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Simplified Model for Nonlinear Frequency-Dependent Soil with
Shallow Foundation
by
Seok hyeon Chai
A thesis submitted in conformity with the requirements
for the degree of Master of Applied Science
Graduate Department of Civil Engineering
University of Toronto
© Copyright by Seok hyeon Chai 2016
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Simplified Model for Nonlinear Frequency-Dependent Soil with
Shallow Foundation
Seok hyeon Chai
Master of Applied Science
Department of Civil Engineering
University of Toronto
2016
Abstract
This thesis introduces a new approach to seismic Soil-Structure Interaction (SSI) analysis method
for shallow foundation using simplified models. The study is inspired by the increasing number of
interest amongst engineers and researchers in the nonlinear behavior of shallow foundation as the
behavior is unavoidable. The conventional methods of capturing this phenomenon includes a
detailed Finite Element Method (FEM) model. However, due to large computational effort in
modeling and analysis, it is not a feasible option in the current practice.
In this thesis, a method is proposed where macro element for static inelastic behavior and a
recursive parameter model for frequency-dependent dynamic characteristics of soil-foundation
system are integrated. The proposed method is verified in the two dimensional parametric space
of frequency and inelasticity. Then, a practical application example of the proposed modelling
method is presented.
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Acknowledgments
6 Be anxious for nothing, but in everything by prayer and supplication, with thanksgiving, let
your requests be made known to God; 7 and the peace of God, which surpasses all
understanding, will guard your hearts and minds through Christ Jesus. - Philippians 4:6-8
First and foremost I would like to thank God for everything, allowing me to be here and giving me
the strength to complete this study. I also would like to give thanks to my parents, my brother, and
my mentor, Thamar Yacoub, for their unconditional love, continuous encouragement and support.
I would also like to thank my supervisor Professor Kwon for providing invaluable support and
guidance throughout this project. Without his help, I could not have been here. Also, I would like
to thank Amirreza Ghaemmaghami for helping me out in this research.
Lastly I would like to thank all my friends in the church and in the office i2c for their continuous
support and encouragement throughout this project.
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Table of Contents
Acknowledgments.......................................................................................................................... iii
List of Tables ................................................................................................................................. vi
List of Figures ............................................................................................................................... vii
Introduction .................................................................................................................................1
1.1 Literature review and problem statement ............................................................................2
1.2 Research objective ...............................................................................................................6
1.3 Outline of the thesis .............................................................................................................7
Methods to model SSI .................................................................................................................8
2.1 Introduction ........................................................................................................................10
2.1.1 Lumped spring approach........................................................................................11
2.1.2 Methods to capture inelasticity in near field soil ...................................................12
2.1.3 Methods to capture the frequency dependency of soil-foundation system ............13
2.2 Macroelement for near-field interaction (Chatzigogos et al. 2011) ..................................15
2.2.1 Formulation of the macroelement in Chatzigogos et al. (2011) ............................15
2.2.2 Implementation of macroelement in MATLAB ....................................................42
2.2.3 Verification of the implementation ........................................................................45
2.3 Recursive parameter model (Nakamura, 2006a) ...............................................................59
2.3.1 Introduction ............................................................................................................59
2.3.2 Formulation of the recursive parameters by Nakamura (2006) .............................60
2.3.3 Implementation of the recursive parameter model in structural analysis ..............65
2.3.4 Verification of the implemented lumped parameter ..............................................69
Proposed method to model SSI of shallow foundation .............................................................73
3.1 Proposed method ................................................................................................................74
3.2 Verification of the proposed method .................................................................................80
3.2.1 Analysis cases ........................................................................................................80
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3.2.2 FEM model approach .............................................................................................80
3.2.3 Quasi-static loading ...............................................................................................96
3.2.4 Dynamic loading ..................................................................................................128
Application example ...............................................................................................................148
Conclusion ..............................................................................................................................155
5.1 Summary of the findings ..................................................................................................157
5.2 Limitations and future studies and future studies ............................................................158
References ....................................................................................................................................160
Appendices A ...............................................................................................................................164
Appendices B ...............................................................................................................................167
Appendices C ...............................................................................................................................172
Appendices D ...............................................................................................................................178
Appendices E ...............................................................................................................................182
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List of Tables
Table 1. Suggested values of the bounding surface parameters for various footing types
(Chatzigogos et al., 2011) ............................................................................................................. 25
Table 2. Model parameter inputs for the verification examples ................................................... 46
Table 3. Structure and soil property for 10 DOF verification model............................................ 69
Table 4. Material properties for homogeneous infinite soil domain ............................................. 96
Table 5. OpenSees and theoretical results .................................................................................... 98
Table 6. Macroelement bounding surface coefficients ............................................................... 103
Table 7. Material properties for homogeneous soil with rigid rock layer .................................. 112
Table 8. Static stiffness of vertical, horizontal, and rocking direction for soil with stratum (Gazetas,
1983) ........................................................................................................................................... 113
Table 9. Theoretical values with OpenSees results .................................................................... 113
Table 10. Material properties for Gibson soil with rigid rock layer ........................................... 120
Table 11. Material properties for homogeneous infinite soil domain ......................................... 128
Table 12. Structural properties of FEM model for Kobe excitation ........................................... 131
Table 13. Parameters of structure and soil for realistic bridge pier example (Chatzigogos et al.,
2009) ........................................................................................................................................... 149
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List of Figures
Figure 2.1. Schematic illustration of complete Soil-Structure Interaction analysis using finite
element methods (Kramer, 1996) ................................................................................................... 8
Figure 2.2. Transformation method using cone model a) imposed displacement and wave
propagation; b) impulse response on multi-layered soil (Nakamura, 2006b) ............................... 14
Figure 2.3. Generalized force and displacement diagram ............................................................. 16
Figure 2.4. Elastic soil with uplift of the foundation with theory and FEM model ...................... 22
Figure 2.5. 3D Bounding surface plot for values in QN, QV, and QM. ........................................... 24
Figure 2.6. Vertical and horizontal bounding surface with current force vector (Q) to image point
(IQ) ................................................................................................................................................ 27
Figure 2.7. Stress and Strain relationship for typical plastic behavior of material in compression
....................................................................................................................................................... 29
Figure 2.8. Cohesive soil combined with general interface (Chatzigogos et al., 2011) ............... 35
Figure 2.9. Bounding surface of soil with purely cohesive interface element (Chatzigogos et al.,
2011) ............................................................................................................................................. 36
Figure 2.10. Multi-mechanism plasticity of frictional interface (Chatzigogos et al., 2011) ........ 39
Figure 2.11. Sliding mechanism of interface with frictional soil a) using the combined mechanism
b) using non-associative rule (Chatzigogos et al., 2011) .............................................................. 40
Figure 2.12. Macroelement analysis flowchart ............................................................................. 43
Figure 2.13. Nonlinear solution algorithm diagram for a) Code Aster b) MATLAB .................. 44
Figure 2.14. Bounding surface plot for macroelement 3D plane view, QV-QN view, and QM-QN
view from top to left and right (Case A-1) ................................................................................... 47
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Figure 2.15. Linear elastic (left) and nonlinear (right) vertical monotonic load and displacement
plot (Case A-1) .............................................................................................................................. 48
Figure 2.16. Load history applied to macroelement (Case A-2)................................................... 49
Figure 2.17. Bounding surface plot for macroelement (Case A-2) 3D plane view, QV-QN view, QM-
QN view from top and bottom, left to right ................................................................................... 50
Figure 2.18. Linear elastic (left) and nonlinear (right) vertical monotonic load and displacement
plot (Case A-2) .............................................................................................................................. 51
Figure 2.19. Bounding surface plot for macroelement (Case B-1) 3D, QV-QN , QM-QN view ..... 52
Figure 2.20. Linear elastic (left) and nonlinear (right) vertical monotonic load and displacement
plot (Case B-1) .............................................................................................................................. 53
Figure 2.21. Bounding surface of macroelement, 3D plane view, QV-QN and QM-QN plane view
(Case C-1) ..................................................................................................................................... 54
Figure 2.22. Linear elastic and nonlinear load and displacement plot (Case C-1) ....................... 55
Figure 2.23. Bounding surface plot for macroelement (Case D-1) 3D plane view, QV-QN and QV-
QN plane view ............................................................................................................................... 56
Figure 2.24. Linear elastic (left) and nonlinear (right) analysis result for macroelement (Case D-1)
....................................................................................................................................................... 57
Figure 2.25. Idealized two-DOF soil-structure system (Duarte-Laudon, A., Kwon, O. and
Ghaemmaghami, 2015) ................................................................................................................. 66
Figure 2.26. Dynamic impedance function of soil model with 10 nodes (real and imaginary terms)
....................................................................................................................................................... 70
Figure 2.27. Nakamura’s coefficients capturing dynamic impedance function of soil ................ 70
Figure 2.28. Sinusoidal response of MDOF system at top node of soil ....................................... 71
Figure 3.1. Schematic diagram of macroelement with extension to dynamic load ...................... 75
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Figure 3.2. Figure of structure and foundation with five degrees of freedom .............................. 76
Figure 3.3. Soil structure system with Macroelement and Nakamura’s model ............................ 78
Figure 3.4. OpenSees pressure independent multi-yield material with user-defined parameters . 82
Figure 3.5. Octahedral stress and strain at material level for OpenSees ....................................... 84
Figure 3.6. FEM model with a) 100 m by 100m soil model with 2 m foundation b) homogeneous
soil with stratum, c) heterogeneous soil layer with rigid rock layer (Gibson soil) ....................... 85
Figure 3.7. Illustration of soil domain boundary condition for the FEM modesl ......................... 86
Figure 3.8. Diagram of foundation and soil node connection using floating soil node ................ 87
Figure 3.9. Time step analysis plot of the foundation uplift from soil, with height and width of the
soil domain .................................................................................................................................... 88
Figure 3.10. Foundation geometry and excitation conditions and Finite domain and absorbing
boundary (Zhang & Tang, 2007). ................................................................................................. 90
Figure 3.11. Rayleigh wave absorption (Lysmer & Kuhlemeyer, 1969)...................................... 91
Figure 3.12. Hysteresis loop (load vs displacement) for excitation frequency ω = 10.075 in 250m
by 250m FE model ........................................................................................................................ 92
Figure 3.13. Deformation plot for 250 m by 250 m FEM model with angular excitation frequency
ω = 10.075 ..................................................................................................................................... 93
Figure 3.14. C11 vs. dimensionless frequency, ao for FEM models ............................................ 94
Figure 3.15. D11 vs. dimensionless frequency, ao for FEM models ............................................ 94
Figure 3.16. Vertical load and displacement plot for FEM model and macroelement for 100m by
100m soil model ............................................................................................................................ 99
Figure 3.17. Deformed mesh plot in OpenSees for vertical loading case ..................................... 99
Figure 3.18. Slope difference for vertical monotonic load ......................................................... 100
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Figure 3.19. Horizontal maximum load for half-space soil in OpenSees and macroelement .... 101
Figure 3.20. Slope difference for horizontal monotonic load ..................................................... 102
Figure 3.21. Moment maximum load for half-space soil in OpenSees ...................................... 102
Figure 3.22. Slope difference for moment monotonic load ........................................................ 103
Figure 3.23. The bounding surface generation using OpenSees and macroelement (positive
horizontal force) .......................................................................................................................... 104
Figure 3.24. The bounding surface generation using OpenSees and macroelement (negative
horizontal force) .......................................................................................................................... 105
Figure 3.25. Combined loading case (vertical load = -250 KN) with monotonic horizontal load in
OpenSees and macroelement ...................................................................................................... 106
Figure 3.26. 3D plot of bounding surface generated in macroelement and OpenSees ............... 107
Figure 3.27. 3D plot of bounding surface generated in macroelement and OpenSees in moment-
vertical force coordinate ............................................................................................................. 107
Figure 3.28. Vertical constant load (620KN) and horizontal cyclic load for half-space infinite soil
domain......................................................................................................................................... 108
Figure 3.29. Macroelement and OpenSees model results for moment cyclic hysteretic loop .... 110
Figure 3.30. Moment cyclic analysis in homogeneous half-space soil with OpenSees and
macroelement .............................................................................................................................. 111
Figure 3.31. Vertical monotonic load for homogeneous soil with stratum for RS 2.0, OpenSees,
Plaxis, and macroelement model ................................................................................................ 114
Figure 3.32. Vertical load and displacement plot for FEM model and macroelement ............... 115
Figure 3.33. Horizontal maximum force for FEM and macroelement model ............................ 116
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Figure 3.34. Macroelement and FEM model comparison for cyclic moment load on shallow
foundation without uplift ............................................................................................................ 117
Figure 3.35. Macroelement and FEM model comparison for cyclic moment load on shallow
foundation with uplift ................................................................................................................. 118
Figure 3.36. 2D FEM model using Dynaflow for Gibson soil (Cremer et al., 2002) ................. 119
Figure 3.37. Gibson soil model created using OpenSees with refined mesh around foundation 121
Figure 3.38. Moment cyclic analysis plot for the Gibson soil model in OpenSees and paper results
by Cremer et al. (2001) ............................................................................................................... 122
Figure 3.39. Horizontal cyclic analysis plot for Gibson soil model in OpenSees and paper FE
results by Cremer et al. (2001) ................................................................................................... 122
Figure 3.40. Correction factors for rough and smooth footings (Booker & Davis, 1974).......... 123
Figure 3.41. Vertical loading case for OpenSees, RS2.0 and Plaxis for Gibson soil ................. 124
Figure 3.42. Horizontal loading case for OpenSees and macroelement for Gibson soil ............ 126
Figure 3.43. Moment cyclic analysis in OpenSees and macroelement for Gibson soil without uplift
..................................................................................................................................................... 126
Figure 3.44. Moment cyclic Analysis in OpenSees and macroelement for Gibson soil with uplift
of the foundation ......................................................................................................................... 127
Figure 3.45. Horizontal displacement of foundation with Kobe excitation applied to massless
foundation; comparison with FEM analysis and FFT analysis result ......................................... 129
Figure 3.46. Rotation of foundation with Kobe excitation applied to massless foundation;
comparison with FEM analysis and FFT analysis result ............................................................ 129
Figure 3.47. Vertical displacement of foundation with Kobe excitation applied to massless
foundation; comparison with FEM analysis and FFT analysis result ......................................... 130
Figure 3.48. Structure and foundation degrees of freedom for 1m beam example .................... 131
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Figure 3.49. Dynamic impedance of 100m by 100m soil domain (Vertical) ............................. 133
Figure 3.50. Dynamic impedance of 100m by 100m soil domain (Horizontal) ......................... 134
Figure 3.51. Dynamic impedance of 100m by 100m soil domain (Rotational) ......................... 134
Figure 3.52. Dynamic impedance of 100m by 100m soil domain (Coupling with rotation and
horizontal) ................................................................................................................................... 135
Figure 3.53. Kobe excitation applied to structure and the horizontal response of foundation; result
comparison with OpenSees, FFT, and Macroelement+Nakamura’s model ............................... 136
Figure 3.54. Rotation at the foundation with Kobe excitation on structure; OpenSees, FFT analysis,
and Macroelement+Nakamura’s model comparison .................................................................. 137
Figure 3.55. Vertical displacement of foundation with Kobe excitation on structure; OpenSees,
FFT analysis and Macroelement+Nakamura’s model comparison ............................................ 137
Figure 3.56. Parametric study of varying frequency and amplitude without uplift .................... 139
Figure 3.57. Parametric study of varying frequency and amplitude with uplift of foundation .. 140
Figure 3.58. 1000 KNm moment applied at the foundation without uplift at 4Hz excitation .... 141
Figure 3.59. 1000 KNm moment applied at the foundation with uplift at 4Hz excitation ........ 141
Figure 3.60. Uplift of foundation with 1500 KNm moment ....................................................... 142
Figure 3.61. Dynamic impedance of varying intensity without uplift ........................................ 144
Figure 3.62. Dynamic impedance with uplift (uplift occurs at M = 1000 KNm) ....................... 145
Figure 3.63. Dynamic impedance without uplift for FEM, macroelement and the proposed method
(macroelement and Nakamura’s model) ..................................................................................... 146
Figure 4.1. Dynamic analysis example with realistic bridge pier and footing dimension
(Chatzigogos et al., 2009) ........................................................................................................... 149
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Figure 4.2. Time history analysis of foundation with Kobe excitation applied to the foundation
horizontally with OpenSeesOpenSees, FFT and Nakamura’s model ......................................... 150
Figure 4.3. Time history analysis of foundation with Kobe excitation applied to the structure
horizontally with OpenSeesOpenSees, FFT, and Nakamura’s model. ....................................... 151
Figure 4.4. Vertical force and displacement monotonic curve for bridge pier foundation ......... 152
Figure 4.5. Moment cyclic force-displacement plot using OpenSees and MATLAB for bridge pier
example ....................................................................................................................................... 153
Figure A.0.1. Finite element cells of unbounded medium (Wolf & Song, 1996) ...................... 164
Figure B.0.1. Illustration of Newton-Raphson nonlinear solution algorithm ............................. 183
1
Introduction
The Soil Structure Interaction (SSI) is one of the major subjects in earthquake engineering that is
gaining comprehensive attention in the recent decade. There are many papers that address the
issues and the importance in consideration of SSI when the structure is subjected to seismic loads.
The SSI effects become significant in various types of infrastructure under different conditions
with seismic excitation; bridge piers, low to high rise buildings in various types of foundation, and
specific structures such as nuclear power plants. The response of the overall structure can be
broken down into three folds when it is subjected to seismic load; the structure, the foundation and
the soil underlying and surrounding the foundation. When the foundation is subjected to dynamic
excitation due to earthquakes or internal sources such as mechanical equipment, understanding the
dynamic response of these structural system due to SSI effects become pivotal in design procedure
of foundation.
In many civil engineering applications for shallow foundation, consideration of SSI effect with
nonlinearity of soil and structure system is pivotal. For instance, the design of off-shore platform
requires consideration of SSI effect where a large cyclic horizontal and moment is applied to the
structure due to sea wave action. This also applies to earthquake-resistant design of structures for
tall buildings and bridges where the entire structure-foundation soil system is subjected to seismic
excitation (Chatzigogos et al., 2011). The current practices of shallow foundation design now
consider rocking foundations as one of the major component to be included in design of shallow
foundation. Also, the research is pushing towards this phenomenon as the rocking behavior is
unavoidable.
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1.1 Literature review and problem statement
With the growing interest in performance-based approaches to seismic design, there is now
increasing awareness of the effect of interaction between foundation and structure and its role on
overall seismic capacity of the system (Paolucci et al. 2008). The performance-based design
concept is reflected on the building performance where the target objective consists of a
performance level of the overall structure and earthquake hazard levels. The primary performance
levels include Immediate Occupancy (IO), Life Safety (LS) and Collapse Prevention (CP)
(Hakhamaneshi et al. 2015). The earthquake hazard levels are based on their probability of
exceedance with respect to specified time period. For instance, a ground motion with a 2%
probability of exceedance in 50 years or 50% probability of exceedance in 50 years.
Shallow foundations exhibit nonlinear behavior resulting from irreversible nonlinear soil behavior
and soil-foundation interface conditions which leads to sliding and rocking of the foundation. In
ASCE 41-13 (Kutter et al., 2015) the provision has included rocking shallow foundations as part
of their provisions in Seismic Evaluation and Retrofit of Existing Buildings. The rationale for
shallow foundation rocking provisions in ASCE 41-13 is to replace the provisions that were
outdated and to require more explicit assessment of the foundation deformation effects on the
structure (Hakhamaneshi et al. 2015). Also, the motivation of the inclusion of rocking is to increase
the accuracy of the building models with structural component assessment with the inclusion of
SSI effects rather than the unrealistic definition of soil with elastic spring which has uncapped
strength and assumed infinite soil ductility without consequences of rocking (Hakhamaneshi et al.
2015).
In ASCE 41-13, a new modeling parameter and acceptance criteria for rocking shallow
foundations are provided. The shallow foundation is subjected to vertical load, horizontal load at
the structure above the base of the footing. These loads, due to seismic forces and gravity load,
can cause the footing to rotate, slide and settle, where the soil then exerts a resultant force on the
footing. Taking consideration of these interactive loads at the footing, the code has summarized
the findings of moment-rotation behavior of shallow foundations and the design criteria and
modeling parameters are provided. These values are obtained from experimental studies and
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numerical models on rocking foundations. Thus, the primary measure to assess the foundation
performance is residual settlement or uplift. Subsequently, more extensive model tests
investigating the effect of footing shape and rocking and embedment of foundation has been
completed by Hakhamaneshi and Kutter (2015). Therefore, ASCE 41-13 sees the need include
rocking of shallow foundation and provides the acceptance limits for rocking of shallow
foundation for various shapes of footings for engineers and researchers.
In ASCE 41, four types of analysis methods are accepted for evaluation of performance: Linear
Static Procedure (LSP), Linear Dynamic Procedure (LDP), Nonlinear Static Procedure and
Nonlinear Dynamic procedure (NDP). The NDP approach provides the most accurate results
which involves analysis of nonlinear building system subjected to ground motion. This analysis
methods takes into account the hysteretic energy dissipation of the building. Radiation damping
occurring from SSI may also be accounted for by adding radiation dashpots to linear components
of the springs connected to the foundation (Hakhamaneshi et al. 2015). For the numerical model
to accommodate for an accurate wave propagation and radiation damping, a fully non-linear
dynamic analysis in the time domain for a three-dimensional configuration would have to be
analyzed. These numerical models require sophisticated soil constitutive models with complex
geometry configurations. The computational time and the modeling method that is required to
carry out the analysis is too expensive in practice (Finn et al., 2011; Kabanda et al., 2015). Also,
Kabanda et al. (2015) have modelled a full 3D FEM structure-soil model for Hualien Large-Scale
seismic Test which took approximately one month of computational time to analyze a single
dynamic time-history analysis.
Nevertheless, there has been increasing number of interest amongst engineers and researchers in
the nonlinear behavior of foundation as the behavior is unavoidable, but may also be beneficial to
structural design (Anastasopoulos et al. 2011). Although there are many studies and research
findings regarding nonlinear load-displacement response of shallow foundation, in order to
provide a practical methods to adapt to this design procedure, it would require reliability and
capability in realistic modeling of the foundation (Anastasopoulos et al., 2011).
In regards to sophisticated models, there are many simplified models for foundation-soil system
behavior from the literature. One of the method is referred to ‘macroelement’ where the
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nonlinearity of the soil, uplift and sliding of the foundation is analyzed in a lumped node. In the
case of macroelement, the footing and soil are represented with a single node with three degrees
of freedom at the foundation in vertical, horizontal, rotational directions. Another simplified model
is based on Winkler decoupling hypothesis spring model. For a Winkler spring model, the soil is
represented with number of decoupled horizontal and vertical springs where each spring has its
corresponding constitutive law (elastoplastic, contact-breaking etc). The simplicity of Winkler
spring method is that the global response of model can be easily obtained by the summation of the
local spring responses. However, there are some limitations in this model as the calibration of the
spring parameters and describing the coupling terms between vertical, rotational and horizontal
degree of freedom (DOF) of the footing is complex and challenging (Chatzigogos & Figini, 2011).
Therefore, in order to capture all these essential features of dynamic soil-structure interaction
problem, macroelement concept has become more and more popular (Paolucci et al., 2008).
However, a macroelement has limitations in dynamic loading application, which became the
motivation of this study.
Upon dynamic loading application, the soil domain can be separated to two sub-domains: the near
field and the far-field. The near-field soil domain is identified as the domain where nonlinearity of
the soil occurs in the vicinity of the footing. Far-field is the soil sub-domain where the response
remains purely linear. The frequency-dependency of the soil is defined by the dynamic
characteristic of soil. Even at low intensity loading, such as foundations with vibrating equipment,
the soil-foundation system shows dynamic characteristics at different frequencies of the load
applied. The dynamic response of the system with various excitation frequencies are represented
with frequency-dependent stiffness and damping in frequency domain, which is referred to as
dynamic impedance function. Because the intensity of the excitation is small, the response is
assumed to be linear-elastic (Elnashai et al. 2015). Thus, a macroelement can be used to represent
this ‘near-field’ soil domain where majority of the nonlinearity of the soil occurs. The far-field
effect, on the other hand, exhibits linear elastic behavior thus dynamic impedance of foundation
can be used for the ‘far-field’ domain.
5
The current macroelement model uses a constant elastic stiffness and damping term related to a
specific frequency of choice to represent this far-field domain of soil. Chatzigogos et al. (2011)
suggests users to use a specific characteristic frequency of the system, such as predominant
frequency of excitation or fundamental Eigen-frequency of the structure. However, this option
does not fully capture the frequency dependency of the soil.
6
1.2 Research objective
In this thesis, it is proposed to integrate frequency dependent characteristics of soil to
macroelement by including dynamic impedance of foundation in time domain using a recursive
parameter model proposed by Nakamura (2006).
The integrated model is then verified with FEM models with various loading cases to consider the
nonlinearity of the soil with varying excitation of frequencies and intensities of the excitation and
intensities of the excitation applied at the foundation. The proposed model will capture the
dynamic impedance of soil at low magnitude of the force. Also, the verification of the model is
provided for high excitation frequency with high intensity of the load.
7
1.3 Outline of the thesis
The thesis is outlined as followed; in chapter two, existing methods to model SSI effect is
introduced. With brief overview and background information regarding current available methods
to model SSI effect, the strengths and limitations for each of the methods are presented in this
chapter. Then, macroelement is discussed in details as the element analyzes nonlinear behavior of
soil and foundation in the near-field domain of soil. Then recursive parameter models are discussed
in order to capture the frequency-dependency of the soil in far-field domain effect of the soil.
The third chapter discuss the proposed method to integrate recursive parameter model with
macroelement model. This chapter elaborates on the methods to combine two models in time
domain. Then, verification of the model is provided in various soil domain and types for quasi-
static loading case scenario. Also, the model is verified with cyclic moment applied to the
foundation at varying excitation of frequencies and intensities with FEM model.
Lastly, in chapter four, a realistic bridge pier design example is analyzed with the proposed model
and the results are compared with FEM model to provide the applicability of the proposed model
in engineering problems. The findings and further discussions on the improvements and future
work are provided in chapter five of the thesis.
8
Methods to model SSI
The overall analysis of structure with SSI effect contains the structure, the foundation, the nearfield
soil of the foundation and far-field soil which collectively interacts with each other in seismic load.
SSI effects are composed of kinematic interaction effects, inertial interaction effects, and radiation
damping effect which accounts for the flexibility of the soil and its capacity to dissipate wave
propagation.
There are two main approaches in evaluating SSI effect. Direct analysis includes all the structure
models in same model and analyzes the model as a complete system. Substructure approach
analyzes distinct parts of the analysis separately and combines the results to yield overall response
of the structure (Kramer, 1996). Direct analysis combines all the structural components of the
analysis in single model as shown in Figure 2.1.
Figure 2.1. Schematic illustration of complete Soil-Structure Interaction analysis using
finite element methods (Kramer, 1996)
All of the components are added in a continuum model in finite elements with nodes representing
different structural elements. Along the boundaries, viscous boundary dashpots are assigned to
dissipate the wave propagation occurring from the structure to soil. The foundation elements
9
connect the soil elements and super-structure altogether. This can also be modelled in 3D using
FEM. Although this approach can address all of the SSI effects in single model, it is difficult to
carry out the analysis due to large computational effort in building the appropriate model and the
analysis. Also, the challenge of this approach is apparent when the system is complex in geometry
with nonlinearity of the soil and structural model. Due to these significant factors, this approach is
seldom used in practice (Harris et al., 2012).
Substructure method considers SSI effect in separate analysis components in the following
manner: evaluation of free-field motion and the corresponding soil behavior, conversion of free-
field motion to transfer function which can be applied as foundation input motion, then
incorporation of springs and dashpot to represent dynamic characteristics of soil foundation
interface. Lastly, the response of the combined structure with springs and dashpot representing soil
foundation system. The underlying assumption in substructure approach is the linear elastic
behavior of soil. In order to take nonlinearity of soil, equivalent linear shear modulus is often used
in this method.
