Simplified Model for Nonlinear Frequency-Dependent Soil with … · 2016-03-15 · Simplified Model...

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


plot (Case A-2) .............................................................................................................................. 51

Figure 2.19. Bounding surface plot for macroelement (Case B-1) 3D, QV-QN , QM-QN view ..... 52


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


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


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


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


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

)


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


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.


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

)


Code aster

MATLAB

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0

x 10-18

-3

-2

-1

0

1x 10


Displ (m)

Forc

e (

MN

)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5


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

)


Code aster

MATLAB

-5 -4 -3 -2 -1 0

x 10-17

-3

-2

-1

0

1x 10


Displ (m)

Forc

e (

MN

)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5


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

)


Code aster

MATLAB

-5 0 5 10 15 20

x 10-4

-1

0

1

2

3


Displ (m)

Forc

e (

MN

)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5


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

)


Code aster

MATLAB

-0.5 0 0.5 1 1.5 2 2.5

x 10-3

-1

0

1

2

3


Displ (m)

Forc

e (

MN

)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5


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

)


Code aster

MATLAB

0 0.5 1 1.5 2 2.5 3

x 10-3

0

2

4

6


Displ (m)

Forc

e (

MN

)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5


Rotation (theta)

Mom

ent

(MN

m)

0 2 4 6 8

x 10-3

0

5

10

15

20

Displ (m)

Forc

e (

MN

)


Code aster

MATLAB

0 1 2 3 4 5 6 7

x 10-3

0

2

4

6


Displ (m)

Forc

e (

MN

)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5


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' + Â ��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'� + ,Â Ã� ∗ "�f − f�� z{�Æ{ + Â �� ∗ "- �f − f�� z{

�Æ{ . = Â Ã� ∗ "�f − f�� z{

�Æh + Â �� ∗ "- �f − f�� z{�Æh

(2.3.4)

Then, the impedance function can be expressed as shown in (2.3.5).

È&' = &Ãh + ∗ �h' + ,Â Ã� ∗ �z)É¦Ê �z{�Æ{ + ∗ Â �� ∗ �z)É¦Ê �z{

�Æ{ . = Â Ã� ∗ �z)É¦Ê �z{

�Æh + ∗ Â �� ∗ �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 + ) Â �A#)� ∗ �� z{�Æh

− Â �A#)� ∗ Ã� �z{�Æh + ) Â �\�#)� ∗ �� 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,Êª = Ë �\�#)�−�A#)�Ï

Í�Ü̅Î = , u�Ü̅Þ,gy … u�Ü̅Þ,ºßÞy… … …u�Ü̅M,gy … u�Ü̅M,ºßÞy., ©�Ü̅à,Êª = 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' + , ∗ Â �� ∗ �z)É¦Ê �z>�Æ{ + Â 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 =ÐÑÒÑÓÂ �\�#)� ∗ Ã� �z{

�Æh + ) Â �A#)� ∗ �� z>�Æh − )>Sh

− Â �A#)� ∗ Ã� �z{�Æh + ) Â �\�#)� ∗ �� 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,Êª = Ë �\�#)�−�A#)�Ï

Í�Ü̅åÎ = ,u�Ü̅Þ,gy … u�Ü̅Þ,ºßry… … …u�Ü̅º,gy … u�Ü̅º,ºßæy., ©�Ü̅à,Êª = 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 )¤{ + Â �� ∗ "- )¤{z� �z{�Æ{ + Â �� ∗ ")¤{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)¤{ = Â 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 + �"- + �" = ��)É¦ (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


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



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


20m by 40m 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


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

)


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




G Shear modulus (KN/m3) 1300*Co = 39000

E Elastic modulus (KN/m) 2*G*(1+v) =

2*39000*(1+0.45)

=113100


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)


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


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


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



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


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

)


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


�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)


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

)


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





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


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)


matlab_impedance

131

Table 12. Structural properties of FEM model for Kobe excitation


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�

+ Â �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

0.2

0.4

0.6

0.8

1

1.2


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


0

0.2

0.4

0.6

0.8

1

1.2


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


0

0.2

0.4

0.6

0.8

1

1.2


0

0.2

0.4

0.6

0.8

1

1.2


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

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


0

0.2

0.4

0.6

0.8

1

1.2


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


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


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

)


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

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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


Displ (m)

Forc

e (

MN

)

-1.5 -1 -0.5 0 0.5 1 1.5

x 10-4

-4

-2

0

2


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

)


Code aster

MATLAB

-2 -1 0 1 2 3 4

x 10-3

-5

0


Displ (m)

Forc

e (

MN

)

-1 0 1 2 3 4 5 6

x 10-4

-4

-2

0

2


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

)


Code aster

MATLAB

0 1 2 3 4 5

x 10-19

0

0.5

1

1.5

2x 10


Displ (m)

Forc

e (

MN

)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5


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)


Code aster

MATLAB

0 1 2 3 4 5 6 7

x 10-18

0

0.5

1

1.5

2x 10


Displ (m)

Fo

rce

(M

N)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5


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


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.

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