lower seismic loads in the LBPT at chainage

62+870 compared with those predicted by the analysis 1BPT-AB. Figure 7.52

illustrates the finite element mesh used in the second set of analyses, which

consists of 5274 8-noded solid elements and 31 3-noded beam elements. The

depth of the mesh for the cross section CD is 183.0m, while the width was taken

the same as before (i.e. 219.0m).

20D=100m 20D=100m19m

183m


Metasediments

Fault Gouge Clay

Sandstone, Marl

Fault Brecciaand Fault Gouge Clay

x

z


the tunnel

324

It should be noted that when the earthquake struck, the shotcrete at chainage

62+870 was 8 days old. In this set of analysis the LBPT was constructed at 60%

stress relaxation and at the end of the excavation process was assigned the

material properties that correspond to the RBPT in Table 7.7 (as the RBPT’s

shotcrete at chainage 62+850 had similar age when the earthquake struck). All

other analysis arrangements (i.e. boundary conditions, time integration e.t.c.)

were kept the same as those used in the analyses of the cross section AB.

Figure 7.53 compares the maximum shear strain profiles (caused only by

the dynamic excitation) predicted by the three analyses (i.e. 2BPTs-AB, 1BPT-

AB and 1BPT-CD) at x=70.0m and at x=0.0m. The free-field response (i.e. at

x=70.0m) obtained by the 2BPT-AB and 1BPT-AB analyses is very similar. This

is not surprising, since if the width of the mesh and the lateral boundaries have

been appropriately chosen the free field response should not be affected by the

structure (see Section 7.7.4). On the other hand, the 1BPT-CD analysis predicts

lower free-field response for the fault gouge clay (i.e. layer 4) than the other two

analyses. Hence, although the thickness of the fault gouge clay layer is the same

in all analyses, the response of the gouge clay seems to be affected by the

thickness of the overlaying layer (i.e. metasediments). Conversely, the response

of the other materials does not seem to be significantly affected by the

stratigraphy. Furthermore, in all analyses, the maximum shear strain profile in

the pillar at the level of the tunnel (i.e. the centre of the tunnel is at z=160.0m) is

amplified with respect to the corresponding free-field profile. However, the

1BPT-AB analysis predicts lower amplification than the 2BPTs-AB analysis.

This difference is quite small, but it indicates that some interaction between the

tunnels takes place in the 2BPTs-AB analysis. Besides, the amplification is even

lower in the 1BPT-CD analysis, suggesting that the stratigraphy rather than the

interaction of the tunnels is the crucial parameter.

325

0 0.1 0.2 0.3


200

160

120

80

40

0

De

pth

(m

)

2BPTs-AB

1BPT-AB

1BPT-CD

(a) x=70.0m

0 0.1 0.2 0.3


200

160

120

80

40

0

De

pth

(m

)

(b) x=0.0m

Figure 7.53: Maximum shear strain profile computed with the 2BPTs-AB, the

1BPT-AB and the 1BPT-CD model at x=70.0m (a) and at x=0.0m (b)

Figure 7.54 illustrates the first 20 seconds of the shear strain time

histories recorded at the integration points R (x=69.26m, z=160.7m, i.e. free-

field location) and S (x=0.235m, z=160.7m, i.e. pillar location) for the three

analyses. Figure 7.54a shows that the 1BPT-CD analysis gives consistently the

lowest response, while the 1BPT-AB and 2BPTs-AB analyses predict almost

identical response. In the pillar, the maximum shear strain predicted by the

2BPTs-AB analysis is 17% higher than the one predicted by the 1BPT-AB

analysis and 33% higher than the one predicted by the 1BPT-CD analysis. It

should be noted that all analyses gave approximately the same value of

permanent shear strain at the end of the analysis.

0 10 20 30 40

Time (sec)

-0.3

-0.2

-0.1

0

0.1

0.2

Sh

ea

r S

tra

in (

%)

2BPT-AB

1BPT-AB

1BPT-CD

0 10 20 30 40

Time (sec)

-0.2

-0.1

0

0.1

0.2

Sh

ea

r S

tra

in (

%)

2BPT-AB

1BPT-AB

1BPT-BC

Figure 7.54: Shear strain time history computed with the 2BPTs-AB, the 1BPT-

AB and the 1BPT-CD model at integration point R

326

Figure 7.55 compares the results of the three analyses at t=10.0sec in

terms of accumulated thrust (compression positive), bending moment and hoop

stress distribution around the LBPT lining. All the results show that the LBPT

suffered the maximum loading at shoulder and knee locations (i.e. θ=137˚, 317˚)

and that the stratigraphy rather than the interaction of the tunnels is the governing

reason for the limited damage at cross section CD. In particular, while the

maximum difference in terms of hoop stress between the 1BPT-AB and the

2BPTs-AB analyses is only 7% at θ=190˚, the corresponding difference between

the 1BPT-CD the 2BPTs-AB analyses reaches 23% at θ=227˚.

0 60 120 180 240 300 360


20000

25000

30000

35000

40000

45000

Ho

op

Str

es

s (

kP

a/m

)

0 60 120 180 240 300 360


-300

-200

-100

0

100

200

Be

nd

ing

Mo

me

nt

(kN

m/m

)

0 60 120 180 240 300 360


6000

6400

6800

7200

7600

8000

Th

rus

t (k

N/m

)

2BPTs-AB

1BPT-AB

1BPT-CD

(a)

θ

(b)

(c)


distribution around the tunnels’ lining at t=10.0sec computed with the 2BPTs-

AB, the 1BPT-AB and the 1BPT-CD model

Table 7.15 summarizes the predicted maximum hoop stress at the LBPT

by the three analyses. It is interesting to note that the 2BPTs-AB and 1BPT-AB

327

analyses predict the same total maximum hoop stress while that obtained by the

1BPT-CD analysis is only 10% lower. Overall the 1BPTs-CD analysis’s results

show that the LBPT was subjected to lower loads at chainage 62+870. However,

the predicted maximum hoop stress exceeds the 8-days shotcrete strength which

is estimated to 30.0 MPa. As discussed earlier, since the lining is modelled as a

linear elastic material, it is expected that all three analyses overestimate to some

extent the loads that were actually acting on it.

Table 7.15: Maximum hoop stress developed at the LBPT for various analyses


Analysis Static Earthquake Total

2BPTs-AB 12.5 29.0 41.5

1BPT-AB 12.2 29.1 41.3

1BPT-CD 10.3 26.9 37.2

In conclusion, it was shown that the interaction of the BPTs and any

potential wave trapping in the pillar had only a minor effect on the seismic tunnel

performance. On the other hand, comparison of the 2BPTs-AB and 1BPT-CD

analyses showed that the differences in stratigraphy considerably affect the

tunnel response. However, these differences cannot fully explain the lack of

serious damage in the cross section CD. To further investigate this, a full 3D

model and a more realistic modelling of the tunnel linings are needed.

7.8 Conclusions

This chapter presented a case study of the Bolu highway twin tunnels that


in Turkey. Attention was focused on a particular section of the left tunnel that

was still under construction when the earthquake struck and that experienced

extensive damage during the seismic event. At the time of the earthquake only

the two bench pilot tunnels (BPTs), which were to be back-filled with concrete to

328

provide a stiff foundation for the top heading of the main left tunnel, had been

constructed. The BPTs were only supported by 30cm thick shotcrete and HEB

100 steel rib sets at 1.1m longitudinal spacing. The post-earthquake

investigations showed that the damage was limited to a zone of fault gouge clay

where the two tunnels overlapped. The leading portion of the left BPT (LBPT),

where the adjacent RBPT had not been constructed, in the same material did not

suffer extensive damage. Static and dynamic plane strain 2D FE analyses were

undertaken to investigate the seismic tunnel response at two sections and to

compare the results with the post-earthquake field observations. The analyses of

the first section (section AB) refer to the area that the two BPTs overlap, while

the analyses of the second section (section CD) refer to the area where the

leading portion of the LBPT did not experience severe damage.

To specify an adequate FE model for the case study a series of numerical

tests were first carried out. Hence, drained linear elastic analyses were

undertaken to check the adequacy of the mesh width and lateral boundary

conditions. The 2D analysis with viscous dashpots along the lateral boundaries

failed to reproduce the free-field response, while the predicted motion in the

vicinity of the tunnels was seriously damped. It was shown that the viscous

dashpots, although widely used in engineering practice, lead to a serious

underestimation of the response when placed close to the excitation. On the other

hand, the 2D analysis with the TDOF boundary condition modelled very well the

free-field response. The TDOF method can perfectly model the one-dimensional

soil response, but it cannot absorb any waves radiating away from the structure

and thus it can result in wave-trapping into the mesh. The analysis with the DRM

showed that the waves radiating away from the tunnels are negligible. Therefore,

it was shown that for the present case study the TDOF method was an adequate

boundary condition.

In addition, constitutive modelling is a very important aspect of dynamic

FE analyses. The employed model should be able to capture features of soil

behaviour when subjected to cyclic loading. In order to decide which constitutive

model is appropriate for the present case study, dynamic undrained FE 1D

analyses were undertaken with the modified Cam Clay model in combination

329

with the small strain stiffness model of Jardine et al (1986) (MCCJ) and with the

two-surface kinematic hardening model (M2-SKH) of Grammatikopoulou

(2004). The results of the 1D FE analyses were compared with those obtained by

equivalent linear analysis using the site response software EERA (Bardet et al

2000). It was shown that the plasticity introduced in the MCCJ analysis is

insufficient, as the model has an unrealistically large yield surface. Hence the

inability of this model to mimic hysteretic behaviour leads to unrealistic

predictions. On the other hand, the M2-SKH model appropriately captured

features of the soil behaviour when subjected to cyclic loading such as hysteretic

damping and plastic deformation during unloading. Therefore, the 2D

investigations of the case study were carried out with the M2-SKH model.

Once the spatial discretization, the boundary conditions, the time

integration and the constitutive model were specified, FE plane strain analyses

were carried out for the cross section AB. Initially 2D static analyses were

undertaken to simulate the construction of the tunnels and to establish the static

stresses that were acting on the lining at the time of the earthquake. The static

analyses results were in agreement with the observed behaviour of the tunnels

reported by Menkiti (2001b). Furthermore, the subsequent dynamic analyses

showed an ovaling deformation of the tunnels, with the maxima of the thrust,

bending moment and hoop stress occurring at shoulder and knee locations of the

lining. This is in agreement with post-earthquake field observations at the

collapsed section of the LBPT that show crushing of shotcrete and buckling of

the steel ribs at shoulder and knee locations of the lining (see Figure 7.6).