10
2.1 Introduction
In this chapter, different modeling methods to capture SSI effect is presented. When the structure
experiences abrupt displacement due to ground excitation, the structure is subjected to seismic
waves which occurs directly from the soil. This excitation causes inertial force that is generated
from the induced motion of the super-structure which is reflected back to the soil. This
phenomenon is considered to be the main factor for analyzing seismic structural response and can
be broken into two separate responses which are referred to as Kinematic Interaction (KI) and
Inertial Interaction (II). Kinematic Interaction (KI) is described as an effect of incident seismic
wave to the foundation, meaning wave traveling from soil domain to the foundation. Soil and
structural response behaves as a new system which has different dynamic properties with period
lengthening and radiation damping (Mahsuli & Ghannad, 2009). This is often referred to as Inertial
Interaction effects (II). Inertial Interaction (II) is a response of complete soil-foundation-structure
system which uses the effective force that is obtained from Kinematic Interaction effect by the
acceleration of the superstructure using D’Alembert’s force integration scheme (Mylonakis et al.,
2006).
There are two main analysis methods in analyzing SSI effect of soil-foundation system. One
method is the direct approach where the inertial and kinematic interaction effect is captured
simultaneously in the entire soil-structure system modelled with FEM. Although realistic behavior
of structural components or soil can be modeled using this approach, the analysis requires a
detailed FEM model creation with high computing time. Another method is the multi-step method
where the KI and II effect are separately analyzed. KI is first analyzed with soil and foundation
without mass of structure (Mylonakis et al., 2006). The KI effect captures resultant foundation
motion due to ground motion excitation occurring at the bedrock without mass of the structure.
Then, the KI results are used in II analysis as an input motion applied directly to the foundation.
The response of the overall multistep analysis is obtained by combining the response from KI
effect and response from II effect of soil-foundation system. For shallow foundation, KI effect is
almost negligible as the embedded depth of foundation is small enough for the waves to be
obstructed when the ground excitation propagates from rigid rock layer to the surface of the soil
11
deposit (Mylonakis et al., 2006; Mahsuli & Ghannad, 2009). Thus, free-field surface motion can
be used as a foundation input motion to the II analysis step.
There are variety of modeling methods available for soil-foundation system. Some of these
elements vary in complexity to applicability in practice. For shallow foundation the following
simplified models are often used: lumped elastic springs and dampers, Winkler-type springs,
macroelement, and FEM models. This chapter briefly covers the background of each model and
explain its strengths and limitations.
2.1.1 Lumped spring approach
Lumped spring and damping approach considers soil to behave in linear elastic manner. Then the
soil-foundation system can be represented as linear springs and dashpots. This spring element
represents the compliance of soil and damping element represents dissipation of the energy in soil
through material damping and radiation damping as seismic wave propagates to infinite soil
medium. The stiffness and damping terms can be represented at different excitation frequency,
also known as frequency-dependency of soil, or dynamic impedance function. If the soil and
foundation system is modelled as a linear elastic springs and dampers in time domain, then one
can use the stiffness and damping coefficients at the fundamental period of a structure.
Mylonakis et al. (2006) has compiled dynamic impedance of various foundation shapes and soil
types. The behavior of foundation at each harmonic excitation frequency is characterized with
dynamic stiffness and damping terms. These dynamic coefficient terms are used to formulate
transfer function in frequency domain, which can be used to analyze the response of foundation
subjected to ground motion in time domain using inverse Fourier transformation method.
The impedance function for horizontal displacement becomes Kz which represents the dynamic
stiffness of the supporting soil with rigid foundation. This expression of this component is shown
in Eq. (2.1.1).
12
�� = ������ = ��� + �� (2.1.1)
Eq. (2.1.1) shows ��� as the frequency dependent dynamic stiffness, and �� as the dashpot
coefficient that reflects radiation and material damping. For three translational and three rotational
modes of vibration along x, y, and z axis, the cross-coupling for horizontal-rocking impedance is
usually negligible in shallow foundation due to moments by the base axis (Mylonakis et al., 2006).
Mylonakis et al. (2006) have provided set of parametric studies to compare dynamic impedance
function for shallow foundation with different dimensions and shapes of the footings. There are
equations available in all degree of freedom direction in different geometry of the foundation.
However, analyze strong non linearity effects at near field soil foundation is found to be beyond
the state of the art in seismic SSI (Mylonakis et al., 2006). The parametric studies for bridge piers
by Mylonakis et al. (2006) provides compilation of frequency-dependent soil foundation system
in linear elastic behavior for case-specific examples. Thus, it should not be applied as a generalized
guideline bridge pier designs.
2.1.2 Methods to capture inelasticity in near field soil
As previously discussed in the chapter, lumped spring model neglects the nonlinearity of the soil.
However, the soil exhibits nonlinear behavior upon large magnitude of excitation at the near-field
domain of the foundation. Thus, in this chapter, different modeling methods to capture the
nonlinearity of the soil is discussed. For shallow isolated foundation, there are three main elements
that are commonly used: the uncoupled lumped spring approach, where uncoupled translational
and rotational springs are used; Winkler approach where distributed vertical and horizontal springs
are applied across the foundation; macroelement with formulation of plasticity with
bounding/yield surface; and continuum approach using FEM model. Continuum FEM model
yields the most accurate results for load-deformation behavior, but it is computationally intensive.
The uncoupled approach model captures load-displacement behavior as a simplified analysis, but
this method cannot predict settlement accurately. Winkler model presents settlement and
13
progressive mobilization of plastic capacity accurately but the calibration effort for modeling the
spring elements and capturing uplift is challenging task. The current development of beam-on-a-
nonlinear Winkler foundation (BNWF) approach also provides nonlinear behavior of soil with
foundation but the drawback of this element is the large number of nonlinear spring that is required
to capture the main features of foundation behavior. (Ganainy & Naggar, 2009) The macroelement
approach provides satisfactory agreement in predicting a complete foundation response of
nonlinear behavior of soil with coupling effect of the foundation in all directions in a lumped node.
This element however is limited to specific bounding surface and may not be applicable to wide
range of problems. For this thesis, macroelement has been investigated in details to examine its
strengths in capturing the nonlinear behavior of the shallow foundation at the near-field of soil.
2.1.3 Methods to capture the frequency dependency of soil-foundation system
There are different methods to capture the frequency-dependency of soil. One of the most
commonly used approach is the Wolfs model where the response analysis in time domain is
obtained by impulse response using inverse Fourier transformation with soil impedance function
available in frequency domain. Furthermore, Wolf (1989) has developed a method for recursive
representation of convolution integral from the soil impedance. Another method includes lumped
parameter model where the soil impedance is approximated with system of spring-dashpot-and
mass models. The proposed model has been studied by De Barros (1990) and Wolf has extended
the work, but the practicality of this model for general use is limited and requires further
improvement in the methods. A recursive parameter introduced by Nakamura (2006) provides a
new method to transform the freuqnecy dependent impedance to the impulse response in time
domain. The impulse response is formulated in terms concerning both the past displacement and
velocity (Nakamura, 2006b). This formulation follows the concept of cone model as shown in
Figure 2.2.
14
Figure 2.2. Transformation method using cone model a) imposed displacement and wave
propagation; b) impulse response on multi-layered soil (Nakamura, 2006b)
As shown in the figure above, from the impulse load applied to the foundation, the reflective
reaction force and reflective waves that come back from the lower boundary is obtained. Based on
this delayed time response, the reflected reaction forces can be used to ccalculate the impedance
of soil using the delayed impulse response. A detailed formulation and methodology of this model
is discussed in details in Chapter 2.3 of this thesis. As this method is introduced as an effective
and stable method to transform frequency-dependency of soil to time domain analysis, this method
is described in details in later part of this chapter.
15
2.2 Macroelement for near-field interaction (Chatzigogos et al. 2011)
In this section, macroelement background and formulation is explicitly discussed. As previously
discussed, macroelement is a simplified model which includes nonlinear behavior of soil with rigid
foundation in a lumped node. Numerous mathematical equations and derivations are used in the
analysis to produce a global response of the foundation with nonlinear soil in macroelement.
Intricate details are provided for each of the nonlinear characteristics of the soil-structure model,
including hypoplastic behavior of soil and geometrical nonlinear mechanisms of the footing when
the structure undergoes uplift or sliding along the interface of the foundation. The explanation in
this chapter is decomposed with the failure mechanisms of the model. Firstly, the original
derivation of the linear elastic part of the soil model is briefly discussed. Then, the formulation of
the plasticity of soil is explained. Lastly, the multi-failure mechanism of the interface and soil are
discussed further to provide a complete analytical procedure for macroelement formulation.
2.2.1 Formulation of the macroelement in Chatzigogos et al. (2011)
The original development of macroelement was first seen with development of theory of plasticity.
Roscoe and Scofield (1956) have first studied the behavior of nonlinear response of shallow
foundation with theory of plasticity. The stress-deformation response are replaced by resultant
force and corresponding displacement vectors, where the generalized forced relationship
formulation is obtained. Then, the model was expanded by Nova and Montrasio (1991) to include
a strip footing on sand under monotonic load with isotropic hardening elastoplastic law. Then the
development of the plasticity is obtained using a chosen hardening law. Using hardening rule and
flow rule, the analysis was able to capture accurate approximations of ultimate surface for soil-
foundation system. Paolucci (1997) has worked on expanding this element with capacity of seismic
analysis and Cremer (2001) has added a two distinct nonlinear mechanism of soil and uplift of the
footing in a single element for cohesive soil under seismic load. Chatzigogos et al (2011) has
expanded the application of the model to frictional soil.
16
The generalized force and displacement relationship is used in macroelement where the response
of the structure is defined by the stiffness of the material and the force experienced by the structure.
For the case of macroelement, the footing of the structure is assumed to be rigid. Due to the perfect
rigidity of the foundation, if a single point of displacement is known in the footing, all of the
movements along the footing is known as well. Thus, a single node is used at the center of the
footing to represent the movement of the overall foundation. This node has horizontal, vertical,
and rotational degrees of freedom to represent the response of the footing. Then, the corresponding
forces apply to these respective degrees of freedom. The following force and displacement
parameters are expressed as shown in Eq. (2.2.1) and illustrated in Figure 2.3 .
� = ������� =������� ����� ����� ������ ��
����� , ! = !�!�!�� =
�����"�� "�� #$ ���
�� (2.2.1)
Figure 2.3. Generalized force and displacement diagram
Q and q are the force and displacement parameters respectively. Nmax is the maximum vertical
force resisted by the footing and D is the characteristic dimension of the footing (diameter if it is
circular footing and width if it is strip footing). Normalization is used to provide dimensionless
N
M
V
ux
uy
θy
17
parameters that is easier to work with. The following expression leads to work equation by the dot
product of force and displacement as shown in Eq. (2.2.2).
%&�, !' = � ∙ ! = �)!) = 1����� &�"� + ��"� + �$#$' = +�����
(2.2.2)
The Eq. (2.2.2) shows total work W done in the system, normalized by the fixed constant value of
NmaxD. The generalized force to displacement relationship can also be represented with
generalized stiffness matrix as shown in the Eq. (2.2.3).
,�-��-��-�. = ��� ��� ������ ��� ������ ��� ���� !-�!�-!�- � (2.2.3)
For macroelement formulation, the generalized force and displacements are expressed with
increments, noted by dots on each force and displacement variables. Then, following the consistent
normalization scheme provided from Eq. (2.2.1), the normalized stiffness matrix is given by:
�/ =�������& �����'��� & �����'��� & 1����'���& �����'��� & �����'��� & 1����'���& 1����'��� & 1����'��� & 1�����'������
���� (2.2.4)
Kij in the matrix, where i and j are N, V, M, is the stiffness of the real system while �/ is the
normalized stiffness matrix. Although the generalized force and displacement relationship in this
formulation is described in two dimensions with planar loading, the macroelement model can be
expanded to three dimensions by introducing additional degrees of freedom in the out-of-plane
direction.
18
Elastic soil domain with uplift behavior of the foundation is examined first in order to observe the
geometric nonlinear behavior of the foundation upon uplift effect. The linear elastic stiffness
parameters are defined by the static impedance functions for strip and circular foundation
(Mylonakis et al., 2006). The stiffness matrix for linear elastic soil domain without uplift of the
foundation is shown in Eq. (2.2.5).
� = ��� 0 00 ��� 00 0 ���� (2.2.5)
Where KNN, KVV, and KMM are vertical, horizontal, and rotational static impedance parameters.
For static impedance with a constant gradient shear modulus with depth are expressed as shown in
Eq. (2.2.6) to Eq. (2.2.8).
��� = 0.731 − 5 67&1 + 29' (2.2.6)
��� = 22 − 5 67 :1 + 23 9; (2.2.7)
��� = <2&1 − 5' 67 :=2;> :1 + 13 9; (2.2.8)
Where α is defined by the gradient shear modulus as
6 = 67&1 + 9?', @AB ? = 2C= (2.2.9)
Where z, the depth of soil, and Go, the shear modulus at depth z = 0, the G, shear modulus at depth
z, v is the Poisson ratio, and B is the foundation width.
Due to the planar base which sits on the soil surface, the coupling terms for the foundation in the
stiffness matrix are almost negligible. However, when the uplift of the footing occurs, the coupling
terms are taken into consideration (Chatzigogos et al., 2011). The uplift condition is determined
based on the magnitude of moment applied to the footing. When the moment exceeds certain value
in ratio of vertical force applied to the footing, then uplift occurs.
19
Before uplift: |��| < F��,7F → �� = ��� ∗ !�IJ (2.2.10)
Uplift initiation: |��| = F��,7F → !�,7IJ = K�,7LMM (2.2.11)
The value of ��,7 is the initiation which is determined prior to uplift. This value is calculated by
the ratio of the vertical force applied at the footing. The following Eq. (2.2.12) shows the
relationship between the uplift moment variable ��,7 to vertical load �� for strip foundation. Note
that ��,7 is normalized with maximum load, ����, and width of the foundation, D, and �� is
normalized with maximum load ����.
F��,7F = ± ��4 = ± :14 ���!�IJ; (2.2.12)
The Eq. (2.2.12) can also be derived from the static equilibrium for strip foundation. For instance,
the rigid beam with some length, D, on elastic soil domain transfers the vertical force experienced
by the footing, V to uniformly distributed support from the soil to the rigid beam. This vertical
force will generate maximum moment of the footing, � = �PQ . As mentioned before, the force
variables in Eq. (2.2.12) is normalized with the maximum vertical bearing force of the footing,
Nmax. Using this normalized force parameter with the aforementioned moment to vertical force
relationship, then the following uplift relationship can be derived as shown in (2.2.13).
M = ��4 M� ∗ �S@T = �4 ∗ �S@T
��.7 = 14 ��
(2.2.13)
Where ��,7 = UV∗WXYZ and �� = ��XYZ. Thus, ��,7 is expressed as the following equation as in Eq.
(2.2.12). Thus, the angle of elastic rotation can be calculated at this uplift moment initiation value
as shown in Eq. (2.2.14).
20
F!�,7IJF = 1��� :14 ��� ∗ !�IJ; (2.2.14)
For other footing shapes, calibration is required from numerical analysis of FEM model to obtain
QM,o. The analysis is carried out by fixing the vertical force to be zero while increasing the moment
until uplift occurs.
Based on the Eq. (2.2.3) for incremental force to displacement equation, if the analysis is carried
out with constant vertical force on the footing, the vertical terms become zero for increment
expression. Thus, two approximation can be derived from the analysis after the vertical terms
become zero as shown in Eq. (2.2.15).
�-� = ���!-�IJ + ���!-�IJ = 0 (2.2.15)
Similarly, the increment of the moment is written as:
�-� = ���!-�IJ + ���!-�IJ = 0 (2.2.16)
Two equations lead to the following approximate relationship
����,7 = 2 − !�,7IJ!�IJ , [\] |��| > F��,7F (2.2.17)
!-�IJ!-�IJ = − 12 _1 − !�,7IJ!�IJ ` (2.2.18)
The expressions above gives coupling effect of the vertical force to moment during uplift. There
are two additional assumptions that are made in this stiffness matrix for simplicity of the
calculation.
a) The elastic stiffness matrix is symmetric. There is no physical meaning behind this
assumption but it is particularly helpful for numerical treatment of the analysis
b) The element KNN remains constant during uplift. Thus, all of the vertical force and vertical
displacement of footing will be taken into account in the coupling term of the stiffness
(Chatzigogos et al., 2011)
21
Thus, the overall increment elastic-uplift soil domain is then updated as the following expression
in Eq. (2.2.19).
,�-��-��-�. = ��� 0 ���0 ��� 0��� 0 ���� ,!-�IJ!-�IJ!-�IJ. (2.2.19)
Where KNN, KVV, KMM stays the same as static stiffness obtained from the static impedance
function of the footing. The coupling terms are expressed as shown in Eq. (2.2.20) and Eq. (2.2.21).
��� = ��� = a0, , [ F!�,7IJF ≤ F!�,7IJF12 ��� _1 − !�,7IJ!�IJ ` , [ F!�,7IJF > F!�,7IJF c
(2.2.20)
���= a���, , [ F!�,7IJF ≤ F!�,7IJF
��� _!�,7IJ!�IJ `> + 14 ��� _1 − !�,7IJ!�IJ `> , [ F!�,7IJF > F!�,7IJF c (2.2.21)
This specific derivation covers uplift of the footing on elastic soil domain. The coupling effect of
uplift and plasticity of soil domain will be further discussed in the multi-failure mechanisms
section.
For verification of this formulation, FEM model has been created with constant vertical force
applied to the foundation at 600KN to homogeneous elastic soil domain with 20 meters in height
by 60 meters in width. This vertical force causes the increment vertical force becomes zero. The
two approximations have been made from numerical model as the following:
����,7 = 2 − !�,7IJ!�IJ , [\] |��| > F��,7F &�d\A "de[f' (2.2.22)
!-�IJ!-�IJ = − 12 _1 − !�,7IJ!�IJ ` (2.2.23)
22
Applying this approximation yields analytical result for monotonic pushover curve of elastic soil
domain with uplift as shown in Figure 2.4.
Figure 2.4. Elastic soil with uplift of the foundation with theory and FEM model
Where the uplift occurs at !�,7IJ = �gL�� , @AB �h = �iQ . Since the elastic stiffness of macroelement
has been covered, the next section focuses on the formulation of plasticity of soil using constitutive
law of soil. The explanation is further discussed in the next section.
0
500
1000
1500
2000
2500
3000
3500
0.00E+00 1.00E-03 2.00E-03 3.00E-03 4.00E-03 5.00E-03 6.00E-03
Mo
men
t (K
Nm
)
Rotation (ϴ)
FEM results_uplift
Paper_theoretical_uplift
23
2.2.1.1 Bounding surface
The material nonlinear behavior of soil can be illustrated by the plasticity of soil that is limited by
strength criterion. For a specific case of undrained condition in cohesive soil, the strength criterion
is often described as Tresca criterion. For frictional soil, different failure criterion is covered in the
latter discussion. Also, the coupling effect of geometric nonlinearity, such as uplift, is not covered
at this point. The soil develops plasticity when the shear strength it experiences exceeds cohesive
strength of soil and dissipates energy.
In contrast to uplift condition, this nonlinear behavior is purely material origin, dissipative and
irreversible. Thus, the response can be described by generalized forces and displacements. In
reality, plasticity of soil occurs even at the initial loading and continuous loading shows
progression of plasticity as it deforms. Thus, bounding surface hyperplastic model is presented to
capture the continuous plastic response in virgin loading and also in reloading of the soil model.
This theory was first developed by Dafalias and Hermann (1982). They have replaced a yield
surface of classical plasticity with a bounding surface denoted as fbs. The ultimate surface is used
in defining the failure criterion of soil with rigid foundation as the overall system of the structure.
It represents failure criterion in generalized force domain with combination of forces along normal,
vertical, and rotational direction of the force. This bounding surface applies to the global ultimate
loads of the system and evaluates the magnitude of plasticity at the current force step of the
analysis. Eq. (2.2.24) shows the simplified approximation of ultimate surface which is an ellipsoid
centered at the origin. Figure 2.5 shows the contour plot of bounding surface when the bounding
surface is plotted for the values in these coordinates, QN, QM and QV.
[ij&��' = ��> + & ����,���'> + & ����,���'> − 1 = 0 (2.2.24)
24
Figure 2.5. 3D Bounding surface plot for values in QN, QV, and QM.
Where QN, QV, and QM are the normalized vertical, horizontal, moment forces that is experienced
by the foundation respectively. QV,max is the QM,max are the normalized maximum horizontal and
moment force of the footing. Salencon et al. (1982) has worked on the vertical bearing capacity of
shallow foundation with heterogeneous soil types. Gourvenec (2007) has studied the maximum
moment capacity of shallow foundation for various foundation shapes using numerical results. The
results are summarized as bounding surface parameters as shown in Table 1.
-1
-0.5
0
0.5
1
-0.2
-0.1
0
0.1
0.2
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
QN
QV
QM
25
Table 1. Suggested values of the bounding surface parameters for various footing types
(Chatzigogos et al., 2011)
Nmax
QN,max
QV,max
QM,max
klmmn = 0
Strip 5.14coa 1 1/5.14=0.195 0.57/5.14=0.111
Circular 6.05cp q�rQ 1 1/6.05=0.165 0.67/6.05=0.111
klmmn = 2
Strip 8.01coa 1 1/8.01=0.125 0.95/8.01=0.119
Circular 7.63cp q�rQ 1 1/7.63=0.131 0.88/7.63=0.115
klmmn = 6
Strip 10.29coa 1 1/10.29=0.097 1.35/10.29=0.131
Circular 9.68cp q�rQ 1 1/9.68=0.103 1.25/9.68=0.129
For different values of a∇c/co, calibrated parameters for Nmax, Qv,max, and QM,max are defined,
where ∇c is the vertical cohesion gradient per depth, and co is the initial cohesion at the surface.
Nmax is determined from solution presented in the paper (Salencon et al., 1982). QV,max is obtained
from the condition of sliding along the interface due to soil strength criterion. QM,max is obtained
for strip footing (Gourvenec & Barnett, 2011) and circular footing (Gourvenec, 2007) respectively.
For simplified macroelement modelling, ellipsoidal ultimate surface at the origin is provided. This
ultimate bounding surface defines the maximum bearing capacity of the foundation. At the interior
of this bounding surface, continuous plastic response is obtained as a function of the distance
between actual force increment �- and an image point I(Q) which lies along the bounding surface.
The image point is the projection of the current force increment �- to the bounding surface. The
image point I(Q) is thus defined by the following expression:
t&��' = uv��|t ∈ x[ij, v > 1y (2.2.25)
This lambda value in the Eq. (2.2.25) is the scalar parameter for measuring proportional loading
of the current force vector to the bounding surface. The equation for the lambda is defined by the
Eq. (2.2.26) and Eq. (2.2.27).
26
vz{ = |��> + & ����,���'> + & ����,���'> (2.2.26)
v = 1|��> + : ����,���;> + : ����,���;>
(2.2.27)
The equations above are derived from the general ellipsoidal equations. For instance, a general
equation of ellipsoid in Cartesian coordinates x, y and z is shown below.
T>@> + }>~> + C>�> = 1 (2.2.28)
Where a, b, and c are the maximum values of each corresponding coordinates, x, y and z
respectively. For simplicity, ellipsoid equation is further discussed where only x and y coordinates
are defined.
T>@> + }>~> = 1 (2.2.29)
The parametrization of coordinates x and y is given by the following.
T = asin&#' } = bcos&#'
(2.2.30)
Where –π<ϴ< π transforms x and y in the spherical coordinates in radians. Then, the radius of
ellipsoid is expressed by the following variable, r in Eq. (2.2.31).
] = @~�&~�\�#'> + &@�A#'> (2.2.31)
Where ϴ is the angle of the radius along the ellipse. If there is a current force, Q, inside the
bounding surface, the magnitude of Q then becomes the expression in Eq. (2.2.32). From this force
step, lambda value can be verified by checking the ratio of the radius to the magnitude of the
current force step as shown in Eq. (2.2.33).
27
|�| = ���> + ��> (2.2.32)
v = ]|�| (2.2.33)
This equation shows that lambda is simply the ratio of the current force step to the radius of the
ellipse, which is the maximum value of the combined forces at the footing represented with the
bounding surface. Thus, as shown in Eq. (2.2.33), it is possible to obtain the projected force vector
IQ with the calculated lambda value and the current force step. Figure 2.6 shows an example of
image point (IQ) on the bounding surface for a single step force (Q).
Figure 2.6. Vertical and horizontal bounding surface with current force vector (Q) to image
point (IQ)
This is also referred to as mapping rule. As the actual force reaches close to the bounding surface,
plasticity becomes more pronounced as defined in this mapping rule, as the lambda value will get
-1 -0.5 0 0.5 1-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15N V
QN
QV
IQ
Q
28
close to one. The image point defines the direction of plastic displacements and magnitude of
plastic modulus.
Depending on the condition of this image point with force increment �- and normal vector n, three
loading conditions can be determined. The normal vector can be written as the following
expression in Eq. (2.2.34). The reason for the use of normal vector will be further explained in
plasticity theory known as flow rule.
A� = _x[ijx�� ��`�x[ijx�� ���
(2.2.34)
The three loading conditions with normal vector n, and normalized force vector Q are:
Pure loading: �- ∗ A > 0, �Ae\@BA�: �- ∗ A < 0, A�"f]@e e\@BA� �- ∗ A = 0
(2.2.35)
Pure loading is followed by the plastic deformation of the soil. In case of neutral and unloading
the response becomes purely elastic. This behavior is the underlying assumption of flow rule which
is discussed in details in the next section.
2.2.1.2 Flow rule plasticity
Flow rule plasticity theory is applicable to various types of material that behaves in inelastic
manner. The flow plasticity theory has certain characteristics in determining the amount of
plasticity in the material. The general assumption of this plasticity theory is that the total strain in
the body can be decomposed into elastic and plastic part. The total response of material is thus
explained by the superposition of elastic and plastic response of the material (Lubliner, 2005).
29
Figure 2.7. Stress and Strain relationship for typical plastic behavior of material in
compression
Three main stages of material behavior that is included in the flow rule.
1. Elastic range: upon small loading the material is assumed to behavior in elastic manner
2. Beyond the elastic limit loading, upon loading (f≥0) the material develops plasticity. It is
assumed that the direction of plastic strain is identical to the normal of the yield surface
(df/dσ). Thus, Eq. (2.2.36) explains the loading condition.
B�: x[x� � 0 (2.2.36)
3. Unloading where increment of stress is less than zero (σ<0) the material has no additional
plastic strain, thus, it behaves elastically.
In flow rule, it is assumed that plastic strain increment and deviatoric stress tensor have the same
principal directions. Deviatoric stress tensor is the remaining stress left after the hydrostatic stress.
The original derivation behind flow rule is explained in details by Bland (1957).
The flow rule hypothesis does not mention the irreversible strain that occurs when yield criterion
is satisfied. In general hypothesis of flow rule, the increments of generalized force-displacement
relationship has to be in incremental plastic strain, not the total plastic strain. Also, the ratio of
30
plastic strain is independent of stress increment. The Eq. (2.2.37) describes the following
relationship.
B�)�� = @)�Bv (2.2.37)
Where the term aij depends on σij. There exist a function g such that the plastic strain increment
are given by Eq. (2.2.38).
B�)�� = x�x�)� Bv (2.2.38)
This equation is identical to the previous equation, but the expression aij is replaced with a function
g taken derivative with respect to the stress increment σij. This is called the plastic potential. dλ is
the non-negative infinitesimal which can depend only upon space co-ordinates and time. This
equation shows the original derivation of ‘non-associated flow rule’.
