development of time integration schemes and advanced boundary
TRANSCRIPT
1
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
6
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
8
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
10
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
Figure 3.7: Evaluation of the various terms of the equilibrium equation of motion
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
12
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
13
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
14
Figure 4.27: Fourier amplitude spectra of the horizontal acceleration time history
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
15
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
Figure 5.24: Comparison of the response for M10x10 and M15x15 for vertical
excitation, Tp=0.2s (ICFEP results) ...........................................................192
Figure 5.25: Comparison of the stress response for vertical excitation and
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
Figure 5.27: Comparison of the response at surface points for horizontal
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
16
Figure 5.30: Displacement normal to the cavity (ρ∞ =0.4, D=10m) ..................199
Figure 5.31: Displacement normal to the cavity at points A, C (ρ∞ =0.4, D=10m)
....................................................................................................................201
Figure 5.32: Displacement normal to the cavity (ρ∞ =0.4, D=15m) ..................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
17
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
18
Figure 6.16: Comparison of the displacement response at nodes C, D for a pulse
of To =2.0s..................................................................................................251
Figure 6.17: Comparison of the displacement response at nodes C, D for a pulse
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
19
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
Figure 7.21: Shear strain time history (a) and maximum shear strain profile (b)
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
Figure 7.23: Shear strain time history (a) and maximum shear strain profile (b)
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
Figure 7.33: 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
the M2-SKH model for the first 6.48sec of the earthquake. ......................302
Figure 7.34: 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 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
21
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
Figure 7.45: Accumulated thrust (a), bending moment (b) and hoop stress (c)
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
Figure 7.47: Accumulated thrust (a), bending moment (b) and hoop stress (c)
distribution around the tunnels’ lining at the end of the static analysis
computed with MCCJ model .....................................................................315
22
Figure 7.48: Pore water pressure (a) and shear strain (b) time histories for
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
Figure 7.51: Accumulated thrust (a), bending moment (b) and hoop stress (c)
distribution around the tunnels’ lining at the end of the quasi-static analysis
....................................................................................................................319
Figure 7.52: FE mesh configuration for chainage 62+870 after the excavation of
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
Figure 7.55: Accumulated thrust (a), bending moment (b) and hoop stress (c)
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
Table 7.11: Maximum hoop stress at shoulder and knee locations of the BPTs’
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
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.
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
Figure 3.7: Evaluation of the various terms of the equilibrium equation of motion
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'
QuadraticAcceleration
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
QuadraticAcceleration
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
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
Figure 4.22: Fourier amplitude spectra of the horizontal acceleration time history
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
initial settlementdue to working load
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
initial settlementdue to working load
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
initial settlementdue to working load
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
Figure 4.27: Fourier amplitude spectra of the horizontal acceleration time history
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
∂∂
+
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
∂
∂+
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
Figure 5.22: Comparison of the stress response for vertical excitation and
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.
Figure 5.23: Comparison of the response for M10x10 and M15x15 for vertical
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
Figure 5.24: Comparison of the response for M10x10 and M15x15 for vertical
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
Figure 5.25: Comparison of the stress response for vertical excitation and
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.
Figure 5.26: Comparison of the response at surface points for horizontal
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
Figure 5.27: Comparison of the response at surface points for horizontal
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
(a) Point A, r=3.4 m
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
Figure 5.30: Displacement normal to the cavity (ρ∞ =0.4, D=10m)
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
(b) Point B, r=9.1 m
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
(a) Point A, r=3.4 m
Figure 5.32: Displacement normal to the cavity (ρ∞ =0.4, D=15m)
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
(a) Point A, r=3.4 m
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
(b) Point B, r=9.1 m
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
(a) Point A, r=3.4 m
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
(b) Point B, r=9.1 m
Figure 5.35: Displacement normal to the cavity for 5α0 =
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
(a) Point A, r=3.4 m
0 0.2 0.4 0.6 0.8 1
Time (s)
-8E-006
-4E-006
0
4E-006
8E-006
1.2E-005
(b) Point B, r=9.1 m
Figure 5.36: Displacement normal to the cavity for 50α0 =
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)(ωcosz)k0.3624(exp1.5609z)k0.8829(exp0.8829Av
t)(ωsinz)k0.3624(exp05656z)k0.8829(expAu
−+−−=
−−−= 5.70
[ ]
[ ] t)(ωcosz)k0.3372(exp1.6579z)k0.9232(exp0.9232Av
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
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)
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
Vertical Stress (kPa)
0 0.4 0.8 1.2 1.6
Time (s)
-0.02
0
0.02
Horizontal Stress (kPa)
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
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
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)
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
Vertical Stress (kPa)
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
Horizontal Stress (kPa)
0 0.4 0.8 1.2 1.6
Time (s)
-0.008
-0.004
0
0.004
0.008
Vertical Stress (kPa)
0 0.4 0.8 1.2 1.6
Time (s)
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
Horizontal Stress (kPa)
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
Horizontal displacement (m)
Background model
Reduced model
0 10 20
Time (s)
-0.02
-0.01
0
0.01
0.02
Horizontal displacement (m)
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
Horizontal acceleration (m/sec2)
Point A(a) Point B(b)
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
Horizontal displacement (m)
Background model
Reduced model
0 10 20
Time (s)
-0.04
-0.02
0
0.02
0.04
Horizontal displacement (m)
Point A(a) Point B(b)
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
Horizontal acceleration (m/sec2)
Background model
Reduced model
0 10 20
Time (s)
-8
-4
0
4
8
Horizontal acceleration (m/sec2)
Point A(a) Point B(b)
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
Horizontal displacement (m)
0 1 2 3 4 5 6
Time (s)
-0.02
-0.01
0
0.01
0.02
Vertical displacement (m)
0 1 2 3 4 5 6
Time (s)
0
0.1
0.2
0.3
0.4
Horizontal displacement (m)
Extended Mesh
Viscous BC
Cone BC
0 1 2 3 4 5 6
Time (s)
-0.008
-0.004
0
0.004
0.008
Vertical displacement (m)
Point DTo=1s
(c)Point DTo=1s
(d)
Point CTo=1s
(a) Point CTo=1s
(b)
Figure 6.15: Comparison of the displacement response at nodes C, D for a pulse
of To =1.0s.