Besides, the predicted maximum total hoop stress values exceed the strength of

shotcrete in both tunnels and they thus match favourably with the observed

failure. However, since the cracking of the lining was not modelled in the present

study, the predicted loads might overestimate to some extent the loads that were

actually acting on it.

The analyses of section AB were repeated with the MCCJ model to

evaluate by how much the choice of constitutive model can affect the predicted

tunnel response. The results verified the above-mentioned conclusions of the 1D

numerical tests. In particular, although the MCC model predicted very well the

330

static response of the BPTs, its inability to mimic hysteretic behaviour led to a

significant overestimation of the seismic loads acting on the tunnels lining.

Furthermore the dynamic analysis results of section AB were compared

with those obtained by a quasi-static method. While significant differences were

observed in the thrust and bending moment distributions around the lining, the

resulting hoop stress distributions were in reasonable agreement.

The results of the dynamic analyses of section AB were also compared

with those obtained by the simplified analytical methods of Wang (1993) and

Penzien (2000). It was shown that the Wang (1993) method, assuming no-slip

between soil and lining, predicts hoop stresses that match quite well with the

dynamic FE analyses and the post-earthquake field observations. On the other

hand, the Penzien (2000) method underestimated the maximum hoop stress

developed due to the earthquake in the BPTs.

Finally, FE analyses were undertaken at section CD, to investigate why

the leading portion of the LBPT tunnel did not experience severe damage. The

different stratigraphy of the cross section CD initially was not modelled in order

to isolate the effect of the dynamic interaction of the two BPTs. It was shown

however, that the interaction of the BPTs and thus any wave trapping in the pillar

had only a minor effect on the seismic tunnel performance. On the other hand,

when the differences in the stratigraphy were taken into the account, the LBPT

response in section CD was considerably lower than it was in section AB.

However, the predicted maximum hoop stress exceeded the shotcrete strength at

section CD. Therefore the FE analyses cannot fully explain the lack of serious

damage in the cross section CD. To further investigate this, a full 3D model with

a more realistic modelling of the tunnel linings would be needed.

331

Chapter 8:

CONCLUSIONS AND RECOMMENDATIONS

8.1 Introduction

The aim of this research was to further develop the existing dynamic

capabilities of the geotechnical finite element program ICFEP and then to apply

them to a geotechnical earthquake engineering case study. Chapter 2 details the

finite element theory for static geotechnical problems, while the essential

extensions to this theory to perform dynamic analyses are given in Chapter 3. A

literature review of some of the available time marching schemes in Chapter 3

identified an efficient time integration scheme, the generalized-α algorithm (CH)

of Chung & Hulbert (1993), which is able to perform accurately and

economically dynamic finite element analyses. This method was extended to deal

with coupled consolidation problems and was then implemented into ICFEP. In

Chapter 4 the newly implemented algorithm is first validated and then compared

with other commonly used time marching schemes in a geotechnical boundary

value problem.

The second development of the program involved the incorporation of

absorbing boundary conditions, which can model the radiation of energy towards

infinity in a truncated domain. The first part of Chapter 5 presents a literature

review of some of the available boundary conditions for solving wave

propagation problems in unbounded domains. Based on the conclusions of this

review, two well-established absorbing boundary conditions, the standard

viscous boundary and the cone boundary, were chosen for implementation into

ICFEP. In the second part of Chapter 5, the results of validation exercises of the

newly implemented boundaries highlight important features of these boundaries

and identify their limitations. To overcome some of the identified shortcomings,

it was decided to use the newly implemented boundaries in conjunction with the

domain reduction method (DRM). Hence Chapter 6 explores the dual role of the

332

DRM, as this method not only reduces the domain that has to be modelled

numerically, but in conjunction with the standard viscous boundary or the cone

boundary also serves as an advanced absorbing boundary condition. In addition,

Chapter 6 discusses the original formulation of the DRM, its extension to deal

with coupled consolidation problems, its implementation into ICFEP and finally

the validation of method in a boundary value problem.

Chapter 7 presents the final task of the thesis which was to use the

modified dynamic version of ICFEP to analyse a case study. For this purpose it

was chosen to investigate the case of the Bolu highway twin tunnels that


in Turkey.

This chapter summarises the main conclusions reached in the previous

chapters and makes recommendations for related further research.

8.2 Direct integration method

Fundamental aspects of dynamic finite element theory were addressed,

including a presentation of the finite element formulation of dynamic equilibrium

and brief discussions of the various constitutive procedures to model soil

behaviour under cyclic loading and of the special spatial discretization

requirements in wave propagation problems. Attention was then focused on some

of the various time integration techniques that approximate the solution of the

equation of motion with a set of algebraic equations in a step-by-step manner.

After conducting a comparative study of some of the most popular integration

schemes, the CH algorithm was chosen to be implemented into ICFEP. This

method, which was also extended to deal with coupled consolidation problems,

was validated using analytical solutions and published numerical solutions.

Finally the behaviour of the CH scheme was compared with other commonly

used time marching schemes by analysing a problem of a deep foundation

subjected to various seismic excitations.

333

8.2.1 Selection and implementation of time integration scheme

The main requirements of an efficient time marching algorithm are

unconditional stability for linear problems, second order accuracy and

controllable numerical dissipation in the high frequency modes. The

unconditional stability makes the convergence of the solution independent of the

size of the time step ∆t. On the other hand, the accuracy of the solution always

depends on the size of the time step, but for a given time step it exclusively

depends on the adopted integration method. Furthermore, the role of numerical

damping is to eliminate spurious high frequency oscillations that are introduced

into the solution due to poor spatial representation of the high-frequency modes.

It is therefore desirable to preferentially “filter” the inaccurate high frequency

modes without affecting the important low frequency ones.

The accuracy and the numerical behaviour of some of the various

available integration schemes were compared in an analytical way. For linear

problems the most common procedure is to undertake an eigenvalue analysis of a

single-degree of freedom (SDOF) system subjected to a free vibration. The

results of this analysis, which is often referred to as a spectral stability analysis,

showed that among the considered unconditionally stable dissipative schemes,

the CH algorithm method is the most accurate and possesses the best numerical

dissipation characteristics. In particular, the great advantage of the CH method is

that it allows the user to control the amount of numerical dissipation at the high

frequency limit, without significantly affecting the lower modes.

The fundamental idea of the CH method is the evaluation of the various

terms of the equation of motion at different points within the time step. This

algorithm employs Newmark’s equations for the displacement and velocity

variations within the time step and it introduces two additional parameter mα , fα

into the equation of motion. The CH algorithm, adopting the Modified Newton

Raphson method to solve iteratively the governing finite element equation, was

formulated for dynamic nonlinear analysis. Finally, the CH method, employing a

“u-p” formulation which uses as primary variables the solid phase displacement

and the pore fluid pressure, was extended to deal with coupled consolidation

problems.

334

8.2.2 Validation

The closed form solution of a SDOF system subjected to a harmonic

oscillation was employed to verify the uncoupled dynamic formulation of ICFEP

for both solid and beam elements. This example was also used to compare the

behaviour of the CH scheme with two commonly used variations of Nemark’s

method (i.e. NMK1, NMK2). It was demonstrated that for the same time step, the

CH and the NMK1 methods achieve better agreement with the closed form

solution than the NMK2 method. Furthermore, the analytical solution of

Zienkiewicz et al (1980a) for the steady state response of a consolidating soil

column subjected to harmonic loading, was used to verify the formulation of the

CH algorithm for dynamic coupled consolidation problems. It was also shown

that the inclusion of the inertia term in the pore fluid equation of continuity

improves the accuracy of the numerical solution for events that lie on the limit up

to which the “u-p” approximation provides sufficient accuracy. Finally numerical

tests by Prevost (1982), Meroi et al (1995) and Kim et al (1993) were used to

validate ICFEP’s dynamic coupled consolidation formulation for both small and

large deformation analysis.

8.2.3 Conclusions from the deep foundation analysis

The behaviour of the CH scheme was compared with more commonly

used schemes (NMK1, NMK2, HHT and WBZ) in a boundary value problem. In

particular, two-dimensional plane strain analyses of a deep foundation for

various earthquake loadings and for various soil properties were undertaken. In

this study the emphasis was placed on the behaviour of the different integration

schemes and not on a thorough investigation of the seismic response of deep

foundations. Hence, a simple elastic perfectly plastic constitutive model was

used.

In the first set of analyses, the foundation response to a seismic excitation

was compared for various levels of high frequency dissipation (i.e. ρ∞ equal to

1.0, 0.818, 0.6, 0.42 and 0.0). The displacement response, which is dominated by

the low-frequency components of an excitation, appeared to be relatively

insensitive to the level of high frequency dissipation. On the other hand,

335

comparing the acceleration response, which is dominated by the high-frequency

components of an excitation, for the various levels of high frequency dissipation

it was shown that the introduction of numerical damping in the CH scheme

eliminates the high frequency noise. Hence the parametric study indicated that

the CH scheme maintains in elasto-plastic analyses its ability to filter the high

frequency modes without significantly affecting the low frequency response, at

least for the problems considered herein.

The second set of analyses investigated the effect of the frequency

content of the excitation on the behaviour of five algorithms (CH, HHT, WBZ,

NMK1 and NMK2). The CH algorithm was found to be insensitive to the

predominant frequency of the input motion and to give similar results to that of

the NMK1 scheme in terms of displacements. The predominant frequency of the

excitation affected more the performance of the NMK2 than the HHT and WBZ

algorithms. Furthermore the acceleration response showed that spurious

oscillations dominate the results of the NMK1, whereas the α schemes (i.e. CH,

HHT and WBZ) perform satisfactorily.

The last set of analyses investigated the effect of the numerical model’s

natural frequencies on the performance of integrations schemes. Hence the

dynamic analyses for one of the seismic excitations (VELS recording) were

repeated for various values of soil stiffness. The CH algorithm was found to be

less sensitive than the HHT and the NMK2 schemes to the fundamental

frequency of the numerical model. However the accuracy of all three schemes

deteriorates in the case that the fundamental frequency of the soil layer is equal

to the predominant frequency of the excitation (i.e. at resonance). Finally

regarding the relative computational costs, the CH was found to be the most

efficient method, whereas the NMK1 was the most expensive.

Overall the results of the deep foundation analysis showed that the CH

scheme maintains in elasto-plastic analyses its favourable features. It was also

shown in a practical application that the choice of integration scheme can

seriously affect the accuracy of the predictions and the computational cost.