The dependency of the function g with invariant of σij shows that principal axis of stress and
plastic-strain increment coincide. Further hypothesis suggests that if a convex yield function f(σij)
and g(σij) are identical, the general equation now becomes Eq. (2.2.39).
B�)�� = x[x�)� Bv (2.2.39)
The flow rule which corresponds to this particular yield condition is called the ‘associated flow
rule’ (Bland, 1957). The equation shows that the plastic strain increment vector is normal to the
yield surface of f(σij).
Experiment with materials satisfying von Mises failure criterion show they also satisfy the
associated flow rule, such as Reuss equation. In the context of macroelement with ellipsoidal
bounding surface of the footing, the normal vector of the yield surface is the same as gradient of
the force. As previously shown in Eq. (2.2.24), the bounding surface of the footing is formulated
with normalized force and maximum load capacity of the foundation in three degrees of freedom
as shown in Eq. (2.2.40). Then, the yield surface that is derived with respect to the stress now
becomes:
31
[ij&��' = ��> + & ����,���'> + & ����,���'> − 1 = 0
x[x��,�,� =������2 ∗ ��2 ∗ �������>�������>
������ (2.2.40)
This yield surface equation is similar to the normal vector in Eq. (2.2.34). For the case of undrained
cohesive soil, the failure criterion is often described by classical Tresca strength criterion as
mentioned before. Therefore, associated flow rule is applied to the direction of plastic loading
using bounding surface formulation for purely cohesive soil with perfectly bounded interface. The
normal vector which represents the direction of plastic loading, with associated flow rule is given
by Eq. (2.2.41).
Aij = � x[ijx��x[ijx� � �
(2.2.41)
Non-associative flow rule is applicable for interface strength criterion where plastic potential
surface g is defined. The direction of normal vector for plasticity loading for this specific case is
expressed as Eq. (2.2.42).
A�� = _x�x���� `�x�x���� �
(2.2.42)
As mentioned before, g is the interface strength criterion. Then, based on this normal vector, plastic
modulus is obtained by the ‘consistency condition’ which requires that during yield, stress point
should always remain on the yield surface (Pastor, Zienkiewicz, & Chan, 1990).
32
2.2.1.3 Plastic modulus of soil
From the original derivation of generalized plasticity and modeling of soil behavior (Pastor et al.,
1990), stress and strain relationship can be specified by the following equation, Eq. (2.2.43).
B� = �: B� (2.2.43)
Where dσ is the stress increment, dε is the strain increment and D is the uniquely defined tangent
stiffness of soil. The inverse relationship is given by Eq. (2.2.44).
B� = �: B� (2.2.44)
Where C is the constitutive tensor which is the inverse of the tangent modulus D. As mentioned
previously in the flow rule, the increment of strain is caused by two deformation components, i.e.
elastic or plastic, which is expressed as the following Eq. (2.2.45).
B� = B�I + B�� (2.2.45)
From the generalized force and displacement relationship in the case of pure loading, a generalized
plastic modulus H can be defined by the following expression Eq. (2.2.46).
�-� = �� ∗ !- �J (2.2.46)
For this case, the plastic displacement is described by the inverse of the plastic modulus.
!- �J = ��z{�-� (2.2.47)
Then, the relationship can be written as the inverse of the plastic modulus in the form Eq. (2.2.48).
��z{ = 1ℎ �A� ⊗ A[� (2.2.48)
In Eq. (2.2.48), h is a constant scalar, and n is the unit normal vector in the direction of plastic
loading. The variables ng and nf are discussed in details in the Eq. (2.2.41) and Eq. (2.2.42). Thus,
dyadic multiplication is applied to normal vectors ng and nf. This means that ngnf are expressed as
shown in Eq. (2.2.49).
33
A� ⊗ A[ = A�A[� = A��A��A��¡ &A[� A[� A[�' = A��A¢� A��A¢� A��A¢�A��A¢� A��A¢� A��A¢�A��A¢� A��A¢� A��A¢��
(2.2.49)
This equation is used such that multi-axial stress and load/unload direction are clearly defined. The
Eq. (2.2.48) is explained further by Pastor (Pastor et al., 1990). The form of the matrix H-1 is
related by the continuity condition on neutral loading condition. Thus, the equivalent relation
between unloading H-1 and neutral loading condition H-1 exist because nfT = 0.
The magnitude of plasticity is defined by scalar variable h which is function of the distance
between current forces to the image point. As mentioned previously, Eq. (2.2.27) shows the
derivation of λ. Then, the scalar variable h can be calibrated with numerical or experimental results.
During pure loading of footing under concentric vertical force, the particular simple expression
can be used.
ℎ = ℎ7ln &v' (2.2.50)
Where ho is the numerical parameter. Eq. (2.2.50) is used for monotonic load. When cyclic load is
applied, different plasticity model is defined.
Isotropic hardening explains the behavior of solid when the material increases in yield stress when
it is loaded, unloaded and reloaded again. In simpler terms, if the solid reaches beyond its yield in
tension and unload it by applying compression, the solid does not yield in compression until it
reaches the yield that was reached by loading in tension. The yield stress increase due to hardening
in tension increases the yield point in compression. Kinematic hardening explains the realistic
behavior of materials subjected to cyclic loading. This hardening effect takes consideration of
Bauschinger effect where increase in tensile yield strength reduces its initial compressive yield
strength. This is different from isotropic hardening rule, because the greater the tensile strain
hardening, the lower compressive yield strength. The soil plasticity is defined by combination of
kinematic and isotropic hardening plasticity model, original presented by Prevost in 1978 using
34
clays (Cremer et al., 2001). For the case of loading history where kinematic and isotropic hardening
is applied, additional λ term is added to account for history of the maximum plasticity loading.
ℎ = ℎ7ln _v�¤{ v�)¥� ` (2.2.51)
Where ho and p are numerical parameters calibrated from footing subjected to vertical load. The
λmin is the minimum value obtained during loading history. Since λ is the ratio of the current force
to the yield surface, when the maximum load is applied close to the bounding surface and reloaded,
kinematic hardening effect is applied by this λmin value. Thus, for the case of monotonic loading,
λ will always replace λmin as the progression load will always yield lower λ value at each stress
increment, as expressed in Eq. (2.2.50). The variable ho is the initial plastic stiffness defined by
the user given by:
ℎ7 = d1 0 00 d1 00 0 d1� ��� 0 00 ��� 00 0 ���� (2.2.52)
The scalar factor matrix is used to multiply a factor to elastic term in order to make plastic initial
stiffness. Chatzigogos et al. (2011) states that there are lack of numerical or experimental results
that is pertained to specific soil, thus, some characteristic values are initially limited to work with,
in order to reflect qualitative description of the system behavior. Thus, the p1 value from Eq.
(2.2.52) is often assumed to be 0.1 while p value from Eq. (2.2.51) suggested value is around 5. It
is clear that the simulation is not restricted to those values. Also, the lognormal relation shown in
Eq. (2.2.51) is also not restrictive to this function. It can be improved or replaced with other
equation or function with varying values of ho by including additional terms to reflect variation of
material characteristic of soil.
2.2.1.4 Cohesive soil and general interface
So far, the analysis of failure criterion was limited by the bounding surface of the foundation as
shown in Figure 2.8.
35
Figure 2.8. Cohesive soil combined with general interface (Chatzigogos et al., 2011)
In this analysis the interface elements have the capacity to handle much larger force than the
bounding surface of the cohesive soil. Thus, all plasticity and failure would be determined solely
by the irreversible and dissipative mechanism of the soil plasticity underneath the foundation. The
figure above shows the diagram of the specified case. However, when the interface strength is
weaker than the bounding surface of soil, the global plasticity mechanism is governed by the
interface strength criterion as shown in Figure 2.9.
In Figure 2.9, the interface strength criterion follows general Mohr-Coulomb strength criterion.
The introduction to new failure criterion interacts with the bounding surface if the interface is
under the bounding surface, truncating the ellipsoidal shape of the bounding surface as shown in
the Figure 2.9.
QN
QVFint(Q)=0
0 1
QoN,int
QoN,int Fbs(Q)=0
36
Figure 2.9. Bounding surface of soil with purely cohesive interface element (Chatzigogos et
al., 2011)
The equation of the interface failure criterion is shown in Eq. (2.2.53).
[)¥¦&�, §' = �"du|§| − �)¥¦ − �f@A∅)¥¦, �)¥¦7 − �y ≤ 0 (2.2.53)
Where Cint is the interface cohesion, σ0int is the allowable tensile force by the footing, Φint is the
interface friction angle. This corresponds to generalized force in macroelement expressions in Eq.
(2.2.54).
[)¥¦&��' = �"d©|��| − ��7)¥¦ − ��f@A∅)¥¦, ��,)¥¦7 − ��ª ≤ 0
��7,)¥¦ = «¬®¯W°±²
��,)¥¦7 = �)¥¦7 ∗ ³����
(2.2.54)
Where A is the area of the footing. Notice that moment is not required in the equation. This
equation explains that for effective shear strength of the interface, the applied shear force is
subtracted from the maximum shear force allowed by the interface (QoV int) and frictional resistance
of the interface (QNtanφint). The equation sets a limit for horizontal loading where it cannot exceed
maximum shear force resisted by the footing and frictional resistance of the footing to the soil.
QN
QV
Fint(Q)=0
0 1
Angular points
QVmaxFbs(Q)=0
QoVmax
nbsnint
nx
37
The expression ‘sup’ in the equation is supremum. The supremum is the least upper bound of any
set, defined by the specific quantity such that no member in the set exceed this specified value
(Jeffereys 1988). In simpler terms, supremum limit is also called as upper limit.
Infimum and supremum are identical as minimum and maximum in the context of finite set of
numbers respectively. For infinite sets of numbers, smallest upper bound set is supremum. Thus,
in this equation, whichever is the smallest value of the two equation (shear strength or tensile
strength) will determine the interface strength.
Different types of failure mechanism can occur depending on the characteristic values of the
interface such as Qvo, QNoint and φint. For instance:
1. In Figure 2.8, the interface strength parameters are large enough that the system would
yield exclusively by soil parameters only.
2. When the friction angle of the interface is zero and tensile force limit is set to be infinite,
then intersection between bounding surface which has cohesive soil with purely cohesive
interface exist as shown in Figure 2.9. Now the maximum allowable force that the
foundation can carry incorporates the intersection of the two strength criterion: bounding
surface fBS of cohesive soil and interface strength fint. This interaction effect can be handled
using the multi-mechanism plasticity theory. The main idea behind this theory is that the
direction of the plastic displacement of cohesive soil denoted by nBS and interface plastic
displacement nint can be combined to yield plastic displacement of the intersecting surfaces
using the Eq. (2.2.55).
A� = ´{Aij + ´>A)¥¦ (2.2.55)
With μ1, μ2 are scalar quantities which is determined by the yield and consistency condition of the
two plastic mechanisms.
Therefore, in macroelement formulation, two independent mechanisms are combined to reflect the
response of soil and soil-footing interface properties of the foundation. The soil is described by the
bounding surface hypoplastic formulation to describe the plastic behavior, while the interface is
described by simple elastic-perfectly plastic Tresca model. In summary, the plasticity of the overall
38
analysis is carried out using hypoplastic formulation of soil, and the interface strength limits the
maximum force the foundation can withstand by truncating the bounding surface of the footing.
In order to perform this type of analysis in numerical setting, multi-mechanism plasticity requires
a two-stage iterative algorithm. Firstly, the classical plasticity analysis is performed using soil
model. Then, for each increment, iterative procedure checks and updates whether the plasticity
mechanisms are violated. Then, appropriate generalized force to displacement relationships are
calculated based on the plastic mechanisms of the macroelement model.
Normally, elastic soil with uplift parameters are combined with plastic response of soil. This means
that the simplest ways of calculating both effects would consider solving plasticity of the footing
iteratively with implicit scheme and uplift problem with explicit scheme in single iteration.
Although this iterative process reduces computation cost efficiently, treating the uplift without
plasticity leads to accumulated error when dealing with large loading history or repeated cyclic
loading (Chatzigogos et al., 2011). Thus, additional elements for uplift-plasticity coupling terms
are added on macroelement to resolve this problem. Notice that the following case considers
cohesive soil with perfectly rough tensionless interface such that
�@A&µ)¥¦' → ∞
�)¥¦ → ∞
�)¥¦ = 0
a) In the plane of QN-QV, the perfectly rough tensionless interface limits cut-off at QN = 0
b) When the uplift is combined with plasticity, the surface of uplift initiation is no longer
associated with linear part of the analysis. Instead, the formulation of uplift of the footing
on elastoplastic soil model is proposed by Cremer et al. (2002).
��,· = ± ��∝ �z¹Kº (2.2.56)
(C. Cremer, Pecker, & Davenne, 2002) suggested a value of range between ζ=1.5 and 2.5, which
is derived from FEM numerical models. Beyond the curve of uplift initiation, elastic part of the
39
response which was obtained by static impedance function of the footing becomes non-linear and
the following uplift equations are applied.
For the fictional interface, the angular points that arise from two Mohr-Coulomb branches in QN-
QV plane would be handled within multi-mechanism plasticity as previously described. Figure
2.10 shows frictional interface with bounding surface of soil.
Figure 2.10. Multi-mechanism plasticity of frictional interface (Chatzigogos et al., 2011)
The only exception to this combined bounding surface is that frictional interface will exhibit a
non-associated behavior. This does not cause complication in the analysis as the direction of
interface plastic displacement nint is uniquely determined.
2.2.1.5 Frictional soil
Purely ‘frictional soil’ with friction angle φ and non-associated behavior for simply bounded
interface follows the stress failure criterion of classical Mohr-Coulomb inside soil volume.
[&�, §' = |§| − �f@Aµ ≤ 0 (2.2.57)
QN
Fint(Q)=0
01
Angular points
QVmaxFbs(Q)=0
nbs
nint nx
nbs
Φint
Uplift
40
Figure 2.11. Sliding mechanism of interface with frictional soil a) using the combined
mechanism b) using non-associative rule (Chatzigogos et al., 2011)
There is sliding mechanism along the interface which violates Mohr-Coulomb criterion in plane
below the footing by assuming flow rule with zero dilation. This mechanism is predominant when
horizontal force Qv is significantly larger than vertical force QN and moment QM. The Mohr-
Coulomb branches with non-associated flow rule in the plane of Qn-QV as described in case of
frictional interface. The difference is driven by the combination of frictional soil parameter with
frictional interface. Thus, if φint < φ, the sliding mechanism is defined by φint.
QN01
Sliding
QV
Fbs(Q)=0
ng
Φint
QN01
Sliding
Fbs(Q)=0
ng =g(Q)=0
nbs
ng
Non-associativity
QV
Detachment
Detachment
ng
Combined
mechanism
a)
b)
41
The mechanism which describes irreversible plastic behavior is analogous to the characteristics
that were discussed for cohesive soil. In particular, the plastic response has to yield in continuous
manner and become more significant as QN increases. This effect is predominant for significant
values of QN and less for QV and QM. This suggests that hypoplastic formulation of the bounding
surface and radial mapping rule introduced in cohesive soil may be retained for frictional soil as
well, except that non-associative flow rule has to be implemented in order to capture realistic
behavior of soil at the vicinity of the footing. Then, the inverse plastic modulus which was used
for cohesive soil now becomes as the following Eq. (2.2.58) and Eq. (2.2.59).
�z{ = 1ℎ A ⊗ A, �\ℎ��5� �\e (2.2.58)
�z{ = 1ℎ A ⊗ A�, »]�f\A@e �\e (2.2.59)
The non-associative term is implicitly defined in the variable ng where the variable represents unit
normal vector of plastic displacement increment, but does not coincide with its original unit normal
vector n.
Once the coupling non-associated plastic mechanisms are combined, the non-smoothness of elastic
domain at point of intersection for both of the mechanism will be questionable. In order to smooth
out this term, study from upper bound yield design theory has been implemented where the
intersection of the mechanisms are equipped with smoother flow rule. The following results in
Figure 2.11 a). The combined bounding surface follows same shape as the general ultimate surface.
In summary, the plasticity mechanism of macroelement is mainly composed of three independent
mechanisms (sliding, uplift, and soil plasticity). The advantage of the element is that it allows all
of the unique characteristics of plasticity mechanisms to be combined to analyze total response of
the shallow foundation. Note that the non-associative parameter is referring to plastic potential and
plastic multiplier of the bounding surface with interface element. For non-associative parameter
value of 1, the model is fully associated and is governed by the bounding surface explicitly. The
authors found that ζ = 0.65 is in agreement with corresponding values they have used in the
experiment.
42
2.2.2 Implementation of macroelement in MATLAB
The open source code containing macroelement is provided in Linux system in FORTRAN
programming language. This code allows macroelement model to be analyzed inside a free FEM
software called Code Aster. The post-process results can be obtained using python.
MATLAB code has been written to create a stand-alone function which allow the users to analyze
the response of shallow foundation with cohesive or frictional soil by defining the load history, the
characteristic values of the footing, and the material properties of soil. This new code was
necessary such that the recursive parameter model can be implemented into this element to
consider frequency-dependency of the soil.
The generalized force and displacement relationship is updated with appropriate constitutive laws
for various types of failure mechanism due to the coupling effect of the footing and the plasticity
soil. Figure 2.12 shows the overall macroelement analysis flowchart.
43
Figure 2.12. Macroelement analysis flowchart
Firstly the user inputs either force or displacement history they want to analyze. Then, the
macroelement updates the linear elastic static stiffness using the parameters of the footing and soil,
such as foundation width, cohesion of soil, etc. Then, plastic stiffness is updated based on the ratio
of the current force step to the maximum force it can withstand using the bounding surface. Once
the stiffness is updated, then the next force increment is predicted using this stiffness to check
whether the predicted force has exceeded any of the failure criterion defined by the soil or the
interface. Once the failure criterion is checked, then the macroelement updates its stiffness to
corresponding to the failure modes and next force increment is analyzed.
Check:
Failure due to soil?
Due to interface?
Generalized Force/
User-defined parameters
Elastic Stiffness/
Plastic Stiffness
1st iteration:
Force prediction
Case A: None of the
mechanisms are
violated
Case B: The
bounding surface is
violated
Case C: Only the
interface element is
violated.
Case D: Both of the
interface and the bounding
surface are violated.
Updated stiffness
Generalized
displacement
analysis
2nd iteration
44
Although the analysis procedure is similar for both of the models, the major difference in the
original code and MATLAB implementation is a nonlinear solution algorithm used. In MATLAB
code, implicit Newton-Raphson method is used to solve nonlinear solution of a problem which
aims to converge at each increment of load/displacement. Code Aster also uses a nonlinear solution
scheme. However, for each iteration step of the analysis, the analysis loops back to the beginning
of the analysis to re-calculate the analysis, which perform unnecessary calculations. Figure 2.13
a) and Figure 2.13 b) shows the diagram illustrating the difference in the iteration of nonlinear
solution algorithm in Code Aster and MATLAB respectively.
Figure 2.13. Nonlinear solution algorithm diagram for a) Code Aster b) MATLAB
The displacement controlled and force controlled analysis are available in MATLAB code. Also,
having an element available in MATLAB allows additional elements to be added which is
explained further in Chapter 3 for proposed model. Thus, it is pivotal to extensively check all the
cases of loading scenarios in order to verify the macroelement model has been implemented
correctly. In the next section, verification of various case studies are covered for this purpose.
Nonlinear
solution
algorithm
Obtain
force/displacement
increment
Update stiffness
and check failure
criterion
If not,
iterate until
Nth steps
(defined
by user)
Next force
increment
Newton-
Raphson
nonlinear
algorithm
Obtain
force/displacement
increment
Update stiffness
and check failure
criterion
If yes
If not,
iterate until
convergence
Next force
increment
Target value
<Tol?
Target
value<Tol?
If yes
a) Nonlinear solution algorithm used with
Code Asterb) Newton-Raphson nonlinear solution algorithm
used with MATLAB
45
2.2.3 Verification of the implementation
As mentioned previously, various verifications are made in this section to confirm the MATLAB
code results are in good agreement with Code Aster. Once this verification is checked, then the
code is verified with the FEM software to provide additional verification of this element in the
next chapter. There are four possible combination of failure mechanism in macroelement.
Case A: None of the mechanisms are violated. Hypoplastic model for bounding surface of soil is
applied.
Case B: The bounding surface is violated. The cutting plane algorithm is applied to the bounding
surface failure criterion.
Case C: Only the interface element is violated. The cutting plane algorithm is applied to the
interface failure criterion.
Case D: Both of the interface and the bounding surface are violated. Multi-failure mechanism
plasticity algorithm is used.
Table 2 shows the model parameter inputs for the verification examples.
46
Table 2. Model parameter inputs for the verification examples
Parameter Parameter description Unit Value
B Foundation width (m) 10
Nmax Vertical bearing capacity (MN) 45.52
KNN Vertical static stiffness (MN/m) 3000
KVV Horizontal static stiffness (MN/m) 2500
KMM Moment static stiffness (MN/m) 22898.1
Qvmax Horizontal maximum load (-) 0.16
Qmmax Moment maximum load (-) 0.11
ζ Uplift parameter (-) 0.0
Co Cohesion of soil (-) 0.15
Φ Friction angle (˚ deg) 0.0
Co_int* Cohesion at the interface (-) 0.1
Φ_int* Friction angle at the
interface
(˚ deg) 0.0
Ho Initial plastic stiffness
parameter
(-) 1.0*KNN
p Plasticity parameter (-) 1
*Applied for Case C and Case D. For other verification cases these values remain large value for Co_int and zero for
Φ_int.
The results for each failure mechanisms are provided below. The listed verification are provided
such that both MATLAB and Code Aster follows a specific failure mechanism as intended with
the theoretical values. Further explanation is provided for each case. More examples are covered
and the results are presented in Appendix B.
47
2.2.3.1 Case A: None of the mechanisms are violated
The first loading condition considers vertical monotonic loading with the following load: Vertical
load = 45.52 MN, Horizontal load = 0 MN, Moment load = 0 MNm. The bounding surface plot
and the current force is presented in Figure 2.14.
Figure 2.14. Bounding surface plot for macroelement 3D plane view, QV-QN view, and QM-
QN view from top to left and right (Case A-1)
-1
-0.5
0
0.5
1
-0.2
-0.1
0
0.1
0.2
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
QN
QV
QM
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
QN
QV
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
QN
QM
48
Note the current force step is plotted with blue round circles while its projected force onto the
bounding surface is plotted in red circles. This plot is helpful in visualizing the force history plot
onto the bounding surface, which will be used to formulate the plasticity of the soil.
Different angles of the bounding surface shows the direction of the load applied. In this case, only
vertical force is applied to the foundation which means that the force history will be plotted on QN
coordinate only. The linear elastic results and overall nonlinear results are presented in Figure
2.15.
Figure 2.15. Linear elastic (left) and nonlinear (right) vertical monotonic load and
displacement plot (Case A-1)
0 0.005 0.01 0.0150
10
20
30
40
50
Displ (m)
Forc
e (
MN
)
[ELASTIC Vertical] Displ vs. Force
Code aster
MATLAB
-1.2 -1 -0.8 -0.6 -0.4 -0.2 0
x 10-18
-3
-2
-1
0
1x 10
-15 [ELASTIC Horizontal] Displ vs. Force
Displ (m)
Forc
e (
MN
)
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1[ELASTIC Moment] rotation vs. Force
Rotation (theta)
Mom
ent
(MN
m)
0 0.01 0.02 0.03 0.04 0.050
10
20
30
40
50
Displ (m)
Forc
e (
MN
)
[Vertical] Displ vs. Force
Code aster
MATLAB
-5 -4 -3 -2 -1 0
x 10-17
-3
-2
-1
0
1x 10
-15 [Horizontal] Displ vs. Force
Displ (m)
Forc
e (
MN
)
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1[Moment] rotation vs. Force
Rotation (theta)
Mom
ent
(MN
m)
49
The second case example considers vertical load applied and subsequent horizontal and moment
cyclic load applied to the foundation. The following load history is applied. Vertical load = 20.0
MN, Horizontal load = ± 1.2 MN, 3.0 MN, Moment load = ± 5, 15,20 MNm. The moment is applied
at magnitude which triggers uplift of the foundation.
Figure 2.16. Load history applied to macroelement (Case A-2)
0 10 20 30 40 50 60 70 80 90 1000
10
20Vertical Load
time step
Forc
e (
MN
)
0 10 20 30 40 50 60 70 80 90 100-5
0
5Horizontal Load
time step
Forc
e (
MN
)
0 10 20 30 40 50 60 70 80 90 100-20
0
20Moment Load
time step
Mom
ent
(MN
m)
50
Figure 2.17. Bounding surface plot for macroelement (Case A-2) 3D plane view, QV-QN
view, QM-QN view from top and bottom, left to right
The bounding surface shows multi-direction load plot. This illustrates cyclic load history and
plasticity formulation being formed as the load reaches closes to the bounding surface. The linear
elastic and overall nonlinear analysis results are presented in Figure 2.18.
-1
-0.5
0
0.5
1
-0.2
-0.1
0
0.1
0.2
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
QN
QV
QM
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
QN
QV
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
QN
QM
51
Figure 2.18. Linear elastic (left) and nonlinear (right) vertical monotonic load and
displacement plot (Case A-2)
0 1 2 3 4 5 6 7
x 10-3
0
5
10
15
20
Displ (m)
Forc
e (
MN
)[ELASTIC Vertical] Displ vs. Force
Code aster
MATLAB
-15 -10 -5 0 5
x 10-4
-3
-2
-1
0
1
2[ELASTIC Horizontal] Displ vs. Force
Displ (m)
Forc
e (
MN
)
-1.5 -1 -0.5 0 0.5 1
x 10-3
-20
-10
0
10
20[ELASTIC Moment] rotation vs. Force
Rotation (theta)
Mom
ent
(MN
m)
0 0.002 0.004 0.006 0.008 0.01 0.0120
5
10
15
20
Displ (m)
Forc
e (
MN
)
[Vertical] Displ vs. Force
Code aster
MATLAB
-2 -1.5 -1 -0.5 0 0.5 1
x 10-3
-3
-2
-1
0
1
2[Horizontal] Displ vs. Force
Displ (m)
Forc
e (
MN
)
-3 -2 -1 0 1 2
x 10-3
-20
-10
0
10
20[Moment] rotation vs. Force
Rotation (theta)
Mom
ent
(MN
m)
52
2.2.3.2 Case B: The bounding surface is violated.
The following case considers vertical load applied to the foundation until maximum bearing
capacity is reached. The main purpose of this analysis is to check whether the analysis fails to
converge once the load exceeds the bounding surface. Load condition are the following; vertical
load = 46 MN (exceeds capacity of 45.52 MN), Horizontal load = 0.0 MN, Moment load = 0.0
MNm
Figure 2.19. Bounding surface plot for macroelement (Case B-1) 3D, QV-QN , QM-QN view
-1-0.5
0
0.5
1
1.5
-0.2
-0.1
0
0.1
0.2
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
QN
QV
QM
0.997 0.998 0.999 1 1.001 1.002
-10
-8
-6
-4
-2
0
2
4
6
8
x 10
QN
QV
-1 -0.5 0 0.5 1
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
QN
QV
-1 -0.5 0 0.5 1-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
QN
QM
53
As shown in Figure 2.19 in the 3D view plot (top), the force step, shown in blue, reaches out of
bounding surface maximum value of 1 in QN coordinates. When this normalized value of
maximum bearing capacity force reaches 1, the failure occurs due to bounding surface of soil as
observed in the analysis. The linear elastic analysis results and nonlinear analysis results are
presented in Figure 2.20.
Figure 2.20. Linear elastic (left) and nonlinear (right) vertical monotonic load and
displacement plot (Case B-1)
In this case, the results are similar to Case A-1. However, once the vertical load reaches the
bounding surface, the analysis should not converge which was the case in both of the code. The
same criteria is used to check other failure mechanisms.
