Incorporating Time Domain Representation of Impedance

Incorporating Time Domain Representation of Impedance Functions into Nonlinear Hybrid Modelling

by

Alexander Carlos Duarte Laudon

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

Civil Engineering Department University of Toronto

© Copyright by Alexander Carlos Duarte Laudon 2013

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Analysis of Impedance Function Time-Domain Transformation

Alexander Carlos Duarte Laudon

Master of Applied Science

Civil Engineering Department

University of Toronto

2013

Abstract

A number of methods have been proposed that utilize the time domain transformations of the

frequency dependent impedance functions to perform time-history analysis of structures

accounting for soil-structure interaction (SSI). Though these methods have been available in

literature for a number of years, this study is the first to rigorously examine the limitations and

advantages of these methods in comparison to one another. These methods contain certain

stability issues that required investigating which lead to the formation of an analysis procedure

that assesses a transform method’s stability.

The general applicability of these methods was demonstrated by utilizing them to model

increasingly sophisticated reference problems. Additionally the suitability of these methods to

being incorporated into hybrid simulations of nonlinear inelastic structures considering soil-

structure interaction was confirmed. The modelling of a nonlinear structure considering soil-

structure interaction is an improvement over the most common modelling strategies that model

solely linear-elastic behaviour.

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Acknowledgments

I would like to express my sincerest thanks to my advisor, professor Oh-Sung Kwon, whose

continued support and patience was invaluable for the preparation this thesis. I am grateful to

have had the opportunity to work and learn with him. I would like to acknowledge also the aide

provided by Dr. Naohiro Nakamura. Through our correspondence he was able to help clarify

various issues and provide me with a greater understanding of the methods discussed in this

thesis.

Of course it goes without saying that none of this work could have been accomplished without

the love and support of my parents and my sister, Joanna. Also I must acknowledge my

colleagues and officemates, including but not limited to John Kabanda, Aleksandar Kuzmanovic,

and Islam Mazen, that made my graduate study all the more enjoyable, in addition to assisting

me in some of my struggles. Thank you all for making this experience so rewarding and

fulfilling.

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Table of Contents

Table of Contents ........................................................................................................................... iv

List of Tables ............................................................................................................................... viii

List of Figures ................................................................................................................................ ix

Chapter 1 Introduction .................................................................................................................... 1

1.1. Overview and Objectives .................................................................................................... 1

1.2. Literature Review ................................................................................................................ 3

1.3. Organization of Thesis ........................................................................................................ 6

Chapter 2 Impedance Functions ...................................................................................................... 9

2.1. Impedance Function Definition .......................................................................................... 9

2.2. Producing Impedances ...................................................................................................... 10

2.2.1. Analytical Impedances .......................................................................................... 11

2.2.2. Finite Element Models .......................................................................................... 15

2.2.3. Algebraic Formulation Method ............................................................................. 15

2.2.4. Numeric Formulation ............................................................................................ 20

Chapter 3 Time Domain Transformation Methods ....................................................................... 25

3.1. Evaluation Procedure of the Transformation Methods ..................................................... 27

3.1.1. Procedure Organization ........................................................................................ 27

3.1.2. Problem Statement ................................................................................................ 28

3.1.3. Impedance Functions ............................................................................................ 31

3.1.4. Reproducing the Impedance Functions ................................................................. 33

3.1.5. Combined Convolution and Newmark Time Integration ..................................... 36

3.1.6. Convolution Time Step ......................................................................................... 37

3.1.7. Stability Assessment ............................................................................................. 38

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3.2. Inverse Fourier Transformation (Wolf, 1985) .................................................................. 43

3.2.1. Coefficient Formulation ........................................................................................ 44

3.2.2. Impedance Function Parameters ........................................................................... 45

3.2.3. Reproducing the Impedance Function .................................................................. 46

3.2.4. Simulation Results ................................................................................................ 48

3.2.5. Stability ................................................................................................................. 49

3.3. Nakamura Model .............................................................................................................. 52

3.3.1. Model Definition ................................................................................................... 52


3.3.3. Impedance Function Parameters ........................................................................... 55



3.3.6. Stability Assessment ............................................................................................. 59

3.4. Şafak Model (2005) .......................................................................................................... 62




3.4.4. Stability ................................................................................................................. 72

3.5. Comparison Conclusions .................................................................................................. 75

Chapter 4 Improving Stability ...................................................................................................... 77

4.1. Evaluation of the Examined Models’ Susceptibility to Non-Causal Impedance

Functions ........................................................................................................................... 77

4.1.1. Causal FFT Treatment .......................................................................................... 80

4.1.2. Nakamura Model Based on Partial Data (2008) ................................................... 83

4.2. Method to Overcome Negative Instantaneous Mass Solutions ........................................ 92

4.2.1. Impedance Expansion Procedure .......................................................................... 92

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4.2.2. Example ................................................................................................................ 93

4.3. Conclusion ........................................................................................................................ 95

Chapter 5 Multiple Degree-of-Freedom Interface Validation ...................................................... 97

5.1. Problem Statement ............................................................................................................ 97

5.1.1. Structure Parameters ............................................................................................. 98

5.1.2. Soil Parameters ..................................................................................................... 98

5.1.3. Soil Model ............................................................................................................. 98

5.1.4. Impedance Functions .......................................................................................... 100

5.1.5. External Force ..................................................................................................... 102

5.2. Reference Model ............................................................................................................. 102

5.3. Nakamura Model ............................................................................................................ 103

5.4. Reproducing Impedance Function .................................................................................. 104

5.5. Structure Response Comparison ..................................................................................... 106

Chapter 6 Hybrid Simulation Validation .................................................................................... 109

6.1. Summary ......................................................................................................................... 109

6.2. Problem Description ....................................................................................................... 109

6.3. Hybrid Simulation ........................................................................................................... 112

6.3.1. REMUS Pre-processor ........................................................................................ 114

6.3.2. Algorithm Alterations ......................................................................................... 118

6.3.3. Simulation Procedure .......................................................................................... 120

6.4. Structure Response Comparison ..................................................................................... 122

Chapter 7 Conclusion .................................................................................................................. 128

7.1. Summary ......................................................................................................................... 128

7.2. Contribution .................................................................................................................... 129

7.3. Limitations ...................................................................................................................... 131

7.4. Future Studies ................................................................................................................. 133

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References ................................................................................................................................... 136

Appendix A ................................................................................................................................. 141

Appendix B ................................................................................................................................. 145

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List of Tables

Table 2.1 Ratios and Variable Values Used in Soil Model Example ........................................... 12

Table 2.2 Partial List of Journals that Provide Generated Impedance Functions ......................... 14

Table 3.1 Structural Model Properties .......................................................................................... 31

Table 3.2 Soil Model Properties for the Two Analysis Cases ...................................................... 31

Table 5.1 Natural Frequency of the Reference Structure ............................................................. 98

Table 5.2 Natural Frequencies of the Soil Validation Structure ................................................. 100

Table 5.3 Comparison of the OpenSees model to Literature Values (Gazetas, 1983) ............... 100

Table 6.1 Structure Properties ..................................................................................................... 110

Table 6.2 Description of the REMUS output files ...................................................................... 120

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List of Figures

Figure 2.1 Disc Foundation on Layered Soil Medium (Luco, 1974) ........................................ 12

Figure 2.2 Coefficient Functions for the Horizontal Impedance (Luco, 1974) ......................... 13

Figure 2.3 Plot of Displacement against Force for a sinusoidal loading on a soil system ........ 21

Figure 3.1 The three different models analyzed in this validation example ............................. 29

Figure 3.2 Time History and Frequency Content of the Ground Acceleration (Kobe 1995) .... 30

Figure 3.3 Impedance Function of the Stiff Soil Model displayed a) in a large frequency range

and b) a shorter frequency range .............................................................................. 32

Figure 3.4 Impedance Function of the Soft Soil Model ............................................................ 33

Figure 3.5 The concept of impulse force response .................................................................... 43

Figure 3.6 Reproduced Impedance of the Stiff Soil Case for the Inverse Fourier Transform .. 47

Figure 3.7 Reproduced Impedance of the Soft Soil Case for the Inverse Fourier Transform ... 47

Figure 3.8 Displacement Responses of the Total and inverse Fourier Models of the Validation

Example undergoing the Kobe Earthquake Loading. .............................................. 48

Figure 3.9 Stability Analysis Curves of the Inverse Fourier Model for both Stiff and Soft Soil

Impedance Examples ............................................................................................... 51

Figure 3.10 Reproduced Impedance of the Stiff Soil Case for the Nakamura Method .............. 56

Figure 3.11 Reproduced Impedance of the Soft Soil Case for the Nakamura Method ............... 57

Figure 3.12 Total Displacement Responses of the Total and Nakamura Models of the Stiff Soil

Example undergoing the Kobe Earthquake Loading ............................................... 58

Figure 3.13 Total Displacement Responses of the Total and Nakamura Models of the Soft Soil

Example undergoing the Kobe Earthquake Loading ............................................... 58

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Figure 3.14 Stability Analysis Curve of the Nakamura Model for the Stiff Soil Example ......... 61

Figure 3.15 Stability Analysis Curve of the Nakamura Model for the Soft Soil Example ......... 62

Figure 3.16 Reproduced Impedance of the Stiff Soil Case for the Şafak Method using

Optimization Scheme ............................................................................................... 68

Figure 3.17 Second Reproduced Impedance of the Stiff Soil Case for the Şafak Method using

Optimization Scheme ............................................................................................... 69

Figure 3.18 Reproduced Impedance of the Stiff Soil Case for the Şafak Method using Z-

Transform Procedure ............................................................................................... 70

Figure 3.19 Displacement Responses of the Total and Şafak Models of the Stiff Soil Example

undergoing the Kobe Earthquake Loading. ............................................................. 71

Figure 3.20 Displacement Responses of the Total and Şafak Models of the Soft Soil Example

undergoing the Kobe Earthquake Loading. ............................................................. 71

Figure 3.21 Stability Analysis Curve of the Şafak Model for the Stiff Soil Example ................ 74

Figure 3.22 Stability Analysis Curve of the Şafak Model for the Soft Soil Example ................. 75

Figure 4.1 Examples of Causal and Non-Causal Time Domain Functions ............................... 78

Figure 4.2 Single Cycle of a Non-Causal Function ................................................................... 79

Figure 4.3 Proposed Procedure (Hayashi & Katukura, 1990) ................................................... 81

Figure 4.4 Non-causal Impedance Function Example .............................................................. 88

Figure 4.5 Nakamura Transform Functions of Non-Causal Example ....................................... 89

Figure 4.7 Reconstructed Impedance Based on the Transform Time-Series ............................ 90

Figure 4.6 Nakamura Transform Time-Series Based on Partial Impedance Data .................... 90

Figure 4.8 Total Displacement Response of Structure Simulated Using Frequency Domain

Analysis and Nakamura Transform Model Using Partial Data ............................... 92

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Figure 4.9 Example Impedance ................................................................................................. 93

Figure 4.10 Extension of the Impedance Function In Order to Ensure Positive Mass in

Nakamura Models .................................................................................................... 94


Analysis and Nakamura Transform Model Utilizing Impedance Expansion

Procedure ................................................................................................................. 95

Figure 5.1 FEM Model of the Soil Domain Used in the Impedance Generation ...................... 99

Figure 5.2 a) Horizontal Dynamic Stiffness b) Rocking Dynamic Stiffness c) Vertical Dynamic

Stiffness d) Coupled Rocking-Horizontal Dynamic Stiffness ............................... 101

Figure 5.3 FEM Model of the Complete Validation Example Including Soil and Structure

Domains ................................................................................................................. 103

Figure 5.4 Model of the Structure Domain and Soil Domain Modelled Using Nakamura’s

Transformation Method ......................................................................................... 104

Figure 5.5 Reproduced Impedance Functions from Nakamura Transform Coefficients ........ 105

Figure 5.6 Total Displacement Response of Structure in the Horizontal Degree-of-freedom 107

Figure 5.7 Total Displacement Response of Structure in the Rotational Degree-of-freedom 107

Figure 5.8 Response of Foundation in the Horizontal Degree-of-freedom during a) complete

simulation duration b) between 6.5 and 9 seconds ................................................ 108

Figure 6.1 Model of Soil and Structure System ...................................................................... 111

Figure 6.2 Stress-Strain Relationship ...................................................................................... 112

Figure 6.3 Input File for REMUS ........................................................................................... 115

Figure 6.4 REMUS Graphic User Interface ............................................................................ 115

Figure 6.5 Impedance Comparison in REMUS ....................................................................... 117

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Figure 6.6 Stability Analysis Curve ........................................................................................ 118

Figure 6.7 Total Displacement Response of Topmost Node in Horizontal DOF ................... 123

Figure 6.8 Stress-Strain Curve of the Yielding Column at Topmost Fibre during Simulation124

Figure 6.9 Total Displacement Response of the Foundation in Horizontal DOF ................... 125

Figure 6.10 Relative Displacement Response of the Fixed Base Model and the Hybrid

Simulation including SSI ....................................................................................... 126

Figure 6.11 Total Displacement Response of Structure in Rotational Degree-of-Freedom ..... 126

Figure 6.12 Total Displacement Response of Foundation in Rotational Degree-of-Freedom .. 127

Figure 6.13 Near Field and Far Field Modelling (Tzong & Penzien, 1986) ............................. 135

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Chapter 1 Introduction

Several destructive earthquakes have recently befallen peoples from many parts of the world,

highlighting the vulnerability of existing infrastructure to strong earthquake motion. The 2011

Fukushima earthquake that affected the Fukushima Daiichi nuclear plant was an especially

unfortunate reminder that certain critical buildings should and must remain fully operational after

an earthquake disaster. Such a requirement would necessitate not only a meticulous design

procedure but also comprehensive and realistic simulations of the building and its environment

under a number of potential seismic activities. Though the dynamic modelling of building

elements and materials is quite sophisticated, the modelling of soil systems and the interaction

between the building and the soil is still a developing field that requires much exploration and

investigation.

The modelling of soil-structure interaction has long been considered an emerging and exciting

research field since no concise and universal procedure has yet been established as the most

appropriate or convenient means to model both the soil and structure domains together. In

particular the soil domain presents many challenges to researchers in creating a realistic and

appropriate model due to actual soil’s inhomogeneity, non-linear behaviour and semi-infinite

domain behaviour (Kausel, 2010). Nevertheless many procedures and methods have been

proposed to capture this soil-structure interaction; each possessing its own unique benefits and

complications. This study will explore and investigate methods that allow time-history analysis

of soil-structure interaction systems by transforming the soil domain’s frequency-dependent

impedance function into the time domain.

1.1. Overview and Objectives

Soil-structure interaction is a broad field of research that involves various types of analyses that

attempt to combine the response of a structure and soil system. Soil structure interaction’s broad

scope compels the field to be increasingly interdisciplinary as it draws concepts from soil and

structural mechanics, soil and structural dynamics, earthquake engineering, geophysics and

geomechanics, material science, computational and numerical methods and many other diverse

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technical disciplines (Kausel, 2010). This encompassing nature of soil-structure interaction

problems implores one to precisely delineate the scope and objective of their research.

The mechanics that describe soil-structure interaction can be divided into two distinct

interactions; inertial and kinematic interactions (Stewart et al., 1999). Kinematic interactions are

the deviation of the soil’s response to an earthquake originating at bedrock from the free-field

motion due to the presence of the structure’s foundation. This deviation is often described using

transfer functions relating the free-field motion to the motion at the interface when considering

the disturbance caused by the presence of the structure (Stewart et al., 1999). The inertial

interaction occurs as the structure responds to motion of the soil surface determined during the

kinematic interaction analysis (Wotherspoon, 2009). Base shears and moments develop at the

foundation of the structure due to the acceleration of the inertial mass of the structure

(Wotherspoon, 2009).

The focus of this thesis will solely be on the accurate modelling of the inertial interaction of the

soil and structure system. The structure is modelled conventionally but the soil system’s

contribution can be modelled in a number of ways depending on the analysis method. Avoiding

non-linear soil behaviour, which is a complicated and important research area in itself, much

research is still being conducted in capturing the frequency-dependent stiffness of linear-elastic

soil models. Most commonly this analysis is conducted in the frequency domain provided that

the impedance function (or dynamic stiffness) is available of the soil-foundation system (Wolf &

Obernhuber, 1985). These impedance functions express the response of ideally massless

foundations, resting on a compliant soil system, excited by a harmonic force applied directly

onto the foundation (Kausel, 2010). Though generating these impedance functions provides its

own unique challenges, once they are generated frequency domain analysis will

straightforwardly provide the linear-elastic inertial interaction response of a structure and its

foundation.

The linear-elastic analysis in the frequency domain nevertheless does limit the application of this

analysis tool. Most structures are expected to experience significant damage during a major

seismic event and the stiffness loss due to damage will alter the dynamic properties of the

structure such as the fundamental period. In the nuclear power industry this limitation is less

alarming since structures were intended to remain linear-elastic under most probable seismic

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incidences and the non-linearity of the soil during an earthquake was of greater concern. Any

non-linear degrading analysis must be accomplished in the time domain (Hayashi & Katukura,

1990), though approximate equivalent linear methods have been proposed to obtain more

accurate responses from the frequency domain analysis (Bolisetti, 2010). Since frequency

domain analysis is relatively straightforward, this approximate linear method is commonly used

in the nuclear industry where nonlinear soil behaviour is of interest (Lysmer, et al., 1981)

The advent of nonlinear finite element modelling allowed researchers to run simulations of soil-

structure interaction systems in the time domain considering a greater degree of material

nonlinearity and varied geometric configurations. These simulations continue to be sophisticated

and computationally expensive leading most analysis in industry and to some extent in academia

to rely on purely linear analyses. There exists however a branch of modelling techniques that

allow for linear-elastic soil models represented by impedance functions to be incorporated into

time history analyses. The various methods transform these impedance functions into time

domain components that can be readily incorporated with a nonlinear inelastic structure model

allowing for the inertial interaction to be calculated for a realistic structure.

In this thesis these transformation methods will be investigated in order to establish their

limitations and strengths when attempting to model the inertial interaction between the soil and

structure. Other additional techniques will be introduced, both original and from the literature,

that attempt to eliminate some of the issues involved with these models. Ultimately the objective

of the research is to demonstrate the capabilities of these methods for practical time-history

analysis of soil and inelastic nonlinear structure systems. This is to be accomplished by

conducting a number of comprehensive validation examples that will demonstrate the capability

of these models in obtaining accurate and precise structural responses and in doing so provide the

procedure and steps involved in using these models.

1.2. Literature Review

Eighty years ago researchers began to realize that a structure and it’s the underlying soil

foundation should be considered jointly in dynamic analysis (Wang et al., 2011). This realization

was dawned upon when it was recognized that the underlying soil and the design of foundations

supporting vibrating machinery were affecting how much shaking workers in the machine’s

vicinity were experiencing (Gazetas, 1983). Research done on foundations under machine

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loading was eventually employed in analyzing foundations under seismic loading to consider

soil-structure interaction.

Investigations into the response of the soil, modelled as an elastic half-space, to a prescribed load

have been conducted since the beginning of the twentieth century but these researchers main

interests lied in problems within the field of geophysics concentrating on far field effects of

applied loading (Shah, 1968). For many years nevertheless researchers were developing more

novel and complex analytical models of the soil domain as an elastic half-space with varying

foundation layouts and soil configuration, with each paper resulting in a unique impedance

function for a different foundation-soil problem (Shah, 1968). These problems eventually

became so intricate that they would required numerical approximation in order to solve the

boundary problems and determine the impedance functions (Luco, 1974).

With increasing computer capabilities researchers have been able to determine the impedance

function of increasingly more complex and sophisticated soil systems. However the specificity

and idealized nature of these solutions meant they were seldom used in practical analysis of soil-

structure interaction problems. Furthermore with increasing computational power, researchers

later were able to develop sophisticated ad hoc finite element models by using any of the number

of modelling software suites available rather than having to master mathematical and numerical

integration techniques to determine a soil domain’s impedance function.

Nevertheless the availability of these impedance functions for varied soil and foundation

configuration facilitated linear soil-structure interaction analysis to be conducted by performing

frequency domain analysis (De Barros & Luco, 1990). Research into exploiting these readily

available impedance functions in time domain analysis was also pursued because of the

convenience and familiarity of engineers to time-history analysis. Impedance functions were

initially substituted using simple mass-spring-dashpot systems with rational procedures to define

the element parameters. The procedures adjusted the three parameters to roughly match the

resulting dynamic stiffness with the desired impedance. This discrete method captured the static

component effectively but it was soon recognized that such a frequency independent model could

not adequately represent the soil’s dynamic behaviour (Nakamura, 2006a). Similar models with a

greater number of components have been suggested in order to capture more sophisticated and

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peculiar impedance functions but each being developed for a specific foundation and soil type

(De Barros & Luco, 1990).

The parameters of these discrete models are typically determined partially based on the soil and

foundation properties, such as the shear modulus and foundation radius, rather than the

impedance function itself. On the other hand some models suggest the use of regression analysis

to ensure the impedance function is best represented (Wang et al., 2011). Since these models

only have a relatively small number of components they are especially computationally efficient,

which lends these models to real-time hybrid simulations (Wang et al., 2011) and allow for the

soil domain to be modelled in conjunction with a nonlinear inelastic structural model. However

these methods typically do not provide an appropriate means to model coupled dynamic

stiffnesses of embedded or any other foundation interface with multiple degrees-of-freedom.

Furthermore discrete models produce inaccurate results when reproducing impedance function

with strong frequency dependencies (Nakamura, 2012).

Methods were proposed that transform the frequency dependent impedance functions into time

domain and use this transform to determine the restoring force that the soil provides to the

foundation at each time step. These transform are time domain functions that are typically

generated using a mathematical transformation or another algebraic procedure. Since the data

contained in the impedance function is being transformed these methods generally exhibit

accurate representation of the soil domain.

The most straightforward procedure available for this transformation is to perform inverse

Fourier transform of the dynamic impedance function to directly determine the dynamic stiffness

as a function in the time domain (Wolf & Obernhuber 1985). This function is interpreted as the

force impulse response of the soil to a given displacement. This signifies that the displacement of

the foundation will produce a restoring force at the foundation over a given time duration that is

dictated by the force impulse response and the magnitude of the displacement. These researchers

developed the governing equation of motion and framework for such time domain analysis and

proceeded to replicate the response of a soil domain that is a semi-infinite rod with exponentially

increasing area. Later Wolf and Motosaka (1989) developed a recursive method of representing

the convolution integral from the soil impedance and Meek (1990) would then determine a

general form of the recursive method.

6

Later a transform model was proposed that decomposed the impedance function into two

separate force impulse responses that are resulting from the instantaneous displacement and

velocity of the foundation (Nakamura, 2006a). This method was later expanded to incorporate an

instantaneous mass term and using least square method to minimize the differences between the

actual impedance of the time series and desired impedance (Nakamura, 2006b).

Recently researchers have explored the possibility of transforming the impedance function using

methods developed in the digital signal processing field within electrical engineering. Şafak

(2006) proposed the use of an infinite impulse response filter to model the impedance function in

the time domain that is rather compact since it utilizes a recursive procedure.

1.3. Organization of Thesis

This thesis will investigate the applicability of three different impedance function transformation

methods to soil-structure interactions problems. The investigation will explore the accuracy and

the stability issues of these methods when incorporated into time integration schemes and

demonstrate the method’s ability to model increasingly complex and varied soil-structure

validation examples.

Before assessing these transform methods however the second chapter will focus on the

importance of the impedance functions and how they are formulated. It is an important

discussion since the transformation methods are completely reliant on the impedance functions’

ability to faithfully capture the desired soil domain’s behaviour. This chapter will summarize

how impedance functions are generated, either analytically or numerically, and the limitations

and benefits of each. Some of these topics have been briefly alluded to in the literature review

and objectives sections of this chapter but will be examined in greater detail and focusing on

some of the theory associated with the impedance function that will be needed for subsequent

chapters.

Each transformation method will be presented in chapter three with a thorough description of the

theory and procedures involved with each method. The methods will be used to model a one

dimensional soil and structure model that an exact solution can be determined by established

means. Comparisons in the response of the methods to this validated response will allow an

assessment to be made of each method’s accuracy and susceptibility to instability. Even if these

7

methods are unable to model this particular example does not suggest they may not be able to

model other soil systems. It would however suggest that the method is not generally applicable to

any arbitrary impedance function. A stability assessment procedure will be introduced in this

chapter that will allow others to determine whether a given soil domain transformation, produced

using any method, is stable or not prior to running any analysis.

Chapter four will discuss in greater detail known stability issues that affect a number of these

methods. The mathematical theory behind some of the instability will be introduced here and a

number of proposed mitigation solutions are described here that are both original and proposed

by other researchers.

The fifth chapter attempts to model a two dimensional structure resting on a soil layer modelled

in a computer program using plain-strain quadrilateral elements. This example displays an

increase in complexity in the soil-structure interaction problem since the foundation interface

will contain multiple degrees-of-freedom requiring the use of coupled impedance functions. The

impedance function is extracted from the finite element model and utilized in the transformation

method to determine the soil-structure system’s response. This solution will be then validated

against the response of the same system modelled solely by finite elements. This example

demonstrates the methods applicability to systems with greater real-life applicability.