Equally important was the choice of appropriate algorithmic parameters. Before

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selecting these parameters, one should carefully consider the frequency content

of the excitation and the possibility of a resonance condition.

8.3 Modelling the unbounded medium

One of the major issues in dynamic analyses of soil-structure interaction

problems is to model accurately and economically the far-field medium. The

most common way is to restrict the theoretically infinite computational domain

to a finite one with artificial boundaries. The reduction of the solution domain

makes the computation feasible, but spurious reflections from the artificial

boundaries can seriously affect the accuracy of the results.

Numerous artificial boundaries have been proposed in the literature over

the last 30 years. A literature review on some of the most important boundary

conditions for solving wave propagation problems in unbounded domains

discussed the relative merits and disadvantages of each method. Based on this

review it was decided to implement into ICFEP two well-established boundaries:

the standard viscous boundary of Lysmer and Kuhlemeyer (1969) and the cone

boundary of Kellezi (1998). The standard viscous boundary can be described by

two series of dashpots oriented normal and tangential to the boundary of the FE

mesh, while the cone boundary, in addition to dashpots, consists of springs which

are also oriented normal and tangential to the boundary of the mesh.

To improve the efficiency of the newly implemented boundaries it was

decided to use them in conjunction with the domain reduction method (DRM).

Hence the DRM was also implemented into ICFEP based on the derivation of

Bielak et al. (2003) and it was further developed to deal with dynamic

consolidation problems.

8.3.1 Conclusions from the investigation and implementation of absorbing

boundary conditions

The numerical examples of Kellezi (1998, 2000) and the closed form

solution of Blake (1952) were used to verify the implementation the above-

337

mentioned boundaries into ICFEP for plane strain and axisymmetric analysis

respectively. The validation exercises verified the accuracy of the

implementation, but also highlighted important features of the transmitting

boundaries. It was shown that the reliability of the transmitting boundaries

depends on the size of the model. The observations of the present study agree

with the general suggestion of Kellezi (1998) that an absorbing boundary should

not be placed closer than (1.2-1.5) λS from the excitation source.

It was also shown that the ability of both boundaries to absorb reflected

waves is very similar. This is not surprising since they have the same dashpot

coefficients. The cone boundary shows no improvement compared to the viscous

boundary for high frequencies of excitation, but it appears to have superior

behaviour for low frequencies. The greater advantage of the cone boundary is

that thanks to its “spring” terms it approximates the stiffness of the unbounded

soil domain. Thus, it eliminates the permanent movement that can occur for low

frequencies with the viscous boundary.

In addition, the ability of the transmitting boundaries to absorb Rayleigh

waves was also investigated. Models with both boundaries predicted reasonably

the displacement response for all wave periods and Poisson’s ratios. However

with respect to the stress response, the errors were tolerable only for small

periods or for values of Poisson’s ratio greater than 0.25. Regarding the Rayleigh

wave absorption, the cone boundary did not appear to be more accurate than the

standard viscous boundary.

Finally the performance of the boundaries was examined for the case of

plane strain analysis of a soil layer with vertically varying stiffness. The

dispersive nature of generalized SV/P waves complicates considerably the wave

field of a vertically heterogeneous half-space. However, both boundaries

predicted reasonably well the displacement and the stress response. This implies

that the boundaries might absorb waves travelling at different speeds. Thus, it

can be postulated that even slow-moving nonlinear waves can be absorbed by the

transmitting boundaries.

338

Overall, the behaviour of the newly implemented boundaries was found

satisfactory when they are placed at a sufficient distance from the excitation

source. The cone boundary was found to be more accurate than the viscous

boundary for low frequencies and therefore its use is preferred. A limitation

however of the cone boundary is that its formulation requires as an input

parameter the distance of the boundary from the excitation source. Hence the use

of the cone boundary is restricted to problems with surface excitations (e.g.

dynamic pile loading, moving vehicles) where the distance of a boundary from

the source is known. Conversely, the cone boundary cannot be used in seismic

soil-structure interaction problems as it is difficult to include the seismic source

(fault) in the numerical model.

8.3.2 Conclusions from the implementation and validation of the Domain

Reduction Method

The domain reduction method (DRM) is a two step procedure that aims at

reducing the domain that has to be modelled numerically. In step I of the DRM a

simplified background model is analysed that includes the source of excitation,

but not the area of interest (that contains geotechnical structures or localised

geological features). The aim of the step I analysis is to calculate and store the

incremental free field response of a single layer of finite elements within the

boundaries Гe and Г. The second step is performed on a reduced domain that

comprises of the area of interest Ω and of a small external region Ω+. The nodal

effective forces ∆Peff, calculated from the results of step I, are applied to the

model of step II at the elements located within the boundaries Гe and Г. In the

case of coupled consolidation analysis the effective forces ∆Peff include

additional terms derived from the free field incremental pore pressures. The

perturbation in the external area Ω + is only outgoing and corresponds to the

deviation of the area of interest from the background model.

The development of the DRM for coupled consolidation problems and its

implementation was verified numerically both for linear and nonlinear analyses.

For the numerical test, the internal area Ω0 of the background model was

identical to the internal area Ω of the reduced model. In the step II analysis since

339

there is no deviation from the background model, zero response was calculated in

the external Ω+ area of the reduced model. Furthermore, the computed responses

in steps I and II were found to be identical for the internal areas Ω0 and Ω.

Generally, in seismic soil-structure problems the excitation is applied as

an acceleration time history at the bottom of the mesh. Hence, in conventional

finite element analysis, absorbing boundaries cannot be employed at the bottom

of the mesh together with the excitation. This restricts the applicability of these

boundaries to the lateral boundaries of the mesh. The great advantage of the

DRM is that the excitation is directly introduced into the computational domain,

leaving more flexibility in the choice of appropriate boundary conditions. Taking

advantage of this feature of the DRM, a methodology was suggested which

allows the use of the cone boundary in seismic soil-structure interaction

problems. In particular, the cone boundary was employed on the external

boundary +Γ of the reduced domain in an analysis of a cut and cover tunnel. For

the sake of comparison, the analyses were repeated with the viscous boundary,

while to verify the applicability of the cone boundary, the step II analyses were

also repeated with an extended mesh. The cone boundary was found to be

slightly superior to the viscous boundary. Both boundaries were subjected to a

quite challenging numerical test and they both performed very well. This agrees

well with the conclusion of Yoshimura et al (2003) that absorbing boundaries


energy.

8.4 Case study on seismic tunnel response

This final research topic of this thesis was the investigation of a

geotechnical earthquake engineering case study. The objectives of this study

were the investigation of theoretical issues of the dynamic finite element method

like spatial discretization, absorbing boundary conditions, time integration and

constitutive modelling, on a practical application and the qualitative and

quantitative comparison of finite element analysis results with simplified

analytical methods and with post-earthquake field observations.

340

The Bolu highway twin tunnels that experienced a wide range of damage

severity during the 1999 Duzce earthquake in Turkey, establish a well-

documented case, as there is information available regarding the ground

conditions, the design of the tunnels, the ground motion and the earthquake

induced damage. The focus in this thesis was placed on a particular section of the

left tunnel that was still under construction when the earthquake struck and that

experienced extensive damage during the seismic event. At the time of the

earthquake only the two bench pilot tunnels (BPTs), which were to be back-filled

with concrete to provide a stiff foundation for the top heading of the main left

tunnel, had been constructed. The BPTs were only supported by 30cm thick

shotcrete and HEB 100 steel ribs, set at 1.1m longitudinal spacing. The post-

earthquake investigations showed that the damage was limited to a zone of fault

gouge clay where the two tunnels overlap. Interestingly, the leading portion of

the left BPT (LBPT) in the same material did not suffer extensive damage. Static

and dynamic plane strain 2D FE analyses were undertaken to investigate the

seismic tunnel response in two sections and to compare the results with the post-

earthquake field observations. The analyses of the first section (section AB)

refers to the area that the two BPTs overlap, while the analyses of the second

section (section CD) refers to the area where the leading portion of the LBPT did

not experience severe damage.

8.4.1 Lessons learned from the numerical investigation of the case study

To specify an adequate FE model for the case study a series of numerical

tests were first carried out. The chosen FE configuration modelled the ground

stratigraphy down to the bedrock, at a depth of 195.0m for section AB, while the

tunnels themselves are also located at a great depth (the centre line is at a depth

of z=160.0m). The bedrock was assumed to act as a rigid boundary. Hence, the

relevant acceleration time history was applied incrementally in the horizontal

direction to all nodes along the bottom boundary of the FE model, while the

corresponding vertical displacements were restricted. Drained linear elastic

analyses were undertaken to check the adequacy of the mesh width and lateral

boundary conditions. As mentioned earlier, without making use of the DRM,

absorbing boundaries cannot be employed at the bottom of the mesh together

341

with the excitation and they thus can only be applied along the lateral boundaries

of the mesh. The 2D linear elastic analysis of section AB with viscous dashpots

along the lateral boundaries failed to reproduce the free-field response, while the

predicted motion in the vicinity of the tunnels was seriously damped. Hence it

was shown that the viscous dashpots, although widely used in engineering

practice, lead to a serious underestimation of the response when placed close to

the excitation (i.e. the bottom boundary of the mesh). Thus, the viscous

boundaries were replaced by the tied degrees of freedom (TDOF) boundary

condition. The 2D analysis with the TDOF boundary condition modelled very

well the free-field response. This method can perfectly model the one-

dimensional soil response, but it cannot absorb any waves radiating away from

the structure and thus it can result in wave-trapping into the mesh. The 2D linear

elastic analysis of section AB was finally repeated with the DRM in conjunction

with the standard viscous boundary. In the present case study the DRM could not

be used to reduce the computational domain, as the tunnels are located very close

to the bedrock (“source”), but it was used as an advanced boundary condition

together with the viscous dashpots. The analysis with the DRM showed that the

waves radiating away from the tunnels are negligible. Therefore, it was

demonstrated that for the present case study the TDOF method is an adequate

boundary condition.

A second series of numerical tests were carried out to decide which

constitutive model is appropriate for the case study. Dynamic undrained FE 1-D

analyses were undertaken with the modified Cam Clay model in combination

with the small strain stiffness model of Jardine et al (1986) (MCCJ) and with the

two-surface kinematic hardening model (M2-SKH) of Grammatikopoulou

(2004). The results of the 1D FE analyses were compared with those obtained by

equivalent linear analysis with the site response software EERA (Bardet et al

2000). It was shown that the plasticity introduced in the MCCJ analysis is

insufficient, as the model has an unrealistically large yield surface. Hence the

inability of this model to mimic hysteretic behaviour leads to unrealistic

predictions. On the other hand, the M2-SKH model appropriately captured

features of the soil behaviour when subjected to cyclic loading like hysteretic

342

damping and plastic deformation during unloading. Therefore, the 2D

investigations of the case study were carried out with the M2-SKH model.