0 0.005 0.01 0.0150
10
20
30
40
50
Displ (m)
Forc
e (
MN
)
[ELASTIC Vertical] Displ vs. Force
Code aster
MATLAB
-1.2 -1 -0.8 -0.6 -0.4 -0.2 0
x 10-18
-3
-2
-1
0
1x 10
-15 [ELASTIC Horizontal] Displ vs. Force
Displ (m)
Forc
e (
MN
)
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1[ELASTIC Moment] rotation vs. Force
Rotation (theta)
Mom
ent
(MN
m)
0 0.01 0.02 0.03 0.04 0.050
10
20
30
40
50
Displ (m)
Forc
e (
MN
)
[Vertical] Displ vs. Force
Code aster
MATLAB
-5 -4 -3 -2 -1 0
x 10-17
-3
-2
-1
0
1x 10
-15 [Horizontal] Displ vs. Force
Displ (m)
Forc
e (
MN
)
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1[Moment] rotation vs. Force
Rotation (theta)
Mom
ent
(MN
m)
54
2.2.3.3 Case C: The failure due to interface
The vertical load of 10 MN is applied to the structure initially. Then, subsequent increase in
horizontal load is applied to the foundation until the load exceeds the cohesion strength at the
interface, as described by Case C failure criterion. The load applied to the structure is the
following: Vertical load = 10 MN, Horizontal load = 3.86 MN (exceeds cohesion strength at the
interface, 3.8MN), Moment load = 0.0 MNm.
Figure 2.21. Bounding surface of macroelement, 3D plane view, QV-QN and QM-QN plane
view (Case C-1)
-1
-0.5
0
0.5
1
-0.2
-0.1
0
0.1
0.2
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
QN
QV
QM
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
QN
QV
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
QN
QM
55
The Figure 2.21 shows the force history plotted on the bounding surface. The figure also shows
black line which represents the yield surface of the element. This additional surface truncates the
existing bounding surface and caps the maximum horizontal force resisted by the footing due to
cohesion strength of soil. As previously discussed, if the cohesion strength at the interface is lower
than cohesion strength of soil, then the failure mechanism is driven by loss of contact force at the
interface. It is observed that for both of the codes, the analysis fails to converge with tolerance
value of 10e-8. Thus, the analysis shows failure due to the exceedance value at the truncated
bounding surface. The results are shown in Figure 2.22 for linear elastic and nonlinear analysis
right before the failure develops.
Figure 2.22. Linear elastic and nonlinear load and displacement plot (Case C-1)
0 0.5 1 1.5 2 2.5 3 3.5
x 10-3
0
2
4
6
8
10
Displ (m)
Forc
e (
MN
)
[ELASTIC Vertical] Displ vs. Force
Code aster
MATLAB
-5 0 5 10 15 20
x 10-4
-1
0
1
2
3
4[ELASTIC Horizontal] Displ vs. Force
Displ (m)
Forc
e (
MN
)
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1[ELASTIC Moment] rotation vs. Force
Rotation (theta)
Mom
ent
(MN
m)
0 1 2 3 4 5
x 10-3
0
2
4
6
8
10
Displ (m)
Forc
e (
MN
)
[Vertical] Displ vs. Force
Code aster
MATLAB
-0.5 0 0.5 1 1.5 2 2.5
x 10-3
-1
0
1
2
3
4[Horizontal] Displ vs. Force
Displ (m)
Forc
e (
MN
)
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1[Moment] rotation vs. Force
Rotation (theta)
Mom
ent
(MN
m)
56
2.2.3.4 Case D: Both of the interface and the bounding surface are violated
For the case where failure occurs due to both interface and bounding surface, the loading condition
is described as shown below. The following load condition is applied: vertical load = 20.0 MN,
horizontal load = 6.0 MN (close to the cohesion strength of interface), and moment load = 0 MNm.
Figure 2.23. Bounding surface plot for macroelement (Case D-1) 3D plane view, QV-QN and
QV-QN plane view
-1
-0.5
0
0.5
1
-0.2
-0.1
0
0.1
0.2
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
QN
QV
QM
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
QN
QV
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
QN
QM
57
The bounding surface of this plot contains the bearing capacity due to soil and also the interface
strength which truncates the bounding surface at the shear strength as shown in Figure 2.23. As
previously discussed, the cohesion strength of interface is capped at 6MN. Then, as the combined
load reaches close to the bounding surface and cohesion strength of the soil, Case D nonlinear
failure mechanism governs the failure for the overall model. As expected, the results show non-
convergent solution as the load exceeds the truncated bounding surface. The results are obtained
before the failure as shown in Figure 2.24.
Figure 2.24. Linear elastic (left) and nonlinear (right) analysis result for macroelement
(Case D-1)
0 1 2 3 4 5 6 7
x 10-3
0
5
10
15
20
Displ (m)
Forc
e (
MN
)
[ELASTIC Vertical] Displ vs. Force
Code aster
MATLAB
0 0.5 1 1.5 2 2.5 3
x 10-3
0
2
4
6
8[ELASTIC Horizontal] Displ vs. Force
Displ (m)
Forc
e (
MN
)
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1[ELASTIC Moment] rotation vs. Force
Rotation (theta)
Mom
ent
(MN
m)
0 2 4 6 8
x 10-3
0
5
10
15
20
Displ (m)
Forc
e (
MN
)
[Vertical] Displ vs. Force
Code aster
MATLAB
0 1 2 3 4 5 6 7
x 10-3
0
2
4
6
8[Horizontal] Displ vs. Force
Displ (m)
Forc
e (
MN
)
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1[Moment] rotation vs. Force
Rotation (theta)
Mom
ent
(MN
m)
58
Therefore, checking various cases where the nonlinearity of soil and failure mechanism with SSI
effect is taken into consideration, these verification provide confidence and credibility in
implementation of macroelement code in MATLAB. Further analysis of macroelement has been
computed using MATLAB code in this thesis.
59
2.3 Recursive parameter model (Nakamura, 2006a)
2.3.1 Introduction
Dynamic properties of soil has gained high interest in the field of earthquake engineering. When
the structure is subjected to dynamic load, foundations oscillate according to the deformability and
strength of soil and foundation. Several other factors which affect the foundation oscillation
include geometry of the foundation, inertial effect of superstructure, and nature of the dynamic
excitation (Kramer, 1996). Thus, it is pivotal to understand the nature of foundation response
subjected to dynamic loading. Gazetas (1985) has worked extensively in the area of foundation
vibration and has summarized many findings through the literature and numerical models to
provide a dynamic properties for various soil and foundation conditions for simplified cases.
Many literature have referenced to original publication by Gazetas (1985) for arbitrarily shaped
surface and embedded foundations in a homogeneous half space. The original derivations are
based on (a) some simple physical models calibrated with the results of rigorous boundary-element
formulations and (b) data from the literature in the past, by notable researchers in dynamic analysis
of soil and foundation such as Lysmer, Veletsos, Luco and others (Fang, 1991).
The derivations for the formulas and graphs are briefly mentioned in the handbook, but are not
discussed further in details. Fang (1991) suggests the readers to refer to the original papers in order
to find useful detailed information of the derived formulas. However, most of these original
derivations are heavily based on numerical data from computer codes in the past which is difficult
for the readers to clearly understand how the equations are formulated. Nevertheless, these
equations are still widely used in design of foundation for dynamic loading conditions.
In order to obtain the dynamic impedance function of homogeneous soil with underlying rigid rock
layer, the formulation for dynamic stiffness and dashpot coefficients provided by Gazetas (1985)
are often used in practice.
60
As previously mentioned in Section 2.2.3, the complex stiffness can be obtained numerically using
the methods presented previously in frequency domain. This chapter introduces a new method to
transform dynamic impedance function of soil-structure system in frequency domain to time
domain using recursive parameters. The original proposed method is explained in details by
Nakamura (Nakamura, 2006b). This method will be referred to as Nakamura’s model on this paper.
The background, derivation, and implementation of Nakamura’s model is briefly discussed in the
next section.
2.3.2 Formulation of the recursive parameters by Nakamura (2006)
In order to capture both the nonlinear response of the structure and soil-structure interaction
response, it is important to transform soil impedance function in the frequency domain to the
impulse response in time domain analysis. Although there are other studies regarding
transformation of dynamic impedance function in frequency domain to time domain in the
literature, the method proposed by Nakamura shows a robust and stable algorithm that solves
potential stability issues which occur during transformation in time history analysis. Further
studies on the analysis of stability issues with various models including Nakamura’s has been
extensively studied in Alex Laudon’s thesis paper (Laudon, 2013).
Capturing the overall nonlinear SSI interaction using FEM analysis is possible. However, this is
computationally expensive. Due to this limitation, study has been favored towards approximate
methods such as Nakamura’s model where nonlinear structural response can be combined with
soil response which is frequency-dependent. Previous literature such as Wolf has studied on
converting impedance function in frequency domain to impulse response in time domain by using
inverse Fourier transformation. This approach only captures linear elastic analysis because
frequency domain analysis is based on the principle of superposition. This method, however, is
susceptible to numerical instability arising from the use of impedance functions that are causal
(Nakamura, 2006a). Thus, Nakamura has proposed a new transformation method where the
dynamic impedance function in frequency domain is formulated in time domain using past
displacement and velocity terms. This function describes the value of the reaction forces from soil-
61
foundation system over a time duration reacting to an impulse displacement. The system expresses
the state variable (displacement, velocity, and acceleration) at a given time step and calculates the
next time step analysis using the operator matrix K0. The following expression describes the
numerical integration scheme of the state variables, assuming no external force is applied on the
system.
¼½½-½¾ ¿+1 = ÀÁ ¼½½-½¾ ¿ (2.3.1)
This operating matrix can be extended to include all the previous displacement and velocity terms.
In order to describe how Nakamura’s coefficient terms are formulated from the soil impedance
function, it is pivotal to review the concept of convolution integral.
Convolution is a process which allows the output of the system to be calculated based on any
arbitrary input signals with known impulse response of the system. This general definition of the
term provides a powerful tool in which any signal process response could be calculated with
summation of the delayed impulse response of the system. For instance, if there is a input signal
x(n) and impulse response h(n), then the output response, y(n), can be summation of the delayed
impulse response of the input signal x(k) at kth term. Eq. (2.3.2) illustrates this relationship.
}&A' = T&A' ∗ ℎ&A' = Â T&Ã' ∗ ℎ&A − Ã'ÄÅÆzÄ
(2.3.2)
Following this concept, Nakamura has formulated the following equation impulse response
equation in the context of soil impedance function. The response of the restoring force from soil
is expressed as Eq. (2.3.3).
»&f' = »{&f' +  ��Ç ∗ »{�f − ��� ¥�Æ{
(2.3.3)
The response of the overall restoring force F(t) is the summation of the current impulse response
F1(t) and summation of the previous impulse response. Due to the linearity of the harmonic
excitation, the impulse force response can be expressed with stiffness and damping terms as
62
discussed previously in Section 2.2. Then the corresponding restoring force can be expressed as
Nakamura’s recursive equation which is expressed as the Eq. (2.3.4).
»&f' = �Ãh ∗ "&f' + �h ∗ "- &f'� + , Ã� ∗ "�f − f�� �z{�Æ{ +  �� ∗ "- �f − f�� �z{
�Æ{ . =  Ã� ∗ "�f − f�� �z{
�Æh +  �� ∗ "- �f − f�� �z{�Æh
(2.3.4)
Then, the impedance function can be expressed as shown in (2.3.5).
È&' = &Ãh + ∗ �h' + , Ã� ∗ �z)É¦Ê �z{�Æ{ + ∗  �� ∗ �z)É¦Ê �z{
�Æ{ . =  Ã� ∗ �z)É¦Ê �z{
�Æh + ∗  �� ∗ �z)É¦Ê �z{�Æh
(2.3.5)
Where k0 and c0 represent instantaneous stiffness, damping of soil respectively. As the soil
impedance function contains real and imaginary part, it can be described with the convolution
terms.
È&' = ËÌ�ÍÈ&)'ÎtSÍÈ&)'ÎÏ =ÐÑÒÑÓ Â �\�#)� ∗ Ã� �z{
�Æh + )  �A#)� ∗ �� �z{�Æh
−  �A#)� ∗ Ã� �z{�Æh + )  �\�#)� ∗ �� �z{
�Æh ÔÑÕÑÖ
(2.3.6)
Where ϴij=ωjtj. With the given impedance function S(ωi) and unknown impulse response
components Gk and Gc, the coefficient matrix can be formulated which has the size of 2M*2N. M
is the number of given impedance data and N is the size of total time-delay components.
ÐÑÒÑÓuÈ&{'y...uÈ&�'yÔÑÕ
ÑÖ = ×ÍØ�ÀÎ ÍØ�ØÎ Ù ∗ ËuÚÀyuÚØyÏ (2.3.7)
Where
63
È&)' = ËÌ�ÍÈ&)'ÎtSÍÈ&)'ÎÏ, u6Ly = a ÃhÃ{…Ã�z{c, u6Üy = a �h�{…��z{
c
Í�L̅Î = , u�L̅Þ,gy … u�L̅Þ,ºßÞy… … …u�L̅M,gy … u�L̅M,ºßÞy., ©�L̅à,ʪ = Ë �\�#)�−�A#)�Ï
Í�Ü̅Î = , u�Ü̅Þ,gy … u�Ü̅Þ,ºßÞy… … …u�Ü̅M,gy … u�Ü̅M,ºßÞy., ©�Ü̅à,ʪ = Ë) ∗ �A#)�) ∗ �\�#)�Ï
(2.3.8)
Once the unknown impulse response components {Gk} and {Gc} are calculated based on the soil
impedance function in frequency domain, then these coefficients can be used to calculate restoring
force of the soil model which can be expressed in time domain using past displacement and
velocity terms as defined in equations . However, this approach does not capture hysteretic
damping of the system which is essential for practical purposes. Nakamura has improved this
method by introducing instantaneous mass term at the current step of the analysis. This method
not only captures the hysteretic damping but also improves accuracy and convergence for the
transformation (Nakamura, 2006b). The following expression describes the improved method with
additional term with virtual mass term defined as m0.
È&' = &−> ∗ Sh + ∗ �h + Ãh' + , ∗  �� ∗ �z)É¦Ê �z>�Æ{ +  Ã� ∗ �z)É¦Ê �z{
�Æ{ . (2.3.9)
This equation of motion meets the number of unknowns and equations by replacing one of the
unknown variables in the time delay component of the velocity term C1-CN-2 with the newly
introduced mass term m0. Then the propsed improved model can be formulated as
È&' = ËÌ�ÍÈ&)'ÎtSÍÈ&)'ÎÏ =ÐÑÒÑÓ �\�#)� ∗ Ã� �z{
�Æh + )  �A#)� ∗ �� �z>�Æh − )>Sh
−  �A#)� ∗ Ã� �z{�Æh + )  �\�#)� ∗ �� �z>
�Æh ÔÑÕÑÖ
(2.3.10)
64
ÐÑÒÑÓuÈ&{'y...uÈ&�'yÔÑÕ
ÑÖ = ×ÍØ�ÀáÎ ÍØ�ØáÎ ÍØ�âáÎ Ù ∗ ãuÚÀáyuÚØáyÚâá ä (2.3.11)
Where
È&)' = ËÌ�ÍÈ&)'ÎtSÍÈ&)'ÎÏ, u6Låy = a ÃhÃ{…Ã�z{c, u6Üåy = a �h�{…��z>
c,6�å = Sh
Í�L̅åÎ = ,u�L̅Þ,gy … u�L̅Þ,ºßÞy… … …u�L̅º,gy … u�L̅º,ºßÞy., ©�L̅à,ʪ = Ë �\�#)�−�A#)�Ï
Í�Ü̅åÎ = ,u�Ü̅Þ,gy … u�Ü̅Þ,ºßry… … …u�Ü̅º,gy … u�Ü̅º,ºßæy., ©�Ü̅à,ʪ = Ë) ∗ �A#)�) ∗ �\�#)�Ï
Í��̅åÎ =ÐÑÒÑÓu��̅Þ...��̅º ÔÑÕ
ÑÖ, ç��̅àè = Ë−)>0 Ï
(2.3.12)
The formulation is consistent with previous method, just a change in the time delay response of
velocity term. Then, Nakamura’s coefficients containing past displacement and velocity terms can
be used to represent soil impedance function in time domain.
65
2.3.3 Implementation of the recursive parameter model in structural analysis
Once the Nakamura’s coefficient terms have been formulated, then the restoring force of soil can
be written as:
éê¤ë = Sh"¾ )¤{ + &�h"- )¤{ + �{"- )¤{z{ + �>"- )¤{z> … '+ &�h")¤{ + �{")¤{z{ + �>")¤{z> … '
= Sh"¾ )¤{ +  �� ∗ "- )¤{z� �z{�Æ{ +  �� ∗ ")¤{z� �z{
�Æ{
(2.3.13)
Where i+1 is the current time step of the analysis which is unknown, Ri+1 is the restoring force
occurring from the soil at time step i+1. K0 and C0 and M0 represent instantaneous stiffness,
damping, and mass of soil at i+1 time step of analysis respectively. The instantaneous mass of the
soil takes into account the mass inertial force of soil. Kj and Cj represent the recursive parameters
of the past displacement and velocity terms. Superposition of the Nakamura’s coefficients with
previous impulse terms result in the current step restoring force of the system. In order to
distinguish the stiffness and damping convolution terms with the overall structure and foundation
system, different variables are used to express these convolution terms. Γj is the Nakamura’s
coefficient terms A, B, and C and X is the state variables as shown in Eq. (2.3.14). In simplified
summation notation.
é)¤{ =  ì� í)¤{z� = Â&î�ï½)¤{z� + ð� ï½- )¤{z� + Ø�ï½¾ )¤{z�'�z{�Æh
�z{�Æh (2.3.14)
Figure 2.25 illustrates the soil domain represented as a restoring force applied to the structure with
two DOF system.
66
Figure 2.25. Idealized two-DOF soil-structure system (Duarte-Laudon, A., Kwon, O. and
Ghaemmaghami, 2015)
The equation of motion for the overall system is defined as:
�½¾ )¤{ + C½- )¤{ + �½)¤{ = −é)¤{ + ò)¤{ (2.3.15)
Where R is the restoring force and F is the external force applied to the structure. Substituting Eq.
(2.3.13) yields the Eq. (2.3.16).
�½¾ )¤{ + C½- )¤{ + �½)¤{= −&�h")¤{ + �{")¤{z{ + �>")¤{z> … '+ &�h"- )¤{ + �{"- )¤{z{ + �>"- )¤{z> … ' + Sh"¾ )¤{ + ò)¤{
(2.3.16)
Since the instantaneous Nakamura’s terms are represented at i+1 time step of the analysis, these
terms can be incorporated into the mass, stiffness, and damping of the overall structure system.
This results in Eq. (2.3.17).
â� ½¾ +1 + ó�½- +1 + À�½+1 = − Â�î�ï½+1−ô + ð�ï½- +1−ô + Ø�ï½¾ +1−ô� + ò+1�−1ô=1
where â� = â + Ø�ù , Ø� = Ø + ð�ù , À� = À + î�ù (2.3.17)
Macroelement has three DOFs at the foundation, and soil dynamic impedance function is available
in all three dofs. Thus, influence vector is required to assign the restoring force to the
corresponding dofs of soil dynamic impedance function. Further information regarding this
/2 /2 /2 /2 /2 /2
(a) Soil-structure system (b) Representation of soil-foundation with
a generic restoring force function
(c) Representation of the restoring force
function with recursive formula
67
influence vector is provided in Appendix C. To illustrate how this is implemented in the code, the
formulation of restoring force is shown at each time increment iteration step.
Ì{ = 0 Ì> = ×ú̅{ =�{ �̅{Ù û"{"- {"¾ {ü = ý{þ{ Ì� = ×ú̅{ =�{ �̅{Ù û"{"- {"¾ {ü + ×ú̅> =�> �̅>Ù û">"- >"¾ >ü
= ý{þ{ + ý>þ> = Íý{ ý>Î Ëþ{þ>Ï
ÌQ = Íý{ ý> ý�Î ûþ{þ>þ�ü
Ì) = Íý{ ý> ⋯ ý)z{ Î a þ{þ>⋮þ)z{c
(2.3.18)
There is no restoring force present at the first time step, thus the restoring force is zero. The
restoring force, Ri, can be calculated at ith time step, with Nakamura’s convolution terms, δ, and
previous state variable terms, X. At each iteration, the matrix is updated with new state variable
terms and the previous terms of the state variables are added consecutively at each time step of the
analysis. The time domain analysis in MATLAB uses Newmark’s time integration scheme. This
method calculates increment of acceleration, velocity and displacement with the following
relationship
½- )¤{ = ��ℎ &½)¤{ − ½)' + :1 − �
�; ½- ) + ℎ :1 − �2�; ½¾ ) (2.3.19)
½¾ )¤{ = 1�ℎ> &½)¤{ − ½)' − 1
�ℎ ½- ) + :1 − 12�; ½¾ ) (2.3.20)
The velocity and acceleration terms for time step + 1 in Eq. (2.3.20) can be expressed in terms
of displacement at time step + 1 as well as the displacement, velocity and acceleration at time
step I using Newmark time-stepping method. Thus, time domain analysis can be carried out using
68
Nakamura’s coefficient terms using the formulations presented in this chapter. The verification of
this model against theoretical equation is provided in the next section.
69
2.3.4 Verification of the implemented lumped parameter
A single DOF structure with soil impedance function generated with 10 nodes are used to verify
this model. Firstly, the soil impedance function is derived based on the following soil matrices (K,
C, and M) as discussed in Section 2.2 with Eq. (3.2.7).
"� = 1−�2 + � + � (2.3.21)
The Table 3 shows the structure and soil properties;
Table 3. Structure and soil property for 10 DOF verification model
Properties Structure Soil
Mass (Kg) 15000 300
Stiffness
(KN/m)
1820000 800000
Damping 3000 KN/m/s 2% Rayleigh
damping at first two
natural frequency
With the known impedance function, Nakamura’s coefficients are obtained from this impedance
function in frequency domain. Then, these coefficients are used to back calculate the soil
impedance function using the coefficient terms with frequency domain values as shown in (2.3.12).
Figure 2.26 shows the dynamic impedance function of soil with 10 nodes. Figure 2.27 shows the
Nakamura’s coefficients obtained from the same soil impedance function and the coefficients are
used to back-calculate its impedance in frequency domain using the Eq. (2.3.12). The results are
in good agreement.
70
Figure 2.26. Dynamic impedance function of soil model with 10 nodes (real and imaginary
terms)
Figure 2.27. Nakamura’s coefficients capturing dynamic impedance function of soil
0 20 40 60 80 100-15
-10
-5
0
5x 10
7Impedance Function
Frq. [sec-1
]
Rea
l
0 20 40 60 80 1000
0.5
1
1.5
2x 10
6
Frq. [sec-1
]
Imag
inary
0 20 40 60 80 100-15
-10
-5
0
5x 10
7
Frq. [sec-1
]
Stiff
ness
Matrix Inv
Nakamura Coeff
0 20 40 60 80 1000
0.5
1
1.5
2x 10
6
Frq. [sec-1
]
Dam
pin
g
Matrix Inv
Nakamura Coeff
71
Once the coefficients are obtained in time domain, then two types of analysis are made for
verifying Nakamura’s model. Firstly, the response at the foundation is obtained with frequency
domain analysis using Fast Fourier Transformation (FFT) method to convert the time domain
acceleration in frequency domain. Then, multiplying the soil impedance function to the ground
motion results in the response in frequency domain. Lastly, inverse Fourier transformation (IFFT)
is applied to convert the ground motion response in time domain analysis. In linear elastic analysis,
this method is valid and agrees well with time history analysis of FEM model. For verifying
Nakamura’s model with frequency domain analysis, Newmark’s time integration scheme has been
used to solve linear equation of motion in time domain where previous displacement and velocity
terms are added cumulatively with each of the Nakamura’s coefficients multiplied to those past
state variable terms. The sinusoidal acceleration is applied to the SDOF structure on top of the soil
with 1Hz frequency for duration of 40 seconds. Figure 2.28 shows the response of analysis using
Newmark and FFT method. The results are in good agreement with each other.
Figure 2.28. Sinusoidal response of MDOF system at top node of soil
0 5 10 15 20 25 30 35 40-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
Time (s)
targ
et
DO
F d
rift
(m
)
Time-history Newmark (Linear)
Frequency Domain (Linear)
72
The target DOF refers to the sinusoidal excitation response at the foundation. Thus, Nakamura’s
model has the capacity to perform time-history analysis with frequency dependent characteristics
of soil. The results of methodology has been verified with frequency domain analysis. This method
will be used for implementation of combining macroelement formulation which is discussed in
next section with verification example.
73
Proposed method to model SSI of shallow foundation
In this chapter, a proposed model which incorporates dynamic impedance function of soil into
macroelement is presented. As mentioned before, macroelement is efficient in capturing quasi-
static loading cases for nonlinear soil with detachment of foundation. However, upon dynamic
loading application, the element is faced with limitation of adding frequency-dependency of soil.
As introduced in the thesis, radiation damping of soil and frequency-dependency of soil is critical
in dynamic analysis of footing. However, the challenge remains as to how to include the frequency-
dependency of soil as the dynamic properties are often defined in the frequency domain. After a
comprehensive review on the method of stable transformation of dynamic impedance function
using Nakamura’s coefficient, it is proposed to integrate macroelement with Nakamura’s
coefficient in order to analyze dynamic analysis of SSI effect at each time step. This analysis
method will include the frequency characteristic of soil while capturing nonlinearity of the soil
and geometric nonlinearity of the foundation from soil simultaneously. Derivation of the proposed
model is discussed in details, and verification of the proposed model is made with FEM model for
various range of frequencies with high amplitude of load. Through two parametric study shown in
matrix, the capacity and limitations of this model is presented.
74
3.1 Proposed method
A macroelement is simplified element where the behavior of soil-structure interaction is captured
at the lumped node. It captures the geometric nonlinear behavior of the foundation such as uplift
and sliding of the foundation as well as nonlinear material behavior of the soil which are modelled
simultaneously. The model is defined by non-linear constitutive law and its stiffness is updated at
each computational step using the generalized force-displacement relationship with bounding
surface hypo-plastic model of the soil and foundation as discussed in previous chapter.
The nonlinearity of the soil occurs in the vicinity of the footing. As the wave propagates away
from foundation to soil, the far-field soil domain behaves in linear elastic manner (Cremer et al.,
2002), due to the inertial effect of the soil which is explained in Section 2.1. Since macroelement
captures the nonlinear behavior of the soil only at the near-field, to extend the application of this
element to the domain of dynamic loading condition requires additional information for far-field
effect using dynamic impedance of soil. Elastic stiffness and constant damping terms are currently
used in lieu of frequency dependency of soil in the far-field for simplicity (Chatzigogos et al.,
2009). The justification behind this simplification is that some characteristic frequency of the
system can be captured by choosing dynamic impedance of soil at a specific frequency value such
as its fundamental frequency, excitation frequency, etc. However, this would neglect rest of the
other frequency dependent values of soil and is not the realistic representation of soil in dynamic
loading application. Also, the choice of which frequency to choose would be in question for the
analysis. The schematic diagram in Figure 3.1 illustrates how the elements are combined in the
existing macroelement model.
75
Figure 3.1. Schematic diagram of macroelement with extension to dynamic load
Thus, when analyzing dynamic loading case, the equation of motion is formulated with Eq. (3.1.1).