Chapter six will discuss the incorporation of the transformation method into a hybrid simulation

framework and will attempt to model a nonlinear inelastic structure resting on a linear elastic soil

domain under seismic loading. The validated response will be generated by modelling the total

soil and structure system in a single finite element model. The accurate reproduction of this

response will demonstrate that the transformation method is capable of providing accurate, stable

and efficient soil domain representation that can be incorporated with a structure with any

material and geometrical complexity. Furthermore it will demonstrate the suitability of these

transform methods to be used in hybrid simulations when consideration of the soil-structure

interaction is desired.

These validation examples will clearly demonstrate the appropriateness of these transformation

methods for modelling soil-structure interaction systems in the time domain. Their computational

efficiency and representation in the time domain allow users to perform nonlinear structural

analysis whilst accounting for soil-structure interactions which represents a significant

8

improvement over conventional soil-structure analysis methods such as frequency domain

analysis. Other implications of these validation methods are summarized in the conclusion

chapter as well as a discussion of known limitations and future possible research that still needs

to be conducted.

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Chapter 2 Impedance Functions

This research investigates the suitability of time domain transformation methods of the soil

domain’s impedance function to simulate systems considering soil-structure interactions. Before

exploring these methods however an examination will be presented on the definition and usages

of the impedance functions. Dependence on these functions provides distinct benefits and

drawbacks over other soil model concepts which will be discussed further in this chapter. It is

important to note that any limitations impedance functions possess will persist into the time

domain representation of these impedance functions.

2.1. Impedance Function Definition

Any discussion must be first initiated by formally describing the definition of the impedance

function. It is sometimes termed as the dynamic stiffness of the soil because it is describes the

ratio between an input displacement and output force. In soil-structure interaction the

displacement in this ratio is the displacement of the foundation interface between the soil and the

structure, and the output force is the restoring force acting on the same interface. The

foundation’s displacement response will differ depending on the frequency of the applied force

implying that the soil’s stiffness varies with the frequency being considered. Determining this

stiffness at various frequencies will produce the impedance function of the given soil-foundation

system.

The impedance function contains complex value data that describes the dynamic ratio between

the force and displacement. The real and imaginary data of the impedance function describe

respectively the displacement response in phase and 90° out of the phase in relations to the force

(Dotson & Veletsos, 1990). The imaginary portion of the data is due to the presence of hysteretic

damping in the soil and radiation of radial damping of the soil infinite medium (Dotson &

Veletsos, 1990).

If the soil domain’s impedance function is already defined, soil-structure interaction can be

easily considered by performing frequency domain analysis. This analysis technique is already

used commonly in the industry, especially in the nuclear industry where reactors are typically

designed specifically to remain linear elastic under earthquake loading (Tyapin, 2007). It is

10

sometimes called Fourier synthesis because it requires the amalgamation of the impedance

function with the frequency domain representation of the structure into a single system (F. C. P.

De Barros & Luco, 1990) .

Any arbitrary loading in the time domain can be transformed into the frequency domain using the

Fourier transform. Each data value in the transformed force function represents a constituent

sinusoidal force at a specific frequency. By obtaining all the responses of the system to each

frequency’s corresponding sinusoidal force, one may obtain the complete response of the system

by simply summing the individual responses by the rule of superposition.

The use of superposition in this procedure is only acceptable if both the soil and structure

domains remain linear elastic. Inelasticity of the soil and hysteretic energy dissipation can be

considered however by using equivalent linearization. The system is still analyzed as being

linear-elastic but the soil modulus and damping levels are chosen to correspond to the likely

maximum strain the soil is to experience under the given loading. This is an approximate method

since all the elements of the soil model are given properties based on this new secant modulus

(Yoshida et al., 2002).

Most of the validation examples presented in this document will be evaluating systems that are

linear-elastic. For these examples this frequency domain analysis can be used as a reference

method whose response is considered valid.

2.2. Producing Impedances

Up to now the discussion has assumed that an impedance function existed for the soil domain

that is desired to be modelled in a soil-structure problem. Obtaining a realistic and proper

impedance function can itself be a challenging task and many researchers have conducted studies

solely focused on impedance function generation.

It is important to remember that any model of the soil, be it a finite element model, analytical

model, or just an impedance function representation, is based on a mathematical assumption of

the material behaviour of the soil. From laboratory experiments the behaviour of some soil types

has been discovered under specific loading conditions but nevertheless it is incredibly difficult to

determine a soil’s actual dynamic response because of the nonlinearity and non-homogeneity of

11

the soil. Consequently research in this field is rarely verified against field data but at least these

approximate models are an improvement on analyses that ignore the soil domain contribution.

2.2.1. Analytical Impedances

Initially impedance functions were determined analytically by making a number of assumptions

of the soil system that allowed the soil material and boundary problem to be expressed simply,

allowing a closed form solution for the impedance function to be determined algebraically.

However as more complicated foundation systems began to be considered, the mathematical

formulation of the impedance function became increasingly more complex. Expressing these

systems would often result in a set of dual integrals for which no closed form solution exist.

These equations would need to be transformed into Fredholm’s integrals and then evaluated

numerically (Shah, 1968) to produce approximate analytical solutions. Later system would begin

to utilize empirical values to simplify the equations even further leading to less mathematically

rational solutions (Veletsos & Verbic, 1973).

Numerically generated impedances are readily available in the literature with typically each

study focusing on a single foundation and soil behaviour type. Most studies that deal with

realistic foundation and soil systems do not generate concise expressions for the impedance

functions; instead a family of curves is presented as the solution. A number of curves are needed

because of the great degree of variability that exists in the soil parameters and configuration. The

variables include the Poisson’s ratio, the shear modulus and any geometric assignments to the

soil profile or foundation.

To obtain an accurate impedance function from these sources is impractical for engineers in

industry because the solution is only displayed graphically and because the foundation and soil

types considered are often too simplified compared to realistic foundations. This has led the

discussion of soil-structure interaction research to focus on advances in finite element modelling

of the soil which can be tailored to any soil and foundation type.

2.2.1.1. Literature Example

An example is provided of an impedance function generated analytically from a complex soil

configuration. This study was conducted in 1974 by J.E. Luco and it determined the impedance

functions for a rigid massless disc foundation supported on an elastic layered soil medium. The

12

mixed boundary value problem resulting from this system was reduced to sets of Fredholm

integrals which were solved numerically for a number of parameter values and then the

impedance function was determined. The configuration of the model problem is presented in

Figure 2.1 and the variables used to describe the soil layers are the shear modulus, 𝐺, the soil

density, 𝜌, and the Poisson’s ratio, 𝜍.

Three cases were investigated in Luco’s study but only the first case will be illustrated here. Each

case examined specific ratios between the two soil layer’s parameter and looked at five different

ratios between the depth of the soil layer interface, 𝑕, and the radius of the disc foundation, 𝑎.

This produced five impedance functions that vary over a number of frequencies. The values for

the ratios that are examined in this case are demonstrated in Table 2.1.

Table 2.1 Ratios and Variable Values Used in Soil Model Example

𝑉𝑠,1/𝑉𝑠,2 𝜌1/𝜌2 𝜍1 𝜍2 𝐻/𝑎

Value 0.8 0.85 0.25 0.25 0.2, 0.5, 1, 3, ∞

Figure 2.1 Disc Foundation on Layered Soil Medium (Luco, 1974)

13

It is apparent from Table 2.1 that each of the analysis cases describes a very specific

configuration of the soil. The generated impedance therefore will be extremely limited in

applicability because it only accurately describes a small amount of soil-foundation conditions.

Each impedance function is expressed in a form similar to the one in Eq. (2.1) where all the

terms outside of the square brackets are constants. The 𝑎0 variable is the dimensionless

frequency in relation with the top layer and it is equal to 𝜔 𝜌1/𝐺1 1/2.

𝑆𝐻𝐻 =8𝐺1𝑎

2 − 𝜍1

𝑘𝐻𝐻 𝑎0 + 𝑖𝑎0𝑐𝐻𝐻 𝑎0 ∆𝐻

The variables 𝑘 and 𝑐 are dependent on the frequency and contain the characteristics of the

impedance function. It is important to note that the variable 𝑎0 is being multiplied to the

damping function of the impedance which allows 𝑐 to contain non-zero values at the static

condition, though the impedance function will possess a value of zero at that point. The

coefficient functions are presented in Figure 2.2 and these functions vary significantly between

values of 𝐻/𝑎.

Figure 2.2 Coefficient Functions for the Horizontal Impedance (Luco, 1974)

(2.1)

14

This example impedance shows the difficulty of relying on analytical impedance functions to

conduct soil-structure interaction analysis of physical systems. Even with the many analyses

performed by researchers it is likely that the foundation and soil properties engineers in industry

or elsewhere require will not be available. Performing such analysis is mathematically intensive

and considering that the coupled relationship between horizontal and rocking motions has been

neglected, the accuracy of these impedances is uncertain. For most cases creating a finite element

model of the desired soil system is more attractive to practioners because of the readibility of

software programs and customizability of the simulation model. These analytical solutions still

are used to verify that the finite element model’s results are reasonable since physical

verification is expensive and often impractical.

2.2.1.2. Partial List of Available Analytical Impedance Functions

A list of available journal articles that describe and provide the impedance functions for a

number of specific soil-foundation configurations is provided in Table 2.2. It is however a partial

list with many more impedance functions available elsewhere.

Table 2.2 Partial List of Journals that Provide Generated Impedance Functions

Reference Geometry of Foundation and Soil Layers

(Wong & Luco, 1985) Square Foundations on Layered Media

(Mira & Luco, 1989) Square Foundation Embedded in an Elastic Half-Space

(Bu, 1998) Square Foundations Embedded in an Incompressible

Half-Space

(Ahmad & Rupani, 1999) Square Foundation in Layered Soil

(Vrettos, 1999) Rigid Rectangular Foundations on Soils with Bounded

Non-Homogeneity

(Andersen & Clausen, 2008) Surface Footings on Layered Ground

(Okyay et al., 2012) Slab Foundations with Rigid Piles

(Padrón, Aznárez, Maeso, &

Santana, 2010)

Deep Foundations with Inclined Piles

(Rajapakse & Shah, 1988) Embedded Rigid Strip Foundations

(Lin, 1978) Circular Plates Resting on Viscoelastic Half Space

(Pradhan, Baidya, & Ghosh, 2003) Circular Foundation Resting on Layered Soil

(Barros, 2006) Rigid Cylindrical Foundations Embedded in

Transversely Isotropic Soils

(Liou & Chung, 2009) Circular Foundation Embedded in Layered Medium

(Hatzikonstantinou et al, 1989) Arbitrarily Shaped Embedded Foundations

15

2.2.2. Finite Element Models

Impedance functions can also be obtained from finite element models of soil using two distinct

methods that are discussed in the next two subsections. These models should be able to

adequately reproduce results similar to the analytically generated impedance functions if the

same assumptions of half-space elasticity, boundary conditions and foundation configurations are

utilized. The greatest challenge facing finite element modelling is in capturing accurately the

wave dissipation property of the infinite soil medium. The use of rigid boundaries in the finite

element soil model will result in seismic wave reflections off the boundary that reverberate back

into the soil system leading to spurious reflected wave responses. A great deal of research has

been conducted into creating appropriate energy absorbing and viscous boundaries and other

boundary elements (Yerli, 2003) that accurately reproduce this infinite boundary behaviour of

the soil domain.

Impedance functions can only be determined from linear-elastic finite element models of the soil

domain since any non-linearity or degradation of the soil cannot be properly captured in a

frequency domain function. The reason being that during nonlinear analysis the soil’s dynamic

response not only becomes dependent on the forcing frequency but also on the amplitude of the

applied force.

The next sections will explain the mathematics of extracting an impedance function from a finite

element model that is primarily intended for time-history analysis. The algebraic formulation

method is only possible if the structural matrices of the soil system are available for

manipulation. This is not typically the case when modelling the soil domain using third party

modelling software therefore the numerical formulation method is also presented.

2.2.3. Algebraic Formulation Method

It is possible to obtain the impedance function from a soil model that has been defined explicitly

in terms of static mass, stiffness and damping matrices. These matrices will produce soil systems

that are linear-elastic which lend themselves easily to the formulation of frequency domain

impedance functions without necessitating numerical procedures. The equations of motion for

these systems are expressed in the frequency domain by using complex exponential notation and

16

then using some algebra the impedance function can be ascertained following a simple

procedure.

2.2.3.1. Single degree-of-freedom Impedance

In explaining the theory of this method it is best to first demonstrate its execution on a single

degree-of-freedom system. The procedure relies on being able to express the global system

matrices in frequency domain in order to obtain the global complex flexibility matrix in the

frequency domain. From this flexibility matrix the impedance function at the interface degrees-

of-freedom can be obtained.

The equation of motion for a system that contains a single node with a mass attached to a rigid

base by a spring and a damper subjected to an external harmonic force is as follows:

𝑚𝑢 + 𝑐𝑢 + 𝑘𝑢 = 𝑓

In order to transform this expression into the frequency domain it has to be rewritten with the

external force expressed as a sum of cosine and sin functions at a distinct frequency 𝜔𝑖 . This

force should be then described using an equivalent complex exponential in the following form.

𝑓 = 𝐹 ∙ 𝑒𝑖𝜔 𝑖𝑡

Taking the displacement to be a similar complex exponential function, the velocity and

acceleration of the system can be simply expressed as:

𝑢 = 𝑈 ∙ 𝑒𝑖𝜔 𝑖𝑡 , 𝑢 = 𝑖𝜔𝑖𝑈 ∙ 𝑒𝑖𝜔 𝑖𝑡 , 𝑢 = −𝜔𝑖2𝑈 ∙ 𝑒𝑖𝜔 𝑖𝑡

𝐹 and 𝑈 are the amplitudes of the force and the resultant displacement functions respectively.

These expressions allow the equation of motion to be rewritten in terms of the two amplitude

variables as shown in Eq. (2.5).

−𝑚𝜔𝑖2𝑈 ∙ 𝑒𝑖𝜔 𝑖𝑡 + 𝑖𝑐𝜔𝑖𝑈 ∙ 𝑒𝑖𝜔 𝑖𝑡 + 𝑘𝑈 ∙ 𝑒𝑖𝜔 𝑖𝑡 = 𝐹 ∙ 𝑒𝑖𝜔 𝑖𝑡

This expression can then be simplified by removing the common exponential term and dividing

the amplitude variable to form the stiffness ratio:

𝐹𝑈 = 𝑘 + 𝑖𝑐𝜔𝑖 −𝑚𝜔𝑖

2

(2.2)

(2.5)

(2.6)

(2.3)

(2.4)

17

From Eq. (2.6) it becomes evident that the system will behave significantly differently depending

on the value of the applied frequency. Furthermore when the frequency is zero the static

condition occurs and the displacement becomes simply the division of the applied static force by

the stiffness. The ratio of force and displacement is the dynamic stiffness and the impedance

function is constructed by determining this ratio for a range of frequencies. The resulting series

of dynamic stiffness is data that makes up the impedance function.

2.2.3.2. Multiple Degree-of-freedom Impedances

To consider the formation of impedances based on a system with an arbitrary number of degrees-

of-freedom the equation of motion must be expressed now with matrix and vector notation.

𝑀𝑢 + 𝐶𝑢 + 𝐾𝑢 = 𝑓

The interface degree-of-freedom is specified to be the first degree-of-freedom and that is where

the impedance function will be derived. To consider this arrangement the force vector is defined

such that only the first degree-of-freedom is being loaded.

𝑓 =

𝐹 ∙ 𝑒𝑖𝜔 𝑖𝑡

00⋮

=

𝐹00⋮

∙ 𝑒𝑖𝜔 𝑖𝑡 = 𝐹 ∙ 𝑒𝑖𝜔 𝑖𝑡

Each degree-of-freedom will respond with different amplitudes in response to this specified

loading. This creates a displacement response vector composed of many unknown amplitudes of

the form presented below. The impedance that is of interest is solely the ratio between 𝐹 and 𝑈1.

𝑢 =

𝑈1 ∙ 𝑒

𝑖𝜔 𝑖𝑡

𝑈2 ∙ 𝑒𝑖𝜔 𝑖𝑡

⋮𝑈𝑁 ∙ 𝑒

𝑖𝜔 𝑖𝑡

= 𝑈 ∙ 𝑒𝑖𝜔 𝑖𝑡

Following the same substitution that was conducted in Eq. (2.5) produces the linear system

below. The new vector 𝑈 is a vector of unknown displacement amplitudes and the vector 𝐹 is a

vector composed of the first degree-of-freedom’s force amplitude and subsequent zeros for all

other degrees-of-freedom.

(2.9)

(2.8)

(2.7)

18

𝐹 ∙ 𝑒𝑖𝜔 𝑖𝑡 = 𝐾 + 𝐶𝑖𝜔𝑖 −𝑀𝜔𝑖2 𝑈 ∙ 𝑒𝑖𝜔 𝑖𝑡

In order to isolate the impedance function of the first degree-of-freedom the inverse of the

dynamic stiffness, the dynamic flexibility matrix, needs to be formulated from Eq. (2.10). To do

so Eq. (2.10) is rearranged and the exponentials removed producing Eq. (2.11).

𝐾 + 𝐶𝑖𝜔𝑖 −𝑀𝜔𝑖2 −1𝐹 = 𝑈

The terms within the brackets in Eq. (2.11) are collectively the dynamic stiffness, which since

they are being inversed, are now the dynamic flexibility. Replacing these terms in the bracket

with flexibility component terms allows the system of equations to be presented in the following

manner:

𝐻1,1

𝐻2,1

⋮𝐻𝑁,1

𝐻1,2

𝐻2,2

⋮𝐻𝑁,2

⋯⋯⋱⋯

𝐻1,𝑁

𝐻2,𝑁

⋮𝐻𝑁,𝑁

𝐹0⋮0

=

𝑈1

𝑈2

⋮𝑈𝑁

From Eq. (2.12) the expression for 𝑈1 is formulated by multiplying the first row of the flexibility

matrix and the force vector. Since all force amplitudes other than the first degree-of-freedom are

zero, only the 𝐻1,1 flexibility component is required in the impedance formulation.

𝐻1,1 × 𝐹 = 𝑈1

𝐹𝑈1 = 𝐻1,1

−1

The procedure therefore for determining the impedance function for a system with a single

interface degree is to first formulate the general dynamic stiffness matrix present in Eq. (2.10).

Inversing this matrix would then generate the dynamic flexibility matrix, and to obtain the

desired impedance function one would then inverse the term inside the flexibility matrix that

corresponds to the degree-of-freedom of the interface.

(2.12)

(2.10)

(2.11)

(2.13)

19

2.2.3.3. Impedance Matrix for Interfaces with Multiple Degrees-of-freedom

In order to expand the ideas formulated in the previous subsection to a soil system with multiple

degrees-of-freedom at the foundation interface a number of adjustments have to be implemented

to the previously established equations. In the previous section the external force was applied to

a single degree-of-freedom that corresponds to the interface. The interface being considered in

this section contains multiple degrees-of-freedom and consequently the external force needs to

be applied in each of these directions. The new force vector presented in Eq. (2.14) corresponds

to a system with 𝑘 number of degrees-of-freedom at the interface.

𝑓 =

𝐹 ∙ 𝑒𝑖𝜔 𝑖𝑡

⋮𝐹 ∙ 𝑒𝑖𝜔 𝑖𝑡

00⋮

Following the same procedure in Section 2.2.3.2, the final linear system composed in this

method will of the following form:

𝐻1,1 ⋯ 𝐻1,𝑘 ⋯ 𝐻1,𝑁

⋮ ⋱ ⋮ ⋱ ⋮𝐻𝑘 ,1 ⋯ 𝐻𝑘 ,𝑘 ⋯ 𝐻1,𝑘

𝐻𝑘+1,1 ⋯ 𝐻𝑘+1,𝑘 ⋯ 𝐻1,𝑘+1

⋮ ⋱ ⋮ ⋱ ⋮𝐻𝑁,1 ⋯ 𝐻𝑁,𝑘 ⋯ 𝐻𝑁,1

𝐹⋮𝐹0⋮0

=

𝑈1

⋮𝑈𝑘𝑈𝑘+1

⋮𝑈𝑁

By expanding the matrix multiplication presented in Eq. (2.15), it would become evident that the

first 𝑘 equations are coupled and independent of the remaining equations. The only flexibility

terms that are relevant are the first 𝑘 columns of the first 𝑘 rows forming a reduced flexibility

matrix. Inverting this matrix and transferring it to the right hand side would produce the

following equation.

𝐹⋮𝐹 =

𝐻1,1 ⋯ 𝐻1,𝑘

⋮ ⋱ ⋮𝐻𝑘 ,1 ⋯ 𝐻𝑘 ,𝑘

−1

𝑈1

⋮𝑈𝑘

(2.15)

(2.14)

(2.16)

20

In fact this inverted reduced flexibility matrix defines the family of impedance functions of the

interface. Since the forces and displacement of the various degrees-of-freedom are coupled,

independent equations for the ratio of force and displacement are not possible. Instead each

impedance function is defined as each component of the inverted flexibility matrix.

2.2.4. Numeric Formulation

Often the soil domain is modelled using computer analysis software that does not provide the

user with values of the system matrices required to determine the impedance function

algebraically. It is still possible to obtain the impedance function from these computer models by

determining the soil’s response to sinusoidal loading and obtaining the dynamic stiffness from

the ratio between the applied load and the resulting foundation response. This method is far more

computationally expensive than the algebraic formulation of the impedance function because for

each frequency a time history of the soil’s response must be determined. The length of this

analysis increases significantly as the model gets larger and more complex, especially if the

model is three-dimensional rather than two-dimensional.

2.2.4.1. Response to a Sinusoidal Force

For this first section the example soil system will contain only a single degree-of-freedom at the

foundation interface and later the procedure for interfaces with multiple degrees-of-freedom will

be presented. In this example therefore only a single sinusoidal load is applied at the interface at

a particular frequency and the analysis’ time step must be chosen appropriately for a

representation of the response motion. Typically in this study it was ensured that each cycle of

loading had at least thirty data points, which would obligate the time step to have the following

relationship.

∆𝑡 ≤2𝜋 × 30

𝜔𝑖

Eq. (2.17) is only a suggestion and most often more data points per cycle were utilized to insure

an accurate response. The second parameter that needs to be decided is the duration of the

analysis. Steady-state response of the system is required for the impedance generation so

sufficient amount of time needs to elapse in order for the transient response to dissipate. The

number of cycles required is highly variable and dependent on a number of parameters, but

(2.17)

21

generally in this study anywhere from forty to seventy cycles was sufficient to produce steady-

state responses. Low damping soil systems will require greater elapsed time since the transient

response lingers longer.

Only the last few cycles of the generated response and input force should be used in the dynamic

stiffness determination analysis. The reason for using these last cycles is that they will be

representative of the steady-state response of the system. Plotting the applied force against the

foundation response for these cycles generates an elliptical plot resembling Figure 2.3.

The steady-state interface response will be a sinusoid with the same frequency as the loading

function but the response will possibly have been phase-shifted. This shift is due to the

relationship between the dynamic properties of the soil (density, modulus and damping)

compared to the forcing frequency and also caused by the damping of the soil.

Figure 2.3 Plot of Displacement against Force for a sinusoidal loading on a soil system

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

-1 -0.5 0 0.5 1

Forc

e [

N]

Displacement [m]

𝐸𝑑𝑎𝑚𝑝

22

From the study of a massless single degree-of-freedom’s impedance function, the real and

imaginary data are related to the stiffness and damping coefficient of a system in the following

manner.

𝑆 𝜔𝑗 = 𝐾 + 𝑖𝜔𝑗𝐶

The real portion of the data constitutes the equivalent stiffness and the imaginary portion is the

product of the equivalent damping and the applied frequency. Dividing the amplitude of the force

function by the amplitude of this response will determine the equivalent stiffness of the soil

domain at this forcing frequency.

The area enclosed by the elliptical curve in Figure 2.3 is equal to the damped energy of the

system. A system without damping would exhibit purely linear response and the damped energy

would be zero. Obtaining the damped energy from the area of the graph one can proceed to

calculate the damping value for an equivalent single degree-of-freedom system by using Eq.

(2.19). It is recommended that when calculating the damping energy that the area of a number of

cycles be taken and then the calculated energy value be divided by the number of cycles in order

to obtain an averaged value of damping so as to decrease error.

𝐶 =𝐸𝑑𝑎𝑚𝑝

𝜔𝑗𝜋 ∙ 𝑢𝑚𝑎𝑥 2

Once both equivalent damping and stiffness are determined for a given frequency, the value of

the impedance function at that frequency is easily determined. Repeating this time-history

analysis for each frequency of interest and performing the same procedure as described above

will generate impedance values for each frequency and produce the desired impedance function.

2.2.4.2. Impedance Matrix for Interfaces with Multiple Degrees-of-freedom

For soil systems with multiple degrees-of-freedom at the foundation interface a number of

adjustments have to be made to the soil model before obtaining the responses necessary for the

numerical impedance function generation procedure discussed above. These interfaces may now

possess a coupled behaviour between the degrees-of-freedom which need to be accurately

captured when generating the impedance functions.