8.4.2 Conclusions regarding the comparison of the finite element analyses with

the post-earthquake field observations

Once the spatial discretization, the boundary conditions, the time

integration and the constitutive model were specified, FE plane strain analyses

were carried out for the cross section AB. Initially 2D static analyses were

undertaken to simulate the construction of the tunnels and to establish the static

stresses that were acting on the lining at the time of the earthquake. The static

analyses results were in agreement with the observed behaviour of the tunnels

reported by Menkiti (2001b). Furthermore, the subsequent dynamic analyses

showed an ovaling deformation of the tunnels, with the maxima of the thrust,

bending moment and hoop stress occurring at shoulder and knee locations of the

lining. This is in agreement with post-earthquake field observations at the

collapsed section of the LBPT that show crushing of shotcrete and buckling of

the steel ribs at shoulder and knee locations of the lining. In addition, the

predicted maximum total hoop stress values exceeded the strength of the

shotcrete in both tunnels and they thus matched favourably with the observed

failure. However, since the cracking of the lining was not modelled in the present

study, the predicted loads might overestimate to some extent the loads that were

actually acting on it in the field.

In addition, the analyses of section AB were repeated with the MCCJ

model to evaluate how much the choice of constitutive model can affect the

predicted tunnel response. The results verified the above-mentioned conclusions

of the 1D numerical tests. In particular, although the MCCJ model predicted very

well the static response of the BPTs, its inability to mimic hysteretic behaviour

led to a significant overestimation of the seismic loads acting on the tunnel

linings.

The dynamic analysis’s results of section AB were also compared with

those obtained by a quasi-static method. While significant differences were

343

observed in the thrust and bending moment distributions around the lining, the

resulting hoop stress distributions were in reasonable agreement.

The results of the dynamic analyses of section AB were also compared

with those obtained by the simplified analytical methods of Wang (1993) and

Penzien (2000). It was shown that the Wang (1993) method, under a no-slip

assumption, predicted hoop stresses that match quite well with the dynamic FE

analyses and the post-earthquake field observations. On the other hand, the

Penzien (2000) method underestimated the maximum hoop stress developed due

to the earthquake in the BPTs.

Finally, FE analyses were undertaken for section CD, to investigate why

the leading portion of the LBPT tunnel did not experience severe damage. The

different stratigraphy of the cross section CD initially was not modelled in order

to isolate the effect of the dynamic interaction of the two BPTs. It was shown

however, that the interaction of the BPTs and thus any wave trapping in the pillar

had only a minor effect on the seismic tunnel performance. On the other hand,

when the differences in the stratigraphy were taken into account, the LBPT

response (i.e. stresses) at section CD was considerably lower than it was at

section AB. However, the predicted maximum hoop stress exceeded the shotcrete

strength at section CD. Therefore the undertaken FE analyses cannot fully

explain the lack of serious damage at the cross section CD.

8.5 Recommendations for Further Research

The recommendations for future research are divided into three parts. The

first suggests a direction for further improving the newly implemented time

integration scheme and additional research work that could be carried out to

obtain a better understanding of the behaviour of time integration schemes in

geotechnical earthquake engineering problems. The second part discusses

examples of additional development work that should be undertaken to further

improve the way that ICFEP models the far-field medium in dynamic analysis.

Finally the last part discusses further research work that could be carried out to

344

further improve the FE prediction regarding the seismic behaviour of the

considered sections of the Bolu tunnels.

8.5.1 Time integration

As discussed in Chapter 2 the Modified Newton Raphson method, which

was employed to iteratively solve the nonlinear global equilibrium equation, is

relatively insensitive to the increment size in static analyses. However, Crisfield

(1997) notes that in nonlinear dynamics an error, which depends on the

increment size, is introduced in the solution. This error is associated with the

time integration and can be controlled if an automatic time step algorithm is

employed. As the optimal time step size may change during the computation,

time step control algorithms automatically adjust the time step to maximize

accuracy. Hulbert and Jang (1995) and Chung et al (2003) introduced strategies

for automated adaptive selection of the time step in the CH method. ICFEP has

an automatic step algorithm for static analyses. Taking into account the above-

mentioned approaches the existing automatic step algorithm of ICFEP could be

extended to dynamic analyses to further improve the efficiency of the CH

scheme.

In addition, although the implementation of the CH algorithm was

extensively validated for simple coupled consolidation problems, the behaviour

of the scheme in a full scale geotechnical application involving coupled

consolidation analysis remains to be investigated.

8.5.2 Modelling the unbounded medium

All the analyses in this thesis were restricted to two dimensional

geometries. Hence an obvious direction for future research would be the

extension of the implemented absorbing boundary conditions and the DRM to

three dimensional space. This will obviously require the identification of suitable

analytical solutions and published numerical tests to validate the new

developments in 3D space. The particular challenge with the DRM would be to

345

find a way to deal with the increased storage cost that the 3D implementation

will require.

The numerical investigation of the Bolu tunnels case study showed that

tied degrees of freedom are a suitable boundary condition for the side boundaries

of the mesh in the case that the waves radiating away from the structure are

negligible. Currently, there are no clear guidelines for which cases the dynamic

soil-structure interaction is important and consequently the waves radiating away

from the structure are not negligible. Hence it would be useful to conduct a

parametric study to investigate the effects of the relative stiffness between soil

and structure, of the friction between the soil and structure and of the embedment

depth on the dynamic soil-structure interaction.

Furthermore, the literature review of Chapter 5 showed that the use of

infinite elements and of the perfectly matched layer (PML) method is rapidly

becoming popular in modelling accurately and economically the far-field

medium. An important limitation of these two methods is that the excitation

cannot be prescribed at the same mesh boundary that the absorbing boundary is

applied. Thus these methods can directly deal only with problems that the

excitation is on the free surface (e.g. dynamic pile loading, moving vehicles). As

noted earlier, an important feature of the DRM is that it introduces the excitation

into the computational domain. This allows flexibility in the choice of absorbing

boundary conditions. Therefore, it would be interesting to explore the behaviour

of infinite elements and the perfectly matched layer method in conjunction with

the DRM in problems of seismic soil-structure interaction.

8.5.3 Case study on seismic tunnel response

Generally the results of the undertaken FE analyses matched reasonably

well with the reported post-earthquake observations. It was however postulated

that the FE analyses overestimated to a certain extent the stresses that were

acting on the tunnels’ lining during the earthquake. Furthermore, the undertaken

FE analyses did not fully explain the lack of serious damage to the cross section

CD. In the present study the beam elements, modelling the lining, were assumed

to behave as a linear elastic material and thus the cracking of the lining could not

346

be modelled. It is believed that adopting a more realistic model for the tunnel

would improve the FE analyses predictions. In addition, as discussed in Section

7.7.2 there is a certain degree of uncertainty regarding the input ground motion.

A detailed seismological study resulting in a more realistic input ground motion

could also improve the predictions. Finally, to fully understand the lack of

serious damage to the cross section CD a three dimensional model would be

needed.

347

References

Achenbach J.D. (1973), Wave Propagation in Elastic Solids, North-Holland

Publishing Company.

Akkar S. & Gülkan, (2002), “A critical examination of near-field accelerograms

from the sea Marmara region earthquakes”, Bulletin of the Seismological

Society of America, Vol. 92, No 1, pp. 428-447.

Akyüz H.S., Hartleb R., Barka A., Altunel E., Sunal G., Meyer B. & Armijo R.,

(2002), “Surface Rupture and Slip Distribution of the 12 November 1999

Düzce Earthquake (M 7.1), North Anatolian Fault, Bolu, Turkey”,

Bulletin of the Seismological Society of America, Vol. 92, No 1, pp. 61-

66.

Al-Tabbaa A. & Wood D., (1989), “An experimentally based bubble model for

clay”, NUMOG III, Niagara Falls, pp. 91-99.

Ambraseys N.N., Douglas J., Sigbjörnsson R., Berge-Thierry C., Suhadolc P.,

Costa G. & Smit P. M. (2004), “Dissemination of European Strong-

Motion Data, using Strong-Motion Datascape Navigator”, CD ROM

collection, Engineering and Physical Sciences Research Council, United

Kingdom.

Archuleta, R.J., Steidl, J.H. & Bonilla L.F., (2000), “Engineering insights from

data recorded on vertical arrays” Proceedings of the 12th World

Conference in Earthquake Engineering, Paper 2681, New Zealand.

Argyris J. & Mlenjek H.P. (1991), Dynamics of Structures, Elsevier Science

Publishers, Amsterdam: North Holland.

Argyris J.H., Dunne P.C. & Angelopoulos T. (1973), “Dynamic response by

large step integration”, Earthquake Engineering & Structural Dynamics,

Vol.2, pp. 203-250.

348

Astley R.J. (1983), “Wave envelope and infinite elements for acoustical

radiation”, International Journal for Numerical Methods in Fluids, Vol.3,

pp.507-526.

Astley R.J. (1996), “Transient wave envelope elements for wave problems”,

Journal of Sound and Vibration, Vol.192, No.1, pp.245-261.

Astley R.J., Coyette J.P. & Cremers L. (1998), “Three-dimensional wave-

envelope elements of variable order for acoustic radiation and scattering.

Part II. Formulation in the time domain”, Journal of Acoustical Society of

America, Vol. 103, No. 1, pp. 64-72.

Bardet J.P., Ichii K. & Lin C.H. (2000), “EERA, A computer program for

Equivalent linear Earthquake site Response Analysis of layered soils

deposits” University of Southern California, Los Angeles.

Basu U. & Chopra A.K. (2003a), “Perfectly matched layers for time-harmonic

elastodynamics of unbounded domains: theory and finite-element

implementation”, Computational Methods in Applied Mechanical

Engineering, Vol. 192, pp. 1337-1375.

Basu U. & Chopra A.K. (2003b), “Perfectly matched layers for elastic waves and

applications to earthquake engineering analysis of dams” Proceedings of

the 16th ASCE Engineering Mechanics Conference, University of

Washington, Seattle.

Basu U. & Chopra A.K. (2004), “Perfectly matched layers for transient

elastodynamics of unbounded domains”, International Journal for

Numerical Methods in Engineering, Vol.59, pp.1039-1074.