â���¤�nê ∗ "¾ + Ø�� ∗ "- + +Ø�kêk�ên� ∗ "- + À�� ∗ " + À�km�n ∗ " = ò �� (3.1.1)
Where â���¤�nê refers to the combined mass matrix of soil and structure at the corresponding
degrees of freedom. Ø�� is the damping of the structure while Ø�kêk�ên� refers to imaginary term
of frequency dependency of soil at a specific frequency of interest. À�� refers to elastic stiffness
of the structure whereas À�km�n refers to the updated plasticity of the model with respect to the
yielding of soil at the near-field of the foundation of soil. It is important to note that Kmacroel is a
nonlinear spring element which updates its stiffness based on its current force at each time step of
the analysis. Thus, one can obtain the restoring force occurring from foundation by multiplying
the updated stiffness of soil with À�km�n and displacement vector, u.
With the assigned number of degrees of freedom as shown in Figure 3.2, the equation of motion
is formulated with matrices as shown in Eq. (3.1.2). In this equation, an effective seismic load is
76
applied directly to the structure only in horizontal direction, Fx. This matrix formulation allows
one to apply load in any direction desired by assigning load to the desired DOF.
Figure 3.2. Figure of structure and foundation with five degrees of freedom
������â��� ù ù ù ùù ���� ù ù ùù ù â� ù ùù ù ù �� ùù ù ù ù â��� + â����
����∗�
∗ÐÑÒÑÓ"¾ {"¾ >"¾ �"¾ Q"¾ �ÔÑÕ
ÑÖ +
�ÍØ��Î�∗� ùù ù��∗�
∗ÐÑÒÑÓ"- {"- >"- �"- Q"- �ÔÑÕ
ÑÖ + ù ù ùù ù ùù ù ÍØ�kêk�ên�Î�∗��
�∗�∗
ÐÑÒÑÓ"- {"- >"- �"- Q"- �ÔÑÕ
ÑÖ +
�ÍÀ��Î�∗� ùù ù��∗�
∗ÐÑÒÑÓ"{">"�"Q"�ÔÑÕ
ÑÖ + ù ù ùù ù ùù ù ÍÀ�km�n Î�∗��
�∗�∗
ÐÑÒÑÓ"{">"�"Q"�ÔÑÕ
ÑÖ =�����ò�ùùùù ���
��
(3.1.2)
As previously mentioned, the Ø�kêk�ên� in Eq. (3.1.2) is a constant damping stiffness at frequency
of interest defined by the user. However, in reality, soil has dynamic properties with different
frequency which makes soil frequency dependent (Kramer, 1996). Thus, in this thesis, it is
proposed to integrate Nakamura’s model (Nakamura, 2006b) to the macro element model by
(Chatzigogos et al., 2009) in order to include the dynamic impedance of soil in the far-field domain
of soil. Nakamura’s coefficient terms formulate the dynamic impedance of soil in the frequency
domain to convolution terms of displacement and velocity terms in time domain, which are used
Structure
Foundation
1
2
3
5
4
77
to represent the dynamic stiffness and damping of the soil. By combining these two models, the
analysis can capture the nonlinear near-field effect of soil-structure interaction and the frequency
dependency of soil in the far-field at each time step of the analysis. The first part of the chapter
presents the combined model derivation and the application of the macroelement and Nakamura’s
coefficients. Various verification examples are provided in this chapter including nonlinear quasi-
static loading condition to seismic response of soil-structure interaction analysis.
As previously presented in Nakamura’s model, the restoring force is calculated based on the
recursive parameters obtained with soil impedance function. Assuming that the far-field response
of soil is linear elastic, this soil model can be represented as the following diagram as shown in
Figure 3.3.
78
Figure 3.3. Soil structure system with Macroelement and Nakamura’s model
Both of the models analyze the restoring force at each time increment of the analysis. This allows
the restoring force which occurs from macroelement due to plasticity of soil, to be combined with
far-field elastic response using Nakamura’s coefficients simultaneously at each time step.
However, direct superposition of these two methods results in stiffer soil due to the inherent elastic
static stiffness of soil applied to both of the models. For instance, for the soil model that is purely
linear elastic, dynamic impedance function is sufficient to replicate the behavior of elastic soil
domain as the dynamic impedance function consists of static terms (elastic stiffness) at frequency
of zero. This means that for linear elastic soil domain, Nakamura’s model can be used to replicate
79
the behavior of overall soil domain, as there is no distinct difference in near-field and far-field
effect of soil. However, macroelement also includes linear elastic static stiffness in which the
plasticity of soil and uplift is formulated. Thus, combining those two models results in double
static stiffness of soil domain that is implicitly implemented in both of the models. Therefore, if
macroelement is using flow rule to compound elastic and plastic analysis, then restoring force of
elastic component can be subtracted for each time step of the analysis from macroelement to take
into account the plasticity and uplift of the analysis only. Thus, the restoring force of overall soil
model is presented in Eq. (3.1.3), which is also shown in Figure 3.3.
Ì&f)¤{' = Íò�km�n − ò Î + éê¤ë�k�k�½�k = ×&À ,½�ê��!IJ + À�!�J' − À !IJÙ
+ ∑ &î�ï½)¤{z� + ð�ï½- )¤{z� + Ø�ï½¾ )¤{z���z{�Æh
(3.1.3)
As discussed, ò�km�n which is the overall restoring force calculated with macroelement model,
is subtracted with ò , which is the restoring force calculated with linear elastic static stiffness.
Then, Nakamura’s model is added to take into account the far-field effect of the soil model. In
theory, if the soil domain is purely linear elastic, then the macroelement will exhibit linear elastic
behavior and should be equal to the restoring force from linear elastic force, ò . As these two
terms cancel out, the linear elastic soil model is governed solely by Nakamura’s model. On the
other hand, if the plasticity of soil is introduced in the model, the ò�km�n will be lower than ò , which results in restoring force with plasticity of soil and Nakamura’s model together. This results
in higher displacement of the overall building response subjected to a seismic load.
80
3.2 Verification of the proposed method
3.2.1 Analysis cases
Many verification examples are provided in this chapter. The verification mainly consists of four
folds; linear elastic static load, nonlinear quasi-static load, linear elastic dynamic load, and
nonlinear dynamic load. For each of these cases, FEM modeling approach is provided in details,
and the calibration of the proposed model is also provided in this chapter with theoretical values.
The three soil models are considered in the analysis: infinite soil domain, homogeneous soil with
rigid rock layer, and heterogeneous soil with rigid rock layer. For each of the soil model, these
four cases of loading scenarios are analyzed and used as an example for verification with the
proposed model.
3.2.2 FEM model approach
This chapter includes a detailed discussion on soil model generation using commercial FEM
software. The software that is used in the analysis is RS 2.0, product from Rocscience, which is a
2D FEM software specialized in solving geotechnical problems. Plaxis 2D has also been used to
verify the model, which is also anther software capable of analyzing soil models and geotechnical
problems. OpenSees is a free FEM software which is capable of analyzing nonlinear dynamic
analysis with many material properties in the library. Since these software has difference in
material properties, it is necessary to compare the material properties available in the software
before comparing global model of soil model.
3.2.2.1 Soil material property
The material nonlinearity of the soil is introduced with appropriate soil material models available
in FEM software. The material properties in all FEM software include Elastic Perfectly Plastic
(EPP) behavior with Mohr coulomb failure criterion, Tresca failure criterion, von Mises failure
criterion, Drucker-Prager failure criterion etc. OpenSees contains additional element called
81
PressureIndependentMultiYield which provides material nonlinearity by the calibrated
coefficients defined by the user. This element is investigated and used in the analysis for
comparison as the original macroelement formulation has been compared with FEM model from
Cremer (Cremer et al., 2001), which uses material property that is similar to this element in
OpenSees.
Since the seismic load is applied to the soil in significantly short duration, in order of magnitude
of few seconds, the response will correspond to undrained loading condition. For the saturated
clays under undrained loading conditions, the failure is defined by the Tresca criterion
(Chatzigogos et al., 2011). The Tresca criterion is expressed in terms of principal stresses as shown
in Eq. (3.2.1).
12 max&|�{ − �>|, |�> − ��|, |�� − �{|' = È�$ = 12 È$ (3.2.1)
where È�$ is the yield strength in shear, È$ is the tensile yield strength. Since Mohr-Coulom failure
criterion is often used for soil model and is widely used in FEM software, the comparison between
Mohr coulomb failure criterion and Tresca failure criterion is made. As shown in Eq. (3.2.2), the
failure criterion of Mohr-Coulomb yield surface is:
S + 12 max &|�{ − �>| + �&�{ + �>', |�> − ��| + �&�> + ��', |�{ − ��|+ �&�{ + ��' = È$�
(3.2.2)
Where
S = È$�È$¦ ; � = S − 1S + 1
The variable Syc and Syt represents yield stress of material in uniaxial compression and tension
respectively. Thus, when the yield in compression equals to yield tension (Syc=Syt), K variable
becomes zero and the Eq. (3.2.2) becomes Tresca failure criterion as shown in Eq. (3.2.1). Thus,
Mohr-Coulomb material properties has been used in the FEM models with the same yield stress in
uniaxial compression and tension.
82
OpenSees has PressureIndependentMultiYield material in the material library. This material is
elastic-plastic material where plasticity is only exhibited in the deviatoric stress-strain response.
The volumetric stress-strain response is linear-elastic and it is independent of deviateoric response.
This material is implemented for material where the shear response is insensitive to confinement
change. Such materials include organic soils or clay under undrained loading condition.
This material property assumes linear elastic response upon gravity load. In the subsequent
dynamic load, the stress-strain response becomes elastic-plastic. The plasticity used in this
behavior is formulated based on multi-surface concept with associative flow rule. The yield surface
is defined as von Mises type.
This material requires few user inputs in order to define the backbone curve of soil at the material
level. The following figure shows the required user-input variables plotted on octahedral shear
stress and strain as shown in Figure 3.4.
Figure 3.4. OpenSees pressure independent multi-yield material with user-defined
parameters
From the graph above, Gr is the shear modulus of soil at low strain (elastic), τf is the maximum
shear strength and γmax is the octahedral shear strain when maximum shear strength is reached.
83
The maximum octahedral shear strength, τf is expressed in terms of its initial effective confinement
p’i as shown in Eq. (3.2.3).
§¢ = 2√2 − �Aµ3 − �Aµ dÇ + 2√23 � (3.2.3)
Where c is the cohesive strength of soil. Then, the backbone curve of soil is generated by the Eq.
(3.2.4).
§ = 6�1 + �
�" #dÇ]dÇ �$
(3.2.4)
Following this material behavior of soil, then the backbone curve of the element in OpenSees
should follow the theoretical octahedral stress and strain relationship that is defined in the model.
Thus, a single element with PressureIndependMultiYield material property is created and pushed
in lateral direction in order to verify the elastic-plastic behavior of soil with theoretical solution.
The octahedral shear stress is calculated based on the stress-strain relation using modified
deviatoric stress and strain spaces (Dobry et al., 1991).
Also, another element is created in other FEM software with the same material property but with
Mohr-Coulomb failure criterion where it is pushed laterally with monotonic load. This is used as
a comparison to illustrate the material behavior at the element level response of elastic perfectly
plastic material versus automatic surface generated behavior. The shear modulus (G) of soil is
defined as 39000 kPa, and maximum cohesive strength (τ) of soil is defined as 30 kPa for the
element model with 0 fiction angle (Φ=0o). Figure 3.5 shows the theoretical and OpenSees result
using the Multi-yield material. Mohr-Coulomb material behavior is also plotted in the same graph.
84
Figure 3.5. Octahedral stress and strain at material level for OpenSees
As shown in Figure 3.5, at the material level, OpenSees results are in good agreement with the
theoretical result for the backbone curve of soil. Also, the elastic perfectly plastic material behavior
behaves as expected. As the load reaches close to the yield value, Mohr-Coulomb failure criterion
is reached and perfectly plastic behavior is assumed. Thus, element behavior of the model agrees
well with theoretical value. This can be used to move onto the next step in creating larger soil
domain with more mesh with this verification.
0
5
10
15
20
25
30
35
0 0.005 0.01 0.015 0.02 0.025 0.03
Oct
ah
edra
l S
hea
r S
tres
s (τ
xy
)
Octahedral Shear strain (ϒoct)
Theoretical
Opensees:
PressureIndependent
Mohr-Coulomb failure
criterion
85
3.2.2.2 Soil and foundation geometry
The soil model is generated using OpenSees, RS2.0, and Plaxis 2D. To replicate the behavior of
infinite soil domain, soil model is created with 100 meter by 100 meter model. Although larger
model resembles the behavior of infinite soil domain closer, the verification shows that this model
size, with viscous dashpot boundaries to dissipate the waves, is sufficient to capture the behavior
as infinite soil domain. The mesh size is 1 m which is sufficient for dissipating the waves with the
specified shear modulus, as discussed in Section 2.2 in FEM model creation. For homogeneous
soil domain with rigid rock layer, a 20 m by 60 m in height and width is created in OpenSees.
Since the model is homogeneous, the blue color shows the uniform material property of the model.
The red line shows foundation with 10 m width.
Figure 3.6. FEM model with a) 100 m by 100m soil model with 2 m foundation b)
homogeneous soil with stratum, c) heterogeneous soil layer with rigid rock layer (Gibson
soil)
Figure 3.6 shows the three model description. Figure 3.6 c) shows the heterogeneous soil model
with rigid rock layer in OpenSees. The model is created with increase in cohesion per depth, also
known as Gibson soil layer. Thus, different colors are used to demonstrate the change in material
properties per depth. Also, the mesh is refined near the foundation to capture the nonlinearity of
the soil accurately.
20m
60m
20m
60m
100m
100m
2m 10 m 10 m
a) 100 m by 100 m soil model
with 2 m foundation
b) 20 m by 60 m homogeneous soil
model with 2 m foundation
c) 20 m by 60 m Gibson soil (increase
in cohesion per depth)
model with 2 m foundation
86
For all of the models above, the base is fixed and sides have viscous dashpots applied as shown in
Figure 3.7. The formulation of the dashpot coefficients are provided in Figure 3.11 by Lysmer
and Kuhlemeyer (Lysmer & Kuhlemeyer, 1969).
Figure 3.7. Illustration of soil domain boundary condition for the FEM modesl
As previously mentioned, uplift of the foundation needs to be implemented in the model. Thus,
zero contact element has been used to allow uplift of the footing. From Figure 2.4 in Section 2.3.1,
the FEM model with uplift is in good agreement with the numerical result from the thesis. It was
interesting to observe that Elastic-no-tension element always failed to converge in OpenSees due
to the abrupt change in the stiffness once the element detects tensile stress. Instead, elastic multi-
linear material is used to attach the foundation to soil where the user can define tensile strength
and compressive strength of an element. Thus, small tensile strength value is used and stiffer value
used for compressive strength of an element to resemble the uplift of the foundation. After using
this as a beam element connected to the soil domain, the result matches close to the paper as shown
in Figure 2.4.
Since the foundation is created with beam element, rotation is also included in the model and the
footing has three degrees of freedom. In order to allow uplift of an element, this zero tensile
element is attached to the footing, but since the footing has rotational degrees of freedom,
additional floating soil node has to put in between the foundation and soil model in order to connect
the foundation to the soil domain using this element. The diagram illustrates how this is
implemented in OpenSees in Figure 3.8.
C=ρVpA
C=ρVsAFixed boundary
Load path
Soil domain
87
Figure 3.8. Diagram of foundation and soil node connection using floating soil node
This mitigates the error in the analysis as the connected nodes share same degrees of freedom.
Directly connecting a node with three degrees of freedom to two degrees of freedom creates non-
converging results. Using this method allows foundation to be detached from soil once tensile
force is applied to the structure. Figure 3.9 shows the detachment of the rigid foundation from the
soil.
Uplift element
Foundation (3 DOF)
Floating soil node (2 DOF)
Soil node (2 DOF)
Rigid connection
88
Figure 3.9. Time step analysis plot of the foundation uplift from soil, with height and width
of the soil domain
This modeling approach has been verified with theoretical results. All of the results for the model
creation mentioned in this model is provided in the next chapter. The nonlinearity of the models
are analyzed with J2-Plasiticy material property in OpenSees where the material has EPP behavior
with von Mises failure criterion. Then, for the Gibson soil model, PressureIndependentMultiyield
element has been used to compare the FEM results with the results obtained in the paper by Cremer
(Cremer et al., 2001).
3.2.2.3 Frequency dependent soil foundation system
For the linear elastic soil domain, FEM model has been created to verify the theoretical values of
dynamic impedance available in the literature. For example, Zhang and Tang (2008) studied the
radiation damping effect behavior of SSI using FEM model. They have created a FEM model with
appropriate soil domain and boundary condition to replicate the behavior of infinite soil domain,
and the results are in good agreement with theoretical equation for the infinite soil medium. The
paper provides brief summary on the modeling approach of FEM model. FEM model has been
45 46 47 48 49 50 51 52 53 54 5599.9
99.92
99.94
99.96
99.98
100
100.02
100.04
100.06
100.08
width (m)
heig
ht
(m)
Soil
Foundation
89
generated in this thesis with MIDAS GTS to replicate the results Zhang and Tang has verified with
the theoretical dynamic impedance results.
Firstly, dynamic impedance function is captured using FEM model in elastic half-space soil
medium. Viscous damping boundary has been used to simulate infinite soil medium with energy
absorbing boundary as shown in Figure 3.10. Linear cases are formulated with force to
displacement matrix in general form:
Ë�%&f'�&&f'Ï = <6 '�{{ + B{{ 00 �>> + B>>( Ë�%&f'�&&f'Ï (3.2.5)
Eq. (3.2.5) is derived from complex stiffness method where the viscously damped structure is
simplified to equivalent stiffness k* upon harmonic loading. For instance, from the equation of
motion for SDOF structure when excited with harmonic function �&f' = �h ∗ sin &f', the steady-
state solution for the equation of motion can be written as Eq. (3.2.6).
�"¾ + �"- + �" = ��)ɦ (3.2.6)
From the equation of motion, k and c can be simplified to a combined stiffness term that contains
dynamic stiffness and damping terms. The complex stiffness can be derived based on the frequency
characteristics of soil. The expression below contains the information of the dynamic stiffness and
damping ratio.
"� = 1−�2 + � + � (3.2.7)
Without mass of the structure, the following Eq. (3.2.7) is then simplified to dynamic stiffness, K,
and damping term, C only. This term is referred to as complex stiffness that is expressed in series
as shown in Eq. (3.2.8).
Ã∗ = Ã + � (3.2.8)
This approach is valid for harmonic load for SDOF structure. Using this stiffness notation, the
authors have modeled linear half-space infinite soil medium. The foundation is excited with
90
vertical and horizontal harmonic load while energy absorbing boundaries are applied at the
boundaries to dissipate the waves occurring from the foundation to soil as shown in Figure 3.10.
Figure 3.10. Foundation geometry and excitation conditions and Finite domain and
absorbing boundary (Zhang & Tang, 2007).
From the FEM analysis, the authors obtained a good agreement with analytical solutions for
dynamic impedance terms with various frequency of excitation.
The 2D FEM model has been created using MIDAS GTS to replicate the study results presented
above. The objective of generating the model is to understand the method of obtaining dynamic
stiffness impedance function from the behavior of the foundation and soil medium when it is
excited with harmonic loading. Three linear elastic FEM models have been created for the analysis
to study the effect of size of the soil domain to replicate infinite soil domain; a 20 meter depth by
40 meter soil medium, 100 meter by 200 meter, and 250 meter by 250 meter FEM model. For each
model, the foundation is assumed to be rigid and is attached to the soil medium. Also, the
appropriate boundary conditions are defined with viscously damped system. The study by Lysmer
et al. (1969) have shown that energy absorbing boundary can be obtained for homogeneous,
isotropic linear soil medium using Rayleigh wave viscous boundary condition. Figure 3.11 shows
Rayleigh wave absorption boundary which is 95% effective in absorbing S-waves (Lysmer &
Kuhlemeyer, 1969).
91
Figure 3.11. Rayleigh wave absorption (Lysmer & Kuhlemeyer, 1969)
For the model creation, the soil profile of Shear modulus (G) 64963.6 kPa, Poisson’s ratio (v) of
0.25, and soil density (ρ) of 1600 kg/m3 has been used. The element size is chosen to be 1 meter
which satisfies the element size limit that is restricted from energy absorbing boundary (Lysmer
& Kuhlemeyer, 1969). The foundation is assumed to be rigid, and harmonic excitation with
angular frequency which varies from 10Hz to 140 Hz is applied to the foundation with 0.005
seconds time step that lasts for 5 seconds. The foundation with 2 meters in width is located on top
centre of the soil model. After the analysis, hysteretic behavior of soil foundation is obtained.
Figure 3.12 shows force per displacement hysteric curve obtained at angular frequency of 10 hz
from 250 m by 250 m soil domain FEM model.
92
Figure 3.12. Hysteresis loop (load vs displacement) for excitation frequency ω = 10.075 in
250m by 250m FE model
From the hysteretic graph, the dynamic stiffness is calculated with the slope. Also, the damping
coefficient term is calcuated by energy dissicipated over the hysteretic loop. Eq. (3.2.9) and Eq.
(3.2.10) below show the equation to obtain the coefficients of slope (c11) and damping term (d11)
from Zhang et al. (2008) paper.
�11 = Ã<6 (3.2.9)
B11 = %$<�7> (3.2.10)
Where k is the stiffness term which is determined from slope of linear regression curve fit in the
hysteretic graph, G is the shear modulus, Wd is the energy dissipated in single cycle of hysteresis
loop which is determined by the area inside the loop, and Uo is the maximum amplitude of
displacement.
y = 62154x
-1.5
-1
-0.5
0
0.5
1
1.5
-2.00E-05 -1.00E-05 0.00E+00 1.00E-05 2.00E-05Fo
rce
(KN
)
Displacement (m)
93
Once the model has been created with the appropriate energy absorbing boundaries and time
forcing function, the model is analyzed with linear time history analysis at each time step. Figure
3.13 shows the deformation plot for the model with angular frequency of ω = 10.075 excitation.
Figure 3.13. Deformation plot for 250 m by 250 m FEM model with angular excitation
frequency ω = 10.075
The model shows outgoing wave propagation dissipates along the depth of the soil. This is in effect
because the soil medium is large enough to replicate the behavior similar to an infinite soil medium.
With the energy absorbing boundary conditions have been defined to generate infinite soil domain
behavior, the results are in good agreement with theoretical solution as the soil model increases.
This explains the model may have some wave propagation reflecting from the boundary to the soil
as a complete energy absorption from the viscous dashpots are unobtainable. Figure 3.14 and
Figure 3.15 shows spring coefficient (c11) and dashpot coefficient (d11) obtained from FE models
respectively, and the results are compared with analytical solution for dynamic impedance function
of infinite soil medium provided from Hryneiwicz (1981). The results for 20m by 40 m soil model
94
show significant difference with the analytical solution. However, as the soil model increase in
size, the models have better matching results with the analytical solution as expected. The final
FEM model with 250 m by 250 m results have good agreement with analytical solution for spring
coefficient and dashpot coefficient as shown in the Figure 3.14 and Figure 3.15.
Figure 3.14. C11 vs. dimensionless frequency, ao for FEM models
Figure 3.15. D11 vs. dimensionless frequency, ao for FEM models
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
k
ao
Hryniewicz 1981
250m by 250m FE model
50m by 100m FE model
20m by 40m FE model
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
d11
ao
Hryniewicz 1981
250m by 250m FE model
20m by 40m FE model
50m by 100m FE model
95
This analysis verifies the method of obtaining dynamic impedance function using FEM model.
The procedure is widely used in the area of shallow foundation analysis where the soil behavior
can be represented by frequency dependent spring and dashpot. These impedance functions can be
used to analyze the inertial interaction of the soil and structure response.
Thus, from this modeling approach, FEM model with appropriate energy absorbing boundary has
been created. The wave propagation from structure to soil has been observed in the contour plot
of the FEM model. The model with foundation attached to the soil domain is verified by obtaining
the dynamic impedance function and comparing the results with the analytical solution. Therefore,
from this section, numerical modeling method to obtain dynamic impedance function of the
foundation has been covered by observing behavior of wave dissipation from structure to soil.
96
3.2.3 Quasi-static loading
Three models are used to verify macroelement with FEM models. These models are:
a) Homogeneous infinite soil domain
b) Homogeneous soil with stratum
c) Heterogeneous soil with stratum
For each of the model, soil material and foundation parameters are provided. Also, calibration
parameters for macroelement are discussed for nonlinear analysis.
a) Homogeneous infinite soil domain (J2 Plasticity – OpenSees)
The material properties are presented in Table 4. OpenSees model with size of 100 m by 100 m
has been created with the following material properties. Notice J2 Plasticity material has been used
where the material behaves in elastic perfectly plastic upon yield stress.
Table 4. Material properties for homogeneous infinite soil domain
Parameters Units Values
v Poisson’s ratio 0.25
ρ Density of soil (t/m3) 1.6
Co Cohesion of soil (KN) 30
Vs Shear wave velocity (m/s) 201.5
G Shear modulus (KN/m3) 6 = ) ∗ ��>
6 = 64963.6
E Elastic modulus (KN/m) + = 2 ∗ 6 ∗ &1 + 5' = 162409
B Foundation width (m) 2
H Soil embedment depth (m) ∞
97
Firstly, the vertical maximum bearing capacity is examined for FEM models and theoretical
values. The ultimate bearing capacity of soil is defined as the specific load per unit area when there
is a sudden failure in the soil supporting the foundation, and the failure surface extend to the ground
surface. The general shear failure is defined at the initial sudden failure of soil.
For cohesive soil with footing resting on top of the surface layer, the ultimate bearing capacity of
foundation is expressed as Eq. (3.2.11). The values for these parameters can be easily obtained
from foundation engineering textbooks.
!, = �,&{'S��»��»�$ + ! !, = �,&1'&5.14' :1 + 0.4 ∗ �¢= ; + 0 !, = 5.14 ∗ �,
(3.2.11)
This equation agrees with homogeneous soil with constant cohesion co as defined by Cremer et al.
(2001). For circular foundation, the ��»�� terms become 6.17. This will lead to the bearing
capacity of circular footing on purely cohesive soil as Eq. (3.2.12).
!, = �,&{'S��»��»�$ + ! !, = �,&1'&6.17'&1' + 0 !, = 6.17 ∗ �,
(3.2.12)
In the case of homogeneous soil layer with rigid bedrock underneath, the embedment depth ratio
can be applied to the depth factor. For the model of interest with 20 m depth and 60 m wide soil
domain, the H/B ratio is still greater than 0.5, thus, the m value will still be one. The bearing
capacity of strip footing on purely cohesive soil with the soil domain size will be same as what
was previously calculated from Eq. (3.2.11).
The calculated stiffness and theoretical equation is provided in Table 5. Although larger soil model
generates results that are much closer to theoretical values, the 100 m by 100 m soil model is used
as a verification example of macroelement to FEM model.
98
Table 5. OpenSees and theoretical results
Elastic stiffness
(KN/m)
OpenSees Theoretical equation
Vertical
(KNN)
59690.6 KNN = 0.73*G/(1-v)
=63231.24
Horizontal
(KVV)
61655.6 KVV = 2*G/(2-v)
=74244.11
Moment
(KMM)
201368.4 KMM = pi*G*B^2/(8*(1-v))
= 136059.45
Nmax (KN,
maximum
vertical loading)
~500KN 5.14*Co*B
=308.4
The theoretical static stiffness terms are originally derived by R. Dobry and Gazetas (1988). In
order to compare the result with FEM model, macroelement uses elastic stiffness directly obtained
from OpenSees model. The Figure 3.16 shows the results obtained from OpenSees and parameter
calibration using macroelement. Figure 3.17 shows the deformed mesh plot is 100 m by 100 m
soil model.