(2.19)

(2.18)

23

In Section 2.2.3, pertaining to the algebraic formulation of multiple degrees-of-freedom

interfaces, the following relationship was developed which relates the displacement response to

the applied force in the frequency domain. This expression is true for a system where harmonic

forces are applied to the interface and at all other degrees-of-freedom the force is zero.

𝐹1

⋮𝐹𝑘

=

𝐻1,1 ⋯ 𝐻1,𝑘

⋮ ⋱ ⋮𝐻𝑘 ,1 ⋯ 𝐻𝑘 ,𝑘

−1

𝑈1

⋮𝑈𝑘

For this analysis however to generate a valid impedance function matrix each degree-of-freedom

must be analyzed separately in order to capture the coupled impedance functions. To do so, an

external sinusoidal force is applied to a single degree-of-freedom at the interface and all other

interface degrees-of-freedom are restrained from movement. When executing the time-history

analysis reaction force responses, 𝑅 , will be generated at the location of the restraints. The

presence of these restrained degrees-of-freedom alters the dynamic stiffness expression and

results in Eq. (2.21):

𝐹1

𝑅2

⋮𝑅𝑘

=

𝑆1,1 ⋯ 𝑆1,𝑘

𝑆2,1 ⋯ 𝑆2,𝑘

⋮ ⋱ ⋮𝑆𝑘 ,1 ⋯ 𝑆𝑘 ,𝑘

𝑈1

0⋮0

To generate the dynamic stiffness components in Eq. (2.21), the same analysis that was

developed for a single degree-of-freedom interface in Section 2.2.4.1 is conducted between the

resulting displacement response of 𝑈1 and the external force and each of the restraining forces

separately. The relationship between 𝑈1 and these forces describe the dynamic stiffness values

for the first row of the impedance matrix in Eq. (2.21): 𝑆1,1, 𝑆1,2, ⋯ , 𝑆1,𝑘 . The values in the

remaining rows are determined by repeating this procedure with a different degree-of-freedom

being the location where the external force is applied and all other directions being restrained.

Once all the components of the matrix are determined, a new frequency is specified and this

procedure is repeated anew until all the frequencies of interest have been analyzed. All the

determined dynamic stiffnesses of a given component in the matrix of Eq. (2.21) express a single

impedance function relative to the frequency range considered.

(2.21)

(2.20)

24

It becomes evident from the description of this procedure that a complex interface will

significantly increase the number of analyses required to generate the impedance function matrix.

This includes most interfaces of three-dimensional soil domains and embedded foundations.

25

Chapter 3 Time Domain Transformation Methods

Literature has provided three distinct models that attempt to transform the frequency dependent

data of the impedance function into a force response function that can be used in time-domain

analysis of structural models. In these models, the fixed foundation is unrestricted in the

numerical model allowing the soil-foundation to displace during the earthquake simulation rather

than remain rigid. These released degrees-of-freedom will be referred to as foundation degrees-

of-freedom.

These methods employ the convolution summation operator to determine the restoring force

imparted from the soil domain. The convolution operator produces a new function by performing

the summation of two functions where one function is reversed and shifted. The convolution

summation is presented in generic equation form in Eq. (3.1). In this equation, the functions 𝑓

and 𝑔 are undergoing the convolution operation where the 𝑔 function is reversed and shifted. The

function that is produced by the convolution is the 𝑕 function and it is only defined in this

expression for the value of 𝑛, whereas the input functions are presumed to be finite and defined

for a limited range starting at 0. Consequently the variable 𝑚 must not exceed the value of 𝑛

because the 𝑔 function is undefined for negative values. The variable 𝑚 must also not exceed the

range of the function 𝑓 . Therefore the variable 𝑁 is equal to the minimum of these two

restrictions.

𝑕 𝑛 = 𝑓 ∗ 𝑔 𝑛 ≝ 𝑓[𝑚] ∙ 𝑔[𝑛 −𝑚]

𝑁−1

𝑚=0

In all these transformation methods the reversed and shifted function 𝑔 will be a time history of a

state variable at the Foundation DOF, such as displacement or velocity, and 𝑓 will be the force

response function generated from the transformation of the frequency dependent impedance

function. The coefficients in the force response function will be determined prior to performing

the simulation by transforming the impedance function and it remains constant during the

simulation. Eq. (3.2) presents the convolution with the generic variables replaced by the new

terms that will be used in the model’s convolution.

(3.1)

26

𝑌𝑜𝑢𝑡𝑝𝑢𝑡 [𝑛] = 𝐴𝑐𝑜𝑒𝑓𝑓𝑖 𝑐𝑖𝑒𝑛𝑡𝑠 𝑚 ∙ 𝑋𝑖𝑛𝑝𝑢𝑡 𝑛 − 𝑚

𝑁−1

𝑚=0

𝐹𝑟𝑒𝑠𝑡𝑜𝑟𝑖𝑛𝑔 𝑡 = 𝑎𝑗 ∙ 𝑋𝑖𝑛𝑝𝑢𝑡 𝑡 − 𝑗 ∙ ∆𝑡

𝑁−1

𝑗=0

The first model presented utilizes the inverse Fourier transformation to transform the impedance

function (Wolf & Obernhuber, 1985) to a single force response function and it is considered the

most straightforward method. Since the impedance function represents the dynamic ratio

between the foundation’s restoring force and its displacement in the time domain, the restoring

force calculation will be dependent on the previous displacement history of the Foundation DOF.

This displacement history is the input function in the convolution computation.

The next method that was investigated proposes that the restoring force be determined by the

convolution of both displacement and velocity history as well as the instantaneous acceleration

of the Foundation DOF (Nakamura, 2006b). Since this method requires the convolution to take

inputs other than the displacement, the inverse Fourier transform procedure is no longer

sufficient. Two force response functions are determined by solving a linear system comprised

from the complex exponential definition of the impedance function. The restoring force is

determined as the summation of two convolutions, one dependent on the displacement history

and the other on the velocity history.

The last method investigated utilizes an existing and well recognized digital filter that has long

been studied in the field of digital signal processing, the Infinite Impulse Response filter. This

filter has unique recursive properties that allow it to use only a few number of parameters, but

consequently causes it to be highly unstable and volatile. It utilizes two convolutions, one

dependent on the foundation displacement input and another that uses the output of previous

computed restoring force values as input to the convolution creating potential feedback

instability.

(3.2)

27

3.1. Evaluation Procedure of the Transformation Methods

3.1.1. Procedure Organization

In order to validate the three method’s effectiveness in modelling soil structure interaction

systems, an example problem was developed that could be modelled correctly by an alternate

already established procedure. The solutions generated by the three transform methods can then

be compared to this exact solution and depending on the degree in which they match an inference

can be made on the validity and precision of each method.

The remainder of Section 3.1 will focus on describing thoroughly the example problem so it may

be replicated by others. The soil and structure properties will be selected, the applied load

specified and the manner the damping is implemented in the system will be described. The

resulting impedance functions of the soil model will also be presented and described.

The second part of this Section 3.1 will consist of general information regarding how the

methods will be incorporated into the time integration scheme. This includes a discussion in

Section 3.1.6 pertaining to the implementation of convolution calculations using a time step that

differs from the one utilized in the time integration scheme. Lastly, before the simulations are

conducted, a stability analysis procedure is presented in Section 3.1.7 that was devised in order

to allow for the stability of a combined time integration scheme and convolution algorithm to be

evaluated. The necessity for such a procedure arose due to a lack of discernible stability criteria

for these combined algorithms.

In Section 3.2 through to Section 3.4 each of the three transformation methods are discussed

individually in identical fashion. The general concept of the method is summarized at the

beginning of these sections and is followed by a comprehensive discussion on how the

coefficients of each method are determined. The maximum frequency and frequency step of the

impedance data affects the effectiveness of these transform methods therefore a subsection

indicating the specific frequency parameters utilized is included in these subsequent sections.

Using the determined force impulse function coefficients the impedance functions are

reproduced and compared to the original inputted impedance functions. The two methods

describing the procedure to obtain the impedance function will be presented in general terms in

Section 3.1.4. Then the simulation using the particular transform method is executed and the

28

results displayed alongside the response of the reference model. The method is then evaluated in

discussing how well the response was reproduced and in what manner they differ.

Following the presentation of the results, the mathematical formulation of the stability analysis is

accomplished for each of the specific methods. The stability assessment is then performed and

the stability of the given impedance function transform and time integration determined. The

assessment should correlate to the divergent or stable responses obtained from the simulations

just conducted.

After all these analyses are conducted a summary is provided that outlines which of these

methods was most reliable and stable in replicating the response of the reference example.

3.1.2. Problem Statement

The problem that will be modelled is that of a single degree-of-freedom building structure resting

on a soil model comprised of ten springs in series with lumped masses at each node and

damping. The building and soil structures have drastically different mass and stiffness values.

A model consisting of the combined soil and structure domains with eleven degrees-of-freedom

model will be analyzed as the reference model and it will be onwards referred to as the Total

Model. If the proposed method produces the same response, then that method can be said to be

valid in reproducing this soil-structure system. The Total model will use the Newmark time

integration scheme to determine the response and is represented in Figure 3.1.b.

Figure 3.1.c displays the structural model utilized in the proposed transform models. DOF 2 is

the additional foundation degree-of-freedom that is introduced to the structure and it is attached

to the rigid base by the instantaneous components of the specific force response function. All

three methods utilize an instantaneous stiffness, but only the Nakamura transform method will

have an instantaneous damper and mass appropriated to the foundation. The restoring force that

is applied to the foundation degree-of-freedom represents the force determined by the

convolution calculation involved in the transform models. This force is what accounts for the

frequency dependency of the soil system and it is dependent on the state of the system at

previous time steps.

29

In addition to these models, another reference model that will be employed is that of the same

structure resting on a rigid foundation. This system will not take into account soil-structure

interaction and will only be used to show the contribution of the soil domain to the softening of

the global response. This model will onwards be referred to as the Rigid model and it is

represented in Figure 3.1a.

Figure 3.1 The three different models analyzed in this validation example

3.1.2.1. Treatment of Force

The models will be subjected to the Kobe 1995 earthquake acceleration time history, which is

displayed in both time and frequency domain in Figure 3.2. For the Rigid model, the ground

acceleration is applied as a force to the single degree-of-freedom equal to the negative product of

the ground acceleration history and the mass of the structure. Similarly the external force will be

applied to only the structural degrees-of-freedom for both the Total and Proposed transform

model simulations. The reason for this is that the proposed transform model can only capture the

inertial interaction aspect of the soil-structure interaction phenomenon, signifying that it can only

account for the displacement of the soil and structure caused by vibrations present in the

structure and foundation. Greater analysis and consideration would be required to model

earthquake motion originating from bedrock.

On the other hand for the Total model it would not be difficult to model earthquakes originating

from the bedrock but since the purpose of this exercise is to validate the proposed model, it is of

upmost important to be applying the forces of these two models in an identical manner. This

30

decision does not ensure that the acceleration of the DOF 2 in the Total model is the same as the

recorded ground motion.

The force therefore will be applied as the negative product of the acceleration time history and

the mass of the structure at DOF 1, as presented Figure 3.1b and c. Since the purpose of the

analysis is to validate these methods in substituting more complex modelling techniques, the

application of force in this manner is acceptable. The time history of the acceleration of the Kobe

earthquake and its representation in the frequency domain is presented in Figure 3.2.

3.1.2.2. Damping

The damping matrix of the soil model is determined by using a Rayleigh damping model to

damp the first and second modes of that system at 5% damping ratio. How the damping is treated

here will affect the impedance function that is generated which will be used in the transform

methods. The value used for the damping ratio will affect the stability and accuracy of the

system, with low damping causing instability sensitivity in the response.

Since the structure is a single degree-of-freedom system, its damping coefficient is set as the

product of the damping ratio and the square root of the product of the mass and stiffness

constants of the structure.

𝐶 = 2𝜉 𝑚𝑘

Figure 3.2 Time History and Frequency Content of the Ground Acceleration (Kobe 1995)

(3.3)

31

3.1.2.3. Structural and Soil Models

The structure that is to be modelled in this validation exercise is a single degree-of-freedom

structure consisting of a spring, mass and damper elements. The specific properties of the

structure are displayed in Table 3.1.

Table 3.1 Structural Model Properties

Stiffness [N/m] Mass [kg] Damping Ratio Natural Frequency Natural Period [s]

1 000 000 1000 5% 31.62 rad/s, 5.033 Hz 0.1987

As stated before, the soil substructure will comprise of ten nodes with a total of ten horizontal

degrees-of-freedom with constant mass, stiffness and damping coefficients. Two soil cases will

be evaluated in order to confirm that the proposed models work for different types of soil

conditions. The cases represent a relatively stiff soil condition and a considerably soft soil.

Modelling such extreme cases will show the robustness of the methods in simulating various soil

conditions. The properties of the soil for the two cases are presented in Table 3.2.

Table 3.2 Soil Model Properties for the Two Analysis Cases

Case Stiffness

Constants [N/m]

Mass

Constants [kg]

Damping

Ratio Fundamental Period [s]

1 800 000 300 5 %

0.814, 0.273, 0.1665, 0.1217,

0.0976, 0.083, 0.0736, 0.0675,

0.0637, 0.0615

2 8 000 600 5 % 11.51, 3.866, 2.355, 1.721, 1.380,

1.174, 1.041, 0.955, 0.900, 0.870

3.1.3. Impedance Functions

The impedance function of the ten parallel spring soil model can be ascertained using the

algebraic formulation method described in Section 2.2.3 since the stiffness, mass and damping

matrices are known. Figure 3.3 and Figure 3.4 demonstrates the impedance functions of the stiff

and soft soil case model.

32

The impedance function of stiff soil model is presented twice in Figure 3.3. Figure 3.3.a presents

the impedance using a high frequency range where the low frequency behaviour in the real

portion of the impedance function is unclear. For this reason the same impedance function is

presented again in Figure 3.3.b but for a shorter frequency range up to 10 Hz.

From Figure 3.3.a it is apparent that the real portion of the data follows a negative parabola shape

and the imaginary portion data increases linearly at higher frequencies beyond 20 Hz. This

behaviour occurs predominantly beyond the largest natural period of the soil domain and this

implies that at this range of frequencies the soil domain behaves similarly to a single degree-of-

freedom spring-mass-damper system.

0 10 20 30 40 50-4

-2

0

2x 10

7

Frq. [sec-1]

Real

Impedance Function

0 10 20 30 40 500

1

2x 10

6

Frq. [sec-1]

Imagin

ary

Figure 3.3 Impedance Function of the Stiff Soil Model displayed a) in a large frequency

range and b) a shorter frequency range

0 2 4 6 8 10-1

0

1x 10

6

Frq. [sec-1]

Real

Impedance Function

0 2 4 6 8 100

1

2x 10

6

Frq. [sec-1]

Imagin

ary

a) b)

33

Figure 3.4 Impedance Function of the Soft Soil Model

It is important to note that the soft soil model manifests identical behaviour as the stiff soil

except at much lower frequencies and with a different amplitude. This similarity is due to the fact

that both models have constant stiffness and mass throughout the soil model thus manifesting

similar scaled dynamic responses. These impedances functions will be used in the transformation

methods with perhaps different maximum frequencies being considered.

3.1.4. Reproducing the Impedance Functions

With the impedance functions formulated the transformation methods can be executed and the

force impulse functions generated that will be used in restoring force convolution calculation.

Once the force impulse functions are formulated it can be checked whether the determined

functions reflect an impedance function identical to that of the soil model. One procedure in

reproducing the impedance function is based on the representation of the convolution calculation

in the 𝑍-domain which utilizes the complex exponential concepts presented in previous chapters.

The second technique available is a numerical method reproduction that is similar to the

numerical derivation of the impedance function from a finite element model presented in Chapter

2.

0 0.5 1 1.5 2-10

-5

0

5x 10

4

Frq. [sec-1]

Real

Impedance Function

0 0.5 1 1.5 20

1

2x 10

4

Frq. [sec-1]

Imagin

ary

34

3.1.4.1. Z-Transform Representation

A convenient approach to determine the impedance function that the transform coefficients

produces is to use 𝒵-transformation which allows for a time-domain signal to be represented in a

complex frequency-domain (Nakamura, 2006a). In this section a general framework will be

established by demonstrating how a simple convolution of displacement and an input function 𝑎

can be transformed using this method. In subsequent sections concerning the specific methods,

the 𝒵-transform representation will be established for that specific method using the framework

established here.

Convolutions are utilized in these methods to determine the restoring force during the numerical

integration algorithm and it is presented below with summation terms and in expanded form.

𝐹 𝑛∆𝑡 = 𝑎𝑖 ∙ 𝑥 (𝑛 − 𝑖)∆𝑡

𝑁

𝑖=0

𝐹 𝑛∆𝑡 = 𝑎0𝑥 𝑛∆𝑡 + 𝑎1𝑥 (𝑛 − 1)∆𝑡 + ⋯+ 𝑎𝑁𝑥 (𝑛 − 𝑁)∆𝑡

The 𝒵-transformation is expressed explicitly below in Eq. (3.5) where definition of the 𝑧 variable

is 𝑒𝑖𝜔 .

𝑋 𝑧 = 𝒵 𝑥[𝑛∆𝑡] = 𝑥 𝑛

∞

𝑛=−∞

𝑧−𝑛∆𝑡

It has been derived elsewhere that a time-shift in the discrete time signal would produce the

following identity in the 𝒵-domain.

𝒵 𝑥[𝑛 − 𝑘] = 𝑧−𝑘𝑋(𝑧)

With these definitions being established the convolution expressed in Eq. (3.4b) can now be

rewritten using the 𝒵-tranform of each 𝑥 variable. Each 𝑥 term occurring before time step 𝑛∆𝑡

can be rewritten, using the identity in Eq. (3.6), as a multiple of the 𝒵-tranform of the current

time step (𝑋 𝑧 ) and an exponential with the base 𝑧. The result of this procedure is the following

summation series.

𝐹 𝑧 = 𝑎0𝑋 𝑧 + 𝑎1𝑋 𝑧 ∙ 𝑧−∆𝑡 + 𝑎2𝑋 𝑧 ∙ 𝑧

−2∆𝑡 + ⋯+ 𝑎𝑁𝑋 𝑧 ∙ 𝑧−𝑁∆𝑡

(3.4a)

(3.4b)

(3.5)

(3.6)

(3.7)

35

Dividing out the 𝑋 𝑧 term and rewriting the series using the summation operator the condensed

Eq. (3.8) is formulated where the ratio between force and displacement is the definition of the

dynamic stiffness and impedance function. Lastly the 𝑧 variable is substituted by complex

exponential 𝑒𝑖𝜔 in order to convert the equation into the frequency domain.

𝐹 𝑧

𝑋 𝑧 = 𝑎𝑗

𝑁

𝑗=0

∙ 𝑧𝑗 ∙∆𝑡

𝑆 𝜔 = 𝑎𝑗

𝑁

𝑗=0

∙ 𝑒−𝑖𝜔𝑗 ∆𝑡

Expression (3.9) provides a simple summation calculation that allows for the determination of

the impedance function from the coefficients used in the transform methods. This Eq. (3.9) is

based on a convolution dependent on displacement and will need to be altered for more involved

impedance transform methods.

3.1.4.2. Numerical Generation

Since the 𝒵 -domain representation circumvents the time domain, it will not manifest any

instability issues or inaccuracies related to time-history analysis. For this reason a numerical

method was developed in order to generate the impedance function from the coefficients used in

a transform method. This method uses similar concepts developed in Section 2.2.4 where the

impedance function was generated numerically from a FEM model.

This procedure requires one to specify a prescribed predetermined sinusoidal displacement at a

given frequency of interest and the resulting velocity and acceleration may be calculated from

the derivative and second derivative of this displacement. These predefined variables allow the

convolution calculation to be performed straightforwardly and the restoring force to be

calculated. The restoring force is then compared numerically to the predetermined displacement

to calculate the effective dynamic stiffness of the transform at the specific frequency of the

harmonic displacement. The duration of the time history of the prescribed displacement must be

long enough to encompass enough cycles so that the transient response of the convolution

calculation dissipates. The duration of the analysis should be at least three or four times greater

(3.8)

(3.9)

36

than the number of coefficients in the force impulse response and should provide about ten

cycles of steady state response.

The cycles at the end of the time-history analysis, which represents the steady-state force

response, are to be used in the impedance generation. Plotting the prescribed displacement

against the calculated force will generate a graph similar to Figure 2.3 from which the real and

imaginary portions of the impedance function can be determined by solving for the equivalent

stiffness and damping of the system. Expressions for these values are defined in Chapter 2 in

Section 2.2.4.

3.1.5. Combined Convolution and Newmark Time Integration

For this validation example the convolution calculation associated with each of the

transformation methods will need to be incorporated into the established Newmark time

integration scheme. This incorporation will be different for other types of integration schemes

such as α-Operator Splitting time integration which is often used for pseudo-dynamic testing

(Combescure & Pegon, 1997). The following integration will be implemented for all the

transform methods that are being considered in this validation example.

Any instantaneous components utilized in the convolution of the transform method will be

incorporated into the system matrices of the combined system at the degree-of-freedom of the

foundation. All three models will at least contain an instantaneous stiffness component which

will be incorporated before commencing the time-history analysis. The matrices below display

how the two degree-of-freedom system matrices are altered when using the Nakamura method

which contributes instantaneous stiffness, damping and mass coefficients.

𝐾 = 𝑘 −𝑘−𝑘 𝑘 + 𝑘0

, 𝐶 = 𝑐 −𝑐−𝑐 𝑐 + 𝑐0

, 𝑀 = 𝑚 00 𝑚0

The time history of the displacement and the other state variables of interest at the foundation are

compiled and then used in the convolution calculation of the restoring force. This force is treated

just like any externally applied force so the restoring force value is added to the vector of forces.

The external force for this example is always applied at the structural degree-of-freedom whilst

the restoring force is exclusively being applied at the foundation degree-of-freedom because it

represents the interface between the soil and structure domains. This force vector is then

(3.10)

37

modified to create the effective force and used to calculate the unknown displacement of the

current time step. With the displacement calculated all the other state variables can be

determined for the new time step and this process then begins again for the next time step until

the analysis is complete.

3.1.6. Convolution Time Step

It is unnecessary that the time step of the convolution calculation be identical to that of the time

integration scheme. The convolution time step is defined as the inverse of the largest frequency

considered in the impedance functions which may render it unreasonable to have the two time

steps be identical. Sometimes a smaller maximum frequency is desirable in order to capture low

frequency behaviour without the need for having a large number of data points. Furthermore

there is a physical limit on the frequencies possible in earthquake motion and impedance

functions that consider frequencies beyond this point are doing so unnecessarily.

Having different time steps however needs to be accounted for when implementing the

convolution calculation in the time integration scheme. It is best to have the convolution time

step to be a multiple of the time integration time step so that the convolution never requires the

response of the foundation from a point in time between two discrete time steps in the

displacement history.

When the time steps are identical, the displacement history compiled for the convolution

operation is made of displacements occurring sequentially after each other in the time history

simulation. If the convolution time step is a multiple of the integration time step the compiled

displacement history will comprise of only displacement data that has occurred at intervals of the

convolution time step in the simulation. This premise is presented in the expressions below

where the first is a convolution using the same time step as the integration scheme and second

equation is using a convolution time step that is 𝑘 times greater than the time integration scheme.

𝐹𝑟𝑒𝑠𝑡𝑜𝑟𝑖𝑛𝑔 𝑛∆𝑡 = 𝑘0𝑢 𝑛∆𝑡 + 𝑘1𝑢 𝑛 − 1 ∆𝑡 + 𝑘2𝑢 𝑛 − 2 ∆𝑡 + ⋯

𝐹𝑟𝑒𝑠𝑡𝑜𝑟𝑖𝑛𝑔 𝑛∆𝑡 = 𝑘0𝑢 𝑛∆𝑡 + 𝑘1𝑢 𝑛 − 𝑘 ∆𝑡 + 𝑘2𝑢 𝑛 − 2𝑘 ∆𝑡 + ⋯

(3.11)

38

3.1.7. Stability Assessment

Section 3.1.4 described how the impedance function could be reproduced from the coefficients in

the transform methods allowing the comparison to be made between this effective impedance

function and the actual impedance function of the soil model. However, even if these two

impedance functions are identical, the combination of the numerical integration of the structural

model and the convolution operation may prove to generate an unstable response. Numerical

integration schemes have their own associated stability criteria and the convolutions associated

with the transform methods introduce additional volatility. The combined system may prove to

be unstable even if each component system is individually stable which motivated the creation of

a procedure that could infer the stability of the transform method when incorporated into a time

integration scheme.

The following procedure has the limitation of being only able to determine the stability of a

given model once the coefficients of the transform model have been determined. There exists no

easily determinable general stability constraint for the combined numerical integration algorithm

and the transform model because the transform method’s stability is dependent on the actual data

content of the impedance function. In addition to this the stability is dependent on the few

transform function parameters which includes the maximum frequency and frequency step of the

impedance function.