Bathe K.J. (1996), Finite element procedures in engineering analysis, Prentice

Hall, New Jersey.

Bérenger J.P. (1994), “A perfectly matched layer for the absorption of

electromagnetic waves”, Journal of Computational Physics, Vol. 114,

No. 2, pp. 185-200.

349

Bettess P. & Zienkiewicz O. C. (1977), ‘‘Diffraction and refraction of surface

waves using finite and infinite elements’’, International Journal for

Numerical Methods in Engineering, Vol.11, pp.1271-1290.

Bettess P. (1992), Infinite Elements, Penshaw Press, Sunderland.

Bielak (2005), “Reply to the comment on ‘Domain Reduction Method for Three-

Dimensional Earthquake Modeling in Localized Regions, Part I: Theory’

by J. Bielak, K. Loukakis, Y. Hisada & C. Yoshimura and ‘Part II:

Verification and Applications’ by C. Yoshimura, J. Bielak ,Y. Hisada &

A. Fernández” by E. Faccioli, M. Vanini, R. Paolucci & M. Stupazzini,

Bulletin of the Seismological Society of America, Vol. 95, No 2, pp. 770-

773.

Bielak J. & Christiano P. (1984), “On the effective seismic input for nonlinear

soil-structure interaction systems”, Earthquake Engineering & Structural

Dynamics, Vol. 12, pp. 107-119.

Bielak J., Loukakis K., Hisada Y. & Yoshimura C. (2003), “Domain Reduction

Method for Three-Dimensional Earthquake Modeling in Localized

Regions, Part I: Theory”, Bulletin of the Seismological Society of

America, Vol. 93, No 2, pp. 817-824.

Biot M.A. (1956a), “Theory of propagation of elastic waves in a fluid-saturated

porous solid. I. Low frequency range”, Journal of the Acoustical Society

of America, Vol. 28, No. 2, pp 168-178.

Biot M.A. (1956b), “Theory of propagation of elastic waves in a fluid-saturated

porous solid. II. Higher frequency range”, Journal of the Acoustical

Society of America, Vol. 28, No. 2, pp 179-191.

Biot M.A. (1962), “Mechanics of deformation and acoustic propagation in

porous media”, Journal of Applied Physics, Vol. 33, No. 4, pp 1482-

1498.

Blake F. G. (1952), “Spherical Wave Propagation in Solid Media”, Journal of

Acoustical Society of America, Vol. 24, No 2, pp.211-215.

350

Bonelli A., Bursi, O.S., Erlicher S. & Vulcan, L. (2002), “Analyses of the

Generalized-α method for linear and non-linear forced excited systems.”

Proceedings of EURODYN 2002, Vol. 2: 1523-1528.

Booker J.R. & Small J.C. (1975), “An investigation of the stability of numerical

solutions of Biot’s equations of consolidation”, International Journal of

Solid Structures, Vol. 11, pp. 907-917.

Boore D.M. & Bommer J.J., (2005), “Processing of strong-motion

accelerograms: needs, options and consequences”, Soil Dynamics and

Earthquake Engineering, Vol. 25, pp.93-115.

Brown D.A., O’Neill M.W., Hoit M., McVay M., El Naggar M.H. &

Chakraborty S., (2001), “Static and Dynamic Lateral Loading of Pile

Groups”, NCHRP report 461, National Academy Press, Washington,

D.C.

Castellani A. (1974), “Boundary conditions to simulate an infinite space”,

Meccanicca, Vol. 9, pp.199-205.

Chan A.H.C. (1988), A unified finite element solution to static and dynamic

geomechanics problems, PhD thesis, University College Swansea.

Chan A.H.C. (1995), User manual for DIANA SWANDYNE –II, School of Civil

Engineering, University of Birmingham.

Chopra A.K. (1995), Dynamics of Structures: Theory and Applications to

Earthquake Engineering, Prentice Hall, Englewood Cliffs, New Jersey.

Chung J. & Hulbert, G.M. (1993), “A time integration algorithm for structural

dynamics with improved numerical dissipation: the generalized-α

method”, Journal of Applied Mechanics, Vol. 60, pp. 371–375.

Chung J., Cho E.H. & Choi K. (2003), “A priori error estimator of the

generalized-α method for structural dynamics”, International Journal for

Numerical Methods in Engineering, Vol. 57 pp. 537-554.

351

Clayton R.W. & Engquist B. (1977), “Absorbing boundary conditions for

acoustic and elastic wave equation”, Bulletin of the Seismological Society

of America, Vol. 67, No 6, pp. 1529-1541.

Clayton R.W. & Engquist B. (1980), “Absorbing boundary conditions for wave

equation migration”, Geophysics, Vol. 45, No 5, pp. 895-904.

Cohen M. & Jennings P.C. (1983), “Silent boundary methods for transient

analysis” in Belytschko T. & Hughes T.J.R eds., Computational Methods

for transient analysis, pp.301-360.

Cohen M. (1980), “Silent boundary methods for transient wave analysis” PhD

Thesis, California Institute of Technology.

Cremonini M.G. & Christiano P. & Bielak J. (1988), “Implementation of the

effective seismic input for soil-structure interaction systems”, Earthquake

Engineering & Structural Dynamics, Vol. 16, pp. 615-625.

Crisfield M.A (1997), Non-linear Finite Element Analysis of Solids and

Structures, Volume two: Advanced Topics. Chichester, 1997.

Cundall P.A., Kunar R.R., Carpenter P.C. and Marti J. (1978), “Solution of

infinite dynamic problems by finite modelling in time domain”

Proceedings 2nd Int Conference on Applied Numerical Modelling, Madrid,

Spain.

Dahlquist, G. (1963). “A special stability problem for linear multistep methods”

BIT, Vol.3, pp.27-43.

Day R.A. & Potts D.M. (1990), “Curved Mindlin beam and axi-symmetric shell

elements – A new approach”, International Journal for Numerical

Methods in Engineering, Vol. 30, pp 1263-1274.

Day R.A. (1990), “Finite element analysis of sheet pile retaining walls”, PhD

thesis, Imperial College, London.

Deeks A.J. & Randolph M.F. (1994), “Axisymmetric time-domain transmitting

boundaries”, Journal of Engineering Mechanics, ASCE Vol. 120, No. 1.

352

Engquist, M. & Majda, A. (1977), “Absorbing boundary conditions for the

numerical simulation of waves” Journal of Mathematics of Computation,

Vol.31, No.139, pp.629-651.

Erlicher S., Bonavetura, O. & Bursi O.S. (2002), “The analysis of the

Generalized-α method for non-linear dynamic problems”, Computational

Mechanics, Vol. 28, pp. 83-104.

Federal Highway Administration (FHWA), (2000), “Seismic retrofitting manual

for highway structures, part II.” Prepared by Multidisciplinary Centre for

Earthquake and Engineering Research, Buffalo, New York.

Fung T.C. (2003), “Numerical dissipation in time-step integration algorithms for

structural dynamic analysis”, Progress in Structural Engineering and

Materials, Vol.3, No.3, pp. 167-180.

Goudreau G.L & Taylor R.L. (1972), “Evaluation of numerical integration

methods in structural dynamics”, Computer Methods in Applied

Mechanical Engineering, Vol.2, pp.69-97.

Grammatikopoulou A. (2004), “Development, implementation and application of

kinematic hardening models for overconsolidated clays”, PhD thesis,

Imperial College, London.

Hardy S. (2003), “The implementation and application of dynamic finite element

analysis in geotechnical problems”, PhD thesis, Imperial College,

London.

Hashash Y.M.A., Hook J.J., Schmidt B. & Yao J.I-C., (2001) “Seismic design

and analysis of underground structures”, Tunnelling and Underground

Space Technology, Vol.16, pp. 247-293.

Herrera I. & Bielak J. (1977), “Soil-structure interaction as a diffraction

problem”, Proceedings of the 6th World Conference in Earthquake

Engineering , New Delhi, Vol. 4, pp. 19-24.

353

Higdon R.L. (1986), “Absorbing boundary conditions for difference

approximations to the multi-directional wave equation”, Journal of

Mathematics of Computation, Vol.47, No 176, pp.437-459.

Higdon R.L. (1987), “Numerical absorbing boundary conditions for the wave

equation”, Journal of Mathematics of Computation, Vol. 49, No 179,

pp.65-90.

Higdon R.L. (1991), “Absorbing boundary conditions for elastic waves”,

Geophysics, Vol. 56, No 2, pp. 231-241.

Hilber H.M., Hughes T.J.R. & Taylor R.L. (1977), “Improved numerical

dissipation for time integration algorithms in structural dynamics”,

Earthquake Engineering & Structural Dynamics, Vol.5, pp. 283-292.

Hoeg K. (1968), “Stresses against underground structural cylinders”, Journal of

Soil Mechanics and Foundations Division, ASCE, Vol.94, No.4, pp. 833-

858.

Houbolt J.C. (1950), “A recurrence matrix solution for the dynamic response of

elastic aircraft”, Journal of the Aeronautical Sciences, Vol. 17, pp. 540-

550.

Hughes T.J.R & Hilber H.M. (1978), “Collocation, dissipation and overshoot for

time integration schemes in structural dynamics”, Earthquake

Engineering & Structural Dynamics, Vol.6, pp. 99-117.

Hughes, T.J.R (1983), “Analysis of transient algorithms with particular reference

to stability behaviour”, in Belytschko, T & Hughes, T.J.R eds.,

Computational Methods for transient analysis, pp.67-155.

Hulbert G. & Jang I. (1995), “Automatic time step control algorithms for

structural dynamics”, Computer methods in applied mechanics and

engineering, Vol. 126, pp.155–178.

354

Jardine R.J., Potts D.M., Fourie A.B. & Burland J.B. (1986), “Studies of the

influence of nonlinear stress-strain characteristics in soil-structure

interaction”, Geotechnique, Vol. 36, No. 3, pp 377-396.

Katona M.G. & Zienkiewicz O.C. (1985), “A unified set of single step

algorithms, Part 3: the beta-m method, a generalization of the Newmark

scheme”, International Journal for Numerical Methods in Engineering,

Vol. 21 pp. 1345-1359.

Kausel E. & Tassoulas J. L. (1981), “Transmitting Boundaries: A closed-form

comparison”, Bulletin of the Seismological Society of America, Vol. 71,

No 1, pp. 143-159.