99
Figure 3.16. Vertical load and displacement plot for FEM model and macroelement for
100m by 100m soil model
Figure 3.17. Deformed mesh plot in OpenSees for vertical loading case
-600
-500
-400
-300
-200
-100
0
-0.025 -0.02 -0.015 -0.01 -0.005 0
Ver
tica
l fo
rce
(KN
)
Vertical displacement (m)
Opensees
macroel_p=5000
macroel_p=5
macroel_p=3
Maximum Vert. load
100
For the FEM soil model, punching shear of soil is observed where the failure occurs in the vicinity
of the foundation. In punching shear of soil, if the load-settlement relationship shows steep and
elastic slope as the load is applied, the ultimate bearing capacity load can be assumed to be at the
point where this constant slope occurs. Thus, the slope of the vertical load-displacement plot has
been used to obtain Nmax of the macroelement coefficient, which is used to formulate the
bounding surface. The slope difference for vertical load progression is shown in Figure 3.18.
Figure 3.18. Slope difference for vertical monotonic load
As shown in the Figure 3.18, the slope difference is almost negligible during load range of 0 KN
to 150 KN where elastic behavior is observed. However, as the load progress, the slope difference
increases. Then, the slope difference between the points becomes smaller as the slope of the
monotonic curve reaches close to linear line from the Figure 3.16. Thus, it is suggested to pay
attention to not only the monotonic load-displacement plot, but also the slope difference between
the points to approximate the maximum force where linear slope occurs after the plasticity of the
curve.
The vertical force-displacement profile in Figure 3.16 can be used to obtain the plasticity of soil
coefficient, P. If the p value is large, the analysis behaves in linear elastic manner and if the p value
-8%
-7%
-6%
-5%
-4%
-3%
-2%
-1%
0%
-600 -500 -400 -300 -200 -100 0
Slo
pe
dif
fere
nce
(%
)
Vertical load (KN)
101
is small, more plasticity of soil model contributes to the model. Further description of this value is
previously discussed in Eq. (2.2.51).
Two other direction of load in horizontal and rotation is required to obtain the bounding surface
of the soil. Thus, monotonic analysis similar to the vertical load case has been analyzed in
horizontal and rotational direction as shown in Figure 3.19 and Figure 3.21 respectively. It is
interesting to note that only one value of p is required to represent plasticity behavior of soil (p=3
in this case) for both vertical and horizontal loading, which only the vertical load analysis was
required to calibrate this coefficient. Similarly, in order to define the maximum load in horizontal
and moment directions, the slope difference between the data points is used as shown in Figure
3.20 for horizontal load and Figure 3.22 for moment load.
Figure 3.19. Horizontal maximum load for half-space soil in OpenSees and macroelement
-300
-250
-200
-150
-100
-50
0
-0.025 -0.02 -0.015 -0.01 -0.005 0
Ho
rizo
nta
l F
orc
e (K
N)
Horizontal displacement (m)
Opensees
macroel_p=5000;
macroel_p=3;
Maximum Horiz. load
102
Figure 3.20. Slope difference for horizontal monotonic load
Figure 3.21. Moment maximum load for half-space soil in OpenSees
-35%
-30%
-25%
-20%
-15%
-10%
-5%
0%
-350 -300 -250 -200 -150 -100 -50 0
Slo
pe
dif
fere
nce
(%
)
Horizontal load (KN)
-500
-450
-400
-350
-300
-250
-200
-150
-100
-50
0
-0.025 -0.02 -0.015 -0.01 -0.005 0
Mo
men
t lo
ad (
KN
m)
rotation (ϴ)
Maximum Moment load
103
Figure 3.22. Slope difference for moment monotonic load
The reason why there is no comparison in the moment direction is due to the fact that there needs
to be a vertical force applied to the macroelement to analyze the three degrees of freedom. Not
having vertical force with pure moment applied to the macroelement results in ill-conditioned
matrix. Nevertheless, the moment-displacement plot has been used to generate the bounding
surface which the macroelement uses to formulate the plasticity of the soil. Thus, these three
direction of loads are used to obtain the macroelement input for bounding surface as shown in
Table 6.
Table 6. Macroelement bounding surface coefficients
Load direction Maximum load Macroelment input
Vertical 480 KN Nmax = 480 KN
Horizontal 250 KN* Qvmax = 250/480
= 0.5208*
Rotational 400 KNm Qmmax = 400/(480*B)
= 0.41667
-40%
-35%
-30%
-25%
-20%
-15%
-10%
-5%
0%
-500 -450 -400 -350 -300 -250 -200 -150 -100 -50 0
Slo
pe
dif
fere
nce
(%
)
Moment (KNm)
104
The formulation of bounding surface as ellipsoid is a simplification. It is important to check
whether the soil model generated using FEM model actually forms ellipsoid ultimate bearing
capacity in the three plane of loading. Thus, FEM model has been used to check whether different
loading paths follow a failure shape that is close to an ellipsoid bounding surface.
Figure 3.23. The bounding surface generation using OpenSees and macroelement (positive
horizontal force)
0
50
100
150
200
250
300
0 100 200 300 400 500 600
Ho
rizo
nta
l fo
rce
(KN
)
Vertical force (KN)
Bounding surface (positive)
Maximum forces [Opensees]
105
Figure 3.24. The bounding surface generation using OpenSees and macroelement (negative
horizontal force)
With the ellipsoid bounding surface and coefficient parameter calibrated with vertical load,
different loading path is examined where vertical load of 250KN is applied and horizontal load is
applied in monotonic curve. The results in OpenSees and macroelement are in good agreement as
shown in Figure 3.25.
0
50
100
150
200
250
300
-600 -500 -400 -300 -200 -100 0
Ho
riz
forc
e (K
N)
Vertical force (KN)
Bounding surface (negative)
Maximum forces [Opensees]
106
Figure 3.25. Combined loading case (vertical load = -250 KN) with monotonic horizontal
load in OpenSees and macroelement
As previously assumed the maximum load was chosen at a point where the load-displacement
curve behaves in elastic manner as load increases. As shown in Figure 3.25, the macroelement
maximum force for combined load intersects at a point where the OpenSees curve shows linear
slope. Figure 3.26 and Figure 3.27 Shows the bounding surface generation in macroelement and
the maximum capacity of the foundation generated using OpenSees in different viewing angles.
Overall, they are in good agreement with each other.
-300
-250
-200
-150
-100
-50
0
-0.015 -0.013 -0.011 -0.009 -0.007 -0.005 -0.003 -0.001
Ho
rizo
nta
l F
orc
es (
KN
)
Horizontal displacement (m)
Opensees
Macroel, p=3
107
Figure 3.26. 3D plot of bounding surface generated in macroelement and OpenSees
Figure 3.27. 3D plot of bounding surface generated in macroelement and OpenSees in
moment-vertical force coordinate
By obtaining p value from the monotonic curves in vertical and horizontal direction, the same
value is used to verify its cyclic behavior. Figure 3.28 shows the constant vertical force applied to
the foundation and cyclic horizontal load is applied. The results in macroelement agrees well with
OpenSees.
-500
0
500
-400
-200
0
200
400-400
-200
0
200
400
QN
QV
QM
-500 -400 -300 -200 -100 0 100 200 300 400 500-400
-300
-200
-100
0
100
200
300
400
QN
QM
108
Figure 3.28. Vertical constant load (620KN) and horizontal cyclic load for half-space
infinite soil domain
There are two calibrated parameters that define the plasticity of overall model. The parameters are
Pl_1 and Pl_2, where Pl_1 defines the initial plasticity of the foundation upon loading, and Pl_2
defines the plasticity of the foundation upon re-loading of the model. In OpenSees model, the
behavior is more symmetric as the uplift of the foundation is neglected. Thus, the following values
shown below allows macroelement to yield results that are close to OpenSees model. Further
parametric study can be carried out, but the combination of values gives the results that are closest
to OpenSees as shown in Figure 3.29.
Pl_1 = 0.7;
Pl_2= 0;
The following the formulation of plasticity shows the reasoning behind the choice of this specific
values.
Pl_1 represents initial plasticity of the model, (by this value, it determines the monotonic curve
where the cyclic loading follows)
-100
-80
-60
-40
-20
0
20
40
60
80
100
-2.00E-03 -1.50E-03 -1.00E-03 -5.00E-04 0.00E+00 5.00E-04 1.00E-03 1.50E-03 2.00E-03
Ho
rizo
nta
l fo
rce
(KN
)
horizontal displ (m)
Opensees
macroel (Pl_1 =3,Pl_2=0)
109
ℎ = ℎ7ln v (3.2.13)
where ho =Pl_1*Elastic_stiffness to define initial plastic stiffness of the model.
Pl_2 represents plasticity of the model with all the other loading cases (reloading phase, upon
unloading linear elastic behavior is assumed under flow rule)
ℎ = ℎ7 ln _ v.J_>¤{v�)¥.J_>` (3.2.14)
In this case, if Pl_2 =0, the plasticity stiffness becomes Eq. (3.2.15).
ℎ = ℎ7 ln _v{1 ` (3.2.15)
The following equation allows the unloading behavior of the model to follow initial loading
condition of the analysis (plasticity defined by Pl_1 upon unloading as well), allowing symmetric
behavior of the model as predicted from OpenSees model with no uplift initiation and small
moment deformation.
110
Figure 3.29. Macroelement and OpenSees model results for moment cyclic hysteretic loop
Different parameters are chosen which will fit the curve more closely with large moment cyclic
load. The results are shown in Figure 3.30.
-1.5 -1 -0.5 0 0.5 1 1.5
x 10-3
-200
-150
-100
-50
0
50
100
150
200
rotation (theta)
Mom
en
t fo
rce
(K
Nm
)
Opensees
Macroel
111
Figure 3.30. Moment cyclic analysis in homogeneous half-space soil with OpenSees and
macroelement
Uplift model fails in OpenSees because the foundation has width of 2 meters with the mesh size
of 1m. This means that when uplift is allowed, one end of beam node will experience uplift when
the moment is applied. However, due to this detachment, the beam does not carry over the moment
to other nodes because there are not enough mesh from soil to foundation to transfer this
detachment experienced at the edge of the foundation. Therefore, macroelement agrees well with
OpenSees when the coefficients are calibrated with the existing model. With just two to three
parameters, macroelement captures nonlinearity of soil and foundation at the near-field.
Next case considers homogeneous soil with rigid rock layer, also referred to as soil layer with
stratum.
-500
-400
-300
-200
-100
0
100
200
300
400
500
-2.00E-02 -1.50E-02 -1.00E-02 -5.00E-03 0.00E+00 5.00E-03 1.00E-02 1.50E-02 2.00E-02
Mo
me
nt
(KN
m)
rotation (theta)
Opensees
macroel_ (Pl_1=0.6,Pl_2=-0.6)
112
b) Homogeneous soil with stratum (J2 Plasticity – OpenSees)
The material property for homogeneous soil with rigid rock layer is provided as shown in Table 7.
Table 7. Material properties for homogeneous soil with rigid rock layer
Variables Units Values
v Poisson’s ratio 0.45
ρ Density of soil (t/m3) 1.9
Co Cohesion of soil (KN) 30
G Shear modulus (KN/m3) 1300*Co = 39000
E Elastic modulus (KN/m) 2*G*(1+v) =
2*39000*(1+0.45)
=113100
B Foundation width (m) 10
H Soil embedment depth (m) 20
The soil material properties assigned case a) homogeneous infinite soil domain remain the same,
but the foundation width and soil embedment depth has been changed in this analysis. The overall
model has been created with 20 m by 60 m in height and width soil model attached to 10 meter
width foundation with uplift element attached. Using the formulations suggested by Gazetas in
Table 8, Table 9 shows the results in OpenSees with the theoretical values. The results are in good
agreement with theoretical values for 20 m by 60 m soil model as shown in the results.
113
Table 8. Static stiffness of vertical, horizontal, and rocking direction for soil with stratum
(Gazetas, 1983)
Table 9. Theoretical values with OpenSees results
Elastic
stiffness
(KN/m)
OpenSees Theoretical
values
Difference (%)
KNN 173078 163534.09 6%
KVV 95773 100645.16 -5%
KMM 3043315 3063052.83 -1%
Nmax (KN,
maximum
vertical
loading)
1500 1540 -3%
The other FEM models are used for 20 m by 60 m model. Both of the FEM software, RS 2.0 and
Plaxis 2D, matches well with the OpenSees results as shown in Figure 3.31.
114
Figure 3.31. Vertical monotonic load for homogeneous soil with stratum for RS 2.0,
OpenSees, Plaxis, and macroelement model
As shown in Figure 3.31, the legend in the graph ‘macroel theory’ refers to static stiffness function
obtained from Table 8. For all the FEM models, they all reach maximum bearing capacity of
around 1540 KN, which is the maximum bearing capacity of soil domain as calculated with Eq.
(3.2.11). Similar to the vertical load analysis in infinite soil domain, the same calibration procedure
is taken for this model as well.
-0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0-1600
-1400
-1200
-1000
-800
-600
-400
-200
0
200
Forc
e (K
N)
Dipslacement (m)
Opensees
Plaxis
RS2
macroel theory
115
Figure 3.32. Vertical load and displacement plot for FEM model and macroelement
From the vertical monotonic load, the nonlinear coefficient, p, with value of 3 fits the curve with
FEM model. This calibrated coefficient value is used throughout the other load cases for
verification in other loading-path conditions. Figure 3.33 shows the horizontal monotonic load
case.
-1800
-1600
-1400
-1200
-1000
-800
-600
-400
-200
0
200
-0.02 -0.015 -0.01 -0.005 0V
erti
cal
load
(K
N)
Vertical displacement (m)
RS2.0_load
controlled
macroel_p=5000;
116
Figure 3.33. Horizontal maximum force for FEM and macroelement model
Then, cyclic moment load for OpenSees and macroelement are compared without uplift allowed
at the foundation. To distinguish the plasticity calibration parameter of pure loading and re-loading
parameter, Pl_1 is used to represent the p value obtained from monotonic vertical load, and Pl_2
is labeled for plasticity upon the re-loading phase of the analysis. The results are shown in Figure
3.34.
-700
-600
-500
-400
-300
-200
-100
0
-0.01 -0.008 -0.006 -0.004 -0.002 0
Ho
rizo
nta
l fo
rce
(KN
)
horiz displacmeent (m)
Opensees_Vmax
macroel_p=5
macroel_p=3
117
Figure 3.34. Macroelement and FEM model comparison for cyclic moment load on shallow
foundation without uplift
Uplift of the foundation is initiated and cyclic moment load is compared for OpenSees and
macroelement results as shown in Figure 3.35.
Therefore, with calibrated value of p value in vertical monotonic load, other loading cases
including horizontal load to moment cyclic loading with uplift agree well with FEM model. The
only other parameter that needs calibration is the Pl_2 value which is the plasticity of re-loading
phase of cyclic analysis.
-3000
-2000
-1000
0
1000
2000
3000
-3.00E-03 -2.00E-03 -1.00E-03 0.00E+00 1.00E-03 2.00E-03 3.00E-03
Mo
men
t lo
ad (
KN
m)
rotation (ϴ)
Opensees_Moment_cyclic
Opensees_Moment_monotonic
Macroelement (Pl_1=3,Pl_2=0)
118
Figure 3.35. Macroelement and FEM model comparison for cyclic moment load on shallow
foundation with uplift
Based on these models, the coupling effect of nonlinearity of soil and uplift of the foundation is
taken into consideration. Both FEM model and macroelement model captures the behavior and
agree well with each other. In order to verify whether the FEM model is modelled correctly, the
next case considers 20 meters by 60 meters in height and width of the soil model which increases
in cohesion per depth, also known as Gibson soil is verified with the paper results presented by
Cremer (Cremer et al., 2002).
-2000
-1500
-1000
-500
0
500
1000
1500
2000
-0.002 -0.001 0 0.001 0.002
Mo
men
t lo
ad (
KN
m)
Rotation (ϴ)
Opensees_cyclic_uplift
macroel_cyclic_uplift
(Pl_1=3,Pl_2=0)
119
c) Heterogeneous soil with stratum (J2 plasticity and PressureIndependentMultiyield
material)
This model was created by (Cremer et al., 2002) for verifying the original macroelement
derivation against FEM model. Figure 3.35 shows the 2D FEM model with mesh created using a
FEM software called Dynaflow.
Figure 3.36. 2D FEM model using Dynaflow for Gibson soil (Cremer et al., 2002)
As shown in the figure, the mesh is refined near the foundation to capture nonlinearity of the soil
more closely, while the base is fixed for rigid rock layer. The cohesion profile is also shown in
right side of Figure 3.35. The cohesion is formulated as Eq. (3.2.16).
� = �\ + 3 ∗ 0 (3.2.16)
Where Co is the cohesion at the surface, Z is the soil depth (m). Based on this cohesion variable,
shear modulus and elastic modulus is calculated. The material properties of this model is presented
in Table 10. Similar model has been created using OpenSees using the same material properties
used in the paper as shown in Figure 3.37, which is also shown in Figure 3.6.
120
Table 10. Material properties for Gibson soil with rigid rock layer
Variables Units Values
v Poisson’s ratio 0.45
ρ Density of soil (t/m3) 1.9
γmax Maximum shear strain 0.016
Co Cohesion of soil (KN) 30+g*depth of soil
g Cohesion gradient per
depth (KN/m)
3
G Shear modulus (kPa)
(at the surface)
1300*C
= 39000 at the surface
E Elastic modulus (KN/m)
(at the surface)
2*G*(1+v)
= 2*39000*(1+0.45)
=113100 at the surface
B Foundation width (m) 10
H Soil embedment depth (m) 20
121
Figure 3.37. Gibson soil model created using OpenSees with refined mesh around
foundation
Also, the maximum shear strain of the soil material has been provided in the model. Thus,
PressureIndependentMultiyield material property has been used in OpenSees where the user can
define automatic surface generation plot of a material with given maximum shear strain as
discussed in details in ‘Section 3.1.2.1 Soil material property’ of this paper. The following results
with paper and OpenSees results are shown in Figure 3.38 and Figure 3.39.
0 10 20 30 40 50 60
-10
-5
0
5
10
15
20
25
30
122
Figure 3.38. Moment cyclic analysis plot for the Gibson soil model in OpenSees and paper
results by Cremer et al. (2001)
Figure 3.39. Horizontal cyclic analysis plot for Gibson soil model in OpenSees and paper
FE results by Cremer et al. (2001)
-2500
-2000
-1500
-1000
-500
0
500
1000
1500
2000
2500
-0.002 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015 0.002 0.0025
Mom
ent
(K
Nm
)
rotation (ϴ)
Opensees_FE_model_cyclic
Opensees_monotonic
Paper result_cyclic
Paper result_monotonic
-250
-200
-150
-100
-50
0
50
100
150
200
250
-0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008
Ho
rizo
nta
l fo
rce
(KN
)
Horizontal displacement (m)
Opensees_cyclic
Opensees_monotonic
Paper_cyclic
Paper_monotonic
123
As shown in these figures, the OpenSees model and the results provided by Cremer et al. (2001)
are in good agreement. This builds confidence in OpenSees model regarding nonlinearity of the
soil which is taken into consideration in the paper FE model. Then, using the same material
properties, but changing the failure criterion to Von mises failure criterion, the OpenSees FEM
model has been compared with other FEM software (RS 2.0 and Plaxis). Also, macroelement is
then verified using the same model.
Initially, the vertical bearing capacity of foundation is calculated for this specific soil type. Where
the theoretical equation is described Eq. (3.2.17).
!��� = ´��7 :5.14 + ∇�=4�7 ; (3.2.17)
Where ∇� defines the c cohesion gradient, � = �7 + ∇�C with z, depth, co the cohesion at depth
z=0, c, the cohesion at depth z, B, the width of the foundation, μc coefficient depending on ∇�i�2
and
B/h, with h, the height of soil layer. This μc is also referred to as variable F in the original derivation
by Davis and Booker. (Booker & Davis, 1974) The F value is provided as shown in Figure 3.40.
Figure 3.40. Correction factors for rough and smooth footings (Booker & Davis, 1974)
Using the variables from the given parameters, the parameters yield maximum bearing capacity,
Nmax of 1940.4 KN as provided in the Eq. (3.2.18).
124
!��� = ´��7 :5.14 + ∇�=4�7 ; = » ∗ :5.14 + 3 ∗ 104 ∗ 30; = 1.2 ∗ :5.14 + 3 ∗ 104 ∗ 30; = 194.04 ��/S
(3.2.18)
Since the qmax is the normalized bearing capacity of footing per unit width of foundation,
multiplying the value to the width, B, results in vertical maximum bearing capacity of the footing
as shown in Eq. (3.2.19).
�S@T = !��� ∗ = = 194.04��/S ∗ 10S = 1940.4 ��
(3.2.19)
The results with other FEM software, such as RS2.0, Plaxis and OpenSees are provided in Figure
3.41. All of the software has Elastic Perfectly Plastic (EPP) behavior with failure following Mohr-
Coulomb failure criterion.
Figure 3.41. Vertical loading case for OpenSees, RS2.0 and Plaxis for Gibson soil
-4000
-3500
-3000
-2500
-2000
-1500
-1000
-500
0
500
-0.15 -0.13 -0.11 -0.09 -0.07 -0.05 -0.03 -0.01
Ver
tial L
oad
(K
N)
Vertical displacement (m)
Opensees_EPP
RS2.0
Plaxis
Theoretical
macroel (Pl_1=3)
125
For both of the FEM models in RS 2.0 and Plaxis, results match well with the theoretical result.
However, OpenSees model actually follow close to the value obtained from maximum value
proposed by Cremer (2001), which is Nmax = 2400 KN. The reason why there is a difference
between the FEM results is due to lack of available functions in OpenSees to model this material
properties. For the soil model with increase in shear modulus per depth, both RS 2.0 and Plaxis
have the capacity to assign the soil domain to have increase in shear modulus with respect to a
datum point, which is with respect to surface in this case. However, OpenSees does not have this
function and the user has to manually create an array of material properties that increases per depth
and assign the property to the appropriate mesh coordinates. Thus, this results in difference in the
analysis between the FEM models. Finer mesh refinement may be able to obtain the results close
to the theoretical value, but since OpenSees does capture the maximum vertical bearing capacity
that is close to the paper value, along with the other loading path scenarios, the macroelement will
be compared with this specific model for further comparison with different loading paths,
including moment cyclic load analysis with uplift of the foundation.
As shown in Figure 3.41 Pl_1 value of 3 is chosen because it gives closest result to the OpenSees
model, with Nmax of 2400 KN. Using this value, horizontal monotonic load is compared with
macroelement and OpenSees as shown in Figure 3.42. The moment monotonic and cyclic load
comparison is shown in Figure 3.43 without an uplift of the foundation.
126
Figure 3.42. Horizontal loading case for OpenSees and macroelement for Gibson soil
Figure 3.43. Moment cyclic analysis in OpenSees and macroelement for Gibson soil without
uplift
-600
-500
-400
-300
-200
-100
0
-0.015 -0.013 -0.011 -0.009 -0.007 -0.005 -0.003 -0.001 0.001
Ho
rizo
nta
l fo
rce
(KN
)
Horizontal displacement (m)
Opensees_monotonic_horiz
macroel (Pl_1=3)
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
-2.50E-03 -1.50E-03 -5.00E-04 5.00E-04 1.50E-03 2.50E-03
Mo
men
t lo
ad (
KN
m)
Rotation (ϴ)
Opensees_moment_cyclic
Opensees_monotonic
macroel (Pl_1=3,Pl_2=0)
macroel_monotonic_moment
127
Figure 3.44. Moment cyclic Analysis in OpenSees and macroelement for Gibson soil with
uplift of the foundation
Therefore, with just a few parameters to calibrate from FEM model, the macroelement is in good
agreement with the numerical results. By verification, this simplified model accurately captures
the nonlinear behavior of soil with uplift of the footing. Thus, throughout various examples
provided in this chapter, macroelement agrees well with FEM model in quasi-static loading
scenario for uplift of foundation with nonlinear soil. The next chapter provides verification
example regarding dynamic loading of the structure with FEM models.
-4000
-3000
-2000
-1000
0
1000
2000
3000
-2.50E-03 -2.00E-03 -1.50E-03 -1.00E-03 -5.00E-04 0.00E+00 5.00E-04 1.00E-03 1.50E-03 2.00E-03
Mo
men
t L
oad
(K
Nm
)
rotation (theta)
Opensees_moment_uplift
macroel (Pl_1=0,Pl_2=0);
128
3.2.4 Dynamic loading
For dynamic analysis of footing, the infinite soil domain model using OpenSees have been used.
The soil domain is modeled with 100 meter by 100 meter with fixed base and viscous boundaries
on the side. The same material properties have been used as shown in Table 4.
Table 11. Material properties for homogeneous infinite soil domain
Parameters Units Values
v Poisson’s ratio 0.25
ρ Density of soil (t/m3) 1.6
Co Cohesion of soil (KN) 30
Vs Shear wave velocity (m/s) 201.5
G Shear modulus (KN/m3) 6 = ) ∗ ��>
6 = 64963.6
E Elastic modulus (KN/m) + = 2 ∗ 6 ∗ &1 + 5' = 162409
B Foundation width (m) 2
H Soil embedment depth (m) ∞
As previously mentioned, this model has been used to create dynamic impedance function which
agrees well with infinite soil domain dynamic impedance function generated by Gazetas
(Mylonakis et al., 2006) also shown in Figure 3.14 and Figure 3.15.
The first verification model examines Kobe excitation applied horizontally to the rigid massless
foundation only. Figure 3.45, Figure 3.46, and Figure 3.47 shows the horizontal, rotational, and
vertical displacement of foundation subjected to Kobe excitation in horizontal direction
respectively. Fast Fourier transformation method has been used with the dynamic impedance
function obtained from 100 meter in height by 100 meter in width with sinusoidal sweep analysis
as discussed in Section 2.2.
129
Figure 3.45. Horizontal displacement of foundation with Kobe excitation applied to
massless foundation; comparison with FEM analysis and FFT analysis result
Figure 3.46. Rotation of foundation with Kobe excitation applied to massless foundation;
comparison with FEM analysis and FFT analysis result
-8.00E-05
-6.00E-05
-4.00E-05
-2.00E-05
0.00E+00
2.00E-05
4.00E-05
6.00E-05
0 5 10 15 20 25 30 35 40
Ho
rizo
nta
l d
isp
lace
men
t (m
)
time (s)
Opensees_foundation only
FFT_impedance
-4.00E-06
-3.00E-06
-2.00E-06
-1.00E-06
0.00E+00
1.00E-06
2.00E-06
3.00E-06
4.00E-06
5.00E-06
6.00E-06
0 5 10 15 20 25 30 35 40
rota
tio
n (
ϴ)
time (s)
Opensees applied @ foundation
matlab_impedance
130
Figure 3.47. Vertical displacement of foundation with Kobe excitation applied to massless
foundation; comparison with FEM analysis and FFT analysis result
The horizontal displacement of foundation subjected to Kobe excitation agrees well with FFT
analysis and FEM analysis. Figure 3.47 does have small discrepancy between FEM and FFT
analysis, but the magnitude of difference is almost negligible as seen in the figure. After the
verification of soil impedance function with time history analysis, then the structure is added on
top of the structure with the following properties as shown in Table 12.
-2.50E-19
-2.00E-19
-1.50E-19
-1.00E-19
-5.00E-20
0.00E+00
5.00E-20
1.00E-19
1.50E-19
2.00E-19
2.50E-19
0 5 10 15 20 25 30 35 40
Ver
tica
l d
isp
lace
men
t (
m)
time (s)
Opensees applied @ foundation
matlab_impedance
131
Table 12. Structural properties of FEM model for Kobe excitation
Parameters Units Values
M Mass of structure (tons) 50
E Elastic modulus (KN/m) 30*10^6
l Length of a column (m) 1
Ir Moment of inertia of column t] = < ∗ ]Q4
r Radius of column (m) 0.6
With the updated model, the Kobe excitation is applied to the structure and the response of the
horizontal displacement of the footing is analyzed. With the degrees of freedom assigned as shown
in Figure 3.48, the matrix formulation of this model is shown in Eq. (3.2.20).