The stability analysis conducted will be performed on the simplest model able to describe the

combined system, which is a single structural node attached to the foundation node with the

restraining force applied to it determined by one of the three transform methods.

3.1.7.1. Theory

The stability assessment will require that the time integration scheme be written as a single

operator where the system state (acceleration, velocity and displacement variables) is expressed

solely in terms of the variables of the system state at the previous time step.

𝑢𝑢 𝑢

𝑖+1

= 𝐾0 × 𝑢𝑢 𝑢

𝑖

(3.12)

39

Since the simplest system that is required to be considered in this stability assessment is a two

degree-of-freedom system, 𝑢 will have two components. This results in the operator matrix, 𝐾𝑂,

being a six-by-six matrix.

𝑢𝑖 = 𝑥1

𝑥2 𝑖

Any point of time in the simulation can then be determined by raising the operator matrix to the

power of the number of time steps that have transpired and multiplying it to the initial system

state vector. Eq. (3.12) assumes free vibration is occurring and no external force is present and as

such the motion of the system should approach zero after sufficient time has transpired. This

dissipation in motion would require that the elements of the operator matrix approach values of

zero when raised to a sufficient power. The raised power operator matrix will approach zero only

if the eigenvalues of the original operator matrix are less than the value of one.

The reason why the eigenvalues of the matrix are indicative of whether or not the values of the

matrix approach zero is explainable using some linear algebra. The eigenvalues of a matrix can

be determined by multiplying this matrix by the inverse of a matrix of eigenvectors on the left

hand side and the same matrix not inverted on the right hand side. This will produce a diagonal

matrix containing all the eigenvalues of the input matrix.

𝑃−1𝐾𝑂 𝑃 =

𝜆1 0 ⋯ 00 𝜆2 ⋯ 0⋮ ⋮ ⋱ ⋮0 0 ⋯ 𝜆𝑚

Raising the left hand side of this equation to a power will generate the following expression.

𝑃−1𝐾𝑂 𝑃 𝑛 = 𝑃−1𝐾𝑂 𝑃 × 𝑃−1𝐾𝑂 𝑃 × 𝑃−1𝐾𝑂 𝑃⋯ = 𝑃−1𝐾𝑂𝑛 𝑃 =

𝜆1

𝑛 0 ⋯ 0

0 𝜆2𝑛 ⋯ 0

⋮ ⋮ ⋱ ⋮0 0 ⋯ 𝜆𝑚

𝑛

This demonstrates that the eigenvalues of the operator matrix raised to a positive integer power

will generate the original eigenvalues raised to the same power. If the free vibration is to

dissipate the eigenvalue of the operator matrix need to be a value under one so that they do not

(3.13)

(3.14)

40

increase with each iteration. The stability assessment will rely on determining these eigenvalues

and checking whether or not they tend towards a value of zero after a large number of iterations.

3.1.7.2. Procedure

To begin determining the operator matrix, the existing time integration equations need to be

expressed along with the equation of motion.

𝑢 𝑖+1 = 𝑢 𝑖 + 𝑕 1 − 𝛾 𝑢 𝑖 + 𝛾 ∙ 𝑢 𝑖+1

𝑢𝑖+1 = 𝑢𝑖 + 𝑕 ∙ 𝑢 𝑖 +𝑕2

2 1 − 2𝛽 𝑢 𝑖 + 2𝛽 ∙ 𝑢 𝑖+1

𝐾 ∙ 𝑢𝑖+1 = 𝑃𝑖+1 −𝑀 ∙ 𝑢 𝑖+1 − 𝐶 ∙ 𝑢 𝑖+1

In this stability analysis 𝑕 is the time step of the time integration scheme, and beta and gamma

are the parameters that define the integration scheme. 𝐾, 𝑀 and 𝐶 are the stiffness, mass and

damping matrix of the two degree-of-freedom respectively. The instantaneous coefficients from

the transform methods are not included in the system matrix in this section.

𝐾 = 𝑘 −𝑘−𝑘 𝑘

, 𝐶 = 𝑐 −𝑐−𝑐 𝑐

, 𝑀 = 𝑚 00 0

From the three equations above, the following relationship is developed:

𝐾 + 𝐶𝛾

𝑕𝛽+ 𝑀

1

𝑕2𝛽 𝑢𝑖+1 = 𝐶

𝛾

𝑕𝛽+ 𝑀

1

𝑕2𝛽 𝑢𝑖 + 𝐶

𝛾

𝛽− 1 + 𝑀

1

𝑕𝛽 𝑢 𝑖 + 𝐶𝑕

𝛾

2𝛽− 1 + 𝑀

1

2𝛽− 1 𝑢 𝑖

This equation is then expressed again with these new terms introduced.

𝐾 𝑢𝑖+1 = 𝑅 𝑢𝑖 + 𝐶 𝑢 𝑖 + 𝑀 𝑢 𝑖

The force term was removed since the stability of the system is independent of the external force.

With the equation of motion and integration scheme integrated into a compact form in Eq. (3.19),

the convolution operation associated with the transformation methods needs to be introduced.

However since the convolution in this example only determines a restoring force at the

foundational degree-of-freedom, the parameters of the convolution need to be rewritten as

matrices.

(3.15a)

(3.15b)

(3.16)

(3.18)

(3.19)

(3.17)

41

𝐾 𝑢𝑖+1 = 𝑅 𝑢𝑖 + 𝐶 𝑢 𝑖 + 𝑀 𝑢 𝑖 + 𝐴𝑗 ∙ 𝑋𝑖+1−𝑗

𝑁

𝑗=0

where 𝐴𝑗 = 0 00 −𝑎𝑗

The 𝑋 variable is an example input parameter describing a system state variable (displacement,

velocity or acceleration) and the 𝑎𝑗 variable represents the coefficient of the force response

function used in the convolution calculation. Typically the first two coefficients of the above

convolution can be incorporated into the other established matrices because they are being

multiplied to the already existing variables, 𝑢𝑖 and 𝑢𝑖+1. After expanding the convolution in Eq.

(3.20) and incorporating the first two terms into the other matrices produces the following

expression.

𝑢𝑖+1 = 𝐾 −1 × 𝑅 𝑢𝑖 + 𝐶 𝑢 𝑖 + 𝑀 𝑢 𝑖 + 𝐴2𝑋𝑖−1 + ⋯+ 𝐴𝑁𝑋𝑖+1−𝑁

Now that the 𝑖 + 1 displacement term has been isolated, it may be substituted into Eq. (3.15a)

and (3.15b) in order to determine the 𝑖 + 1 velocity and acceleration terms. This allows for the

computation of the operator matrix, 𝐾𝑂, however the inclusion of the convolution terms in the

time integration scheme no longer allows the combined algorithm to be expressed as a single

matrix multiplication. This is due to the presence of state variables associated to time steps

before the current and past time steps in the expression. The determination of displacement,

velocity and acceleration terms at time step 𝑖 + 1 allows for the assembling of the operator

matrix which is being multiplied to the general state variable 𝑈 as displayed in Eq. (3.22). The

𝐴0 matrix contains the information pertaining to the remaining convolution coefficients present

in Eq. (3.20).

𝑈𝑖+1 = 𝑢𝑢 𝑢

𝑖+1

= 𝐾0 × 𝑈𝑖 + 𝑎2 ∙ 𝐴0 × 𝑈𝑖−1 + 𝑎3 ∙ 𝐴0 × 𝑈𝑖−2 + ⋯+ 𝑎𝑁 ∙ 𝐴0 × 𝑈𝑖+1−𝑁

When describing the stability of just the time integration scheme the algorithm can be described

by a single matrix multiplication. All future time steps can be therefore be expressed as a product

of the operator matrix 𝐾0 raised to a power positive integer and multiplied by the initial

displacement as demonstrated below.

(3.20)

(3.21)

(3.22)

42

𝑈1 = 𝐾0 × 𝑈0

𝑈2 = 𝐾0 × 𝑈1 = 𝐾02 × 𝑈0

𝑈3 = 𝐾03 × 𝑈0

⋮

𝑈𝑚 = 𝐾0𝑚 × 𝑈0

As stated before as long as the eigenvalues of the operator matrix remains below one then when

the matrix is raised to an exponential, it will decrease in value over time which is indicative that

the system response is dissipating and approaching zero. For the combined algorithm containing

the numerical time integration and the transform convolution, the matrix that expresses the

current state in terms of the original state changes at each time step and it needs to be computed

at each iteration. This matrix will be onwards referred to as the compounded iteration matrix,

𝐾𝐶, since it is no longer equivalent to the previously described single operator matrix, 𝐾0. The

eigenvalues of this matrix vary over time but if they eventually tend to zero after many iterations

have transpired than the algorithm can be considered to be stable. The equations below show

how the compounded iteration matrix changes at each iteration and how it deviates from the

operator matrix 𝐾0.

𝑈1 = 𝐾0 × 𝑈0

𝑈2 = 𝐾0 × 𝑈1 + 𝑎2 ∙ 𝐴0 × 𝑈0 = (𝐾02 + 𝑎2 ∙ 𝐴0) × 𝑈0

𝑈3 = 𝐾0 × 𝑈2 + 𝑎2 ∙ 𝐴0 × 𝑈1 + 𝑎1 ∙ 𝐴0 × 𝑈0 = (𝐾03 + 2 ∙ 𝑎2 ∙ 𝐾0 × 𝐴0 + 𝑎1 ∙ 𝐴0) × 𝑈0

⋮

𝑈𝑚 = (𝐾0𝑚 + ⋯ + 𝑎1 ∙ 𝐴0) × 𝑈0

Therefore,

𝐾𝐶1 = 𝐾0

𝐾𝐶2 = (𝐾02 + 𝑎2 ∙ 𝐴0)

𝐾𝐶3 = (𝐾03 + 2 ∙ 𝑎2 ∙ 𝐾0 × 𝐴0 + 𝑎1 ∙ 𝐴0)

⋮

𝐾𝐶𝑚 = (𝐾0𝑚 + ⋯ + 𝑎1 ∙ 𝐴0)

The value of 𝑚 is the number of time steps that have transpired which is the same as having ran

that many iterations in a simulation. It was decided that the value of 𝑚 should be chosen to be at

least three times larger than the number of parameters in the convolution in order to determine

the lasting numerical stability. The maximum eigenvalue of the compounded iteration matrix at

the 𝑚th iteration should be less than one and approaching zero.

(3.23)

(3.24)

43

3.2. Inverse Fourier Transformation (Wolf, 1985)

The most straightforward method to represent a soil domain’s frequency dependent impedance

function in the time domain would be to determine the inverse Fourier transform of the

impedance function. The transform solution generated from this method is a function that is the

directly analogous to the impedance function but in time domain. This solution function is the

force response function and it describes the value of reaction force over a time duration reacting

to an impulse displacement. In the numerical model the foundation displacement at a given time

step will have lasting ramification on restoring forces in the future as described by the force

response impulse function and this idea is displayed visually in the figure below. This idea is

similar to the concept of a displacement impulse response responding to an impulse force used in

the Duhamel’s integral.

Figure 3.5 The concept of impulse force response

The restoring force therefore is determined by multiplying the previous displacement history by

the respective term in the force impulse response; provided that the time step in the time

integration and impulse response are the same. This calculation is equivalent to performing a

convolution of the force impulse response coefficients 𝑎𝑗 and the displacement response history.

𝐹𝑟𝑒𝑠𝑡𝑜𝑟𝑖𝑛𝑔 𝑡 = 𝑎𝑗 ∙ 𝑢 𝑡 − 𝑗 ∙ Δ𝑡

𝑁

𝑗=0

(3.25)

44

3.2.1. Coefficient Formulation

The coefficients in the force response function were at first determined by performing the inverse

Fourier transform on the impedance function; however this led to inaccuracies in the resulting

solution when compared to the reference solution. A more rigorous and precise method to

determine the parameters was developed following a similar procedure conducted in Nakamura’s

method (Nakamura, 2006a).

This procedure requires the displacement to be expressed as a combination of a sinusoidal

function and an imaginary cosine function. The imaginary portion constitutes the phase shift of

the displacement response. The combination of sinusoidal and cosine functions can be expressed

simply as a complex exponential as demonstrated in Eq. (3.26).

𝑢 𝑡 = sin 𝜔𝑡 + 𝑖 ∙ cos 𝜔𝑡 = 𝑒𝑖𝜔𝑡

Eq. (3.25) can be rewritten using this displacement definition and simplified by factoring out the

complex exponential. This exponential in front of the summation term is in fact the displacement

definition introduced in Eq. (3.26) which can be divided out in order to formulate the expression

for the impedance function in frequency domain as presented in Eq. (3.28).

𝐹𝑟𝑒𝑠𝑡𝑜𝑟𝑖𝑛𝑔 𝜔 = 𝑎𝑗 ∙ 𝑒𝑖𝜔 𝑡−𝑗 ∙Δ𝑡

𝑁

𝑗=0

= 𝑒𝑖𝜔𝑡 𝑎𝑗 ∙ 𝑒−𝑖𝜔 ∙𝑗Δ𝑡

𝑁

𝑗=0

= 𝑢 𝜔 𝑎𝑗 ∙ 𝑒−𝑖𝜔∙𝑗Δ𝑡

𝑁

𝑗=0

𝑆 𝜔 = 𝑎𝑗

𝑁

𝑗=0

∙ 𝑒−𝑖𝜔𝑗 ∆𝑡

Using the exponential complex trigonometric identities, the above equation can be written once

more as a summation of cosines and sine terms and subsequently the real and imaginary part of

the impedance function can be separated. The summation term is replaced by the matrix

multiplication of those trigonometric functions and a vector of the undetermined coefficients.

𝑆 𝜔𝑖 = 𝑅𝑒𝑎𝑙 𝑆 𝜔𝑖

𝐼𝑚𝑎𝑔 𝑆 𝜔𝑖 =

𝑎𝑗 ∙ 𝑐𝑜𝑠 𝜃𝑖𝑗

𝑁−1

𝑗=0

− 𝑎𝑗 ∙ 𝑠𝑖𝑛 𝜃𝑖𝑗

𝑁−1

𝑗=0

(3.26)

(3.27)

(3.28)

(3.29)

45

Where 𝜃𝑖𝑗 = 𝜔𝑖 ∙ 𝑡𝑗 = 𝜔𝑖 ∙ 𝑗 ∙ Δ𝑡

𝑅𝑒𝑎𝑙 𝑆 𝜔𝑖


𝑐𝑜𝑠 𝜃𝑖0 𝑐𝑜𝑠 𝜃𝑖1 ⋯

−sin 𝜃𝑖0 −sin 𝜃𝑖1 ⋯

𝑐𝑜𝑠 𝜃𝑖 ,𝑁−1

−sin 𝜃𝑖 ,𝑁−1 ×

𝑎0

𝑎1

⋮𝑎𝑁−1

As of Eq. (3.30) the matrix represents a linear system of two equations with 𝑁 − 1 unknown

coefficients. This equation can be repeated for each frequency used in the impedance function

resulting in 2𝑀 number of equations where 𝑀 is the number of impedance function data points.

Selecting the number of coefficients, 𝑁, to be half the number of impedance data points will

generate a square matrix and allow a solution for the parameters to be determined through the

use of matrix inversion.

𝑆 𝜔0

𝑆 𝜔1 ⋮

𝑆 𝜔𝑀

=

𝑅𝑒𝑎𝑙 𝑆 𝜔0

𝐼𝑚𝑎𝑔 𝑆 𝜔0

𝑅𝑒𝑎𝑙 𝑆 𝜔1

𝐼𝑚𝑎𝑔 𝑆 𝜔1 ⋮

𝑅𝑒𝑎𝑙 𝑆 𝜔𝑀

𝐼𝑚𝑎𝑔 𝑆 𝜔𝑀

=

+𝑐𝑜𝑠 𝜃0,0 +𝑐𝑜𝑠 𝜃0,1 ⋯ 𝑐𝑜𝑠 𝜃0,𝑁−1

− sin 𝜃0,0 − sin 𝜃0,1 ⋯ − sin 𝜃0,𝑁−1

+𝑐𝑜𝑠 𝜃1,0 +𝑐𝑜𝑠 𝜃1,1 ⋯ 𝑐𝑜𝑠 𝜃1,𝑁−1

− sin 𝜃1,0 − sin 𝜃1,1 ⋯ − sin 𝜃1,𝑁−1

⋮ ⋮ ⋱ ⋮+𝑐𝑜𝑠 𝜃𝑀,0 +𝑐𝑜𝑠 𝜃𝑀 ,1 ⋯ 𝑐𝑜𝑠 𝜃0,0

− sin 𝜃𝑀,0 − sin 𝜃𝑀 ,1 ⋯ − sin 𝜃𝑀 ,0

×

𝑎0

𝑎1

⋮𝑎𝑁−1

This procedure however will result in only half the impedance function data being used in the

formation of the coefficients. This is not a problem as long as that half of the impedance function

data constitutes a frequency range that includes the frequencies of interests for the seismic

analysis.

3.2.2. Impedance Function Parameters

The impedance function utilized in the two validation example cases use different parameters

since critical impedance data occurs in different frequency ranges. The stiff soil’s natural

frequencies all occur within the first 20 Hz while the soft soil’s natural frequencies occur in

frequencies smaller than 1 Hz. It is important to characterize these regions adequately with a

small enough frequency step to capture the behaviour around these natural frequencies.

It was decided that both impedance functions are to contain 500 data points and consequently the

stiff soil was selected to range from 0 to 100 Hz, resulting in a 𝑑𝑡 of 0.01 s and a frequency step

(3.30)

(3.31)

46

of 0.2 Hz. The soft soil case used a frequency range from 0 to 5 Hz which results in a 𝑑𝑡 of 0.2 s

and a frequency step of 0.01 Hz.

Having defined the impedance functions the coefficients for the inverse Fourier Transform

method can be determined by the procedure described in Section 3.2.1. Consequently the force

response function contains 250 coefficients, which corresponds to half the number of the data

present in the impedance function.

3.2.3. Reproducing the Impedance Function

In order to assess whether the coefficients determined using this method accurately represents the

impedance functions, the effective impedance functions will be generated from the coefficients.

This effective function will be compared to the desired impedance and if the coefficients are

found to be acceptable then they may be utilized in the combined integration scheme.

The impedance function may be determined using the 𝒵-tranform or by numerical means as

described in Section 3.1.4.1 of this chapter. However performing the 𝒵-tranform will not provide

significant information on the accuracy of the coefficients in the time domain since it strictly

performs analysis in the frequency domain. The numerical generation of the impedance function

will on the other hand demonstrate whether the convolution of the prescribed displacement and

the force response coefficients generate the desired dynamic stiffness behaviour in the time

domain. Numerical time integration is not involved in this analysis because the displacement

history is defined in advance. Therefore any stability problems this transform may have when

combined with the time integration scheme will not be exposed here however this procedure

does ensure that the convolution operation itself is stable.

Both the desired impedance function and the numerically generated impedance functions are

presented below in Figure 3.6 and Figure 3.7, and they correspond meaningfully. In these figures

the desired impedance function is labelled as the ‘actual’ impedance. The maximum frequency in

these graphs is half of the maximum frequency specified previously because only half the data

was used in the formulation of the coefficients.

47

Figure 3.6 Reproduced Impedance of the Stiff Soil Case for the Inverse Fourier Transform

Figure 3.7 Reproduced Impedance of the Soft Soil Case for the Inverse Fourier Transform

0 10 20 30 40 50-3

-2

-1

0

1x 10

7

Impedance [

Real]

Impedance Comparison

Actual

Numerical

0 10 20 30 40 500

0.5

1

1.5

2x 10

6

Frq. [sec-1]

Impedance [

Imagin

ary

]

0 0.5 1 1.5 2 2.5-15

-10

-5

0

5x 10

4

Impedance [

Real]


Actual

Numerical

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2x 10

4

Frq. [sec-1]

Impedance [

Imagin

ary

]

48

3.2.4. Simulation Results

Having successfully reproduced the impedance functions numerically; the simulation of the

reference structural example can now be conducted. The reference structure that was previously

described consists of a single storey structure resting on a soil model consisting of ten nodes. In

the investigation simulation the soil domain’s influence on the structure is captured by an applied

restoring force. This force determined by the convolution calculation as described in Section 3.2

using the determined coefficients in the force response function.

The simulation of the system using the inverse Fourier Transformation model exhibited

divergent unstable response for both soil cases and the results of the simulation are presented

below in Figure 3.8. The soft soil structural response increased somewhat exponentially and the

stiff soil response increased rapidly but alternating between positive and negative values. This is

indicative that the developed transform coefficients are incompatible with the given time

integration scheme resulting in an unstable response behaviour. This does not necessarily imply

that this method is unable to transform every impedance function; however for this system it was

inadequate and the method proved to lack general applicability.

0 10 20 30 40 50-4

-3

-2

-1

0

1

2x 10

300

Time [sec]

Dis

pla

cem

ent

[m]

Kobe Earthquake - Stiff Soil

Rigid

Inverse Fourier

Total

Figure 3.8 Displacement Responses of the Total and inverse Fourier Models of the

Validation Example undergoing the Kobe Earthquake Loading.

0 10 20 30 40 50-2

0

2

4

6

8

10

12

14x 10

31

Time [sec]

Dis

pla

cem

ent

[m]

Kobe Earthquake - Soft Soil

Rigid

Inverse Fourier

Total

49

3.2.5. Stability

From the failed simulation of the soil domain it becomes clear that the stability assessment

described in Section 3.1.7 of this chapter is necessary to determine if this method is stable when

utilized to transform other impedance functions. This assessment procedure will allow

researchers and other practitioners to be able to determine whether a determined set of transform

coefficients will be stable when used in the convolution calculation in a simulation. Conducting

this assessment on the coefficients used in the simulations in Section 3.2.4 and demonstrating the

given algorithm is unstable, will corroborate that the resulting divergent response was due to the

coefficient’s inherent instability rather than error in implementation.

The procedure was introduced in Section 3.1.7.2; however the equations found there need to be

altered to represent the convolution used in this transformation method. Matrix manipulation has

to be performed to determine an expression for the operator matrix, 𝐾0, and the compounded

iteration matrix, 𝐾𝐶. These alterations begin by replacing the general convolution variable 𝑋 in

Eq. (3.20) with the displacement which is used in this method’s convolution.

𝐾 𝑢𝑖+1 = 𝑅 𝑢𝑖 + 𝐶 𝑢 𝑖 + 𝑀 𝑢 𝑖 + 𝐴0𝑢𝑖+1 + 𝐴1𝑢𝑖 + 𝐴2𝑢𝑖−1 + ⋯+ 𝐴𝑁𝑢𝑖+1−𝑁

The first two terms of the convolution can be factored into the 𝐾 and 𝑅 matrices since they are

multiplied by the same displacement terms. The new effective system matrices are defined now

as the following.

𝐾 = 𝐾 + 𝐶𝛾

𝑕𝛽+ 𝑀

1

𝑕2𝛽− 𝐴0

𝑅 = 𝐶𝛾

𝑕𝛽+ 𝑀

1

𝑕2𝛽+ 𝐴1

𝐶 = 𝐶 𝛾

𝛽− 1 + 𝑀

1

𝑕𝛽

𝑀 = 𝐶𝑕 𝛾

2𝛽− 1 + 𝑀

1

2𝛽− 1

In order to express the time integration scheme as a single matrix operation the 𝐾 matrix is

inverted and transferred to the right hand side, and the displacements terms on the right hand side

of Eq. (3.32) have to be replaced by the general system variable 𝑈 which is a vector of

displacement, velocity and acceleration. To accomplish this the terms in front of the

(3.32)

(3.33)

50

displacement variable need to be reconstructed as new matrices that correspond to this variable

exchange.

𝑢𝑖+1 = 𝕊𝑢 ∙ 𝜌 × 𝑈𝑖 + 0 −𝑕2𝛽 𝑕𝑘𝛽 + 𝑐𝛾

0 −𝑕𝛽 𝑚 + 𝑕 𝑕𝑘𝛽 + 𝑐𝛾

0 0 0 0

0 0 0 0 𝜌 ∙ 𝑎2 × 𝑈𝑖−1 + ⋯

𝕊𝑢 is a dense 2×6 matrix that is defined in Appendix A and 𝜌 is expressed below.

𝜌 =1

𝑚 ∙ 𝑕𝑘𝛽 + 𝑐𝛾 + 𝑎0 ∙ 𝑕𝛽 ∙ 𝑚 + 𝑕 ∙ 𝑕𝑘𝛽 + 𝑐𝛾

The matrix and 𝜌 term in front in Eq. (3.34a) in front of the 𝑖 − 1 system state variable term

would appear in front of system state variable at time steps further in the past. The transform

coefficient 𝑎𝑖 however is different for each time step which does not allow further simplification

of Eq. (3.35).

Having expressed the displacement at the next time step in isolation, it may be substituted into

Eq. (3.15a) and (3.15b) in order to determine the expressions for the velocity and acceleration of

the time step 𝑖 + 1 in terms of previous state variables. Doing so produces the two equations in

Eq. (3.34) presented below. The definitions of the matrices 𝕊𝑣 and 𝕊𝑎 have been presented in

Appendix A rather than in this section because their expressions are large and cumbersome to

demonstrate.