Kausel E. (1988), “Local transmitting boundaries” Journal of Engineering

Mechanics, ASCE, Vol.114, No. 6, pp. 1011-1027.

Kausel E. (1992), “Physical interpretation and stability of paraxial boundary

conditions” Bulletin of the Seismological Society of America, Vol. 82, No

2, pp. 898-913.

Kausel E. (1994), “Thin-layer method: formulation in time domain”,

International Journal for Numerical Methods in Engineering, Vol.37,

pp.927-941.

Kausel E. (2000), “The in layer method in seismology d earthquake

engineering”, in Wave Motion in Earthquake Engineering, Kausel E. &

Manolis G. (eds),, (Chapter 5),, WIT Press: Southampton, U.K., Boston,

U.S.A..

Kellezi L. (1998), “Dynamic soil-structure interaction transmitting boundary for

transient analysis”, PhD Thesis, Department of Structural Engineering

and Materials, Technical University of Denmark.

Kellezi L. (2000), “Local transmitting boundaries for transient elastic analysis”,

Soil Dynamics and Earthquake Engineering, Vol.19, pp. 533-547.

355

Kim C.S., Lee T.S., Advani S.H. & Lee J.K. (1993), “Hygrothermomechanical

evaluation of porous media under finite deformation: Part II- model

validations and field simulations” International Journal for Numerical

Methods in Engineering, Vol. 35 pp 161-179.

Kim D.K. & Yun C.B. (2000), “Time domain soil-structure interaction analysis

in two-dimensional medium based on analytical frequency-dependent

infinite elements”, International Journal for Numerical Methods in

Engineering, Vol.47, pp.1241-1261.

Kim D.K. & Yun C.B. (2003), “Earthquake response analysis in the time domain

for 2D soil-structure systems using analytical frequency-dependent

infinite elements”, International Journal for Numerical Methods in

Engineering, Vol.58, pp.1837-1855.

Kramer S.L. (1996), Geotechnical earthquake engineering, Prentice Hall, New

Jersey.

Kuhlemeyer R.L. & Lysmer J. (1973), “Finite element method accuracy for wave

propagation problems”, Technical Note, Journal of Soil Mechanics and

Foundation Division, ASCE, Vol.99, No5 pp. 421-427.

Kunar R.R. & Marti J. (1981), “A non-reflecting boundary for explicit

calculations” Computational Methods for Infinite Domain Media-

Structure Interaction, ASME, AMD, Vol. 46, pp. 183-204

Liao Z.P. & Wong H.L. (1984), “A transmitting boundary for the numerical for

the numerical simulation of elastic wave propagation” Soil Dynamics and

Earthquake Engineering, Vol. 3, pp.174-183.

Loukakis K. & Bielak J. (1994), “Seismic response of to-dimensional sediment-

filled valleys to oblique incident SV-waves calculated by the finite

element method”, Proceedings of the 5th U.S. National Conference on

Earthquake Engineering, Chicago, Vol. III, pp. 25-34.

356

Luco J.E., Hadjian A.H. & Bos H.D (1974), “The dynamic modeling of the half-

plane by Finite Elements”, Nuclear Engineering and Design, Vol. 31, pp.

184-194.

Lysmer J. & Kuhlemeyer R.L. (1969), “Finite dynamic model for infinite media”

Journal of the Engineering Mechanics Division, ASCE, Vol.95, No.4, pp.

859-877.

Lysmer J. & Kuhlemeyer R.L. (1969), “Finite dynamic model for infinite

media”, Journal of the Engineering Mechanics Division, ASCE, Vol.95,

No.4, pp.:859-877

Lysmer J. & Waas G. (1972), “Shear waves in plane infinite structures”, Journal

of Engineering Mechanics Division, ASCE Vol. 98, No. 1, pp.85-105.

Lysmer J. (1970), “Lumped mass method for Rayleigh waves”, Bulletin of the

Seismological Society of America, Vol. 60, No 1, pp. 89-104.

Maheshwari B.K., Truman K.Z., El Naggar M.H. & Gould P. L. (2004), “Three-

dimensional finite element nonlinear dynamic analysis of pile groups for

lateral transient and seismic excitations” Canadian Geotechnical Journal,

Vol. 41, pp. 118-133.

Meek J.W. & Wolf J.P. (1993), “Why cone models can represent the Elastic

Halfspace”, Earthquake Engineering and Structural Dynamics, Vol. 22,

No. 8, pp. 649-664.

Menkiti C.O., Mair R.J. & Miles R. (2001b), “Tunneling in complex ground

conditions in Bolu, Turkey” Proceedings of the Underground

Construction 2001 Symposium, London.

Menkiti C.O., Sanders P., Barr J., Mair R.J., Cilingir & James S. (2001a),

“Effects of the 12th November 1999 Duzce earthquake on the stretch 2 the

Gumusova-Gerede Motorway in Turkey”, Proceedings of the

International Road Federation 14th World Road Congress, Paris.

357

Meroi E.A., Schrefler B.A. & Zienkiewicz O.C. (1995), “Large strain static and

dynamic semi-saturated soil behaviour”, International Journal for

Numerical and Analytical Methods in Geomechanics, Vol. 19, pp 81-106.

Naimi M., Sarma S.K., & Seridi A. (2001), “New inclined boundary conditions

in seismic soil–structure interaction problems”, Engineering Structures,

Vol. 23, pp. 966–978.

Newmark N.M. (1959), “A method of computation for structural dynamics”,

Journal of Engineering Mechanics Division, ASCE ,Vol. 85, pp. 67-94.

Novak M. & Mitwally H. (1988), “Transmitting boundary for axisymmetric

dilation problems”, Journal of Engineering Mechanics Division, ASCE

Vol. 104, No. 4, pp.953-956.

Novak M., Nogami T. & Aboul-Ella F. (1978), “Dynamic so reactions for plane

strain case”, Technical Note, Journal of Engineering Mechanics Division,

ASCE, Vol.112, No. 3, pp. 363-368.

O’Rourke T.D., Goh S.H., Menkiti, C.O. & Mair R.J., (2001), “Highway tunnel

performance during the 1999 Duzce earthquake” Proceedings of the

Fifteenth International Conference on Soil Mechanics and Geotechnical

Engineering, August 27-31, Istanbul, Turkey.

Owen G.N. & Scholl R.E. (1981), “Earthquake engineering of large underground

structures”, Report no. FHWA/RD-80/195, Federal Highway

Administration and National Science Foundation.

Papastamatiou D.J. (1971) “Ground Motion and Response of Earth Structures

Subjected to Strong Earthquakes”, PhD thesis, Imperial College, London.

Park K.C. (1975), “An improved stiffly stable method for direct integration of

nonlinear structural dynamic equations”, Journal of Applied Mechanics,

ASME, Vol.42, pp. 464-470.

358

Pegon P. (2001) “Alternative characterization of time integration schemes”,

Computer Methods in Applied Mechanical Engineering, Vol. 190, pp.

2707-2727.

Penzien, J. (2000) “Seismically induced racking of tunnel linings”, International

Journal of Earthquake Engineering & Structural Dynamics, Vol.29, pp

683-691.

Potts D.M. & Zdravković L.T. (1999), Finite element analysis in geotechnical

engineering: theory, Thomas Telford, London.

Potts D.M. & Zdravković L.T. (2001), Finite element analysis in geotechnical

engineering: application, Thomas Telford, London.

Prakash S. (1981), Soil Dynamics, McGraw-Hill.

Prevost J.H. (1982), “Nonlinear transient phenomena in saturated porous media”,

Computer Methods in Applied Mechanics and Engineering, Vol. 20,

pp.3-18.

Roscoe K.H. & Burland J.B. (1968), “On the generalised stress-strain behavir of

‘wet’ clay”, Engineering plasticity, Cambridge University Press, pp. 535-

609.

Sarma S.K. & Mahabadi S.G. (1995), “Dynamic response of earth and rock-fill

dams with radiating boundary condition using finite element method”,

Proceedings of 10th European Conference on Earthquake Engineering,

pp. 1975-1980, Rotterdam : A.A. Balkema.

Sarma S.K. (1990), “A two dimensional radiating boundary condition for seismic

response of large structures” Proceedings of 9th European Conference on

Earthquake Engineering, pp. 115-124, Moscow.

Sarma S.K. (2000), “Basic Dynamics”, MSc Lecture Notes, Department of Civil

& Environmental Engineering, Imperial College, London.

Schnabel P. B., Lysmer J. & Seed H. B., (1972), “SHAKE: A Computer Program

for earthquake response analysis of horizontally layered sites”, Report

359

No.UCB/EERC-72/12, Earthquake Engineering Research Center,

University of California, Berkeley.

Seed H. B., Wong R. T., Idriss I. M. & Tokimatsu K. (1986) “Moduli and

damping factors for dynamic analyses of cohesionless soils”, Journal of

the Geotechnical Engineering Division, ASCE, Vol. 112, No.GTI1,

,pp.1016-1032.

SeismoSoft, (2004), “SeismoSignal - A computer program for signal processing

of strong-motion data”, (online), Available from URL:

http://www.seismosoft.com.

Smith I.M. (1994), “An overview of numerical procedures used in the VELACS

project”, in Verifications of Numerical Procedures for the Numerical

Analysis of Soil Liquefaction Problems, Arulanandan & Scott (eds),

Balkema, Rotterdam.

Smith W.D. (1974), “A nonreflecting plane boundary for wave propagation

problems”, Journal of Computational Physics, Vol.15, pp.492-503.

St. John C.M. & Zahrah T.F. (1987), “Aseismic design of underground

structures”, Tunnelling and Underground Space Technology, Vol.2, No.2,

pp. 165-197.

Stacey (1988), “Improved transparent boundary formulations for the elastic wave

equation”, Short note in Bulletin of the Seismological Society of America,

Vol. 78, No 6, pp. 2089-2097.

Stolaski H.K. & D’Costa J.F. (1997), “On consistent discretization of inertia

terms for Co structural elements with modified stiffness matrices”,

International Journal for Numerical Methods in Engineering, Vol. 40,

pp. 3299-3312.

Strang G. & Fix G.J. (1973), An analysis of the finite element method, Prentice

Hall, Englewood Cliffs, New Jersey.

360

Sucuoğlu H., (2002), “Engineering characteristics of the near-field strong

motions from the 1999 Kocaeli and Düzce Earthquakes in Turkey”,

Journal of Seismology, Vol.6, pp. 347-355.

Thomas J.N. (1984), “An improved accelerated initial stress procedure for elasto-

plastic finite element analysis.” International Journal for Numerical and

Analytical Methods in Geomechanics, Vol. 8, pp 359-379.