251
0 1 2 3 4 5 6
Time (s)
0
0.4
0.8
1.2
1.6
Horizontal displacement (m)
0 1 2 3 4 5 6
Time (s)
-0.014
-0.007
0
0.007
0.014
Vertical displacement (m)
0 1 2 3 4 5 6
Time (s)
0
0.4
0.8
1.2
1.6
Horizontal displacement (m)
Extended Mesh
Viscous BC
Cone BC
0 1 2 3 4 5 6
Time (s)
-0.014
-0.007
0
0.007
0.014
Vertical displacement (m)
Point DTo=2s
(c)Point DTo=2s
(d)
Point CTo=2s
(a) Point CTo=2s
(b)
Figure 6.16: Comparison of the displacement response at nodes C, D for a pulse
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
Horizontal displacement (m)
0 1 2 3 4 5 6
Time (s)
-0.016
-0.008
0
0.008
0.016
Vertical displacement (m)
0 1 2 3 4 5 6
Time (s)
0
2
4
6
Horizontal displacement (m)
Extended Mesh
Viscous BC
Cone BC
0 1 2 3 4 5 6
Time (s)
-0.016
-0.008
0
0.008
0.016
Vertical displacement (m)
Point DTo=4s
(c)Point DTo=4s
(d)
Point CTo=4s
(a) Point CTo=4s
(b)
Figure 6.17: Comparison of the displacement response at nodes C, D for a pulse
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
Vertical acceleration (m/s2)
Extended Mesh
Viscous BC
Cone BC
0 1 2 3 4 5 6
Time (s)
-0.2
-0.1
0
0.1
0.2
Vertical acceleration (m/s2)
0 1 2 3 4 5 6
Time (s)
-0.06
-0.03
0
0.03
0.06
Vertical acceleration (m/s2)
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
Horizontal displacement (m)
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
Horizontal displacement (m)
Extended Mesh
Viscous BC
Cone BC
0 1 2 3 4 5 6
Time (s)
-0.014
-0.007
0
0.007
0.014
Vertical displacement (m)
0 1 2 3 4 5 6
Time (s)
-0.016
-0.008
0
0.008
0.016
Vertical displacement (m)
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
Horizontal displacement (m)
0 10 20 30 40 50 60
Time (s)
-0.02
-0.01
0
0.01
0.02
Vertical displacement (m)
0 10 20 30 40 50 60
Time (s)
-0.2
-0.1
0
0.1
0.2
Horizontal displacement (m)
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
Vertical displacement (m)
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
perform better when incorporated in the DRM, as they are required to absorb less
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
Calcareous Sandstone
Metasediments
Fault Gouge Clay
Sandstone, Marl
Quarzitic Rock (bedrock)
Fault Breccia
and Fault Gouge Clay
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
Calcareous Sandstone
Metasediments
Fault Gouge Clay
Sandstone, Marl
Fault Breccia
and Fault Gouge Clay
x
z
Figure 7.13: FE mesh configuration for chainage 62+850 after the excavation of
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 2D viscous, x=70.0m
ICFEP 2D viscous, x=90.0m
ICFEP 1D
EERA
(b)
depth=157.9m
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
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
Maximum Shear Strain (%) (only due to e/q)
200
160
120
80
40
0
De
pth
(m
)
ICFEP 2D TDOF, x=70.0m
ICFEP 2D TDOF, x=90.0m
ICFEP 2D TDOF, x=50.0m
ICFEP 2D TDOF, x=13.0m
ICFEP 2D TDOF, x=0.0m
ICFEP 1D
EERA
(b)
depth=157.9m
Figure 7.21: Shear strain time history (a) and maximum shear strain profile (b)
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
Maximum Shear Strain (%) (only due to e/q)
200
160
120
80
40
0
De
pth
(m
)
ICFEP 2D DRM & viscous BC, x=70.0m
ICFEP 2D DRM & viscous BC, x=90.0m
ICFEP 1D
EERA
(b)
depth=157.9m
Figure 7.23: Shear strain time history (a) and maximum shear strain profile (b)
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)
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.