𝑢 𝑖+1 = 𝕊𝑣 ∙ 𝜌 × 𝑈𝑖 + 0 −𝑕𝛾(𝑕𝑘𝛽 + 𝑐𝛾)0 −𝛾(𝑚 + 𝑕(𝑕𝑘𝛽 + 𝑐𝛾))

0 0 0 0

0 0 0 0 𝜌 ∙ 𝑎2 × 𝑈𝑖−1 + ⋯

𝑢 𝑖+1 = 𝕊𝑎 ∙ 𝜌 × 𝑈𝑖 +

0 −𝑕𝑘𝛽 − 𝑐𝛾

0 −𝑚 + 𝑕(𝑕𝑘𝛽 + 𝑐𝛾)

𝑕

0 0 0 0

0 0 0 0 𝜌 ∙ 𝑎2 × 𝑈𝑖−1 + ⋯

The displacement, velocity and acceleration terms of the step 𝑖 + 1 have now been explicitly

expressed in Eq. (3.34) allowing the expression for the system state variable 𝑈 for step 𝑖 + 1 to

be assembled. Doing so results in computing the operator matrix, 𝐾0, from the matrices 𝕊𝑢 , 𝕊𝑣

and 𝕊𝑎 . New matrices need to be assembled from the matrices present in Eq. (3.34) in front of

the state variables of past time steps. The new expression formulated has the following form.

(3.34a)

(3.34b)

(3.34c)

(3.35)

51

(3.36) 𝑈𝑖+1 = 𝐾0 × 𝑈𝑖 + 𝑎2 ∙ 𝐴0 × 𝑈𝑖−1 + 𝑎3 ∙ 𝐴0 × 𝑈𝑖−2 + ⋯

Having assembled the operator matrix and the 𝐴0 matrix, all the variables needed to perform the

stability analysis have been determined. The analysis requires the compounded iteration matrix,

𝐾𝐶, to be determined at each iteration and the eigenvalues of this matrix will be monitored.

Since the compounded iteration matrix is of the order of six, there exist always six unique

eigenvalues which correspond to the different lines in Figure 3.9. Should the largest eigenvalue

increase above the value of one as the iterations continue, this will be indicative that the given

the transform coefficients produce an unstable algorithm if incorporated into the time integration

scheme.

The above eigenvalue curves were calculated using the transform coefficients used in the

previous simulation and 1000 iterations were performed in order to encompass the 500

coefficients used in each force response function. It was expected that the result of this analysis

would indicate the resultant algorithm is unstable since the simulations too exhibited divergent

behaviour.

The stability analysis plots demonstrate that the eigenvalues increase with each iteration and thus

suggest the combined convolution and time integration algorithm is unstable. The plots allow

one to infer the relative instability of the two sets of coefficients by comparing how large the

eigenvalues became by the 1000th

iteration. The soft soil analysis produce a maximum

Figure 3.9 Stability Analysis Curves of the Inverse Fourier Model for both Stiff and Soft

Soil Impedance Examples

0 200 400 600 800 10000

0.5

1

1.5

2

2.5

3

Iterations

Eig

envalu

e A

mplit

ude

Stability Analysis - Soft Soil Case

0 200 400 600 800 10000

2

4

6

8

10

12

14x 10

108

Iterations

Eig

envalu

e A

mplit

ude

Stability Analysis - Stiff Soil Case

52

eigenvalue within an order of magnitude of ten while the stiff soil transform coefficients

produced eigenvalues that were exponentially growing with each iteration. This indicates that

perhaps the soft soil impedance transformation was not too far from being a stable system but

this provides no indication of how the coefficients need be changed in order to obtain that

stability.

Specific stability issues affecting the inverse Fourier method are known in literature are

discussed in depth in chapter four since they also afflict other methods as well. These stability

concerns are linked to the characteristics of the impedance functions used in the analysis which

affects the stability of the impedance’s transform in the time domain.

3.3. Nakamura Model

While the inverse Fourier Transform method possesses the most straightforward implementation

since it only involves a single convolution, its stability issues left researchers seeking

alternatives. The method investigated in this section was first proposed by Nakamura in 2006 and

it proposes to represent the impedance function in the time domain with a model similar to real

physical systems by including mass, stiffness and damper components. This method proved to be

the most reliable of the ones investigated in this study and was used for subsequent validation

examples.

3.3.1. Model Definition

Building on previous research Nakamura had done involving the cone-model, his transform

model proposes that the restoring force be calculated using two convolutions involving both the

displacement history and the velocity history. This would require that the impedance function be

decomposed into two separate force response functions responding to impulse displacement and

velocity. The restoring force is therefore dependent on the displacement history and on the

history of velocity (Nakamura, 2006a). This method was improved by introducing an

instantaneous mass component which would require the restoring force being also dependent on

the current acceleration at the foundation degree-of-freedom (Nakamura, 2006b). This is the

model that is used in this chapter and in the remainder of this thesis.

The restoring force is defined in Eq. (3.37) where the instantaneous variables have been

separated from the rest of the convolution calculation.

53

𝐹𝑟𝑒𝑠𝑡 𝑡 = 𝑘0 ∙ 𝑢 𝑡 + 𝑐0 ∙ 𝑢 𝑡 + 𝑚0 ∙ 𝑢 𝑡 + 𝑘𝑗 ∙ 𝑢 𝑡 − 𝑡𝑗

𝑁−1

𝑗=1

+ 𝑐𝑗 ∙ 𝑢 𝑡 − 𝑡𝑗

𝑁−1

𝑗=1


In order to determine the transform coefficients in this method the convolution needs to be

expressed in the frequency domain similar to how the parameters of the inverse Fourier

transform method were determined. This is accomplished by expressing the displacement,

velocity and acceleration terms by using the complex exponential function, 𝑒𝑖𝜔𝑡 . Substituting

these new functions and dividing out the common displacement exponential forms the dynamic

stiffness term on the left hand side and produces the following expression in the frequency

domain.

𝑆 𝜔𝑖 = −𝜔𝑖2 ∙ 𝑚0 + 𝑖𝜔𝑖 ∙ 𝑐0 + 𝑘0 + 𝑘𝑗 ∙ 𝑒

𝑖𝜔 𝑖𝑡𝑗

𝑁−1

𝑗=0

+ 𝑖𝜔 ∙ 𝑐𝑗 ∙ 𝑒𝑖𝜔 𝑖𝑡𝑗

𝑁−2

𝑗=0

Separating the real and imaginary components of the impedance function data and of the right

hand side of Eq. (3.38) allows this expression to be decomposed into two separate equations.

This new equation is presented below with the complex exponential being replaced with

equivalent sinusoidal and cosine functions.

𝑆 𝜔𝑖 = 𝑅𝑒𝑎𝑙 𝑆 𝜔𝑖


−𝜔𝑖

2 ∙ 𝑚0 + 𝑘𝑗 ∙ 𝑐𝑜𝑠 𝜃𝑖𝑗

𝑁−1

𝑗=0

+ 𝜔𝑖 ∙ 𝑐𝑗 ∙ 𝑠𝑖𝑛 𝜃𝑖𝑗

𝑁−2

𝑗=0

− 𝑘𝑗 ∙ 𝑠𝑖𝑛 𝜃𝑖𝑗

𝑁−1

𝑗=0

+ 𝜔𝑖 ∙ 𝑐𝑗 ∙ 𝑐𝑜𝑠 𝜃𝑖𝑗

𝑁−2

𝑗=0


The above equation is rewritten by replacing the summation terms with matrix multiplication.

𝑅𝑒𝑎𝑙 𝑆 𝜔𝑖 = 𝑐𝑜𝑠 𝜃𝑖0 𝑐𝑜𝑠 𝜃𝑖1 ⋯ × 𝑘0

𝑘1

⋮

+ 𝜔𝑖 ∙ 𝑠𝑖𝑛 𝜃𝑖0 𝜔𝑖 ∙ 𝑠𝑖𝑛 𝜃𝑖1 ⋯ × 𝑐0

𝑐1

⋮ + −𝜔𝑖

2 × 𝑚0

(3.37)

(3.38)

(3.39)

54

𝐼𝑚𝑎𝑔 𝑆 𝜔𝑖 = −𝑠𝑖𝑛 𝜃𝑖0 −𝑠𝑖𝑛 𝜃𝑖1 ⋯ × 𝑘0

𝑘1

⋮

+ 𝜔𝑖 ∙ 𝑐𝑜𝑠 𝜃𝑖0 𝜔𝑖 ∙ 𝑐𝑜𝑠 𝜃𝑖1 ⋯ × 𝑐0

𝑐1

⋮

The summation of these matrices can be simplified into a single matrix multiplication presented

below:


𝐼𝑚𝑎𝑔 𝑆 𝜔𝑖 = 𝐶𝑘𝑖 ,0 ⋯ 𝐶𝑘𝑖 ,𝑁−1 𝐶𝑐𝑖,0 ⋯ 𝐶𝑐𝑖,𝑁−2 𝐶𝑚𝑖 ×

𝑘0

⋮𝑘𝑁−1𝑐0

⋮𝑐𝑁−2𝑚0

Where:

𝐶𝑘𝑖 ,𝑗 = 𝑐𝑜𝑠 𝜃𝑖𝑗

−𝑠𝑖𝑛 𝜃𝑖𝑗 , 𝐶𝑐𝑖,𝑗 =

𝜔𝑖 ∙ 𝑠𝑖𝑛 𝜃𝑖𝑗

𝜔𝑖 ∙ 𝑐𝑜𝑠 𝜃𝑖𝑗 , 𝐶𝑚𝑖 = −𝜔𝑖

2

0

This equation can be assembled for each frequency in the impedance function data in order to

form a linear system of 2𝑀 equations, with 2𝑁 unknowns:

𝑆 𝜔0

𝑆 𝜔1 ⋮

𝑆 𝜔𝑀

=

𝐶𝑘0,0

𝐶𝑘1,0

⋮𝐶𝑘𝑀,0

⋯⋯⋱⋯

𝐶𝑘0,𝑁−1

𝐶𝑘1,𝑁−1

⋮𝐶𝑘𝑀,𝑁−1

𝐶𝑐0,0

𝐶𝑐1,0

⋮𝐶𝑐𝑀,0

⋯⋯⋱⋯

𝐶𝑐0,𝑁−2

𝐶𝑐1,𝑁−2

⋮𝐶𝑐𝑀,𝑁−2

𝐶𝑚0

𝐶𝑚1

⋮𝐶𝑚𝑀

×

𝑘0

⋮𝑘𝑁−1𝑐0

⋮𝑐𝑁−2𝑚0

Selecting the total number of parameters to be the same as the number of data points ensures that

the matrix in Eq. (3.42) is square and allows the coefficients to be determined by solving that

linear system using matrix inversion.

It was advised that when formulating the impedance function data that the static zero frequency

condition not be included (Nakamura, 2006a). The reason for this is that the inclusion of the zero

frequency data creates a row of zeros in the square matrix in Eq. (3.42). The row of zeros causes

the matrix to be singular and no longer invertible and the coefficients unsolvable. Nakamura

(3.42)

(3.40)

(3.41)

55

(2006a) suggests that instead of using the static case one should input near zero frequency data,

and it was found that it acceptable to use a tenth of the frequency step for this purpose.

3.3.3. Impedance Function Parameters

The parameters of the frequency range of the impedance function will have a direct effect on the

quality of the transform method. The maximum frequency selected in the impedance function

will dictate the time step used in the convolution calculation of the restoring force and the

number of data points in the impedance will determine the number of coefficients in the force

response function.

The stiff soil case can be modelled using the same time step as the simulation so that the

convolution calculation can be implemented simply in the time integration scheme. The number

of data points required for a stable and accurate simulation was found to be around 500 data

points.

The natural periods of the soft soil model are exceptionally long with the largest fundamental

period being 11.5 seconds corresponding to a frequency of 0.0869 Hz. In order to capture this

fundamental period in an impedance function with a maximum frequency of 100 Hz,

corresponding to a time step of 0.01s, the impedance function would require a very small

frequency step and a large number of data points. The stiff soil impedance function utilizes a

frequency step of 0.2 Hz which is too coarse of a resolution to capture the soft soil’s fundamental

period. The parameters that were decided to model the soft soil dynamic stiffness was a

maximum frequency of 5 Hz, corresponding to a time step of 0.2 seconds, and maintaining the

constant 500 data points. These parameters result in a frequency step of 0.01 Hz which proved to

be acceptable.


Using the coefficients calculated by the parameter determining procedure described in Section

3.3.2, the effective impedance function can be calculated and compared to the inputted

impedance function. The confirmation that these two impedance functions are identical will

demonstrate that the coefficients are valid time domain representations of the soil domain. This

reproduction will be conducted using the numerical generation technique described in Section

3.1.4.2 of this chapter.

56

The predescribed displacement, velocity and acceleration histories will be used in the

convolution to calculate the resulting restoring force. This force is then analyzed in relation to

the prescribed displacement and the effective impedance function is determined for each

frequency in the impedance function. The result of this analysis is displayed in Figure 3.10 and it

shows the desired and produced impedance functions are identical. The desired impedance

function is labelled in these figures as ‘Actual’.

In comparison to the impedance generated when using the coefficients from the inverse Fourier

transform method, these coefficients produce an impedance function that has the same frequency

range as the original impedance function. This is because this method contains twice as many

coefficients than the first method.

Figure 3.10 Reproduced Impedance of the Stiff Soil Case for the Nakamura Method

0 20 40 60 80 100-15

-10

-5

0

5x 10

7

Frq. [sec-1]

Impedance [

Real]


Actual

Numerical

0 20 40 60 80 1000

0.5

1

1.5

2x 10

6

Frq. [sec-1]

Impedance [

Imagin

ary

]

57

Figure 3.11 Reproduced Impedance of the Soft Soil Case for the Nakamura Method


The simulations of the reference structure were conducted and compared to the Total model, the

system where the structure and soil are modelled together in the time domain using spring,

damper and mass elements, and the proposed model that uses the Nakamura transform to capture

the effect of the soil domain.

The results of the simulation are presented in Figure 3.12 and Figure 3.13. These plots show that

the Nakamura method excelled in being able to replicate the results of the Total model. There

exists some discrepancies between the amplitudes of the two responses but they are not

disparaging and the overall structure response appears to be in agreement. The soft soil case

generates a response with far greater amplitude and longer periods which is in accord with the

larger periods of the soft soil case as presented in Table 3.2.

0 1 2 3 4 5-6

-4

-2

0

2x 10

5

Frq. [sec-1]

Impedance [

Real]


Actual

Numerical

0 1 2 3 4 50

0.5

1

1.5

2x 10

4

Frq. [sec-1]

Impedance [

Imagin

ary

]

58

Figure 3.12 Total Displacement Responses of the Total and Nakamura Models of the Stiff

Soil Example undergoing the Kobe Earthquake Loading

Figure 3.13 Total Displacement Responses of the Total and Nakamura Models of the Soft

Soil Example undergoing the Kobe Earthquake Loading

59

The accurate representation of these two cases show that with the correct parameters defining the

impedance function, the soil domain can be accurately represented using Nakamura’s transform

method.

3.3.6. Stability Assessment

Though the simulation proved to be stable it is important to compose the stability analysis for

Nakamura’s method in order to allow others in the future to assess this transform’s stability when

implemented using a completely different impedance function.

This method contains instantaneous stiffness, damping and mass coefficients that need to be

incorporated into the two node stability model’s system matrices. The combined structural and

soil stiffness and combined damping coefficients have been replaced by a single variable to make

the equations in this section more compact.

𝐾 = 𝑘 −𝑘−𝑘 𝑘 + 𝑘0

, 𝐶 = 𝑐 −𝑐−𝑐 𝑐 + 𝑐0

, 𝑀 = 𝑚 00 𝑚0

𝐾 = 𝑘 −𝑘−𝑘 𝑘

, 𝐶 = 𝑐 −𝑐−𝑐 𝑐 + 𝑐

, 𝑀 = 𝑚 00 𝑚0

These new definitions of the matrices lead to the following equation of motion with the

convolution summations starting at 𝑗 = 1 rather than at zero.

𝐾 ∙ 𝑢𝑖+1 = 𝑃𝑖+1 −𝑀 ∙ 𝑢 𝑖+1 − 𝐶 ∙ 𝑢 𝑖+1 − 𝐾𝑗 ∙ 𝑢𝑖+1−𝑗 − 𝐶𝑗 ∙ 𝑢 𝑖+1−𝑗

𝑁−1

𝑗=1

𝑁−1

𝑗=1

Using Eq. (3.15) to substitute the values of the acceleration and velocity of time step 𝑖 + 1 the

displacement of the time step 𝑖 + 1 can be wholly expressed in terms of the variables of previous

time steps. The external force is removed in order to only consider the response of the system in

free vibration. These actions result in the formulation of Eq. (3.45).

𝐾 𝑢𝑖+1 = 𝑅 𝑢𝑖 + 𝐶 𝑢 𝑖 + 𝑀 𝑢 𝑖 − 𝐾𝑗 ∙ 𝑢𝑖+1−𝑗 − 𝐶𝑗 ∙ 𝑢 𝑖+1−𝑗

𝑁−1

𝑗=2

𝑁−1

𝑗=2

(3.44)

(3.45)

(3.43)

60

Where:

𝐾 = 𝐾 + 𝐶𝛾

𝑕𝛽+ 𝑀

1

𝑕2𝛽

𝑅 = 𝐶𝛾

𝑕𝛽+ 𝑀

1

𝑕2𝛽− 𝐾1

𝐶 = 𝐶 𝛾

𝛽− 1 + 𝑀

1

𝑕𝛽− 𝐶1

𝑀 = 𝐶𝑕 𝛾

2𝛽− 1 + 𝑀

1

2𝛽− 1

The displacement, velocity and acceleration terms in the right hand side of Eq. (3.45) are

substituted with the general state variable 𝑈 which is a vector that contains those three variables

for the two degrees-of-freedom system. In order to perform this substitution all the terms of the

matrices in front of variables acting at the same time step need to be assembled together to form

new matrices. Performing this variable substitution and matrix manipulation generates Eq.

(3.47a). With the displacement term for step 𝑖 + 1 being established expressions for velocity and

acceleration at time step 𝑖 + 1 are generated and presented as Eq. (3.47b) and (3.47c).

𝑢𝑖+1 = 𝕊𝑢 ∙ 𝜌 × 𝑈𝑖 + 𝕂𝑢 ∙ 𝜌𝑘2 × 𝑈𝑖−1 + ⋯+ ℂ𝑢 ∙ 𝜌𝑐2 × 𝑈𝑖−1 + ⋯

𝑢 𝑖+1 = 𝕊𝑣 ∙ 𝜌 × 𝑈𝑖 + 𝕂𝑣 ∙ 𝜌𝑘2 × 𝑈𝑖−1 + ⋯+ ℂ𝑣 ∙ 𝜌𝑐2 × 𝑈𝑖−1 + ⋯

𝑢 𝑖+1 = 𝕊𝑎 ∙ 𝜌 × 𝑈𝑖 + 𝕂𝑎 ∙ 𝜌𝑘2 × 𝑈𝑖−1 + ⋯+ ℂ𝑎 ∙ 𝜌𝑐2 × 𝑈𝑖−1 + ⋯

The double lined characters in these equations represent 2×6 matrices whose definitions are

found in Appendix A. The left hand side of these three equations creates the general state

variable, 𝑈, at time step 𝑖 + 1. These three expressions therefore can be united into a single

equation by combining the various matrices into larger 6×6 matrices which results in the

following desired expression that characterizes the combined integration scheme.

𝑈𝑖+1 = 𝐾0 × 𝑈𝑖 + (𝑘2 ∙ 𝐷0 + 𝑐2 ∙ 𝑉0) × 𝑈𝑖−1 + (𝑘3 ∙ 𝐷0 + 𝑐3 ∙ 𝑉0) × 𝑈𝑖−2 + ⋯

Starting with time step 𝑖 = 0, the compounded iteration matrix, 𝐾𝐶, can now be determined

using Eq. (3.48) for 𝑚 number of iterations. The constant 𝑚 is taken to be three times larger than

(3.47a)

(3.47b)

(3.47c)

(3.48)

(3.46)

61

the number of parameters, resulting in 1500 iterations in this stability analysis since the

Nakamura models used 500 coefficients.

Figure 3.14 and Figure 3.15 present the eigenvalues of the compounded iteration matrix plotted

against the number of iterations for both the stiff and soft soil simulation cases. In these figures

the different coloured lines represents each of the six eigenvalues though some may not be

visible since their values are approach zero or they are overlapped by other lines. It is evident

that the combined time integration scheme and convolution algorithm produce stable algorithm

results for the given transform coefficients, analysis time step and time integration parameters

since the eigenvalues remain below a value of unity.

Figure 3.14 Stability Analysis Curve of the Nakamura Model for the Stiff Soil Example

0 500 1000 15000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Eig

envalu

e A

mplit

ude

Iterations

Stability Analysis - Case 1

62

Figure 3.15 Stability Analysis Curve of the Nakamura Model for the Soft Soil Example

3.4. Şafak Model (2005)

Recently researchers have recognized that analogous concepts exist between seismic simulations

and digital signal processing leading to the proposal of transform methods already established in

that electrical engineering field. The last method considered is one such transform method and it

relies on implementing an infinite impulse response filter (Şafak, 2006). This transform method

generates reaction forces that are dependent on the value of the reaction force at previous time

steps in addition to being dependent on the foundation displacement history. Due to this

recursive nature, the convolution calculation can produce a force impulse response of infinite

duration which is why the filter is called an infinite impulse response.

The convolution requires the formation of two force response functions; one function is

analogous to stiffness and is dependent on the displacement history of the foundation degree-of-

freedom and is represented with the variable 𝑏𝑘 . The other function describes the dependency of

the current restoring value to the value of previous restoring forces and is labelled 𝑎𝑘 . This

second function has little physical meaning since it is a purely mathematical construct. The

convolution that calculates the restoring force is as follows:

0 500 1000 15000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Eig

envalu

e A

mplit

ude

Iterations


63

𝐹𝑟𝑒𝑠𝑡𝑜𝑟𝑖𝑛𝑔 𝑛∆𝑡 = − 𝑎𝑖𝐹𝑟𝑒𝑠𝑡𝑜𝑟𝑖𝑛𝑔 𝑛∆𝑡 − 𝑗∆𝜏

𝑀

𝑗=1

+ 𝑏𝑖𝑢 𝑛∆𝑡 − 𝑗∆𝜏

𝑁

𝑗=0

The previous methods were somewhat recursive because the current foundation displacement

was dependent on the restoring force being applied which in turn was calculated using previous

foundation displacements. This method however is explicitly recursive since the calculation of

the restoring force is dependent on the value of the previous restoring forces. This method in fact

can easily produce coefficients that violate Bounded-Input-Bounded-Output (BIBO) stability

(Şafak, 2006). This signifies that even if the input is well define and bounded, the output of the

convolution may not. It is possible therefore that the convolution calculation produces unstable

results even when not incorporated into the time integration scheme. This BIBO stability can be

checked by attempting to produce the impedance function numerically from the transform

coefficients.

There exists a definite criterion for the BIBO stability condition (Şafak, 2006) which is

formulated by transforming all the variables in Eq. (3.49) into the 𝒵-domain. The terms within

the convolution can be expressed anew using the time shift identity presented in Section 3.1.4.1

in Eq. (3.6). Executing these steps and transferring the force terms to the left hand side of the

equation produces the following expression.

1 + 𝑎1𝑧−1 + 𝑎2𝑧

−2 + ⋯+ 𝑎M𝑧−𝑀 𝐹 𝑧 = 𝑏0 + 𝑏1𝑧

−1 + 𝑏2𝑧−2 + ⋯+ 𝑏N𝑧

−𝑁 𝑈 𝑧

In this equation the 𝑧 variable represents the same complex exponential as in previous chapters,

𝑒𝑖𝜔 . The ratio between the force and displacement can now be isolated and this ratio is the

definition of the impedance function in the frequency domain.

𝑆 𝑧 =𝐹 𝑧

𝑈 𝑧 = 𝑏0 + 𝑏1𝑧

−1 + 𝑏2𝑧−2 + ⋯+ 𝑏N𝑧

−𝑁

1 + 𝑎1𝑧−1 + 𝑎2𝑧−2 + ⋯+ 𝑎M𝑧−𝑀 =𝐵(𝑧)

𝐴(𝑧)

The transform model’s convolution will be BIBO stable provided that the poles of this

expression are within unity (Şafak, 2006). This is equivalent to ensuring that the absolute zeros

of the denominator of Eq. (3.51) are less than a value of one.

(3.49)

(3.51)

(3.50)

64


The recursive nature of this method provides some challenges in generating a procedure to

determine the transform coefficients. Şafak (2006) proposed a method where the two force

response functions coefficients were determined by performing an optimization technique to

determine the best possible coefficients. A second procedure was developed based on the ideas

and procedures developed by Nakamura (2006b) for his method based on creating a linear

system of equations using the 𝒵-transform of the convolutions.