Tsatsanifos C.P. (1982), “Effective stress method for dynamic response of

horizontally layered soils”, PhD thesis, Imperial College, London.

Underwood P., & Geers T.L (1981), “Doubly asymptotic, boundary element

analysis of dynamic soil-structure interaction” International Journal of

Solids & Structures, Vol.17, pp.687-697.

Urlich C.M. & Kuhlemeyer R.L. (1973), “Coupled rocking and lateral vibrations

of embedded footings” Canadian Geotechnical Journal, Vol. 10, pp. 145-

160.

Valliappan S., Favaloro J.J. & White W. (1977), “Dynamic Analysis of

Embedded Footings”, Technical Note, Journal of Geotechnical Division,

ASCE, Vol. 103, pp.129-133.

Vrettos C. (1990), “In-plane vibrations of soil deposits with variable shear

modulus: I. Surface Waves”, International Journal for Numerical and

Analytical Methods in Geomechanics, Vol. 14, pp. 209-222.

Vrettos C. (2000), “Waves in vertically inhomogeneous soils”, in Wave Motion

in Earthquake Engineering, Kausel E. & Manolis G. (eds),, (Chapter 7),,

WIT Press: Southampton, U.K., Boston, U.S.A..

Vucetic M. & Dobry R., (1991), “Effect of soil plasticity on cyclic response”,

Journal of Geotechnical Engineering, ASCE,Vol. 117, No.1, pp. 89-107.

Wang J.N. (1993), “Seismic Design of Tunnels: A State-of-the-Art Approach”,

Monograph 7, Parsons, Brinckerhoff, Quade and Douglas Inc, New York.

361

White W., Somasundaram V. & Lee I.K. (1977), “Unified Boundary for Finite

Dynamic Models”, Journal of Engineering Mechanics Division, ASCE,

Vol. 103, No. 5, pp.949-964.

Whitham G.B (1974), Linear and Nonlinear Waves, Wiley, New York.

Wilson E.L., Farhoomand I., & Bathe K.J. (1973), “Non-linear dynamic analysis

of complex structures”, Earthquake Engineering and Structural

Dynamics, Vol. 1, pp. 241-252.

Wolf J.P. & Deeks A.J. (2004), Foundation Vibration Analysis: A Strength -of –

Materials Approach, Elsevier.

Wolf J.P. & Song C. (1995), “Doubly asymptotic multi-directional transmitting

boundary for dynamic unbounded medium-structure interaction analysis”,

Earthquake Engineering & Structural Dynamics, Vol.24, pp. 175-188.

Wolf J.P. & Song C. (1996), Finite-element modeling of unbounded media, New

York, Wiley eds.

Wolf J.P. (1988), Soil-Structure Interaction Analysis in Time Domain, Prentice-

Hall, Englewood Cliffs, New Jersey.

Wolf J.P. (1994), Foundation Vibration Analysis Using Simple Physical Models,

Prentice-Hall, Englewood Cliffs, New Jersey.

Wood W.L. (1990), Practical Time-Stepping Schemes, Clarendon Press, Oxford.

Wood W.L., Bossak M. & Zienkiewicz O.C. (1981), “An alpha modification of

Newmark’s method”, International Journal for Numerical Methods in

Engineering, Vol.15, pp.1562–1566.

Woodward P.K. & Griffiths D.V. (1996), “Influence of viscous damping in the

dynamic analysis of an earth dam using simple constitutive models”,

Computers and Geotechnics, Vol. 19, No.3, pp. 245-263.

Yoshimura C., Bielak J., Hisada Y. & Fernández A. (2003), “Domain Reduction

Method for Three-Dimensional Earthquake Modeling in Localized

362

Regions, Part II: Verification and Applications”, Bulletin of the

Seismological Society of America, Vol. 93, No 2, pp. 825-840.

Zerwer A., Cascante G. & Hutchinson J. (2002), “Parameter estimation in finite

element simulations of Rayleigh waves”, Journal of Geotechnical and

Geoenvironmental Engineering, Vol. 128, No.3, pp.250-261.

Zienkiewicz O.C. & Shiomi T. (1984), “Dynamic behaviour of saturated porous

media; the generalised Biot formulation and its numerical solution”,

International Journal for Numerical and Analytical Methods in

Geomechanics, Vol. 8, pp 71-96.

Zienkiewicz O.C., Bicanic N. & Shen F.Q. (1988), “Earthquake input definition

and the transmitting boundary condition”, Conf: Advances in

computational non-linear mechanics, Editor: St. Doltnis I., pp. 109-138.

Zienkiewicz O.C., Chan A.H.C., Pastor M., Schrefler B.A. & Shiomi T. (1999),

Computational geomechanics with special reference of earthquake

engineering, John Wiley & Sons, Chichester.

Zienkiewicz O.C., Chang C.T. & Bettress P. (1980a), Drained, undrained,

consolidating and dynamic behaviour assumptions in soils, Geotechnique,

Vol. 30, No. 4, pp 385-395.

Zienkiewicz O.C., Wood W.L. & Taylor R.L. (1980b), “An alternative single-

step algorithm for dynamic problems”, Earthquake Engineering and

Structural Dynamics, Vol. 8, pp. 31-40.

363

Appendix A: Spectral stability analysis of the CH method

The equation governing the dynamic response of a linear multi-degree of

freedom (MDF) system subjected to a free vibration at t=tk+1 is expressed as:

[ ] ( ) [ ] ( ) [ ] ( ) 0tuKtuCtuM 1k1k1k =++ +++ &&& A1

The displacement u of this system can be expressed as the sum of modal

contributions:

( ) ( )∑=

==N

1r

rr tqΦ(t)qφtu A 2

where φr is the deflected shape of the rth mode and (t)q r are the unknown modal

coordinates. Employing the above expression, Equation A1 can be transformed

to set of uncoupled equations with modal coordinates with (t)q r as the

unknowns. As mentioned in Chapter 3, when the uncoupled modal equations are

integrated with the same time step ∆t, then the superposition of the modal

solutions is completely equivalent to a direct integration analysis of the complete

system using the same time step ∆t and the same integration scheme (Bathe,

1996). To investigate the stability characteristics of an integration scheme in the

linear regime, it is common practice to consider the modes of a system

independently with a common time step ∆t instead of considering Equation A1.

Employing the property of orthogonality of the eigenvetors, the problem can be

reduced down to a SDOF system. The governing equation of the CH method for

a free vibration of a SDOF is given by Equation A3.

( ) ( ) ( )

0)u(tkα)(tucα)(tumα

)u(tkα-1)(tucα-1)(tumα-1

kfkfkm

1kf1kf1km

=+++

++ +++

&&&

&&& A3

It is convenient to write the above equation as:

( ) ( ) ( )

0)u(tαω)(tuαωξ2)(tuα

)u(tα-1ω)(tuα-1ωξ2)(tuα-1

kf

2

kfkm

1kf

2

1kf1km

=+++

++ +++

&&&

&&& A4

364

where the natural frequency m

kω = and

2k

ωcξ = is the damping ratio.

Furthermore, Newmark’s recurrence expressions for displacement and

acceleration are given by Equations A5 and A6.

( ) ( ) ( ) ( ) ( ) 2

1k

2

kkk1k ∆ttuα∆ttuα2

1∆ttututu ++ +

−++= &&&&& A5

( ) ( ) ( ) ( ) ( )∆ttuδ∆ttuδ1tutu 1kkk1k ++ +−+= &&&&&& A6

Substituting Equations A5 and A6 into Equation A4 and rearranging to put all of

the known terms (which refer to the previous time step) on the right hand side

yields Equation A7:

( ) ( ) ( ) ( )k33k32k311k tuatuatuatu &&&&& ++=+ A7

where

( ) ( ) ( )[ ]

( )

( ) ( ) ( )[ ]

( )( ) ( ) ( )

( ) ( ) ( )[ ]αΩα1δΩξα12α1

δ1Ωξα12Ωα-1/2α1αa

∆tαΩα1δΩξα12α1

Ωα1Ωξ2a


Ωa

2

ffm

f

2

fm33

2

ffm

2

f32

22

ffm

2

31

−+−+−

−−−−−−=

−+−+−

−−−=

−+−+−

−=

and ∆tωΩ = . Furthermore, substituting Equation A7 into Equation A6, the

velocity at time 1ktt += can also be expressed as a function of the known

displacement, velocity and acceleration of the previous time step.

( ) ( ) ( ) ( )k23k22k211k tuatuatuatu &&&& ++=+ A8

where

365

( ) ( ) ( )[ ]

( ) ( )( )

( ) ( ) ( )

( ) ( ) ( )[ ]( ) ( ) ( ) αΩα1δΩξα12α1

∆tδ/2αΩα1α-δ-1a

αΩα1δΩξα12α1

Ωξδα2Ωδαα1α1a


δΩ-a

2

ffm

2

fm23

2

ffm

f

2

fm22

2

ffm

2

21

−+−+−

−−+=

−+−+−

−−−+−=

−+−+−=

Finally, substituting Equation A7 into Equation A5 the displacement at time

1ktt += can also be related to the known displacement, velocity and acceleration

of the previous time step.

( ) ( ) ( ) ( )k13k12k111k tuatuatuatu &&& ++=+ A9

where

( ) ( )

( ) ( ) ( )

( ) ( ) ( )[ ]

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) αΩα1δΩξα12α1

2

∆tδ-1αΩξα14α2-1δΩξα12α2α-1

a


∆tαΩξα2α-δΩξα12α1a


Ωαα-δΩξα12α1a

2

ffm

2

ffm

13

2

ffm

ffm12

2

ffm

2

ffm11

−+−+−

−−−+−=

−+−+−

−−+−=

−+−+−

−+−=

Hence the amplification matrix of the CH method can be determined combing

Equations A7, A8 and A9:

[ ]

=

=

+

+

+

k

k

k

k

k

k

333231

232221

131211

1k

1k

1k

u

u

u

A

u

u

u

ααα

ααα

ααα

u

u

u

&&

&

&&

&

&&

& A10

The characteristic equation of the amplification matrix is:

366

0AλAλA2λλI)det(A 32

2

1

3 =−+−=−− A11

where

( )

3223112231133221133123123321123322113

3113322321123311332222112

3322111

ααα-αααααααααααααααA

ααααααααααααA

ααα2

1A

−++−=

−−−++=

++=

As mentioned in Chapter 3, the spectral radius of an algorithm is defined as:

( ) 321 λ,λ,λmaxAρ = A12

The procedure to calculate the roots of a cubic equation is described in many

textbooks (e.g. Press et al, 1986). It is convenient to set:

54

A27AA18A16R

9

A3A4Q

321

3

1

2

2

1

−+−=

−=

A13

When R2≥Q

3 the eigenvalues of the characteristic polynomial has two complex

conjugates (λ1,2) and one real (λ3). The complex eigenvalues are the principal

roots whereas the real one is the so-called spurious root.