Figure 3.48. Structure and foundation degrees of freedom for 1m beam example
Structure
Foundation
1
2
3
5
4
Kst, Cst
Kf, Cf
132
��¦"¤�7)J ∗ "¾ + ��¦ ∗ "- +�"�$)�¦)7¥ ∗ "- + ��¦ ∗ " + �S@�]\�e ∗ "= »I�¦ + »"I�¦7")¥� ¢7"�I
�������¦ 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 ��¦ + �¢���
�� ∗ÐÑÒÑÓ"¾ {"¾ >"¾ �"¾ Q"¾ �ÔÑÕ
ÑÖ +
u@1 ∗ Í��¦Î + @> ∗ Í��¦Îy ∗ÐÑÒÑÓ"- {"- >"- �"- Q"- �ÔÑÕ
ÑÖ +�����0 0 0 0 00 0 0 0 00 0 ��� ��� 00 0 ��� ��� 00 0 0 0 ������
�� ∗ÐÑÒÑÓ"- {"- >"- �"- Q"- �ÔÑÕ
ÑÖ +
��������� 12+te�
6+te> − 12+te�6+te> 06+te> 4+te − 6+te> 2+te 0
− 12+te� − 6+te> 12+te� − 6+te> 06+te> 2+te − 6+te> 4+te 00 0 0 0 0���������
∗ÐÑÒÑÓ"{">"�"Q"�ÔÑÕ
ÑÖ +�����0 0 0 0 00 0 0 0 00 0 ��� ��� 00 0 ��� ��� 00 0 0 0 ������
�� ∗ÐÑÒÑÓ"{">"�"Q"�ÔÑÕ
ÑÖ
=�����»�0000 ���
�� + �&��e,"de[f!�e + �de!de' − ��e!�e�
+  �î� ï½+1−ô + ð� ï½- +1−ô + Ø�ï½¾ +1−ô��−1ô=0
(3.2.20)
Where the damping of the structure is formulated based on Rayleigh formulation where C matrix
is proportional to the mass and stiffness matrix as shown below.
��¦ = @{Í��¦Î + @>Í��¦Î (3.2.21)
The terms a1 and a2 can be calculated based on the constant damping terms with frequency of
interest as shown in Eq. (3.2.22).
'?�?¥ ( = ���� 1[� [�1[¥ [¥ ��
�� ©@7@{ª (3.2.22)
133
Thus, the structural frequency of interest in this case is chosen at 1st mode and 3rd mode of its
natural frequencies at structural damping coefficient of 5%. Also, the damping and stiffness matrix
of soil is derived based on the dynamic impedance function of soil using Nakamura’s coefficient
values. And lastly, the restoring coefficients are derived based on Nakamura’s recursive
parameters defined by dynamic impedance function of soil. The dynamic impedance function of
soil domain with dimensions 100 meters in height by 100 meters in width is calculated with
sinusoidal sweep analysis with different excitation of frequency. The dynamic impedance
functions of soil are shown in four plots: vertical, horizontal, rotational and the coupling degrees
of freedom with horizontal and rotational direction as shown in Figure 3.49 to Figure 3.52
respectively.
Figure 3.49. Dynamic impedance of 100m by 100m soil domain (Vertical)
0 20 40 60 80 100-2
-1
0
1x 10
5
Frq. [sec-1
]
Stiff
ness
Matrix Inv
Nakamura Coeff
0 20 40 60 80 1000
2
4
6x 10
5
Frq. [sec-1
]
Dam
pin
g
Matrix Inv
Nakamura Coeff
134
Figure 3.50. Dynamic impedance of 100m by 100m soil domain (Horizontal)
Figure 3.51. Dynamic impedance of 100m by 100m soil domain (Rotational)
0 20 40 60 80 100-3
-2
-1
0
1x 10
5
Frq. [sec-1
]
Stiff
ness
Matrix Inv
Nakamura Coeff
0 20 40 60 80 1000
5
10x 10
5
Frq. [sec-1
]
Dam
pin
g
Matrix Inv
Nakamura Coeff
0 20 40 60 80 100-5
0
5x 10
5
Frq. [sec-1
]
Stiff
ness
Matrix Inv
Nakamura Coeff
0 20 40 60 80 1000
1
2
3x 10
5
Frq. [sec-1
]
Dam
pin
g
Matrix Inv
Nakamura Coeff
135
Figure 3.52. Dynamic impedance of 100m by 100m soil domain (Coupling with rotation and
horizontal)
Initially, the excitation of sinusoidal force of 20 Hz is applied to the FEM model and its dynamic
impedance function is calculated. The frequency of interest may extend further beyond this
excitation frequency as the maximum frequency range of motion extend up to the inverse of time
increment. Nakamura’s model fails to converge and faces stability issue as the coefficients try to
interpolate a dynamic impedance outside the given range of data, resulting in instantaneous
negative mass. Laudon (2013) has worked on resolving this issue by introducing extension of the
provided dynamic impedance of range of frequency by introducing a negative parabola in the
stiffness and linear increasing function in damping terms as this avoids creating the negative
instantaneous mass which undoubtedly causes the Newmark integration scheme to produce divergent
results (Laudon, 2013). More details on this stability issue has been provided by Laudon (2013).
0 20 40 60 80 100-2
0
2
4x 10
4
Frq. [sec-1
]
Stiff
ness
Matrix Inv
Nakamura Coeff
0 20 40 60 80 1000
2000
4000
6000
Frq. [sec-1
]
Dam
pin
g
Matrix Inv
Nakamura Coeff
136
Therefore, after applying macroelement coefficient to be perfectly linear elastic by assigning the
plasticity coefficient of large value, the macroelement and linear static stiffness should cancel out
as shown in Eq. (3.2.20) and Nakamura’s model should govern the analysis in linear elastic range.
The results for OpenSees analysis, the combined model, along with the FFT analysis are compared
in Figure 3.53, Figure 3.54, and Figure 3.55 for horizontal, rotational, and vertical displacement
of foundation respectively when the structure is excited with Kobe ground motion.
Figure 3.53. Kobe excitation applied to structure and the horizontal response of
foundation; result comparison with OpenSees, FFT, and Macroelement+Nakamura’s
model
-1.500E-04
-1.000E-04
-5.000E-05
0.000E+00
5.000E-05
1.000E-04
0.000E+00 5.000E+00 1.000E+01 1.500E+01 2.000E+01 2.500E+01 3.000E+01
Hori
zonta
l dis
pla
cem
ent
(m)
Time (s)
Opensees_with structure
FFT analysis
Nakamura+macroel
137
Figure 3.54. Rotation at the foundation with Kobe excitation on structure; OpenSees, FFT
analysis, and Macroelement+Nakamura’s model comparison
Figure 3.55. Vertical displacement of foundation with Kobe excitation on structure;
OpenSees, FFT analysis and Macroelement+Nakamura’s model comparison
-4.00E-05
-3.00E-05
-2.00E-05
-1.00E-05
0.00E+00
1.00E-05
2.00E-05
3.00E-05
4.00E-05
5.00E-05
6.00E-05
0.00E+00 5.00E+00 1.00E+01 1.50E+01 2.00E+01 2.50E+01 3.00E+01
Ro
tati
on (
thet
a)
time (s)
FFT analysis
Opensees_Mass on structure only
Nakamura+macroel
-6.00E-19
-4.00E-19
-2.00E-19
0.00E+00
2.00E-19
4.00E-19
6.00E-19
0.000E+00 5.000E+00 1.000E+01 1.500E+01 2.000E+01 2.500E+01 3.000E+01
Ver
tica
l dis
pla
cem
ent (m
)
time (s)
Opensees_Mass on structure only
Macroelement+Nakamura
138
The horizontal and rotational displacement of foundation when the structure is excited with Kobe
ground motion agrees well with the OpenSees FEM model and the proposed model with
Macroelement and Nakamura’s model. As mentioned before, the vertical movement in OpenSees
occurs due to numerical error in the analysis but the difference is almost negligible as shown in
Figure 3.55. This verifies the proposed model in the linear elastic range and the results are in good
agreement with FEM model as expected. The next chapter contains an application example with
realistic bridge pier dimensions with soil foundation that is infinite in soil domain. The nonlinearity
of the soil with foundation is also introduced in the next chapter.
The nonlinearity of soil with varying frequency of excitation with varying amplitude of load is
analyzed. The purpose of this analysis is to create parametric space of frequency and inelasticity
where the model captures the inelastic behavior of soil with the frequency-dependent soil. The soil
model is created with 100 meter by 100 meter model with beam width of 10 meters. The constant
vertical force of 385.5 KN is applied to the foundation (25% of the maximum bearing capacity of
the foundation) while cyclic moment is applied at the foundation. Two cases are considered: a)
foundation attached to the soil and b) foundation which undergoes uplift, detached from the soil.
FEM model has been analyzed to produce batch analysis of the cases.
• Increase in amplitudes: M = 100KNm, M = 1000KNm, M = 1500KNm,
M = 2000KNm,
*Note: Maximum moment capacity <2700 KNm
• Increase in excitation frequency: 1Hz ~ 20 Hz
The results are compared for foundation without uplift and with uplift as shown in Figure 3.56 and
Figure 3.57 respectively. Also, in order to illustrate that the imposed moment introduces
nonlinearity of the soil, or whether the detachment of the foundation to soil is occurring at the
analyzed moment amplitude, FEM contour plots of excitation at 2Hz with magnitude of 1000 KNm
is shown in Figure 3.58 for case without uplift and Figure 3.59 for uplift.
139
Figure 3.56. Parametric study of varying frequency and amplitude without uplift
No
Uplift
M=1KNm
(0.04% of Mmax)
M=100KNm
(4% of Mmax)
M=1000KNm
(37% of Mmax)
M=1500KNm
(56% of Mmax)
M=2000KNm
(74% of Mmax)
1Hz
2Hz
4Hz
6Hz
8Hz
10Hz
12Hz
15Hz
18Hz
20Hz
-3.0E-07 0.0E+00 3.0E-07
-3.0E-07 0.0E+00 3.0E-07
-3.0E-07 0.0E+00 3.0E-07
-3.0E-07 0.0E+00 3.0E-07
rotation (ϴ)
-3.0E-07 0.0E+00 3.0E-07
-3.0E-07 0.0E+00 3.0E-07
-3.0E-07 0.0E+00 3.0E-07
-3.0E-07 0.0E+00 3.0E-07
-3.0E-07 0.0E+00 3.0E-07
-3.0E-07 0.0E+00 3.0E-07
-0.00005 0 0.00005
-0.00005 0 0.00005
-0.00005 0 0.00005
-0.00005 0 0.00005
-0.00005 0 0.00005
-0.00005 0 0.00005
-0.00005 0 0.00005
-0.00005 0 0.00005
-0.00005 0 0.00005
-0.00005 0 0.00005
-0.0005 0 0.0005
-0.0005 0 0.0005
-0.0005 0 0.0005
-0.0005 0 0.0005
-0.0005 0 0.0005
-0.0005 0 0.0005
-0.0005 0 0.0005
-0.0005 0 0.0005
-0.0005 0 0.0005
-0.0005 0 0.0005
-0.001 0.001
-0.001 0 0.001
-0.001 0.001
-0.001 0 0.001
-0.001 0 0.001
-0.001 0 0.001
-0.001 0 0.001
-0.001 0 0.001
-0.001 0 0.001
-0.001 0.001
-0.001 0 0.001
-0.001 0 0.001
-0.001 0 0.001
-0.001 0 0.001
-0.001 0 0.001
-0.001 0 0.001
-0.001 0 0.001
-0.001 0 0.001
-0.001 0 0.001
-0.001 0 0.001
FEMMacro
140
Figure 3.57. Parametric study of varying frequency and amplitude with uplift of foundation
Uplift M=1KNm
(0.067% of Mmax, uplift)
M=100KNm
(6.7% of Mmax, uplift)
M=1000KNm
(67% of Mmax, uplift)
M=1500KNm
(100% of Mmax, uplift)
1Hz
2Hz
4Hz- FEM fails to
converge
6Hz-
8Hz -
10Hz -
12Hz -
15Hz -
18Hz -
20Hz --3.0E-07 0.0E+00 3.0E-07
-3.0E-07 0.0E+00 3.0E-07
-3.0E-07 0.0E+00 3.0E-07
-3.0E-07 0.0E+00 3.0E-07
rotation (ϴ)
-3.0E-07 0.0E+00 3.0E-07
-3.0E-07 0.0E+00 3.0E-07
-3.0E-07 0.0E+00 3.0E-07
-3.0E-07 0.0E+00 3.0E-07
-3.0E-07 0.0E+00 3.0E-07
-3.0E-07 0.0E+00 3.0E-07
-0.00005 0 0.00005
-0.00005 0 0.00005
-0.00005 0 0.00005
-0.00005 0 0.00005
-0.00005 0 0.00005
-0.00005 0 0.00005
-0.00005 0 0.00005
-0.00005 0 0.00005
-0.00005 0 0.00005
-0.00005 0 0.00005
-0.0005 0 0.0005
-0.0005 0 0.0005
-0.0005 0 0.0005
-0.0005 0 0.0005
-0.0005 0 0.0005
-0.0005 0 0.0005
-0.0005 0 0.0005
-0.0005 0 0.0005
-0.0005 0 0.0005
-0.0005 0 0.0005
-0.001 0 0.001
-0.001 0 0.001
FEMMacro
141
Figure 3.58. 1000 KNm moment applied at the foundation without uplift at 4Hz excitation
Figure 3.59. 1000 KNm moment applied at the foundation with uplift at 4Hz excitation
142
For the case where the moment of 1500 KNm is applied to the foundation with uplift, the
foundation is detached to the soil in excessive manner where the center of the foundation detaches
from the soil as shown in Figure 3.60.
Figure 3.60. Uplift of foundation with 1500 KNm moment
Therefore, the analysis beyond 2 Hz of excitation with this magnitude of sinusoidal moment fails
to converge since the excess portion of the foundation has detached from the soil.
In order to demonstrate the trend of overall analysis, dynamic impedance function has been used
to plot all the result findings in frequency domain. As previously discussed, within the hysteretic
graph, dynamic stiffness can be obtained by taking the slope of the hysteretic loop while energy
dissipation can be analyzed as a dynamic damper of the system. To normalize these values, the
dynamic stiffness terms are normalized with rotational stiffness of the foundation. The equivalent
damping terms are achieved by taking the energy dissipated in hysteretic loop, Ed, and normalizing
it to the area under the maximum force and displacement, ESo, as shown in (3.2.23)
ζ = E62 ∗ < ∗ +È\ = Energy dissipated
2 ∗ < ∗ :�@T[\]�� ∗ �@TB�de2 ; (3.2.23)
143
Figure 3.61 shows the results in dynamic impedance of the proposed model and FEM model at
different magnitude of moment applied to the foundation. Note that these load and excitation
frequency values are used for this specific examples with user defined geometry and material
properties. Thus, the proportion of the load to maximum load capacity of this specific example is
also shown in the results.
From the analysis, the results are in good agreement for low frequency range with all magnitudes
of moment. Thus, the nonlinearity in quasi-static loading scenario is well captured. Also, at low
magnitude of cyclic moment, the proposed model match well with FEM results, demonstrating
that at low amplitude of load, the frequency dependency of soil is captured. There are deviation in
the analysis results when it comes to high intensity of moment, with high frequency of excitation.
This behavior is quite difficult to capture using the simplified model, but the model can capture
most of the nonlinearity with limited range of excitation with small loss of accuracy as summarized
in the Figure 3.61.
144
Figure 3.61. Dynamic impedance of varying intensity without uplift
No uplift Dynamic Stiffness coefficient Equivalent Damping
M = 1 KNm
(0.04% of Mmax)
M = 100 KNm
(4% of Mmax)
M = 1000 KNm
(37% of Mmax)
M = 1500 KNm
(56% of Mmax)
M = 2000 KNm
(74% of Mmax)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 5 10 15 20D
ynam
ic s
tiff
nes
s co
effi
cien
tFrequency (Hz)
FEM
Macroel+Nakamura
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 5 10 15 20
Equiv
alen
t D
amp
ing
Frequency (Hz)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 5 10 15 20Frequency (Hz)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20Frequency (Hz)
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20Frequency (Hz)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20Frequency (Hz)
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20Frequency (Hz)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20Frequency (Hz)
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20Frequency (Hz)
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20Frequency (Hz)
145
Figure 3.62. Dynamic impedance with uplift (uplift occurs at M = 1000 KNm)
The case for uplift results are presented with moment applied up to 1000 KNM. The results are
presented in Figure 3.62. As shown in the figure, the results are in good agreement from load
intensity of 1 KNm to 100 KNm because uplift of the foundation has not occurred at these loads.
At the magnitude of load at 1000 KNm, the FEM and proposed model do have similar dynamic
stiffness but the equivalent damping is slightly different as the frequency increases. The
detachment of foundation to soil with nonlinearity of the soil with high excitation of frequency in
numerical model is quite difficult to capture, but the model does a decent job in predicting the
overall behavior with these nonlinearities as shown in Figure 3.62.
Comparing the proposed method with the existing model for macroelement with frequency
independent soil, macroelement analysis has been carried out without the frequency dependency
Uplift Dynamic Stiffness coefficient Equivalent Damping
M = 1 KNm
(0.067% of
Mmax, uplift)
M = 100 KNm
(6.7% of
Mmax, uplift)
M = 1000 KNm
(67% of
Mmax, uplift)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 5 10 15 20D
ynam
ic s
tiff
nes
s co
effi
cien
tFrequency (Hz)
FEM
Macroel+Nakamura
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 5 10 15 20
Eq
uiv
alen
t D
amp
ing
Frequency (Hz)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 5 10 15 20Frequency (Hz)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20Frequency (Hz)
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20Frequency (Hz)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20Frequency (Hz)
146
of soil. A specific radiation damping of soil is defined at 10Hz of the soil impedance function used
in the analysis. Thus, a dynamic stiffness and damping terms are frequency-independent and the
results are compared for foundation with moment load of 2000KNm as shown in Figure 3.63.
Figure 3.63. Dynamic impedance without uplift for FEM, macroelement and the proposed
method (macroelement and Nakamura’s model)
As predicted, the macroelement model without frequency-dependency of soil does not capture the
dynamic stiffness characteristics of soil at different loading excitation. The material damping and
radiation damping at a specific frequency in macroelement model does provide some damping
characteristic of soil, but the proposed model provides results that agree with FEM model with
frequency dependency of soil.
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20
Dynam
ic s
tiff
nes
s co
effi
cien
t
Frequency (Hz)
FEM
macroelelement
Macroel+Nakamura
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20
Eq
uiv
alen
t d
amp
ing
Frequency (Hz)
147
Therefore, from the parametric study of varying excitation with amplitude of moment, the results
shows that in range of low frequency with high amplitude the results are in good agreement with
FEM model. On the other hand, there are still some deviation of the results at high frequency with
high intensity of the load, introducing large plasticity deformation with frequency dependency of
the soil taken into an effect.
Another criterion of the analysis comparison is the computation time it takes to analyze each of
the model. For the proposed model, each analysis takes about approximately 30 seconds to analyze
a nonlinear frequency-dependent analysis with Intel ® Core ™ i7-4810MQ CPU @ 2.80 GHz,
and 8.00 GB of RAM. With the same computer setting, it takes approximately 1.5 to 2 hours to
analyze a full 2D FEM model using OpenSees for these case study examples. In John et al. (2015),
it took approximately one month to analyze a full 3D FEM model with nonlinearity and wave
propagation. Thus, the proposed model greatly reduces the computation time for each of the
analysis, making parametric study more feasible than the FEM model.
148
Application example
In order to demonstrate the applicability of this method, realistic bridge pier example has been
used to illustrate how this model can be used to model the SSI effect, including nonlinearity of the
soil and geometric nonlinearity such as uplift of the foundation. Also, the results are compared
with FEM model to verify this combined method with realistic bridge pier example.
The same soil domain is used for this specific example as the dynamic analysis example from
Section 3.1.4, and the structure and foundation material properties are provided by Chatzigogos et
al., (2011). The model is first created in OpenSees using FEM model, and sinusoidal sweep
analysis without structure is analyzed in order to obtain dynamic impedance function of soil for
Nakamura’s recursive parameters. Then, the combined method with macroelement is analyzed in
the section. Details of the example is provided in this section. Overall, the modeling approach of
structure-soil with the proposed model matches well with numerical FEM model in linear elastic
range.
A bridge pier example is provided from Chatzigogos et al. (2011) as shown in Figure 4.1. The
material properties of structure, foundation, and soil is provided in Table 13. 2D FEM analysis is
carried out using OpenSees to compare the time-history analysis of a SSI effect with the same
material properties with the proposed model. The first verification is provided in linear elastic
dynamic analysis. Then, quasi-static loading case of the foundation to nonlinear dynamic analysis
is covered. These verification provides a full analysis comparison of a bridge pier example with
the nonlinearities with SSI effect.
In linear elastic dynamic analysis, the Kobe excitation is applied to the foundation without
structure. The OpenSees, FFT analysis, and Nakamura’s analysis are compared for 10 m
foundation width footing in order to verify the dynamic impedance function of the soil. The
horizontal displacement of foundation results are shown in Figure 4.2.
149
Figure 4.1. Dynamic analysis example with realistic bridge pier and footing dimension
(Chatzigogos et al., 2009)
Table 13. Parameters of structure and soil for realistic bridge pier example (Chatzigogos et
al., 2009)
150
Figure 4.2. Time history analysis of foundation with Kobe excitation applied to the
foundation horizontally with OpenSeesOpenSees, FFT and Nakamura’s model
-1E-07
-8E-08
-6E-08
-4E-08
-2E-08
0
2E-08
4E-08
6E-08
8E-08
0 5 10 15 20 25 30
Ho
rizo
nta
l d
isp
lace
men
t (m
)
Time (s)
Frquency-domain analysis Nakamura Opensees
151
Figure 4.3. Time history analysis of foundation with Kobe excitation applied to the
structure horizontally with OpenSeesOpenSees, FFT, and Nakamura’s model.
Therefore, the model with realistic bridge pier design in linear elastic dynamic analysis comparison
agrees well for the proposed model with FEM model and also with FFT analysis. The nonlinear
quasi-static case has also been analyzed where the foundation is subjected to monotonic vertical
pushover analysis to calibrate its plasticity of the foundation. The vertical monotonic load and
displacement plot is shown in Figure 4.4.
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0 5 10 15 20 25 30
Ho
rizo
nta
l d
isp
lacem
ent
(m)
Time (s)
Frquency-domain analysis Macroelement+Nakamura Opensees
152
Figure 4.4. Vertical force and displacement monotonic curve for bridge pier foundation
As shown in the graph above, the Pl_1 value of 5 to 6 matches well with the OpenSees result for
vertical monotonic load condition. Using this calibrated parameter, moment cyclic load is applied
to the structure to analyze its nonlinear effect of soil and geometric uplift of a foundation. The
results are shown in Figure 4.5.
-1800
-1600
-1400
-1200
-1000
-800
-600
-400
-200
0
200
-5.00E-02 -4.00E-02 -3.00E-02 -2.00E-02 -1.00E-02 0.00E+00
Ver
tica
l fo
rce
(KN
)
Vertical displacement (m)
Opensees
macroel_p=5000
macroel_p=5
macroel_p=6
153
Figure 4.5. Moment cyclic force-displacement plot using OpenSees and MATLAB for
bridge pier example
Then the nonlinear quasi-static analysis of soil and foundation is provided as shown in Figure 4.5.
As proposed, the imposed model provides good agreement with the FEM model regarding
nonlinearity of soil and foundation.
For the dynamic nonlinear loading condition with uplift of the foundation, the overall structural
response experienced excess moment at the foundation due to the rocking motion. Therefore, both
of FEM model and the proposed method failed to converge for this analysis with Kobe excitation
applied to the structure. As previously provided with verification, if the load exceeds the bounding
surface of the foundation, the analysis will fail to converge. Thus, one should be aware of the
maximum bearing capacity of the foundation in seismic analysis with rocking of shallow
foundation.
-0.015 -0.01 -0.005 0 0.005 0.01 0.015-5000
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
5000
rotation (theta)
mom
en
t (K
Nm
)
Opensees
MATLAB
154
Therefore, the proposed model with macroelement and Nakamura’s recursive parameters
represented as a restoring force to represent nonlinearity of soil and foundation accurately captures
the response of overall structure when subjected to a seismic excitation. Also, the analysis method
follows non-convergent solution when the load exceeds bearing capacity of the foundation. Not
only does this model greatly simplify the modeling approach and parameters that are required to
analyze the nonlinear behavior of foundation, but it also reduces significant amount of time it
requires to analyze the model. Due to the generalized force to displacement relationship, the
analysis can handle various types of analysis at a computationally efficient speed.
155
Conclusion
In this chapter, the summary of the findings and limitation of the studies, along with future studies
and recommendations are provided. The study aims to provide a simplified method in analyzing
nonlinear behavior of shallow foundation with frequency dependent characteristic of soil
considered when subjected to seismic load. In order to build credibility of this method, verification
examples have been provided with theoretical results from literature and FEM models.
As there has been increasing number of interest and awareness regarding nonlinear behavior of
shallow foundation upon large seismic excitation by researchers and engineers, there has been
various methods in modeling shallow foundation. Lumped spring approach provides a simple way
to represent soil with spring and dashpot, which can be used to model a linear elastic soil domain.
However, this method is not applicable to seismic analysis of soil as the soil exhibits nonlinearity
at large magnitude of load due to earthquake load. Winkler type foundation replaces soil-
foundation interface with series of springs and dashpots which can be used to simulate the
nonlinear behavior of soil. Although this methods captures nonlinearity of soil-foundation system,
the calibration efforts that are required to model is complex. Another method is to create the soil
domain and foundation using FEM. This continuum model allows analysis to capture nonlinearity
of the soil, material damping and radiation damping as the incident wave propagates to infinite
soil medium, and the detachment of foundation. This analysis accurately captures SSI effect of
shallow foundation but the computation time and modeling effort is too expansive and it is not
feasible in practice. Macroelement was introduced as a simplified method where the nonlinearity
of soil and detachment of the foundation is modeled in a lumped node. This method uses
generalized force and displacement relationship where plasticity of the soil-foundation is defined
based on the ratio of the load it experiences to the maximum load the foundation can carry.
However, this element has limitation to certain types of soil based on the calibration and frequency
dependent characteristic of soil is not captured in seismic analysis.
Thus, the purpose of this study is to provide a simplified method to capture the nonlinear behavior
of shallow foundation when subjected to earthquake load with frequency dependency of the soil.
Macroelement has been used in this thesis to model the nonlinearity of the soil as the method
156
requires one to model shallow foundation with a few parameters that needs to be calibrated. In
order to combine the frequency characteristic of soil, dynamic impedance of soil domain is
required. Thus, a recursive parameter model has been used to convert the dynamic impedance in
frequency domain to time domain using a delayed response of the soil in velocity and displacement.
By combining these two methods, nonlinear behavior of shallow foundation with frequency
dependency of soil is captured. This method has been verified with FEM model and are in good
agreement for low magnitude with wide range of frequency of excitation. Also, the model agrees
well with FEM model for quasi-static loading scenario with high magnitude of excitation.
However, the model is in satisfactory agreement for high magnitude and frequency of excitation.
Discussion on the result finding is provided in the next section, and the limitations and
recommended future studies are provided in the following section.
157
5.1 Summary of the findings
In this thesis, a method is proposed to integrate frequency-dependency soil by adding a recursive
parameter model proposed by Nakamura (2006) with macroelement (Chatzigogos et al, 2011).
The results are verified for static load, quasi-static nonlinear load, and nonlinear load at different
frequency of excitation. The proposed model captures nonlinear behavior of soil-foundation
system with frequency-dependency of soil. The results show excellent agreement with FEM model
with low-magnitude moment with all of the excitation frequency, and high-magnitude with low
frequency of load. However, the results are in satisfactory agreement for high magnitude of load
with high frequency range of excitation, around 10 Hz to 20 Hz. For its simplification as a lumped
node to capture nonlinearity of the soil with frequency dependency of the soil, the nonlinear
behavior of the soil with wave propagation and accumulation of plasticity with high number of
mesh and nodes, it is quite complex to capture all of these phenomenon with just a single node
presented with a macroelement.