3.4.1.1. Optimization Scheme

Şafak suggests that a least square optimization scheme be adopted in order to obtain the most

appropriate choice of coefficients. The number of coefficients in the force convolution, 𝑀, is

termed the ‘filter order’ because the filter’s accuracy and stability is much more sensitive to this

convolution that of the displacement convolution. The higher the filter order, the better the

accuracy of the filter but it is more likely to result in an unstable convolution operation. It is

suggested that the optimal filter order is either four or five with anything higher being too

volatile and lower filter orders would generate results insufficiently accurate. In Şafak’s

examples 𝑁 − 1 is always taken to be the same values as 𝑀 , leading to only around ten

coefficients in total that need to be determined.

For a given iteration in the optimization algorithm the effective impedance function

corresponding to the current configuration of coefficients needs to be determined using Eq.

(3.51). This expression is the 𝒵-domain definition of the impedance function produced by the

transform coefficients. This effective impedance function will then be compared to the desired

impedance function. The error associated with a set of given coefficient is the sum of the square

differences of these two impedance functions for all frequencies of interest. The minimization of

this error is the objective function of the optimization algorithm and it is presented below.

𝐸 𝜔𝑖 = 𝑆𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝜔𝑖 − 𝑆𝑒𝑓𝑓𝑒𝑐 𝑡𝑖𝑣𝑒 𝜔𝑖 2

Şafak (2006) suggests that the optimization scheme be performed iteratively, rather than directly.

Using this modified least-square representation in equation (3.53) as the objective function, the

parameters may be more efficiently determined.

(3.52)

65

(3.54)

𝐸 𝜔𝑖 𝑗 = 𝑊 𝜔𝑖

𝐴 𝜔𝑖 𝑗−1 𝐵 𝜔𝑖 𝑗 − 𝐴 𝜔𝑖 𝑗 ∙ 𝑆𝑑𝑒𝑠𝑖𝑟 𝑒𝑑 𝜔𝑖

2

The 𝐴(𝜔) and 𝐵(𝜔) functions are respectively the numerator and denominator functions of Eq.

(3.51). The previous iterations, 𝑗 − 1, values for the 𝐴 functions are used in the least square

determination of the current iteration. The 𝑊 function is a weighting function introduced so that

the errors can be scaled allowing the optimization scheme to be more biased towards lower

frequencies.

In addition to the least square function being the objective of the optimization scheme, the BIBO

stability of this transform method incorporated. At each iteration of the optimization scheme the

coefficients can be checked whether the absolute values of the zeros of the 𝐴 function are less

that the value of one. If this is violated the objective function is penalized severely in order to

ensure that the optimization scheme produces coefficients that create a BIBO stable convolution

algorithm.

3.4.1.2. Z-Transform Procedure

An alternative method was developed that is able to determine the exact solution for the

coefficients. It follows a similar procedure that was used in Nakamura’s model with alterations

implemented to account for the recursive nature of this model. This formulation begins by

altering the expression of the coefficients in the 𝒵-domain by reintroducing the summation terms

and substituting the 𝑧 variable with the complex sinusoidal and cosine functions. The value of

the impedance is also then decomposed into real and imaginary data.

𝑆 𝑧 1 + 𝑎1𝑧−1 + 𝑎2𝑧

−2 + ⋯+ 𝑎M𝑧−𝑀 = 𝑏0 + 𝑏1𝑧

−1 + 𝑏2𝑧−2 + ⋯+ 𝑏N𝑧

−𝑁

𝑆𝑅,𝑖 + 𝑖𝑆𝐼,𝑖 1 + 𝑎𝑗

𝑀

𝑗=1

𝑧−𝑗 = 𝑏𝑗 𝑧−𝑗

𝑁

𝑗=0

𝑆𝑅,𝑖 + 𝑖𝑆𝐼,𝑖 1 + 𝑎𝑗

𝑀

𝑗=1

𝑐𝑜𝑠 𝜃𝑖𝑗 − 𝑖𝑠𝑖𝑛 𝜃𝑖𝑗 = 𝑏𝑗 𝑐𝑜𝑠 𝜃𝑖𝑗 − 𝑖𝑠𝑖𝑛 𝜃𝑖𝑗

𝑁

𝑗=0


(3.53)

66

(3. 55)

(3.56)

(3.57)

Expanding the left hand side of the previous equation and grouping the terms into real and

imaginary parts the following expression for the left hand side of the above equation is produced.

𝑆𝑅,𝑖 + 𝑆𝑅,𝑖 𝑎𝑖

𝑚

𝑗=1

𝑐𝑜𝑠 𝜃𝑖𝑗 + 𝑆𝐼,𝑖 𝑎𝑖

𝑚

𝑗=1

𝑠𝑖𝑛 𝜃𝑖𝑗 + 𝑖 𝑆𝐼,𝑖 + 𝑆𝐼,𝑖 𝑎𝑖

𝑚

𝑗=1

𝑐𝑜𝑠 𝜃𝑖𝑗 − 𝑆𝑅,𝑖 𝑎𝑖

𝑚

𝑗=1

𝑠𝑖𝑛 𝜃𝑖𝑗

Just like in previous methods, the expression must be split into imaginary and real components

and the input impedance data has to be partially separated from the parameter expressions:

𝑆𝑅,𝑖

𝑆𝐼,𝑖

+

𝑆𝑅,𝑖 𝑎𝑖

𝑚

𝑗=1

𝑐𝑜𝑠 𝜃𝑖𝑗 + 𝑆𝐼,𝑖 𝑎𝑖

𝑚

𝑗=1


𝑆𝐼,𝑖 𝑎𝑖

𝑚

𝑗=1

𝑐𝑜𝑠 𝜃𝑖𝑗 − 𝑆𝑅,𝑖 𝑎𝑖

𝑚

𝑗=1


=

𝑏𝑖𝑐𝑜𝑠 𝜃𝑖𝑗

𝑛

𝑗=0

− 𝑏𝑖𝑠𝑖𝑛 𝜃𝑖𝑗

𝑛

𝑗=0

The difference between this procedure and previous methods is that the matrix that is being built

has the impedance data embedded in it. This is indicative of the inherent recursive nature of this

method and can easily lead to singular matrix formation. Expressing the above two equations in

matrix and vector multiplication form allows the following equation to be produced:


𝐼𝑚𝑎𝑔 𝑆 𝜔𝑖 = 𝐶𝐵𝑖 𝐶𝐴𝑖

𝑏0

𝑏1

⋮

𝑏𝑁

𝑎1

𝑎2

⋮

𝑎𝑀

Where:

𝐶𝐵𝑖 = 𝑐𝑜𝑠 𝜃𝑖0 𝑐𝑜𝑠 𝜃𝑖1 … 𝑐𝑜𝑠 𝜃𝑖𝑁

−𝑠𝑖𝑛 𝜃𝑖0 −𝑠𝑖𝑛 𝜃𝑖1 … −𝑠𝑖𝑛 𝜃𝑖𝑁

𝐶𝐴𝑖 = −𝑆𝑅,𝑖𝑐𝑜𝑠 𝜃𝑖1 − 𝑆𝐼,𝑖𝑠𝑖𝑛 𝜃𝑖1 ⋯ −𝑆𝑅,𝑖𝑐𝑜𝑠 𝜃𝑖𝑀 − 𝑆𝐼,𝑖𝑠𝑖𝑛 𝜃𝑖𝑀

−𝑆𝐼,𝑖𝑐𝑜𝑠 𝜃𝑖1 + 𝑆𝑅,𝑖𝑠𝑖𝑛 𝜃𝑖1 ⋯ −𝑆𝐼,𝑖𝑐𝑜𝑠 𝜃𝑖𝑀 + 𝑆𝑅,𝑖𝑠𝑖𝑛 𝜃𝑖𝑀

The last step is to assemble the linear system of equations by repeating the above equation for the

complete impedance function data that is to be considered producing Eq. (3.58).

67

(3.58)

(3.59)

(3.60)

𝑆 𝜔0

𝑆 𝜔1 ⋮

𝑆 𝜔𝐿

=

𝐶𝐵0

𝐶𝐵1

⋮

𝐶𝐵𝐿

𝐶𝐴0

𝐶𝐴1

⋮

𝐶𝐴𝐿

×

𝑏0

𝑏1

⋮

𝑏𝑁

𝑎1

𝑎2

⋮

𝑎𝑀

The matrix on the right hand side of Eq. (3.58) will be invertible only if it is a square matrix

which produces the following condition for precise coefficient determination:

𝑀 + 𝑁 = 𝐿

It was mentioned in Section 3.4.1.1 that to obtain coefficients that are stable and accurate, the

‘filter order’ should be set at about four or five. Therefore 𝑁 should be selected in that range and

then 𝑀 should be set to be 𝐿 − 𝑁 , where 𝐿 is the number of data points in the impedance

function.

Though this method will ensure that the impedance function produced by the transform

coefficients is accurate it does not ensure the transform is BIBO stable. There are no restrictions

on the values of the recursive coefficients, 𝑎𝑘 , so there is the possibility that the denominator of

Eq. (3.51) possesses zeros at values greater than one. This will result in divergent behaviour

when trying to execute the convolution calculation even without the incorporation of the

convolution into the time integration scheme.


The effective impedance function of the transform coefficients can be determined in the

frequency domain using the equations that were previously used to express the transform method

in the 𝒵-domain. The 𝑧 variable is replaced by the complex exponential term in order to obtain a

frequency domain expression.

𝐻 𝑧 = 𝑏0 + 𝑏1𝑧

−1 + 𝑏2𝑧−2 + ⋯+ 𝑏N𝑧

−𝑁

1 + 𝑎1𝑧−1 + 𝑎2𝑧−2 + ⋯+ 𝑎M𝑧−𝑀

𝐻 𝜔 = 𝑏0 + 𝑏1𝑒

−𝑖𝜔∆𝑡 + 𝑏2𝑒−2𝑖𝜔∆𝑡 + ⋯+ 𝑒−𝑁𝑖𝜔∆𝑡

1 + 𝑎1𝑒−1𝑖𝜔∆𝑡 + 𝑎2𝑒−2𝑖𝜔∆𝑡 + ⋯+ 𝑎M𝑒−𝑀𝑖𝜔∆𝑡

68

In the first soil model case, it became apparent that the optimization scheme was experiencing

difficulties ensuring the convolution was BIBO stable. As a compromise for the stability, the

requirement for accuracy in the impedance function was reduced. This led the procedure to

convergence on transform coefficients that produced impedance functions that captured the low

frequency distinctive peaks well but possessed great disparity at higher frequencies. This is

evidenced in Figure 3.16 where the discrepancies between the effective and input impedance

functions are visualized. In the proceeding figures the curve labelled ‘Actual’ refers to the input

impedance function.

Figure 3.16 Reproduced Impedance of the Stiff Soil Case for the Şafak Method using

Optimization Scheme

The above graph was produced using only a tenth of the impedance frequency data (up to 10 Hz)

in order to have greater correlation at the lower frequencies. To compensate for this reduced

range, the number of data points in the impedance function was increased by a factor of ten so

that the impedance data that is being used had 100 data points. If the entire data set would have

been used the optimization would produce an impedance function that on average would have

less error but would ultimately contain no region of where the impedance functions were equal.

0 1 2 3 4 5 6 7 8 9 10-1

-0.5

0

0.5

1x 10

6

Frq. [sec-1]

Impedance [

Real]

Actual

Z-transform

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Imagin

ary

]

Actual

Z-transform

69

The previous figure utilized four parameters, however it was observed that a stable filter could be

generated that was slightly more accurate with six parameters. The results of the six parameter

filter is shown below and it evident that the filer is accurate for frequencies under 5 Hz.

Figure 3.17 Second Reproduced Impedance of the Stiff Soil Case for the Şafak Method

using Optimization Scheme

More than ever now, the numerical reproduction of the impedance function will be essential

because the recursive convolution has inherent unstable tendencies. This may lead to an accurate

representation of the transfer function in the frequency domain, but when the impedance function

is generated numerically, it may manifest divergent behaviour and be unable to produce a

realistic solution. The impedance function was reproduced numerically showing that the

convolution using the transform coefficients is numerically stable. This affirms that the

optimization scheme was capable of imposing the stability condition in its objective function.

In order to utilize the second procedure in generating the transform coefficients, the number of

displacement convolution coefficients had to be set to the difference between the number of

frequency data points being considered and the filter order. Six 𝑎𝑗 coefficients were utilized in

this transform, the same number as in the optimization procedure, the number of 𝑏𝑗 coefficients

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Real]

Actual

Numerical

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Imagin

ary

]

Actual

Numerical

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Real]

Actual

Numerical

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Imagin

ary

]

Actual

Numerical

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Real]

Actual

Numerical

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Imagin

ary

]

Actual

Numerical

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Real]

Actual

Numerical

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Imagin

ary

]

Actual

Numerical

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Real]

Actual

Numerical

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Imagin

ary

]

Actual

Numerical

70

must be 94 since there are 100 data points. The reproduced impedance functions were exact and

only limited by the resolution of the frequency data.

Figure 3.18 Reproduced Impedance of the Stiff Soil Case for the Şafak Method using Z-

Transform Procedure


This transform method produced divergent results suggesting that when the time integration

algorithm is combined with the Şafak model, unstable behaviour is likely generated regardless of

which coefficient determining method is utilized.

Figure 3.19 displays the structure response from the simulation using the optimization scheme

generated coefficients and Figure 3.20 is the response from the coefficients generated using the

Z-transform procedure. Both methods displayed divergent results in the figures. Though it cannot

be observed in these figures, the optimized parameter model was found to be able to simulate for

4 seconds longer than that of the other procedure. This is perhaps indicative that the optimized

parameters produced a less unstable combined system.

0 1 2 3 4 5 6 7 8 9 10-1

-0.5

0

0.5

1x 10

6

Frq. [sec-1]

Impedance [

Real]

Actual

Numerical

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2x 10

6

Frq. [sec-1]

Impedance [

Imagin

ary

]

Actual

Numerical

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Real]

Actual

Numerical

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Imagin

ary

]

Actual

Numerical

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Real]

Actual

Numerical

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Imagin

ary

]

Actual

Numerical

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Real]

Actual

Numerical

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Imagin

ary

]

Actual

Numerical

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Real]

Actual

Numerical

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2x 10

6

Frq. [sec-1]

Impedance [

Imagin

ary

]

Actual

Numerical

71

Figure 3.19 Displacement Responses of the Total and Şafak Models of the Stiff Soil

Example undergoing the Kobe Earthquake Loading.

Figure 3.20 Displacement Responses of the Total and Şafak Models of the Soft Soil

Example undergoing the Kobe Earthquake Loading.

0 2 4 6 8 10 12 14 16

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

300

Time [sec]

Dis

pla

cem

ent

[m]

W/O SSI

Total

Safak

0 2 4 6 8 10 12 14 16-16

-14

-12

-10

-8

-6

-4

-2

0

2x 10

297

Time [sec]

Dis

pla

cem

ent

[m]

W/O SSI

Total

Safak

72

3.4.4. Stability

Given that this method was unable to produce a proper stable response when modelling the

reference example the development of a stability assessment procedure is warranted. The

stability assessment procedure was ascertained using the same technique that has already been

determined for the other two transform methods. The Şafak model contains only one

instantaneous variable which corresponds to the instantaneous stiffness of the foundation degree-

of-freedom. This parameter will be incorporated into the structural stiffness matrix, 𝐾, of this

system which leads to the following system matrices.

𝐾 = 𝑘 −𝑘−𝑘 𝑘 + 𝑏0

, 𝐶 = 𝑐 −𝑐−𝑐 𝑐

, 𝑀 = 𝑚 00 0

The new definitions of the matrices are utilized in the following equation of motion that includes

the convolution summation terms that start at 𝑗 = 1 rather than zero.

𝐾 ∙ 𝑢𝑖+1 = −𝑀 ∙ 𝑢 𝑖+1 − 𝐶 ∙ 𝑢 𝑖+1 − 𝐵𝑗 ∙ 𝑢𝑖+1−𝑗 − 𝐴𝑗 ∙ 𝐹𝑖−𝑗

𝑁−1

𝑗=1

𝑁−1

𝑗=1

The 𝐹 variable in the convolution is not the external force but the restoring force determined at

previous time steps by the convolution calculation. Using this equation and Eq. (3.15) that

defines the time integration scheme Eq. (3.63) is produced that relates the next displacement

vector as a function of previous state variables.

𝐾 𝑢𝑖+1 = 𝑅 𝑢𝑖 + 𝐶 𝑢 𝑖 + 𝑀 𝑢 𝑖 − 𝐵𝑗 ∙ 𝑢𝑖+1−𝑗 + 𝐴𝑗 ∙ 𝐹𝑖+1−𝑗

𝑀

𝑗=1

𝑁−1

𝑗=2

This expression is identical to the equation that was developed for the inverse Fourier method

except that the force convolution has been added. The transform coefficients did not change the

structural matrices therefore the resultant 𝐾0 and 𝐴0 matrices will be the identical to the matrices

in the inverse Fourier transform stability analysis in Section 3.2.5.

𝑈𝑖+1 = 𝐾0 × 𝑈𝑖 + 𝑏2 ∙ 𝐴0 × 𝑈𝑖−1 + 𝑏3 ∙ 𝐴0 × 𝑈𝑖−2 + ⋯− 𝑎1 ∙ 𝐴0 × 𝐹𝑖 − 𝑎2 ∙ 𝐴0 × 𝐹𝑖−1 + ⋯

(3.62)

(3.63)

(3.61)

73

The difference between the two method’s stability assessments is the inclusion of the

convolution dependent on the previous restoring force values. In order to execute this

convolution the force vector has to be determined at each time step as defined in Eq. (3.64). The

I(2,2) matrix is necessary because the restoring force is solely dependent on the displacement of

the foundational degree-of-freedom and this matrix allows the calculation to extract the required

displacement from the system variable 𝑈 and apply the restoring force only at this degree-of-

freedom as well.

𝐹𝑖 = 𝑏𝑗 ∙ 𝐼(2,2) × 𝑈𝑖+1−𝑗

𝑁+1

𝑖=0

− 𝑎𝑗 ∙ 𝐼(2,2) × 𝐹𝑖+1−𝑗

𝑀

𝑖=1

Where:

𝐼(2,2) =

0 00 1

0 00 0

0 00 0

0 00 0

0 00 0

0 00 0

0 00 0

0 00 0

0 00 0

Executing this procedure on the coefficients used in the validation example simulation produced

Figure 3.21 and Figure 3.22. The graphs show the eigenvalues of the compounded iteration

matrix, 𝐾𝐶 , as it changes with each iteration. Both plots show that at least one eigenvalue

increases steadily beyond unity, indicating that both sets of coefficients are unstable when

incorporated into the time-integration scheme. This corroborates the instability demonstrated

during the simulation on the validation example.

(3.64)

(3.65)

74

Figure 3.21 Stability Analysis Curve of the Şafak Model for the Stiff Soil Example

The fact that the eigenvalues for the coefficients produced using the optimization scheme reaches

a value that is fifty orders of magnitude less than that of the other method demonstrates it is by

comparison perhaps more stable. This is perhaps due to the fact that the method using the 𝒵-

domain was only concerned with exact representation of the impedance function while the

optimization scheme had a stability criteria built into its objective function.

This stability assessment could be incorporated into the optimization scheme in order to evaluate

potential coefficients on their stability when incorporated into the time integration scheme. This

of course will produce an optimization algorithm that is significantly more computationally

expensive since free vibration simulations would need to be conducted at each optimization

iteration. Furthermore it would be difficult to describe an objective function relating the

importance of stability and accuracy.

0 20 40 60 80 100 120 140 160 1800

2

4

6

8

10

12

14x 10

4

Iterations

Eig

envalu

e A

mplit

ude

75

Figure 3.22 Stability Analysis Curve of the Şafak Model for the Soft Soil Example

3.5. Comparison Conclusions

From the proceeding simulation attempts it becomes evident that the Nakamura model was the

only method that was capable of modelling the reference example. This does not discredit the

other methods ability to model other impedance functions in the time domain but it did

demonstrate they are unable to model any impedance function and thus they lack general

applicability. Furthermore the stability assessment analyses demonstrated that stability issues do

arise in the incorporation of the convolution calculation step into the numerical time integration

scheme.

The inverse Fourier transform may be the simplest to implement and to understand given the

ubiquity of the Fourier transform in a great number of engineering fields; however it was

incapable of modelling this particular impedance function even though it has been documented to

be able to generate solutions for certain analytical impedance functions in the literature (Wolf &

Obernhuber, 1985). Certain instability issues have been addressed elsewhere and some of these

concepts will be presented and addressed in Chapter 4.

0 20 40 60 80 100 120 140 160 1800

1

2

3

4

5

6

7x 10

54

Iterations

Eig

envalu

e A

mplit

ude

76

Nakamura’s model was the only model that did produce stable response, but there still existed

some minor discrepancies between the reference solution and the simulation using Nakamura’s

transform method. Furthermore it should be recognized that creating an accurate and stable

transform requires some trial and error when selecting the number of data points and convolution

time steps. This selection process becomes easier as one garners more experience in creating

these sorts of models.

The Şafak model was perhaps the most ambitious of these models since it relied completely on

concepts established in the field of signal processing. Though a stable transform was not

generated it could potentially be attributed to the optimization implemented not being adequately

sophisticated. Nevertheless this model always presented some uncertainty because Şafak only

provided examples of this transform replicating impedance functions in the frequency domain

(Şafak, 2006). No seismic response analysis examples were provided and furthermore he only

demonstrated that the impedance functions correlated within the first 6 Hz which would

disregard any high frequency content of earthquakes.

Ultimately Nakamura’s model proved to be the most reliable and accurate transform method

since it was the only model that was capable of replicating the validation example. For this

reason in the subsequent example chapters only Nakamura’s model will be utilized and

investigated.

77

Chapter 4 Improving Stability

In the previous chapter it became apparent that the incorporation of the transform of the

impedance function into numerical time integration schemes may exhibit potential stability

problems. The stability is dependent on both the method utilized and data content of the

impedance function. This chapter presents a number of methods from the literature and proposes

an original method that attempt to produce transforms less prone to produce divergent responses.

The transform methods are susceptible to instability when attempting to transform impedance

functions that are non-causal. This can hinder all the transform methods but has the greatest

influence on the Wolf transform method. Nakamura’s method has a unique stability issue which

occurs when the instantaneous mass that is determined is negative. Implementing a negative

mass undoubtedly causes the Newmark integration scheme to produce divergent results and an

original method has been proposed in Section 4.2 to mitigate this issue.

4.1. Evaluation of the Examined Models’ Susceptibility to Non-Causal Impedance Functions

To explain this stability issue it is important to investigate the relationships between the

frequency and time domain representation of the impedance functions. The impedance functions

used in these transform methods are discrete, band-limited in the frequency domain and are

representative of a soil system. Furthermore the impedance functions may be properly defined

for any frequency value, typical impedance functions only consider a band of useful frequencies

because it is computationally not worthwhile to transform frequencies outside of the scope of

interest.

Though the impedance function may be discrete and band limited, the methods implement

transformations that are analogous to the discrete Fourier transform procedure and therefore it is

assumed that the frequency domain data of the impedance function repeats itself periodically and

indefinitely (Brazil, 1995). The transformation of such a function will result in the creation of a

discrete time domain function that is also periodic (Brazil, 1995), therefore the force response

function, the time domain analogy of the impedance function, will also be periodic.

78

The periodicity of the force response function in the time domain is an important fact because it

belies the notion that these functions are implicitly impulses or causal. Causal systems solely

dependent on the values of the system at the current or previous time steps and non-causal

systems are additionally dependent on future values. Non-causal periodic time domain functions

will contain non-zero values near the end of a cycle which prevents it from behaving as true

impulse response. This indicates that there exist meaningful data in the time series before the

start time at zero seconds as presented in Figure 4.1.

Figure 4.1 Examples of Causal and Non-Causal Time Domain Functions

The data before time zero implies that the displacement at a given time step will have

repercussions on restoring forces of a previous time step. This can equivalent to stating that the

current restoring force will be dependent on future displacements. Such a function could only be

implemented in a convolution operation only if the second function in that convolution is fully

defined for all of the time duration of interest: future, past and present. Sinusoidal and other

periodic functions are an example of functions that are fully defined at any time. However this

condition does not represent any real physical systems and is completely incompatible with the

formulations of any time integration scheme used in seismic simulations.

The convolution calculation implemented in the transform methods does not utilize a periodic

force response; instead only use single cycle of the function. In comparison to the repeating

periodic functions in Figure 4.1, the convolution calculation will involve a single period of a

non-causal function such as the example function in Figure 4.2. The instability produced in the

combined model can be attributed to the non-zero values of the time series at the end of the

period. Considering the free vibration example, an impulse displacement should produce a force

79

response that approaches zero over time. In a non-causal example however the restoring force

will be similar at 10 seconds to the initial restoring force produced at time zero. This strong

future restoring force will have a detrimental effect on the stability of the system given the

feedback nature of the foundational displacement and restoring force.

Figure 4.2 Single Cycle of a Non-Causal Function

According to the Kramers-Kroning relationship (Nakamura, 2008), a band-limited frequency

dependent function is causal only if the imaginary and real portions of the frequency domain data

must form a Hilbert transform pair (Bartholdi & Ernst, 1973). Consequently then, when using the

proposed transform methods, an impedance function will only produce a force response function

that is an impulse only if the Hilbert transform of the real impulse function data is identical to the

imaginary data and the reverse is true as well.