( ) ( )

( ) 13

11,2

A3

2DCλ

DC2

3iA

3

2DC

2

1λ

++=

−

±++−=

A14

where

( )[ ]

=

≠=

−+−=

0C0,

0CQ/C,D

QRRRsignC1/3

32

Furthermore when R2<Q

3 the cubic equation has three real roots.

367

13

12

11

A3

2

3

2πθcosQ2λ

A3

2

3

2πθcosQ2λ

A3

2

3

θcosQ2λ

+

−−=

+

+−=

+

−=

A15

All the roots can be evaluated as function of the algorithmic parameters, the

frequency ω and the time step. Thus, employing the above-mentioned

expressions of the eigenvalues in an excel spreadsheet, the spectral radius

(Equation A12), the period elongation error (Equation 3.23) and algorithmic

damping ratio (Equation 3.25) of the CH method can be calculated for any

natural frequency (ω) using a common time step ∆t.

Chung and Hulbert (1993) note that the behaviour of both the principal

roots and the spurious root as a function of the frequency Ω is important. It is

desirable that the principal roots are complex conjugates and that 1,23 λλ ≤ as Ω

increases. For a given level of high frequency dissipation (i.e. for a given value

of ∞ρ ) the low frequency impact is minimized when ∞∞ = 1,23 λλ . It is reminded

that the superscript ∞ denotes the value of the root for Ω→∞. Chung and Hulbert

(1993), fulfilling the conditions 3.69, 3.70 and 3.71, classified the CH method in

the fm αα − space (Figure A1). The shaded area represents the stability region

and is bounded by the lines 1λ1,2 −=∞ , 1λ3 −=∞ that respectively correspond to the

stability conditions fm αα ≤ and 0.5αf ≤ . While unconditional stability is

guaranteed for the whole shaded area, minimum low-frequency impact for a

desired user-controlled level of high-frequency dissipation is only attained along

the line AB (as ∞∞ = 1,23 λλ corresponds to the conditions of Equation 3.71).

Similarly, the optimum algorithmic parameters for the HHT and WBZ methods

lie along the lines OB and OA respectively.

368

Figure A0.1: Classification of the CH, HHT, WBZ methods in fm αα − space

(after Chung and Hulbert, 1993)

369

Appendix B: Material parameters

B.1 Modified Cam Clay parameters

As mentioned in Section 7.7.5, the MCC model requires three

compression parameters (λ, κ and the specific volume at unit pressure v1), one

drained strength parameter (φ΄) and one elastic parameter (the maximum shear

modulus G). The values of these parameters for the different layers are listed in

Table B.1. In the absence of oedometer test data, typical values of compression

parameters for stiff clays/ soft rocks were chosen. Furthermore, the selected

values of φ΄ are based both on the peak strength variation of Table 7.4 and on the

geotechnical in-situ description of the different units for the relevant cross-

sections reported by Menkiti (2005, personal communication). Moreover, Potts

and Zdravković, (1999) showed that the above-mentioned input parameters of

the MCC model and the initial state of stress can be directly related to the

undrained strength Su, as follows:

( ) ( ) ( )[ ] ( )( ) [ ]

λ

κ

2NC

O

OC

O2NC

O

vi

u

B1OCRK21

K212B1K21

6

OCRθcosθg

σ

S

++

+++=

′ B.1

where viσ′ is the initial vertical effective stress, g(θ) is a function defining the

shape of the yield surface in the deviatoric plane (given in Equation 7.17), NC

OK

is the value of the coefficient of earth pressure at rest associated with normal

consolidation, OC

OK is the current value of the coefficient of earth pressure at rest,

θ is the Lode’s angle defined in Equation 7.16, OCR is the overconsolidation

ratio defined as: vi

vm

σ

σOCR

′′

= , where vmσ′ is the maximum vertical effective

stress that the soil has been subjected to and B is defined as:

( )( )NC

o

NC

o

K21)30g(

K13B

+−−

=o

B.2

370

Table B.1: Material properties used for MCC model

Layer λ κ v1 G

(MPa)

φ΄

1 0.2 0.02 3.2 1000.0 30˚

2 0.2 0.02 4.5 750.0 17˚

3 0.2 0.02 3.2 1500.0 30˚

4 0.2 0.02 4.5 850.0 17˚

5 0.2 0.02 3.2 2500.0 30˚

0 400 800 1200 1600

200

160

120

80

40

0

De

pth

(m

)

Su(kPa)


Fault Breccia

Fault Gouge Clay

Metasediments

Fault Gouge Clay

Sandstone, Marl

Quartizic rock

(a) (b)

1 2 3 4 5

200

160

120

80

40

0

OCR

Ko

OC

Figure B.1: Undrained strength (Su) (a), overconsolidation ratio (OCR) and

coefficient of earth pressure at rest ( OC

OK ) (b) profiles

The estimated undrained strength for each layer is listed in Table 7.4. Employing

Equation B.1 and the input parameters listed in Table B.1, the initial stress state

parameters (OCR, Ko) can be selected to match the undrained strength values of

Table 7.4 for the middle of each layer (assuming that the undrained strength

varies linearly with depth in each layer). Figure B.1 plots the assumed variation

of Su with depth and the resulting OCR and OC

OK profiles. It should be also noted

that a linear variation of suction is assumed above the water table.

371

B.2 Small strain stiffness model parameters

The variations of the secant shear modulus, G, and the bulk modulus, K,

with the mean effective stress, p΄, and strain level in the nonlinear elastic region

are given by Equations 7.18 and 7.19 respectively. Tables B.2 and B.3

summarize the parameters used for the secant shear modulus and bulk modulus

equations, respectively. Figures B.2 and B.3 show the resulting stiffness-strain

plots for the secant shear and bulk stiffness, respectively. Employing the OC

OK

profile of Figure B.1, the shear stiffness variations of the clays and

metasediments were matched to data from pressuremeter tests. Since there was

no information available regarding the shear stiffness degradation of the

sandstones (i.e. layers 1 and 5), the two sandstones were assumed to have similar

shear stiffness degradation as the metasediments (i.e. layer 3). Furthermore, for

all layers the bulk modulus was assumed to degrade in a similar way to that of

the shear modulus.

Table B.2: Parameters used in the secant shear modulus equation for different

units

Layer G1 G2 G3

(%)

α γ Ed(min)

(%)

Ed(max)

(%)

Gmin

(MPa)

1 1550.0 1450.0 1.7x10-3 1.2 0.77 0.003 1.0 151.9

2 1080.0 1040.0 6.4x10-5 1.22 0.62 0.003 1.0 53.7

3 980.0 950.0 1.5x10-3 1.105 0.82 0.003 1.0 231.6

4 900.0 750.0 6.4x10-5 1.22 0.62 0.003 1.0 168.4

5 1350.0 1400.0 4.5x10-4 1.11 0.63 0.003 1.0 486.4

372

Table B.3: Parameters used in the secant bulk modulus equation for different

units

Layer K1 K2 K3

(%)

δ µ εv(min)

(%)

εv(min)

(%)

Kmin

(MPa)

1 2300.0 2400.0 1.7x10-4 1.05 0.6 0.003 1.0 540.6

2 1130.0 1110.0 9.0x10-6 0.999 0.66 0.003 1.0 99.4

3 850.0 850.0 6.0x10-4 0.999 0.74 0.003 1.0 558.9

4 900.0 750.0 9.0x10-6 0.999 0.66 0.003 1.0 357.2

5 1350.0 1400.0 4.5x10-4 1.2 0.59 0.003 1.0 865.6

0.0001 0.001 0.01 0.1 1 10

Ed (%)

0

400

800

1200

Gs

ec/p

′

Layer 1

Layer 2

Layer 3

Layer 4

Layer 5

Figure B.2: Secant shear stiffness-strain curves for different materials

373

0.0001 0.001 0.01 0.1 1 10

εv (%)

0

1000

2000

3000

Ks

ec/p

′

Layer 1

Layer 2

Layer 3

Layer 4

Layer 5

Figure B.3: Secant bulk stiffness-strain curves for different materials

B.3 Two-surface kinematic hardening model parameters

As briefly discussed in Chapter 7, the two-surface kinematic hardening

(M2-SKH) model requires in total 7 parameters. Five of them have their origin in

the MCC model and their values for the various layers are therefore listed in

Table B.1, while the remaining 2 parameters (Rb, α) define the behaviour of the

kinematic surface. To derive reliable values for the parameter Rb (i.e. the ratio of

the size of the bubble to that of the bounding surface), test data with

measurements of strains in the very small and small strain region are required.

Since no such data are available, the Rb is assumed to be 0.1 for the two clays

and 0.15 for the soft rock layers. Furthermore, the parameter α, which controls

the decay of stiffness, cannot be measured directly from the experimental data

and is usually determined by trial and error. However due to lack of data, a value

of α equal to 15.0 was adopted for all layers based on Grammatikopoulou (2004).

B.4 Equivalent linear elastic model parameters

Figure B.4 illustrates the shear stiffness degradation curves that were

used for each layer in the equivalent linear analyses. Their derivation is based on

the variation of the secant shear modulus with the mean effective stress, p΄, and

strain level of Figure B.2 for the middle point of each layer. In addition the

374

damping ratio curves of Vucetic and Dobry (1991) for overconsolidated clays

with a plasticity index of 50 were adopted for the two clay layers while for the

remaining rock strata the lower limit of the Seed et al’s (1986) range of damping

ratio curves for sands was employed (Figure B.5).

0.001 0.01 0.1 1

Shear strain (%)

0

0.2

0.4

0.6

0.8

1G

/Gm

ax

Layer 1

Layer 2

Layer 3

Layer 4

Layer 5

Figure B.4: Shear stiffness-strain curves of different materials used in equivalent

linear analyses

0.0001 0.001 0.01 0.1 1 10

Shear strain (%)

0

4

8

12

16

20

Da

mp

ing

ra

tio

(%

)

Seed et al (1986) (lower-bound)

Vucetic & Dobry (1991)

Figure B.5: Damping ratio-shear strain curves used in equivalent linear analyses

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