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
)
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.
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
Figure 7.33: 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 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
Figure 7.34: 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 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
Angle around tunnel lining, θ
-20
-10
0
10
20
Be
nd
ing
Mo
me
nt
(kN
m/m
)
0 60 120 180 240 300 360
Angle around tunnel lining, θ
2800
3200
3600
4000
Th
rus
t (k
N/m
)
LBPT
RBPT
(a)
θ
(b)
(c)
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
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
Maximum Shear Strain (%) (only due to e/q)
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)
Figure 7.44: Pore water pressure (a) and shear strain (b) time histories for
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
Angle around tunnel lining, θ
20000
24000
28000
32000
36000
40000
44000
Ho
op
Str
es
s (
kP
a/m
)
0 60 120 180 240 300 360
Angle around tunnel lining, θ
-300
-200
-100
0
100
200
Be
nd
ing
Mo
me
nt
(kN
m/m
)
0 60 120 180 240 300 360
Angle around tunnel lining, θ
6000
6400
6800
7200
7600
8000
Th
rus
t (k
N/m
)
LBPT
RBPT
(a)
θ
(b)
(c)
Figure 7.45: Accumulated thrust (a), bending moment (b) and hoop stress (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.
Table 7.10: Maximum hoop stress at shoulder and knee locations of the BPTs’
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
Angle around tunnel lining, θ
8000
9000
10000
11000
12000
13000
Ho
op
Str
es
s (
kP
a/m
)
0 60 120 180 240 300 360
Angle around tunnel lining, θ
-8
-4
0
4
8
12
Be
nd
ing
Mo
me
nt
(kN
m/m
)
0 60 120 180 240 300 360
Angle around tunnel lining, θ
2400
2600
2800
3000
3200
3400
3600
Th
rus
t (k
N/m
)
LBPT
RBPT
(a)
θ
(b)
(c)
Figure 7.47: Accumulated thrust (a), bending moment (b) and hoop stress (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)
Figure 7.48: Pore water pressure (a) and shear strain (b) time histories for
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.
Table 7.11: Maximum hoop stress at shoulder and knee locations of the BPTs’
lining computed with the MCCJ model
Maximum Hoop Stress ( Hσ ) (MPa)
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
Angle around tunnel lining, θ
10000
20000
30000
40000
50000
Ho
op
Str
es
s (
kP
a/m
)
0 60 120 180 240 300 360
Angle around tunnel lining, θ
-400
-200
0
200
400
Be
nd
ing
Mo
me
nt
(kN
m/m
)
0 60 120 180 240 300 360
Angle around tunnel lining, θ
4000
5000
6000
7000
8000
Th
rus
t (k
N/m
)
LBPT
RBPT
(a)
θ
(b)
(c)
Figure 7.51: Accumulated thrust (a), bending moment (b) and hoop stress (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
Calcareous Sandstone
Metasediments
Fault Gouge Clay
Sandstone, Marl
Fault Brecciaand Fault Gouge Clay
x
z
Figure 7.52: FE mesh configuration for chainage 62+870 after the excavation of
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
Maximum Shear Strain (%) (only due to e/q)
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
Maximum Shear Strain (%) (only due to e/q)
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
Angle around tunnel lining, θ
20000
25000
30000
35000
40000
45000
Ho
op
Str
es
s (
kP
a/m
)
0 60 120 180 240 300 360
Angle around tunnel lining, θ
-300
-200
-100
0
100
200
Be
nd
ing
Mo
me
nt
(kN
m/m
)
0 60 120 180 240 300 360
Angle around tunnel lining, θ
6000
6400
6800
7200
7600
8000
Th
rus
t (k
N/m
)
2BPTs-AB
1BPT-AB
1BPT-CD
(a)
θ
(b)
(c)
Figure 7.55: Accumulated thrust (a), bending moment (b) and hoop stress (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
Maximum Hoop Stress ( Hσ ) (MPa)
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
experienced a wide range of damage severity during the 1999 Duzce earthquake
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.
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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
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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
experienced a wide range of damage severity during the 1999 Duzce earthquake
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.
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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.
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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,
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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.
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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
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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
perform better when incorporated in the DRM, as they are required to absorb less
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.
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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
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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
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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
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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
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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
∆tαΩα1δΩξα12α1
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
∆tαΩα1δΩξα12α1
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
αΩα1δΩξα12α1
∆tαΩξα2α-δΩξα12α1a
αΩα1δΩξα12α1
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)
Calcareous Sandstone
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