Soil systems with high hysteretic damping will produce impedance functions that are highly non-

causal (Nakamura, 2007). These soil models remain a challenge to model using these transform

methods since altering the impedance function to conform to the Kramers-Kroning relationship

in order to create a causal transform would signify this hysteretic damping behaviour would be

not modelled. The transform model would therefore produce an inaccurate response

The following subsections of Sections 4.1 contain a number of proposed methods that utilize

unaltered impedance functions to produce force response functions that are manipulated to be

impulses. The force responses however are inaccurate because the modifications that have been

implemented will ensure these functions correspond to causal impedance functions in the

frequency domain that nonetheless are different compared to the inputted impedance function of

the desired soil domain.

80

4.1.1. Causal FFT Treatment

Hayashi and Katukura (1990) noted that the best technique to obtain causal impulse responses

from the inverse Fourier transformation method would be to consider an impedance function

with a large frequency range far outside the frequencies of interest for the earthquake history and

structural response. Such an impedance function would be transformed into a long duration force

impulse function that even if it exhibited non-causal data, would be too long to manifest the

instability associated with it. However long duration force impulse functions are computer

inefficient and the frequency range considered by the impulse function contains data that is

largely unimportant. Hayashia et al. (1990) therefore proposed a new approximation method to

ensure a causal transform of the impedance function using fast Fourier transform.

Since the time-domain transform of the impedance is to be real and causal, it can be decomposed

into an even and an odd function. Even functions have identical values for negative and positive

values of the dependent variable. Odd functions will have the opposite value if the input has the

opposite sign and these relationships are described in Eq. (4.1).

𝑒𝑣𝑒𝑛: 𝑓 𝑥 = 𝑓(−𝑥)

𝑜𝑑𝑑: − 𝑓 𝑥 = 𝑓(−𝑥)

For 𝑡 < 0 the values of the even and odd components of the time-series must cancel out to

satisfy causality. Performing the inverse Fourier transform on the real portion of the impedance

function will produce an even time function while performing the transform on the imaginary

data produces an odd time function. Using these functions separately or in summation will

produce a causal force response whose resulting impedance function will not correspond to the

desired impedance function. Eq. (4.2) describes the manipulation that is required to obtain a

causal force response from the real and imaginary data.

𝑆𝑒𝑣𝑒𝑛 𝑡 =1

2𝜋 𝑅𝑒𝑎𝑙 𝑆 𝜔 ∞

−∞

𝑒𝑗𝜔𝑡 𝑑𝜔, 𝑆𝑜𝑑𝑑 𝑡 =1

2𝜋 𝐼𝑚𝑎𝑔 𝑆 𝜔 ∞

−∞

𝑒𝑗𝜔𝑡 𝑑𝜔

𝑆 𝑐𝑎𝑢𝑠𝑎𝑙 ,1 𝑡 = 2𝑈 𝑡 ∙ 𝑆𝑒𝑣𝑒𝑛 𝑡

𝑆 𝑐𝑎𝑢𝑠𝑎𝑙 ,2 𝑡 = 2𝑈 𝑡 ∙ 𝑆𝑜𝑑𝑑 𝑡

𝑆 𝑐𝑎𝑢𝑠𝑎𝑙 ,3 𝑡 = 𝑆𝑜𝑑𝑑 𝑡 + 𝑆𝑒𝑣𝑒𝑛 𝑡

(4.2)

(4.1)

81

Where:

𝑈 𝑡 = 0 𝑡 < 0 , 𝑈 𝑡 = 0.5 𝑡 = 0 , 𝑈 𝑡 = 1 𝑡 < 0

This idea of causal transforms based on partial impedance function data is also proposed by

Nakamura (2008) for ensuring that transform is strictly causal.

Using these three types of causal impulses Hayashi proposes a number of ways to implement an

causal transform that relies on minor changes to the impedance functions (system function) and

the resulting force response function (digital simulators). These methods correspond to concepts

in the field of digital signal processing relating to the implementation of digital simulators.

Hayashi’s based his method on altering the flexibility functions which are the inverse of the

impedance function, but this method should work using the impedance function since these

transformation methods are independent of the input function’s definition.

Figure 4.3 Proposed Procedure (Hayashi & Katukura, 1990)

Figure 4.3 demonstrates the definition and processes in determining the time-series transform

functions. For each method the impedance function is modified as prescribed and transformed

using inverse Fourier transform to obtain the raw discrete time-series, 𝑕[𝑛], which is a force

(4.3)

82

response function. These time-series are then modified as well in order to account for the value at

𝑡 = 0 which the imaginary data cannot account for. This adjustment is requires determining the

constants 𝐶0 and 𝐶1using Eq. (4.4) and (4.5).

𝐶0 = −1

2𝜋 2 𝑅𝑒𝑎𝑙 𝐻(𝑛∆𝜔) −1 𝑛 + 𝑅𝑒𝑎𝑙 𝐻 𝜔𝑁

𝑁/2−1

𝑛=1

∆𝜔

𝐶1 = 𝑅𝑒𝑎𝑙 𝐻 ∆𝜔 /∆𝑡 − 𝑕 𝑛

𝑁/2

𝑛=1

𝑐𝑜𝑠 𝑛∆𝑡∆𝜔

Hayashi investigated the four distinct transform model alternatives which are present in Figure

4.3. These methods include transform method using only the real data (𝑕𝑅 𝑛 ), transform method

using only the imaginary data (𝑕𝐼 𝑛 ), causal transform using combined data (𝑕𝐶 𝑛 ), and a

modified transform method that used the combined data ( 𝑕𝐶𝑀 𝑛 ). These methods were

evaluated by comparing the impedance functions they produced to the original impedance

function and later by conducting time domain analysis on a reference system. The structure that

was to be analyzed was that of a typical nuclear reactor building embedded in stratified soil and

subjected to the El Centro 1940 NS acceleration history (Hayashi & Katukura, 1990). The

structure was represented as stick model and there were five degrees-of-freedom on the interface

of the structure and soil.

Hayashi demonstrated that all four proposed transform methods created causal impulse

transforms in the time domain from the impedance function that was demonstrated to be non-

causal. However when these impulses were converted back into frequency domains, it was

revealed that these methods were inaccurate in replicating the impedance function of the soil

model especially at higher frequencies. No comparison was provided between the imaginary

portions of the input impedance function and the effective impedance function of the transform

that utilized only the real data, (𝑕𝑅 𝑛 ). Since the transform was completely independent of the

imaginary data, the two impedance functions likely were disparagingly different in the imaginary

region. Similarly no comparison was made in the real data portion for the model that only

utilized only the imaginary data. Greatest disparity however existed when comparing the

interaction forces among the simulations that used the combined data transform model, (𝑕𝐶 𝑛 ),

with the results being completely unacceptable at low frequencies.

(4.4)

(4.5)

83

Despite this Hayashi showed that the peak displacement experienced in the structure during the

seismic simulation was accurately predicted. Considering the inaccuracies present in the

interaction forces, this is perhaps indicative that the interaction forces were not significant in

determining the structure response as a result of the structure being stiffer than the analyzed soil

domain.

Hayashi concluded that the proposed methods produced promising results with good correlation

between desired and output impedance functions in the frequency domain. However inaccuracies

existed in the values of the interaction forces, in the time-history analysis, which sheds some

doubt on the suitability of these methods to model SSI behaviour. Furthermore responses that

were determined by models using the imaginary data transform (𝑕𝑅 𝑛 ) and the modified

combined data transform (𝑕𝐶𝑀 𝑛 ) were divergent and unstable even though the transforms were

supposed to be causally stable. Since other stability concerns may exist, further investigation

should be conducted to better understand the susceptibility of these methods to instability when

conducting time-history analyses. Hayashi concluded that the method with the greatest general

applicability is that of the transform based solely on real data.

4.1.2. Nakamura Model Based on Partial Data (2008)

A number of modifications to Nakamura’s transform model were proposed in recent journal

articles (2008) that allow the approximate modelling of non-causal impedance functions. It was

recognized that utilizing either the real or imaginary portion of the data in the coefficient

formulation procedure would produce a causal force response function in the time domain. This

new causal force response reflects an impedance function in the frequency domain that

corresponds exactly to either real or imaginary impedance data that was inputted to make the

force response. The other portion of the reproduced impedance function will be a Hilbert

transform of the input data portion. Ensuring causality by using only half of the data will have

negative repercussions on the accuracy of the method since the Hilbert transform portion of the

data will be arbitrary and not be representative of the soil model behaviour. This compromise in

accuracy may perhaps be justifiable in situations where the soil model contains an impedance

function with strong non-causality that always produces unstable time domain analysis.

84

4.1.2.1. Real Data Method

Recalling the steps performed in the matrix assembly in the Nakamura method in Section 3.3.2,

the real portion of the data can be expressed as a sum of the force response coefficients shown in

Eq. (4.6).

𝑆𝑅 𝜔𝑖 = 𝑅𝑒𝑎𝑙 𝑆 𝜔𝑖 = −𝜔𝑖2 ∙ 𝑚0 + 𝑘𝑗 ∙ 𝑐𝑜𝑠 𝜃𝑖𝑗

𝑁1

𝑗=0

+ 𝜔𝑖 ∙ 𝑐𝑗 ∙ 𝑠𝑖𝑛 𝜃𝑖𝑗

𝑁2

𝑗=1


This equation is identical to the real portion of Eq. (3.39) except that the 𝑗 variable starts at the

value of one since 𝑠𝑖𝑛(𝜃𝑖𝑗 ) will equal zero regardless of the value of 𝑖. Consequently 𝑐0 , the

instantaneous damping, cannot be ascertained solely from the real portion of the impedance

function. This is because the instantaneous damping is solely dependent on the imaginary data

and therefore it will be determined by another means involving a least-square method.

Rewriting the summation operations as vector multiplication, a single linear system of equations

is determined.

𝑅𝑒𝑎𝑙 𝑆 𝜔𝑖 = 𝑐𝑜𝑠 𝜃𝑖0 𝑐𝑜𝑠 𝜃𝑖1 ⋯ × 𝑘0

𝑘1

⋮

+ 𝜔𝑖 ∙ 𝑠𝑖𝑛 𝜃𝑖1 𝜔𝑖 ∙ 𝑠𝑖𝑛 𝜃𝑖2 ⋯ × 𝑐1

𝑐2

⋮ + −𝜔𝑖

2 × 𝑚0

𝑅𝑒𝑎𝑙 𝑆 𝜔𝑖 = 𝑐𝑜𝑠 𝜃𝑖 ,0 ⋯ 𝑐𝑜𝑠 𝜃𝑖 ,𝑁1 𝜔𝑖 ∙ 𝑠𝑖𝑛 𝜃𝑖 ,1 ⋯ 𝜔𝑖 ∙ 𝑠𝑖𝑛 𝜃𝑖 ,𝑁2

−𝜔𝑖2 ×

𝑘0

⋮𝑘𝑁1

𝑐1

⋮𝑐𝑁2

𝑚0

Repeating the above equation for all the frequencies of interest in the impedance function

produces the following linear system. Inverting the matrix and multiplying it to the real data will

determine the coefficients.

𝑆𝑅 𝜔1

⋮𝑆𝑅 𝜔𝑁

= 𝐶𝑘 𝐶𝑐 𝐶𝑚 ×

𝑘0

⋮𝑘𝑁1

𝑐1

⋮𝑐𝑁2

𝑚0

(4.6)

(4.7)

(4.8)

85

Where:

𝐶𝑘 =

𝑐𝑜𝑠 𝜃1,0 ⋯ 𝑐𝑜𝑠 𝜃1,𝑁1

⋮ ⋱ ⋮𝑐𝑜𝑠 𝜃𝑁,0 ⋯ 𝑐𝑜𝑠 𝜃𝑁,𝑁1

, 𝐶𝑐 =

𝜔1 ∙ 𝑠𝑖𝑛 𝜃1,1 ⋯ 𝜔1 ∙ 𝑠𝑖𝑛 𝜃1,𝑁2

⋮ ⋱ ⋮𝜔𝑁 ∙ 𝑠𝑖𝑛 𝜃𝑁,0 ⋯ 𝜔𝑁 ∙ 𝑠𝑖𝑛 𝜃𝑁,𝑁2

, 𝐶𝑚 = −𝜔1

2

⋮−𝜔𝑁

2

It is important to realize that the number of parameters has changed compared to the original

Nakamura method. 𝑁 is the number of frequencies in the impedance data, and Eq. (4.9) and

(4.10) relate the values of 𝑁1 and 𝑁2 to number of frequencies.

𝑁 = 𝑁1 + 𝑁2 + 2

Depending whether 𝑁 is even or odd the number of stiffness and damping coefficients will

change unlike the original Nakamura method whose number of stiffness and dampers were

always equal. Given that the number of coefficients should be almost equal, Nakamura suggests

the following values:

𝑁: 𝑒𝑣𝑒𝑛 𝑁1 = 𝑁2 = 𝑁2 − 1

𝑁: 𝑜𝑑𝑑 𝑁1 = 𝑁 − 1

2 𝑁2 = 𝑁1 − 1

The instantaneous damping coefficient has a frequency domain content equivalent to a dashpot

damper element. In frequency domain the instantaneous coefficient would produce a purely

linear imaginary dynamic stiffness that increases with frequency. This fact enables one to

determine the coefficient’s value by comparing additional impedance function values to the

impedance generated by the already determined coefficients. The discrepancy is then eliminated

with the selection of an appropriate instantaneous damping coefficient through a least-square

method provided by Nakamura in the following equations:

𝑐0 = −𝐸𝐵

Where:

𝐵 = 𝜔𝑖

𝑁𝑎𝑑𝑑

𝑖=1

, 𝐸 = 𝜔𝑖 𝑆𝐼 𝜔𝑖 − 𝐷𝐼 𝜔𝑖

𝑁𝑎𝑑𝑑

𝑖=1

(4.9)

(4.10)

(4.11)

(4.12)

86

𝑆𝐼 is the imaginary portion of the additional impedance value while 𝐷𝐼 is the impedance value

generated from using the already determined coefficients.

4.1.2.2. Imaginary Data Method

Alternatively one could form a causal transform by only utilizing the imaginary data, which

produces the same limitations on accuracy as using only the real data. The instantaneous

damping is determined in the coefficient formation however the instantaneous mass and stiffness

is unknown. The reason for that is that those components manifest solely real dynamic stiffness

values in the frequency domain and must be determined through the use of extra impedance data

and the least-square method.

Only using the imaginary portion of the basis of the Nakamura equation will provide the

groundwork for the determination of the causal coefficients.

𝑆𝐼 𝜔𝑖 = 𝐼𝑚𝑎𝑔 𝑆 𝜔𝑖 = − 𝑘𝑗 ∙ 𝑠𝑖𝑛 𝜃𝑖𝑗

𝑁1

𝑗=1

+ 𝜔𝑖 ∙ 𝑐𝑗 ∙ 𝑐𝑜𝑠 𝜃𝑖𝑗

𝑁2

𝑗=0

Proceeding with a similar vector manipulation, the linear system that defines the coefficient

formulation procedure is revealed to be as follows:

𝑆𝐼 𝜔1

⋮𝑆𝐼 𝜔𝑁

= 𝐶𝑘 𝐶𝑐 ×

𝑘1

⋮𝑘𝑁1

𝑐0

⋮𝑐𝑁2

Where:

𝐶𝑘 =

𝑐𝑜𝑠 𝜃1,1 ⋯ 𝑐𝑜𝑠 𝜃1,𝑁1

⋮ ⋱ ⋮𝑐𝑜𝑠 𝜃𝑁,0 ⋯ 𝑐𝑜𝑠 𝜃𝑁,𝑁1

, 𝐶𝑐 =

𝜔1 ∙ 𝑠𝑖𝑛 𝜃1,0 ⋯ 𝜔1 ∙ 𝑠𝑖𝑛 𝜃1,𝑁2

⋮ ⋱ ⋮𝜔𝑁 ∙ 𝑠𝑖𝑛 𝜃𝑁,0 ⋯ 𝜔𝑁 ∙ 𝑠𝑖𝑛 𝜃𝑁,𝑁2

It is important to be aware of the difference in the initial values of 𝑗 for the stiffness and damping

coefficients in this determination step compared to the method that used only the real data. The

relationship between the number of frequencies and the number of stiffness and damping

coefficients has also changed because one less coefficient is determined in this procedure than in

the real-data method. Therefore the equations for 𝑁1 and 𝑁2 are now:

(4.13)

(4.14)

(4.15)

87

𝑁: 𝑒𝑣𝑒𝑛 𝑁1 = 𝑁2 𝑁2 = 𝑁

2 − 1

𝑁: 𝑜𝑑𝑑 𝑁1 = 𝑁2 = 𝑁 − 1

2

To determine the indeterminate instantaneous components a similar procedure is taken as before.

Additional real impedance data is compared to the impedance generated by the determined

coefficients for those added frequencies. The difference is then eliminated or minimized by the

selection of appropriate instantaneous mass and stiffness values by using the least-square

method. Nakamura has provided the following equations that determine the unknown

coefficients using the described methods.

𝑚0 =𝐶𝑁𝑎𝑑𝑑 − 𝐵𝐷

𝐴𝑁𝑎𝑑𝑑 − 𝐵2, 𝑘0 =

𝐵𝐶 − 𝐴𝐷

𝐴𝑁𝑎𝑑𝑑 − 𝐵2

where

𝐴 = 𝜔𝑖4

𝑁𝑎𝑑𝑑

𝑖=1

, 𝐶 = 𝜔𝑖2 𝑆𝑅 𝜔𝑖 − 𝐷𝑅 𝜔𝑖

𝑁𝑎𝑑𝑑

𝑖=1

𝐵 = 𝜔𝑖2

𝑁𝑎𝑑𝑑

𝑖=1

, 𝐷 = 𝑆𝑅 𝜔𝑖 − 𝐷𝑅 𝜔𝑖

𝑁𝑎𝑑𝑑

𝑖=1

4.1.2.3. Example

In order to demonstrate the effectiveness of this method, an impedance function must be

analyzed that exhibits such significant non-causality that any numerical analysis involving the

model likely becomes unstable. The Nakamura model using partial data will then be used to

transform this non-causal impedance function and the transform model will be incorporated into

the time-history analysis of the reference structure analyzed in Chapter 3. The response

generated by these means can be compared to the response of the same structure and impedance

function system using frequency domain analysis.

The structure that is to be analyzed is the same single storey structure containing a single degree-

of-freedom presented in Section 3.1.2.3. It supports a 1000 kg mass and has a storey stiffness of

(4.16)

(4.17)

(4.18)

88

1000 kN/m resulting in a natural period of 0.1987 seconds. The damping ratio is selected to be

5%. The convolution calculation, that will use the coefficients determined by Nakamura’s part

data transform method, will determine the restoring force acting upon the foundation of this

reference structure thus accounting for the soil domain’s contribution.

An arbitrary non-causal impedance function was generated in the time domain and the equivalent

frequency domain function of the same function was determined using the Fourier transform.

The time and frequency domain of this example impedance is displayed in Figure 4.4.

It is important to notice that the impedance function in frequency domain behaves dissimilar to a

spring–mass–damper system. The real portion does not behave like negative parabolic nor does

the imaginary portion behave linearly. This suggests that the impedance function is a purely

fictitious construct with little physical meaning and thus may be difficult to model.

Performing normal Nakamura transformation presented in Section 3.3 produces the two force

response functions composed of stiffness and damping coefficients in Figure 4.5. The stiffness

coefficients presented in Figure 4.4 and Figure 4.5 should not be identical because the latter force

response function is only partially representative of the impedance function. The force response

function in Figure 4.5 only represents the impedance function example only when utilized

alongside the damping coefficients.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4

-2

0

2

4

6

8

10x 10

5

Time [sec]

Stiff

ness

Impedance Function

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4x 10

5

Frq. [sec-1]R

eal

0 10 20 30 40 50 60 70 80 90 100-4

-2

0

2

4x 10

5

Frq. [sec-1]

Imagin

ary

Figure 4.4 Non-causal Impedance Function Example

89

The Nakamura transform coefficients reproduce the desired impedance function in the frequency

domain with great precision. However the stiffness and damping coefficients exhibit

considerable non-causal behaviour as evidenced by the large non-zero values near the end of the

functions at 1 second. Expectedly the time integration scheme including the convolution based

on these two functions proved to be unstable and the response divergent.

Performing Nakamura transform based solely on the imaginary data produced the stiffness and

damping coefficients presented in Figure 4.6. It is apparent that these functions behave more like

impulses than the previously generated transform functions. It is also important to note that these

functions are half the length of the transform functions determined by the original Nakamura

method because only half the available data is utilized in the formulation of these coefficients.

Figure 4.7 demonstrates the impedance function that is generated from using the stiffness,

damping and mass components of the causal Nakamura transform. As expected the imaginary

data is in identical whereas the real data is completely arbitrary.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3x 10

5

xlabel('Time [sec]')

Stiff

ness [

kN

/m]

Stiffness Coefficients (Ki)

Figure 4.5 Nakamura Transform Functions of Non-Causal Example

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-600

-400

-200

0

200

400

600

Time [sec]

Dam

pin

g [

kN

s/m

]

Damping Coefficients (Ci)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-600

-400

-200

0

200

400

600

Time [sec]

Dam

pin

g [

kN

s/m

]Damping Coefficients (Ci)

90

Figure 4.7 Reconstructed Impedance Based on the Transform Time-Series

It is important to address the fact that the static damping component is negative. Physically this

signifies that the restoring force generated by such a damper would act in the direction of the

velocity rather than in the opposite direction. Another way to imagine this situation is that if a

constant external force would be applied on an object, the object would have a velocity in the

opposite direction than the applied force. This scenario is undesirable in numerical integration

since it will be lead to instability but since this negative damping is being added to the existing

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-1

-0.5

0

0.5

1

1.5x 10

5

Time [sec]

Stiff

ness [

kN

/m]

Stiffness Coefficients (Ki)

0 5 10 15 20 25 30 35 40 45 50-2

0

2

4

6x 10

5

Frq. [sec-1]

Impedance [

Real]

Actual

Z-transform

0 5 10 15 20 25 30 35 40 45 50-3

-2

-1

0

1x 10

5

Frq. [sec-1]

Impedance [

Imagin

ary

]

Actual

Z-transform

Figure 4.6 Nakamura Transform Time-Series Based on Partial Impedance Data

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-600

-400

-200

0

200

400

600

Time [sec]

Dam

pin

g [

kN

s/m

]

Damping Coefficients (Ci)

91

damping matrix of the structure, as long as the total damping coefficient value at the node is not

less than zero there should not be an issue. Generating a negative mass term from the transform

model was a greater stability concern since the structure typically was assumed to have a

massless foundation resulting in a negative mass diagonal term in the mass matrix.

A causal impulse response could have been generated however simply by removing the tail

portion of the stiffness and damping time series presented in Figure 4.5. This alteration would

ensure that the coefficient time series are strictly impulses and causal however the impedance

function generated from them would have no similarity to that of the desired original impedance

function. On the other hand the Nakamura model based on partial data at least ensures there is

correlation between either the real or imaginary data.

Utilizing the new causal stiffness and damping coefficients in the convolution calculation

produces a stable response when combined with the time integrations scheme and a single

degree-of-freedom structure. The response was compared to a frequency domain analysis of the

combined soil and structure system and as expected there is a discrepancy between the results.

Given that soil and structure systems are linear elastic, the frequency domain analysis is

considered the exact solution for the combined system. The discrepancies between the responses

are significant, but given that the regular Nakamura model is incapable conducting this

simulation, a less accurate yet causal method is desirable for an approximate analysis.

92


Analysis and Nakamura Transform Model Using Partial Data

4.2. Method to Overcome Negative Instantaneous Mass Solutions

It has been observed that the Nakamura transform method has a unique stability issue concerning

the value of the instantaneous mass coefficient. A stable response in the time integration scheme

is typically unfeasible should the instantaneous mass coefficient be negative. Such a situation has

little physical meaning and cannot be accounted for in time integration schemes.

An original procedure has been developed that eliminates this instability scenario. It relies on

extending the impedance function into previously undefined frequencies and selecting new

impedance values that would force the mass to be positive during the coefficient formulation

procedure.

4.2.1. Impedance Expansion Procedure

A positive instantaneous mass value exhibits a negative parabolic relationship in the real

frequency domain and it comes to reason that any impedance with a similar behaviour should

produce a positive instantaneous mass using Nakamura’s transformation. This method therefore

93

proposes that any impedance exhibiting negative mass instability should be given a negative

parabolic shape in the real data portion beyond the inputted impedance. The imaginary data

portion in this new frequency should be a linear extension of the original imaginary impedance

function which would replicate the behaviour of a single damper system.

Increasing the number of frequencies of impedance function is only possible if the time step of

the convolution summation is greater than the time step of the integration scheme. Extending the

frequency range will increase the impedance’s maximum frequency and decrease the convolution

time step, and it is advantageous to keep the convolution time step a scalar multiple of the time

integration scheme time step. For this reason it is suggested that the impedance be extended by

doubling the number of frequencies in the impedance and thus the convolution time step would

be halved.