The advantage of using this proposed model is that it does not require much calibration for the
parameters to use. The user would have to provide strength parameters for soil and properties of
foundation with just three parameters to define the overall soil plasticity with uplift behavior of
the foundation. This can be easily used by Engineers and researchers once the element can be
available with suggested values of calibration values for the plasticity parameters. The dynamic
impedance function can be obtained from literature, or SASSI can be used to obtain the dynamic
impedance of various soil types. In addition, the analysis is incredibly efficient in computation
time. For the FEM analysis that usually takes 1.5 to two hours to analyze the SSI effect with
seismic excitation, this model only takes 20 seconds to analyze a single model and accuracy of the
results has been shown in this thesis.
The realistic bridge pier example has been provided in the paper to demonstrate the applicability
of this model in design of bridge pier and shallow foundation. Thus, the user can model shallow
foundation subjected to seismic excitation with the nonlinearity of soil and SSI effect of the
foundation, including uplift of the foundation and frequency-dependency of the soil represented
as a recursive model with great simplicity than FEM model.
158
5.2 Limitations and future studies and future studies
Few limitations remain in this proposed model. Firstly, the determination of bounding surface is
not defined at a specific point and is oftentimes vaguely defined. Unless a theoretical maximum
bearing capacity formulation is provided, FEM analysis result of vertical monotonic load is used
to obtain this value. At this point, Terzhaghi’s bearing capacity concept is used to define the
maximum bearing capacity of the foundation from the FEM analysis result. The concept assumes
bearing capacity occurs where the residual increase in force to displacement relationship is steep
and linear. This type of failure is called local punching shear failure and occurs mostly in loose
sands as the failure occurs near the foundation. Similar type of failure is observed in FEM model
as the mesh around the foundation initially fails first before the stress propagates further meshes.
Thus, there may be improved approach to define the maximum bearing capacity of foundation
with various types of soil domain.
Also, the user would require dynamic impedance function of soil. Although there are available
software such as SASSI which provides the dynamic impedance of soil, the limitation of available
dynamic impedance of various types of soil still exists.
In addition, the analysis is provided only for shallow foundation application. The proposed model
may need some, if not major adjustments in analyzing deep foundation example. The simplicity of
macroelement formulation is only applicable to shallow foundation at this moment.
Furthermore, there are not a wide range of values available for defining plasticity of soil. More
analysis comparison with FEM analysis may be able to cover this limitation. In addition, the
lognormal relationship to define the plasticity stiffness is an assumption to clay soil that follows a
similar trend as proposed in previous literature. This may be limited to clay samples of soil and
the results may differ for other types of soil.
Therefore, there needs more refined work in expanding its limitation to wide range of analysis
options. Future studies regarding some of these issue may be implemented to overcome the
weaknesses of this model.
159
In addition, potential work to introduce a three-dimensional analysis using this model and
comparing the analysis results with numerical model would expand this analysis in plane-strain
analysis to multi-direction analysis. Analysis steps may include expanding degrees of freedom to
consider movement in out-of-plane direction and also adding torsional component at the
foundation. The coupling effect in these extra degrees of freedom may need more calibration and
verification. Once verified, this model would expand the applicability of the model to wide range
of seismic design problems.
160
References
Anastasopoulos, I., Gelagoti, F., Kourkoulis, R., Gazetas, G., & Asce, M. (2011). Simplified
Constitutive Model for Simulation of Cyclic Response of Shallow Foundations : Validation
against Laboratory Tests. American Society of Civil Engineers, (December), 1154–1168.
doi:10.1061/(ASCE)GT.1943-5606.0000534.
Bland, D. R. (1957). The associated flow rule of plasticity. Journal of the Mechanics and Physics
of Solids, 6, 71–78. doi:10.1016/0022-5096(57)90049-2
Booker, J. R., & Davis, E. H. (1974). Discussion: The effect of increasing strength with depth on
the bearing capacity of clays. Géotechnique, 24(4), 690–690. doi:10.1680/geot.1974.24.4.690
Chatzigogos, C. T., Figini, R., Pecker, A., & Salençon, J. (2011). A macroelement formulation for
shallow foundations on cohesive and frictional soils. International Journal for Numerical and
Analytical Methods in Geomechanics, 35(May 2010), 902–931. doi:10.1002/nag.934
Chatzigogos, C. T., Pecker, A., & Salençon, J. (2009). Macroelement modeling of shallow
foundations. Soil Dynamics and Earthquake Engineering, 29(5), 765–781.
doi:10.1016/j.soildyn.2008.08.009
Chatzigogos, & Figini, R. (2011). A macroelement formulation for shallow foundations on
cohesive and frictional soils. … Journal for Numerical …, (May 2010), 902–931.
doi:10.1002/nag
Cremer, C., Pecker, A., & Davenne, L. (2001). Cyclic macro-element for soil-structure interaction:
material and geometrical non-linearities. International Journal for Numerical and Analytical
Methods in Geomechanics, 25(13), 1257–1284. doi:10.1002/nag.175
Cremer, C., Pecker, A., & Davenne, L. (2002). Modelling of Nonlinear Dynamic Behaviour of a
Shallow Strip Foundation With Macro-Element. Journal of Earthquake Engineering, 6(2),
175–211. doi:10.1080/13632460209350414
161
Dobry, R., & Gazetas, G. (1988). Simple method for dynamic stiffness and damping of floating
pile groups. Géotechnique, 38(4), 557–574. doi:10.1680/geot.1988.38.4.557
Dobry, R., Petrakis, E., States, U., Force, A., Air, B., & Base, F. (1991). A Constituitive Relation
for Granular Materials Based on the Contact Law Between Two Spheres. United States Air
Force, Office of Scientific Research (Vol. 1).
Duarte-Laudon, A., Kwon, O. and Ghaemmaghami, A. (2015). Stability of time-domain analysis
of frequency-dependent soil-foundation system. Toronto: Earthquake Engineering and
Structural Dynamics.
El Ganainy, H., & El Naggar, M. H. (2009). Efficient 3D nonlinear Winkler model for shallow
foundations. Soil Dynamics and Earthquake Engineering, 29(8), 1236–1248.
doi:10.1016/j.soildyn.2009.02.002
Elnashai, A. S., & Sarno, L. D. (2015). Fundamentals of Earthquake Engineering: From Source
to Fragility, 2nd Edition.
Fang, H.-Y. (1991). Foundation Engineering handbook (Second Edi.). Van Nostraqnd Reinhold.
Finn, W. D. L., Pandey, B. H., & Ventura, C. E. (2011). Modeling soil – foundation – structure
interaction, (November). doi:10.1002/tal.735
Gazetas, G. (1983). Analysis of machine foundation vibrations: State of the art. International
Journal of Soil Dynamics and Earthquake Engineering, 2(1), 2–42. doi:10.1016/0261-
7277(83)90025-6
Gourvenec, S. (2007). Shape effects on the capacity of rectangular footings under general loading.
Géotechnique, 57(8), 637–646. doi:10.1680/geot.2007.57.8.637
Gourvenec, S., & Barnett, S. (2011). Undrained failure envelope for skirted foundations under
general loading. Géotechnique. doi:10.1680/geot.9.T.027
Hayes, J., McCabe, S. L., & Harris, J. (2012). Soil-Structure Interaction for Building Structures.
Kabanda, J., Kwon, O., & Kwon, G. (2015). Time and Frequency-Domain Analyses of Hualien
162
Large-Scale Seismic Test. Nuclear Engineering and Design. doi:DOI:
10.1016/j.nucengdes.2015.10.011
Kramer, S. L. (1996). Geotechnical Earthquake Engineering. Prentice-Hall international series
in civil engineering and engineering mechanics (Vol. 6). Prentice-Hall civil engineering and
engineering mechanics series.
Kutter, B. L., Moore, M., Hakhamaneshi, M., & Champion, C. (2015). Rationale for shallow
foundation rocking provisions in ASCE 41-13. Earthquake Spectra In-Press.
doi:http://dx.doi.org/10.1193/121914EQS215M
Laudon, A. C. D. (2013). Incorporating Time Domain Representation of Impedance Functions in
Nonlinear Hybrid Modeling by Analysis of Impedance Function Time-Domain
Transformation. University of Toronto.
Lubliner, J. (2005). Plasticity theory (Revised ed.). Courier Dover Publications.
Lysmer, J., & Kuhlemeyer, R. L. (1969). Finite dynamic model for infinite media. Journal of the
Engineering Mechanics Division, ASCE, 95, 859–877.
Mahsuli, M., & Ghannad, M. A. (2009). The effect of foundation embedment on inelastic response
of structures. Earthquake Engineering & Structural Dynamics, 38, 423–437.
doi:10.1002/eqe.858
Mylonakis, G., Nikolaou, S., & Gazetas, G. (2006). Footings under seismic loading: Analysis and
design issues with emphasis on bridge foundations. Soil Dynamics and Earthquake
Engineering, 26(9), 824–853. doi:10.1016/j.soildyn.2005.12.005
Nakamura, N. (2006a). A practical method to transform frequency dependent impedance to time
domain. Earthquake Engineering and Structural Dynamics, 35(2), 217–231.
doi:10.1002/eqe.520
Nakamura, N. (2006b). Improved methods to transform frequency-dependent complex stiffness to
time domain. Earthquake Engineering and Structural Dynamics, 35(8), 1037–1050.
doi:10.1002/eqe.570
163
Paolucci, R., Shirato, M., & Yilmaz, M. T. (2008). Seismic behaviour of shallow foundations :
Shaking table experiments vs numerical modelling, (November 2007), 577–595.
doi:10.1002/eqe.773
Pastor, M., Zienkiewicz, O. C., & Chan, A. H. . (1990). Generalized Plasticity and the Modelling
of Soil Behavior. International Journal for Numerical and Analytical Methods in
Geomechanics, 14(August 1989), 151–190.
Wolf, J., & Song, C. (1996). To radiate or not to radiate. Earthquake Engineering & Structural
Dynamics, 25, 1421–1432.
Zhang, J., & Tang, Y. (2007). Radiation damping of shallow foundations on nonlinear soil
medium. In 4th international conference on Earthquake Geotechnical Engineering.
Thessaloniki, Greece.
164
Appendices A
Wolf and Song have expressed dynamic stiffness function by separating the ability in the soil to
restore its original position while the soil experiences inertial force from ground excitation as
shown in the Eq. (5.2.1).
ÍÈ�&'Î = Í�ÜÎ − >Í�ÜÎ (5.2.1)
Where Kc is represented as static stiffness and Mc is the mass matrices of the cell. When this
expression is applied to the finite element mesh as shown in the Figure A.0.1 below, the formula
is now expressed with radial co-ordinates r at the interior boundary of the soil medium.
Figure A.0.1. Finite element cells of unbounded medium (Wolf & Song, 1996)
With static stiffness and mass matrices of the cell,
Í�ÜÎ = 67]7�z> : ]]7;�¤�z> Í��{Î (5.2.2)
Í�ÜÎ = )7]7� : ]]7;�¤� Í��{Î (5.2.3)
When KC is multiplied by displacement matrix, it will show elastic restoring force and when MC
is multiplied by displacement amplitude it will calculate inertial force of the cell. With these
expressions the dynamic impedance function can be rewritten as the following.
165
ÍÈÜ&-'Î = 67]7�z> : ]]7;�¤�z> &Í��{Î − @>Í��{Î' (5.2.4)
The variable in front of mass matrix, a, is the dimensionless frequency and it shows relative
contribution of inertial force and elastic resorting force within the dynamic matrix impedance
function.
@ = ��7 ]{z#�>�¤&�> ']7z#�>�¤&�> ' ¡ (5.2.5)
Where ro is the 1st cell radius illustrated as structure-medium interface and r is the radial co-ordinate
of the mesh onwards from ro as shown in Figure A.0.1. This dimensionless frequency is used to
describe conditions when the elastic restoring force or inertial force dominate within the soil.
Based on the exponential of the variable r, the expression (1-g/2+m/2) plays a significant role in
determining whether radiation damping effects exist in the finite element model with unbounded
medium.
1 − #�2� + #S2 � > 0 (5.2.6)
In the expression above, a →∞ for r→ ∞ and inertial force will always dominate the restoring
force. On the other hand,
1 − #�2� + #S2 � < 0 (5.2.7)
The restoring force will always dominate the inertial force for r→ ∞. Lastly, for the case where
1 − #�2� + #S2 � = 0 (5.2.8)
Elastic force will dominate for small frequencies while inertial force will dominate for large
frequencies of excitation. This is defined as radiation criterion which applies for unbounded
medium. The soil conditions and settings are explained through the parameters g and m. For
166
instance, homogeneous elastic half-space model is defined with g=m=0 where radiation criterion
implies radiation damping will occur for all frequencies. Also, when g=0 it is considered
homogeneous case and g =1, it is considered linear increase of G.
Further derivations are made in out-of-plane motion, but the main conclusions remain consistent
with respect to the relationship between restoring force, inertial force, and magnitude of frequency
to the radiation damping effects.
Therefore, taking the fundamental concepts of cut-off frequency and how it was derived, this will
assist in understanding the design of bridge piers as discussed before with various parametric study
carried out to find the influence of radiation damping effects in different conditions. Eliminating
radiation damping in the analysis may cause severe consequences for heavy foundations oscillating
vertically or horizontally.
167
Appendices B
Comparison of macroelement analysis using MATLAB and Code Aster are provided in this
Appendix. For each of the case, results are presented in this format:
a) Load history in three degrees of freedom (vertical, horizontal, and rotational)
b) Bounding surface plot (3D dimension, and other plane direction)
c) Results (linear elastic and nonlinear analysis results)
1) Vertical+Horizontal+moment cyclic load
Vertical load = 20.0 MN, Horizontal load = ± 5.0 MN, Moment load = ± 3.0 MNm
0 10 20 30 40 50 600
5
10Vertical Load
time step
Forc
e (
MN
)
0 10 20 30 40 50 60-5
0
5Horizontal Load
time step
Forc
e (
MN
)
0 10 20 30 40 50 60-5
0
5Moment Load
time step
Mom
ent
(MN
m)
168
-1
-0.5
0
0.5
1
-0.2
-0.1
0
0.1
0.2
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
QN
QV
QM
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
QV
QM
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
QN
QV
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
QN
QM
169
0 0.5 1 1.5 2 2.5 3 3.5
x 10-3
0
2
4
6
8
10
Displ (m)
Forc
e (
MN
)[ELASTIC Vertical] Displ vs. Force
Code aster
MATLAB
-2 -1 0 1 2
x 10-3
-5
0
5[ELASTIC Horizontal] Displ vs. Force
Displ (m)
Forc
e (
MN
)
-1.5 -1 -0.5 0 0.5 1 1.5
x 10-4
-4
-2
0
2
4[ELASTIC Moment] rotation vs. Force
Rotation (theta)
Mom
ent
(MN
m)
0 1 2 3 4 5 6
x 10-3
0
2
4
6
8
10
Displ (m)
Forc
e (
MN
)
[Vertical] Displ vs. Force
Code aster
MATLAB
-2 -1 0 1 2 3 4
x 10-3
-5
0
5[Horizontal] Displ vs. Force
Displ (m)
Forc
e (
MN
)
-1 0 1 2 3 4 5 6
x 10-4
-4
-2
0
2
4[Moment] rotation vs. Force
Rotation (theta)
Mom
ent
(MN
m)
170
2) Vertical upwards
Vertical load = -20.0 MN, Horizontal load = 0.0 MN, Moment load = 0.0 MNm
0 5 10 15 20 25 30 35 40 45-20
0
20Vertical Load
time step
Forc
e (
MN
)
0 5 10 15 20 25 30 35 40 450
1
2x 10
-15Horizontal Load
time step
Forc
e (
MN
)
0 5 10 15 20 25 30 35 40 45-1
0
1Moment Load
time step
Forc
e (
MN
)
-1
-0.5
0
0.5
1
-0.2
-0.1
0
0.1
0.2
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
QN
QV
QM
171
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
QN
QV
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
QN
QM
-7 -6 -5 -4 -3 -2 -1 0
x 10-3
-20
-15
-10
-5
0
5
Displ (m)
Forc
e (
MN
)
[ELASTIC Vertical] Displ vs. Force
Code aster
MATLAB
0 1 2 3 4 5
x 10-19
0
0.5
1
1.5
2x 10
-15 [ELASTIC Horizontal] Displ vs. Force
Displ (m)
Forc
e (
MN
)
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1[ELASTIC Moment] rotation vs. Force
Rotation (theta)
Mom
ent
(MN
m)
-0.012 -0.01 -0.008 -0.006 -0.004 -0.002 0-20
-15
-10
-5
0
5
Displ (m)
Fo
rce
(M
N)
[Vertical] Displ vs. Force
Code aster
MATLAB
0 1 2 3 4 5 6 7
x 10-18
0
0.5
1
1.5
2x 10
-15 [Horizontal] Displ vs. Force
Displ (m)
Fo
rce
(M
N)
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1[Moment] rotation vs. Force
Rotation (theta)
Mo
me
nt
(MN
m)
172
Appendices C
This Appendix contains information regarding influence vector applied to recursive parameter
model by Nakamura (2006) to replicate the dynamic impedance function in different degrees of
freedom.
In order to run time history analysis, mass, stiffness and damping terms with user-defined DOF is
constructed as previously discussed. Using the user-defined matrices, then equation of motion can
be expressed as shown below.
�T¾ + �T- + �T + » = d (C.1)
Where F is the restoring force obtained from the far-field soil, and p is the excitation force
subjected to the structure.
Using the frequency dependent characteristics of soil, Nakamura’s coefficient allows users to
obtain the impulse response in time domain using past displacement and velocity.
Nakamura’s original derivation contains SDOF structure model with soil node attached at the base.
This example follows the same routine as the original code but has three dofs that have their own
respective impedance functions. The program is written so that once the Nakamura’s coefficients
are obtained, the instantaneous mass, spring and damping terms are added to the original dynamic
equation of motion as shown below:
A0=[aCon(1)]*infVE;
B0=[bCon(1)]*infVE;
C0=[cCon(1)]*infVE;
K1=K+A0;
C1=C+B0;
M1=M+C0;
173
Due to the different frequency impedance function of soil in in each degrees of freedom, the
influence vectors are created in each degrees of freedom in order to assign its distinct impedance
functions. Then, the influence vectors are:
infVEx =
�����0 0 0 0 00 0 0 0 00 0 1 0 00 0 0 0 00 0 0 0 0���
��
infVEy =
�����0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 1 00 0 0 0 0���
��
infVEr =
�����0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 1���
��
The a, b, and c convolution terms are updated using Nakamura’s coefficient in x, y, and r degrees
of freedom.
Figure B.1. Soil impedance function in x – direction with Nakamura’s coefficient terms
174
Figure B.2. Soil impedance function in y – direction with Nakamura’s coefficient terms
175
Figure B.3. Soil impedance function in r (rotation) – direction with Nakamura’s coefficient
terms
The command line that are used to generate these curves are:
[Kix,Cix, Mix, PLOT, Snew] = NakamuraCoef(fo,Horiz_imp, 1);
[Kiy,Ciy, Miy, PLOT, Snew] = NakamuraCoef(fo,Vert_imp, 1);
[Kir,Cir, Mir, PLOT, Snew] = NakamuraCoef(fo,rot_imp, 1);
Which means that the mass, stiffness, and damping terms are generated with each degrees of
freedom accordingly. (For instance Kix refers to Nakamura’s stiffness terms that is obtained
through soil impedance function in x-direction)
After all the terms are obtained, then mass, stiffness, and damping terms are extracted to
convolution coefficients.
176
for i=1:length(Kix)
aCon(i,1)=Kix(i);
aCon(i,2)=Kiy(i);
aCon(i,3)=Kir(i);
end
for i=1:length(Cix)
bCon(i,1)=Cix(i);
bCon(i,2)=Ciy(i);
bCon(i,3)=Cir(i);
end
for i=1:length(Mix)
cCon(i,1)=Mix(i);
cCon(i,2)=Miy(i);
cCon(i,3)=Mir(i);
end
This generates convolution terms with each columns representing x, y, and r dof. For instance,
û@�\A~�\A��\Aü = ã�)� �)$ �)"�)� �)$ �)"�)� �)$ �)"ä (C.2)
Then, each convolution terms are multiplied to influence vectors.
A0=[aCon(1,1)]*infVEx+[aCon(1,2)]*infVEy+[aCon(1,3)]*infVEr;
B0=[bCon(1,1)]*infVEx+[bCon(1,2)]*infVEy+[bCon(1,3)]*infVEr;
C0=[cCon(1,1)]*infVEx+[cCon(1,2)]*infVEy+[cCon(1,3)]*infVEr;
177
This is where the first term of the coefficients become the instantaneous mass, damping and
stiffness terms and gets added to the original mass, damping and stiffness matrix respectively.
K1=K+A0;
C1=C+B0;
M1=M+C0;
After all of this is updated, then Newmark time integration scheme is used. Because each degrees
of freedom has different impedance functions, same approach is used at each time step of the
analysis.
The last part of the code calculates inter-storey drift at target node, but since the degrees of freedom
in soil domain is x, y, and rotation, inter-storey drift displacement is not calculated unless the
degrees of freedom is greater than 3 (1 – x degrees of freedom, 2-y degrees of freedom, 3 – r
degrees of freedom) in this specific problem set.
After all is updated, the time history analysis of soil system, assuming the excitation is applied to
the lumped mass, and the lateral output of far-field response is obtained as shown in the report.
178
Appendices D
Appendix D contains alternative approach to obtain dynamic impedance function of soil when
force-displacement history profile is provided at each harmonic excitation frequency. In the paper,
the original approach in obtaining dynamic impedance function is discussed in Section 2.2 of the
analysis where the dynamic stiffness term is function of the slope of the force and displacement
plot, and damping term is function of the energy dissipated in the hysteretic loop of the force
displacement plot.
This approach considers theoretically derived transfer function using steady-state equation of
motion with complex stiffness. This allows the verification of methodology used for adding the
inverse matrix of structure and soil. Previously, the transfer function has been theoretically derived
using horizontal and rotational degrees of freedom. The transfer function is derived as: (when
horizontal and moment is taken into consideration because the vertical degrees of freedom is not
coupled):
'�==&' �=�&'�=�&' ���&'( �"�# � + '�==&' �=�&'�=�&' ���&'( '"�-#- ( = ��7�A&f'0 � (D.1)
If the transfer function is derived as TF = u/P, then isolating ux and dividing the equation by Po to
obtain the following simplified expression:
�» = "��7 = 1'�&& + �&& − &�&� + �&�'>��� + ��� (
(D.2)
Then, a ground motion in frequency domain can be multiplied directly to this transfer function to
obtain the displacement, u, in frequency domain. Inverse Fourier transformation can then be used
to convert this displacement to time domain analysis.
Another way to obtain transfer function is to use matrix inverse using the soil impedance function
using the force displacement plot. The impedance matrix can be formulated as:
Soil impedance matrix (including vertical, horizontal and rotational dof of foundation)
179
È)�� = �&& + �&& �&� + �&� 0�&� + �&� ��� + ��� 00 0 �%% + �%%� (D.3)
This impedance matrix represents dynamic stiffness of soil. Thus, the transfer function, which was
previously defined as u/Po, would be the inverse of this matrix, Simp.
�» = È)��z{ = �&& + �&& �&� + �&� 0�&� + �&� ��� + ��� 00 0 �%% + �%%�z{
(D.4)
Then, as previously mentioned before, the matrix will be multiplied to the ground acceleration in
frequency domain resulting in frequency domain displacement which can be converted to time
domain using inverse Fourier transformation. The two following approach show exactly the same
results in horizontal excitation of the foundation. Comparison with time domain analysis using
OpenSees is also provided as shown in figure blow.
Figure D.1. Load applied at foundation comparison with theoretical transfer function,
inverse soil impedance and OpenSees results.
-8.00E-05
-6.00E-05
-4.00E-05
-2.00E-05
0.00E+00
2.00E-05
4.00E-05
6.00E-05
0 5 10 15 20 25 30 35 40
dis
pla
cem
ent
(m)
time (s)
Load applied at foundation
FFT_using Transfer function
FFT_using inverse soil impedance
Opensees applied @ foundation
180
The purpose of this comparison is to check whether the inverse of the soil impedance is the same
as the theoretically derived transfer function. Once this is checked, we know that the transfer
function of soil is simply the inverse of the impedance functions at each dof.
The structural component to the impedance function is (only structural component):
�"¾ + �"- + �" = �7 sin&f' (D.5)
Once the derivative of displacements (u=sin(wt)) are substituted to the equation, the equation of
motion is simplified to:
−&>�'" + &�'" + �" = �7 (D.6)
The stiffness of structure is then:
�7" = −&>�' + &�' + � (D.7)
Thus, adding the structural stiffness and soil impedance function would be
Impedance function matrix overall:
Èf] + �\e)�� = −&>�' + &�' + � + È)��
Èf] + �\e)�� = −&>�' + &�' + �+ �&& + �&& �&� + �&� 0�&� + �&� ��� + ��� 00 0 �%% + �%%
� (D.8)
Then, the transfer function of the overall stiffness would be:
�» = �Èf] + �\e)���z{= ,−&>�' + &�' + �+ �&& + �&& �&� + �&� 0�&� + �&� ��� + ��� 00 0 �%% + �%%
�.z{ (D.9)
181
Adding two components to the respective global dof then would generate displacements at each
degrees of freedom. This approach is verified with FE model as shown in the paper.
182
Appendices E
From the FE principle, the analysis aims to converge a function with time step ut+Δt to zero as
shown in Eq. (5.2.1).
6&"¦¤>¦' = 0 (5.2.1)
For a force-controlled analysis, this function would be force increment of the analysis; for a
displacement-controlled analysis, this function would be displacement increment the analysis
wants to converge. In general notation, the predicted variable, u, is then formulated as shown in
Eq. (5.2.2).
")¤{f+Δf = ")f+Δf − ¼x6�")f+Δf�x" ¿z{ 6&")f+Δf' (5.2.2)
Then, the increment of variable u is calculated based on the tangent stiffness of the function G
which the analysis wants to converge. This is shown in Eq. (5.2.3).
ý")¤{ = ")¤{f+Δf − ")f+Δf = − ¼x6�")f+Δf�x" ¿z{ 6�")f+Δf� = −��"¦¤>¦�−16&"¦¤>¦'
(5.2.3)
Then, this this increment is added onto the function G until this function becomes zero. Figure
B.0.1 shows the illustration of this solution algorithm in a graph.
183
Figure B.0.1. Illustration of Newton-Raphson nonlinear solution algorithm
In the context of macroelement, force controlled analysis can be used as an example to illustrate
how the Newton-Raphson nonlinear solution algorithm is implemented. The function G as shown
in Eq. (5.2.1) is presented as a function the analysis wants to converge to zero. Then, the target
force increment the analysis aims to converge would be shown in Eq. (5.2.4) where the force
history is provided by the user.
Δ» = »)¤{ − ») (5.2.4)
Then, after the elastic stiffness and plastic stiffness is updated, then new force increment can be
calculated with this updated stiffness. This is expressed as the following equation.
6 = �IJ¤�J ∗ Δ! − Δ» (5.2.5)
Where the Δq is the displacement increment calculated with linear elastic stiffness analysis. Then,
the corrected increment of displacement would be calculated using Eq. (5.2.6).
ΔB! = −6z{ ∗ Δ» (5.2.6)
Then, the displacement is updated with this correction of increment as shown in Eq. (5.2.7).
Δ! = Δ! + ΔB! (5.2.7)
184
Through iteration of this updated displacement, the convergence of the function G becomes zero.
This means that using the nonlinear solution algorithm, corrected displacement is calculated with
the plastic stiffness from the force increment input.