4.2.2. Example

Figure 4.9 Example Impedance

The impedance function presented in Figure 4.10 will be used as the soil domain in this example.

It was generated from a finite element model of a soil domain that will be presented in Chapter 5

and it was found to produce a negative mass coefficient when attempting to transform it using

Nakamura’s transformation. This negative mass invariably would cause divergent responses

0 5 10 15 20 254

6

8

10

12x 10

6

Frq. [sec-1]

Real

0 5 10 15 20 250

0.5

1

1.5

2x 10

7

Frq. [sec-1]

Imagin

ary

94

when time-history analysis is attempted. Therefore in order to overcome this instability issue the

proposed impedance expansion procedure will be performed.

The structure that is to be analyzed is the same as the single storey structure containing a single

degree-of-freedom presented in Section 3.1.2.3. It contains a natural period of 0.1987 seconds,

supports a 1000 kg mass and has a storey stiffness of 1000 kN/m. The damping ratio is selected

to be 5%. This storey is attached to the foundation degree-of-freedom where the restoring force

produced by the convolution calculation will act upon and where the instantaneous mass of the

transform will be.

The new expanded impedance function will be defined up to 50 Hz and contain twice as many

data points as the original impedance. Therefore the new convolution time step will be 0.02 s as

oppose to 0.04 s of the previous impedance configuration. The extended impedance function is

presented in the Figure 4.10. Beyond 25 Hz the parabola with negative values is apparent in the

real portion data and the imaginary data follows a linear trend in this range.

Figure 4.10 Extension of the Impedance Function In Order to Ensure Positive Mass in

Nakamura Models

Since the inputted impedance was undefined beyond 25 Hz, specifying dynamic stiffness values

for this region will be just as arbitrary as having it indeterminate. For this reason this procedure

0 5 10 15 20 25 30 35 40 45 50-7

-6

-5

-4

-3

-2

-1

0

1

2

3x 10

7

Frequency

Sxx -

Im

pedance

Extended Impedance Function

real

imag

95

will not decrease the level of precision of the impedance function capturing the soil model

behaviour because the impedance function remains the same in the region of interest. Conducting

this procedure allowed the Nakamura model to produce a positive instantaneous mass

coefficient. Using the determined coefficients allowed the time-history analysis to be conducted

and the structure response was accurate, proving that the higher frequency values were

inconsequential. The comparison of the structural response using the Nakamura model with the

extended impedance function and the frequency domain analysis is presented in Figure 4.11.

Figure 4.11 Total Displacement Response of Structure Simulated Using Frequency

Domain Analysis and Nakamura Transform Model Utilizing Impedance

Expansion Procedure

4.3. Conclusion

This chapter has provided a number of meaningful methods to overcome some stability issues

associated with the transformation methods. These causality mitigation methods provide a stable

convolution for non-causal impedance functions but in turn provide reduced accuracy in

conducting time-history analysis. However, given that non-causal impedance functions are

especially challenging to transform into the time domain, this compromise in accuracy may be

tolerable.

96

The instability that is due to the negative instantaneous mass in the Nakamura model was

demonstrated to be avoidable. One could always remove the instantaneous mass in order to avoid

this instability and consequently have an effective impedance function that is lower than

expected at high frequencies. The impedance expansion procedure however provides an accurate

transform function that also eliminates this instability. The only drawback that this procedure

possesses is that the force response function produced after expanding the impedance function is

significantly elongated which will reduce the efficiency of the convolution calculation of the

restoring force and increase the convolution time step that is utilized in the time integration

algorithm.

97

Chapter 5 Multiple Degree-of-Freedom Interface Validation

Building on the results and conclusions of the previous chapters, Chapter 5 will validate that

Nakamura’s model can be implemented to transform a number of impedance functions

exhibiting coupled behaviour. These impedance functions represent a soil system with multiple

degrees-of-freedom foundation interface. A new reference example will be created that contains

such a soil-foundation system supporting a beam-column structure with three degrees-of-

freedom. From the soil model a number of impedance functions will be generated and

transformed using Nakamura’s method, which will be integrated with the structural model and

used to perform time-history analysis accounting for soil-structure interaction. The responses

generated will be compared to the results of a finite element model that contains both the soil and

structure system, and should the responses be equivalent then Nakamura’s method can be said to

be suitable for this example.

5.1. Problem Statement

The reference problem must be a system where the structure and soil domains can be represented

in a single model and in two separate models. Furthermore the soil system must allow the

generation of impedance functions of the degrees-of-freedom at the interface of the soil and the

structure. The system that will be modelled is a single beam-column structure resting on an

infinite strip foundation. The soil beneath the foundation is a two dimensional soil domain

exhibiting plain-strain behaviour resting on shallow bedrock.

In comparison with the reference problem used in Chapter 3, the system model is two

dimensional with nodes containing three degrees-of-freedom; the displacement in the horizontal

and vertical directions as well as the rotation. The structure exhibits beam deformation behaviour

and the soil model will provide dynamic stiffness in the horizontal, rotational and vertical

displacements as well as a coupled stiffness component between horizontal and rotational

degrees-of-freedom.

98

5.1.1. Structure Parameters

The structure to be modelled is a 3.6 m tall steel column with a second moment of area and

sectional area of 348 × 106 mm

4 and 18,200 mm

2 respectively. The column is to behave similarly

to a vertical cantilever beam. The beam-column will be supporting a nodal mass of 36.7 tons.

With these parameters now defined the stiffness and mass matrix of the structure can be easily

assembled. The natural frequencies of the structure resting on a rigid foundation are provided in

Table 5.1.

Table 5.1 Natural Frequency of the Reference Structure

Natural Frequencies (Hz) 2.54 11.16 91.52

The above specified structure rests on a 2 m wide perfectly rigid and massless foundation. Such a

foundation is relatively straightforward to implement in a finite element model and allow the

transferring of moments from structural nodes that contain three degrees-of-freedom vertical

forces on the soil domain nodes that contain two degrees-of-freedom only. The use of an infinite

strip foundation also allows for the soil-domain to be modelled using a two-dimensional model

and to assume plane-strain conditions in the quadrilateral elements. The external load applied to

this structure is discussed in Section 5.1.5.

5.1.2. Soil Parameters

The soil system considered is an 8m deep soil domain resting on rigid bedrock. This rigid

interface was provided so that static stiffness of the soil could be easily determined in

comparison to a completely unrestrained soil domain that is sometimes utilized in soil-structure

analysis. The soil material has an elastic modulus of 162,409 MPa, poisons ratio of 0.25 and

density of 1.6 t/m3. For plain strain elements these parameters are sufficient to describe the soil

domain, and the parameters correspond to a soil shear velocity of 201.5 m/s.

5.1.3. Soil Model

The soil domain was modelled in Open System for Earthquake Engineering Simulation,

commonly referred to as OpenSees. It is a seismic simulation software suite developed by

researchers at the University of California, Berkley, which is especially capable in running many

types of non-linear analyses and provides a great variety of different modelling elements. Its

capability of running mixed dimensional elements has made it a common modelling tool in the

99

area of soil-structure interaction research which frequently requires various element types when

analyzing both the soil and the structure.

The soil was modelled using two dimensional quadrilateral elements of the 1 m length and width.

They form an 8 m layer and extend 100 m in either horizontal directions from the centre of the

model. This distance is provided so that an appropriate distance is available for the seismic

waves to propagate and dissipate. At the horizontal boundaries of the model viscous dampers

were provided in order to reproduce the infinite medium behaviour that the soil possesses

direction.

Since the soil quadrilateral elements do not contain a rotational degree-of-freedom, moment and

rotation is transmitted to the soil model by attaching a rigid beam connection to the structure at

the location of the interface. Because the beam is rigid, the horizontal and rotational

displacements will be constant for the nodes of the beam. Rotation in the beam will cause

differential vertical displacements in the soil nodes just below the beam. The soil model and rigid

beam interface is displayed in Figure 5.1.

From this model, impedance functions will be generated that will be used in the time-history

analysis using the Nakamura model. The means with which the impedance is generated is

Figure 5.1 FEM Model of the Soil Domain Used in the Impedance Generation

100

described in detail in Section 5.1.4. The first five natural frequencies of the soil domain are

provided in the table below.

Table 5.2 Natural Frequencies of the Soil Validation Structure

Values can be found in the literature for uncoupled static stiffness of a rigid strip foundation on a

soil stratum-over-rigid-base (Gazetas, 1983) The soil model’s determined static stiffness were

compared to Gazeta’s reference value but it should be noted that the soil model considered in this

validation problem lies outside the range of validity for the rocking stiffness. Furthermore the

values presented in the paper do not consider the coupled interaction between horizontal and

rocking movement which signifies that their values may be somewhat erroneous. The values

being within proximity is an indication that the soil domain has been properly modelled in

OpenSees. The discrepancy in vertical stiffness, of approximately 13%, is perhaps indicative that

the quadrilateral elements should have been smaller considering the shallow bedrock boundary,

however since this discrepancy will exist in both the reference model and the model using

Nakamura’s method, the validation exercise can still be executed.

Table 5.3 Comparison of the OpenSees model to Literature Values (Gazetas, 1983)

Stiffness Range of Validity Gazetas Soil Model Discrepancy

Vertical [kN/m] 1 < H/B < 10 153 152 133 295 -12.97 %

Horizontal [kN/m] 1 < H/B < 8 97 445 97 337 -0.11 %

Rocking [kN/rad] 1 < H/B < 3 139 461 208 185 49.3 %

5.1.4. Impedance Functions

The impedance functions were generated numerically, as described in Section 2.2.4, at discrete

frequencies by applying a sinusoidal force at the interface in the degree-of-freedom of interest

with all other motions at that location restrained. The resultant displacement and reaction forces

were then used to determine the discrete stiffness for that frequency. Repeating this procedure for

all degrees-of-freedom of interest and for all the frequencies generated all the necessary

impedance functions.

The frequency range considered in this analysis contains frequencies up to 25 Hz, which will

include most of the significant motion contained within the Kobe 1995 earthquake. The

frequency step utilized here is 1 Hz, which forms a frequency range containing twenty-five data

points, but in addition to this, an impedance value is included for a frequency approaching static

Natural Frequencies (Hz) 6.04 6.12 6.84 7.74 8.82

101

conditions. In this analysis, this near static frequency is taken to be 0.1 Hz because the inclusion

of the static case (0 Hz) into the impedance data would hinder the coefficient formulation

procedure.

For this validation example four unique impedances were formulated and they are presented in

the graphs in Figure 5.5. They include the horizontal, rocking and vertical impedance functions

as well as the coupled rocking-horizontal displacement impedance function.

All the above impedance functions appear to have no imaginary value at near static conditions

which is to be expected since no velocity is present at this point. The horizontal and vertical

impedances have a negative parabolic shape in the initial frequencies which is reminiscent of the

-50

0

50

100

150

200

250

0 10 20

[10

³ kN

/m]

Frequency [Hz]

c)

-20

-10

0

10

20

30

40

50

0 10 20

[10

³ kN

/rad

]

Frequency [Hz]

d)

0

30

60

90

120

150

180

0 10 20

[10

³ kN

/m]

Frequency [Hz]

a)

0

50

100

150

200

250

0 10 20[1

0³

kNm

/rad

]Frequency [Hz]

b)

Figure 5.2 a) Horizontal Dynamic Stiffness b) Rocking Dynamic Stiffness c) Vertical

Dynamic Stiffness d) Coupled Rocking-Horizontal Dynamic Stiffness

102

impedance of spring-mass system. The impedance deviates from this behaviour beyond 6 Hz

which is the location of the lowest natural frequency of the soil domain. This is perhaps an

indication that the impedance functions were ascertained correctly.

5.1.5. External Force

For an accurate simulation, the ground acceleration utilized in the analysis would need to be

imposed as if the rigid bedrock is shaken through the application of equivalent mass-proportional

loads at every node with mass. Furthermore the ground acceleration history utilized in the

analysis should be from field measurements deep in the soil near bedrock to obtain an

appropriate bedrock translation motion.

These transform methods alone however cannot provide a means to model the complete

interaction that occurs in soil-structure interaction analysis. As stated early on in this study the

transformation methods are appropriate only for modelling inertial interaction problems since the

soil model is removed and replace by the convolution calculation which can only respond to the

motions originating from the structure. For this reason the external loads considered in this

analysis will only be applied to the structural degrees-of-freedom.

The load utilized in this analysis is based on recordings of the Kobe 1995 earthquake. The

recorded acceleration history is multiplied by the nodal mass of the structure in order to

determine the equivalent external force which will be applied in the simulations only to the

structure as illustrated in Figure 5.3 and Figure 5.4. This is the same acceleration history that was

used in the Chapter 3 validation example of the single degree-of-freedom interface.

5.2. Reference Model

The reference model was created with both the structure and soil domains modelled jointly. This

model will be used to generate the reference responses which the Nakamura model will attempt

to replicate. It contains the soil domain of the previous model with the structural element and

mass now included to simulate the combined total system that experiences inertial soil-structure

interaction. In Figure 5.3 the reference model, comprised of the structure, rigid interfacing

elements, and soil domain, is displayed.

103

Figure 5.3 FEM Model of the Complete Validation Example Including Soil and Structure

Domains

5.3. Nakamura Model

The simulation involving Nakamura’s transform method will include the single structural node

as well as the foundational node associated with the movement of the interface. The

displacement and velocity of this interface node is utilized by the convolution calculation in the

time integration scheme to generate the interaction force. The instantaneous stiffness, damping

and mass components of the Nakamura transformation will be incorporated into the static system

matrices of the structure to create new system matrices. From these new combined system mass

and spring matrices an approximate period of the combined soil structure system can be obtained

that however does not account for any frequency dependencies of the soil. The combined system

is illustrated Figure 5.4 where the instantaneous coefficients constitute the soil static structure. It

is important to note that the instantaneous mass of the soil model has a unique value in each of

the three degrees-of-freedom and contains coupled mass components. It is equally important to

104

note that the three restoring forces displayed are dependent on four convolutions including the

coupled component between horizontal and rotational degrees-of-freedom.

Figure 5.4 Model of the Structure Domain and Soil Domain Modelled Using Nakamura’s

Transformation Method

The first attempt to perform the Nakamura transformation on these impedances proved to

produce an unstable combined convolution and time integration scheme. The reason for this was

that a negative mass was obtained from the coefficient formulation procedure which caused

instability when incorporated into the system matrices in the time-history analysis. To overcome

this obstacle, the impedance expansion method introduced in Section 4.2 was employed in order

to assure a positive instantaneous mass coefficient is generated. Only the horizontal and coupled

rocking-horizontal impedance functions required expansion to obtain stable coefficients.

5.4. Reproducing Impedance Function

It is important to first discern whether the determined coefficient will represent the frequency

domain impedance function accurately in the time domain. To do so the impedance functions

were determined numerically as previously described in Chapter 3 in Section 3.1.4 by applying

105

discrete sinusoidal displacements and determining the restoring force using the convolution

algorithm. The dynamic ratio between the restoring force and the displacement would be equal to

the dynamic stiffness at that specific frequency and repeating this process for all frequencies of

interest would construct the impedance functions. The impedance functions in Figure 5.5 were

generated from the coefficients and they agree with impedances that were inputted.

As evidenced in the above graphs, the coefficients generated for Nakamura’s method accurately

reproduce the original impedance functions found in Figure 5.2. The extended impedance

function portions of the impedance are evidenced in the horizontal and coupled horizontal-

rocking impedance functions since they are defined up to 50 Hz and display concave down

parabola shapes in the real portions.

Figure 5.5 Reproduced Impedance Functions from Nakamura Transform Coefficients

a) Horizontal b) Coupled Rocking-Horizontal c) Rocking d) Vertical

0 5 10 15 20 251.4

1.6

1.8

2

2.2x 10

5

Frq. [sec-1]

Impedance [

Real]

Actual

Numerical

0 5 10 15 20 250

5

10x 10

4

Frq. [sec-1]

Impedance [

Imagin

ary

]

Actual

Numerical

0 5 10 15 20 25

0

10

20

x 104

Frq. [sec-1]

Impedance [

Real]

Actual

Numerical

0 5 10 15 20 25-1

0

1

2

3x 10

5

Frq. [sec-1]

Impedance [

Imagin

ary

]

Actual

Numerical

0 5 10 15 20 25 30 35 40 45 50-10

-5

0

5x 10

5

Frq. [sec-1]

Impedance [

Real]

Actual

Numerical

0 5 10 15 20 25 30 35 40 45 500

1

2

3x 10

5

Frq. [sec-1]

Impedance [

Imagin

ary

]

Actual

Numerical

0 5 10 15 20 25 30 35 40 45 50-2

0

2

4

6x 10

4

Frq. [sec-1]

Impedance [

Real]

Actual

Numerical

0 5 10 15 20 25 30 35 40 45 50-1

0

1

2

3x 10

4

Frq. [sec-1]

Impedance [

Imagin

ary

]

Actual

Numerical

a) b)

c) d)

106

This consistency in the frequency domain between the input and reproduced impedances

however does not insure that performing time-history analysis using these transforms will

produce stable or accurate responses. This is because the inherent feedback nature of the

combined analysis is not captured in this numerical analysis as explained in Chapter 4.

5.5. Structure Response Comparison

Following the elimination of the negative mass stability issue and the accurate recreation of the

inputted impedance functions, all the tasks prior to executing the simulation had been completed.

The first analysis that is to be executed is the reference model that uses OpenSees to model both

the soil and structure domain simultaneous in a single finite element model. This model will

provide the reference response with which the proposed model will be judged against. The

proposed model will model the same system however the soil domain is modelled using

Nakamura’s transform method which captures the soil system’s contributions using a

convolution calculation.

Comparisons between the responses generated by the reference and proposed models will help

verify that the Nakamura transform method is applicable and accurate in modelling soil-structure

systems that contain multiple degrees-of-freedom interfaces. This is in contrast to the previous

validation that was conducted which verified Nakamura’s method for modelling systems with an

interface containing a single degree-of-freedom.

The force was applied in the horizontal direction which would activate only the horizontal and

rotational degrees-of-freedom of the soil and structure and since the vertical motion is

independent it will not experience movement during the simulation. Below the total displacement

response of the structure in the horizontal and rotational direction are displayed and they exhibit

near identical motions. In comparison with the fixed base model, the simulations that account for

the soil’s contribution exhibit responses that are much greater in amplitude, indicating that the

soil-structure interaction phenomenon is significant for this force loading. The identical response

of the reference OpenSees model and the model using Nakamura’s transform indicates that the

proposed method is effective in reproducing the soil domain’s contribution to the response.

107

Figure 5.6 Total Displacement Response of Structure in the Horizontal Degree-of-freedom

Figure 5.7 Total Displacement Response of Structure in the Rotational Degree-of-freedom

108

The displacement response of the foundation however exhibited slight discrepancies between the

reference model and the proposed model. The location of the peak response and the response

period are similar but the response amplitudes were not identical and this disagreement is

demonstrated visually in Figure 5.8. The difference in the peak displacement value between the

two models is 16.67 % and the local differences in the portion of the response are apparent in

Figure 5.8.b which displays an enlarged segment of the models’ response. This discrepancy can

be attributed to a number of possible issues perhaps due to compounded inaccuracy found in the

transform functions. There however was no noticeable difference between the rotational response

of the foundation node between the Nakamura simulation and the OpenSees simulation.

Ultimately the replication of the reference response by the model using Nakamura’s

transformation method is indicative of the proposed method’s potential as an effective alternative

to costly finite element based soil system modelling. This validation exercise demonstrated that

the analysis using Nakamura’s method is capable of modelling foundations with multiple

degrees-of-freedom interfaces. Though the reference example used one node with multiple

degrees-of-freedom its validity extends to interfaces with any number of degrees-of-freedom

across many nodes.

Figure 5.8 Response of Foundation in the Horizontal Degree-of-freedom during a)

Complete Simulation Duration b) between 6.5 and 9 seconds

a) b)

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

6.5 7 7.5 8 8.5 9

Time [s]

Soil DOF Horizontal Displacement

109

Chapter 6 Hybrid Simulation Validation

The last validation that is to be conducted in this study is the incorporation of Nakamura’s

transformation model into a hybrid simulation framework. This exercise will demonstrate that

Nakamura’s method can be utilized to analyze a structure undergoing a non-linear structural

response whilst accounting for the frequency dependent soil-structure interaction. The analysis in

this example is significant and novel because the proposed method can capture a nonlinear or

inelastic structural behaviour and the soil-structure interaction which is not possible in frequency

domain analysis.

6.1. Summary

The nonlinear behaviour of the structure could have been accounted for by using a similar

approach by including an iterative stiffness evaluation procedure to the time integration scheme.

However it was thought to be a more worthwhile endeavour to demonstrate the transformation

method’s ability to be substructured and separated explicitly from the integration scheme and

incorporated into a hybrid simulation framework. This required the restoring force convolution

calculation to be slightly altered in order to facilitate this integration. The integration alterations

will be discussed further on in Section 6.3.2.

An identical soil model will be used in this chapter to the multiple degree-of-freedom interface

soil model presented in Chapter 5. This was implemented in order to focus on the complications

involved in tailoring the transform procedures to hybrid simulations rather than obfuscating any

issues by introducing errors that could be present in the modelling of the soil and impedance

generation.

6.2. Problem Description

This reference problem will include a soil domain that is supporting a structure that is to undergo

a nonlinear inelastic response when loaded. The soil model is two dimensional with only

translational degrees-of-freedom while the structure is a beam-column element utilizing nodes

that contain both translational degrees-of-freedom and a rotational degree-of-freedom.

110

The combined soil and nonlinear structure are modelled together in a single finite element model

in OpenSees. This is the reference model and its structural response is to be replicated by the

hybrid simulation in order to demonstrate that the transform methods can successfully generate

the correct response.

The hybrid simulation utilizing Nakamura’s transform model is being verified. It contains the

nonlinear structural element in a substructure OpenSees model and the soil domain is represented

in another substructure that utilizes Nakamura’s transform. This substructure module utilizes the

impedance functions of the soil model, from Chapter 5 that was shown in Figure 5.2, to generate

the restoring forces acting on the structure based on the dynamic behaviour of the soil model.

The structure was designed to experience strains causing nonlinearity however in the structural

design it became apparent that the structure in the Chapter 5 validation example was too stiff to

experience yielding. Attempting to create a structure with only a single yielding column element

produced a system that was so soft that modelling the inertial interaction did not excite the soil

system since the deformation was localized at the structural level. This would signify that the

combined soil and structure system behaved almost identically to a fixed base model. In order to

observe both nonlinear structural behaviour and excite the soil domain, a two storey column

system was created. The top storey was given soft properties and was designed to yield while the

bottom column was designed to be relatively stiff in order to transfer displacement to the soil

domain.

Table 6.1 Structure Properties

Element Column

Height

Girder

Height

Web

Thickness

Flange

Thickness

Flange

Width

Units [m] [mm] [mm] [mm] [mm]

Yielding Column 3 251 4.8 5.3 101

Stiff Column 3 365 26.9 44.1 322

In this example the columns in the structure are three meters tall steel I-girder column supporting

a thirty ton mass at the first storey and a ten ton mass at the top of the structure. Both the mass

and stiffness of the structure has been changed from the structure in Chapter 5 and the new

properties are presented in Table 6.1. The two column system and soil domain are presented in

Figure 6.1 which depicts the reference system that is to be modelled in OpenSees.

111

Figure 6.1 Model of Soil and Structure System

The nonlinear behaviour of the steel column is a consequence of the stress-strain relationship that

is specified in the structural model. The steel is specified to yield at 350 MPa and rather than

maintaining a constant stress after yielding, the modulus of elasticity is specified to be 4000 MPa

after yielding. Furthermore the structural model utilizes a steel material behaviour that allows for

plastic deformations to occur, which consequently allows for hysteric damping to be captured in

the model. The stress-strain relationship of the structural component is presented in Figure 6.2.

Greater mass was given to the node above the rigid element than at the top node in order to

insure that the relative displacement in the yielding element was large enough to cause strains

greater than the yielding strain of 0.002. Larger mass at the rigid element would also hopefully

transfer greater force to the soil domain and cause significant soil domain participation in the

112

structural response. The inclusion of these mass systems produced a structural system with a

fundamental period of 0.983 and 0.239 seconds.

Figure 6.2 Stress-Strain Relationship

The structural model allows for the efficient modelling of nonlinear structural behaviour and the

soil-structure interaction phenomenon however it is not a representation of a realistic structure.

This is acceptable for this investigation since the modelling of soil-structure interaction and

nonlinear structural behaviour is of interest and perhaps in future studies, simulations of more

realistic case studies may be conducted.

The soil system considered in this simulation is the 8m deep soil domain resting on rigid bedrock

presented in Chapter 5. The properties and impedance function definitions are found respectively

in Section 3.1.2.3 and Section 3.1.3.

6.3. Hybrid Simulation

In this analysis hybrid simulation is being conducted by utilizing the software suite UI-SimCor

(Kwon & Elnashai, 2008) that allows for the substructure pseudo-dynamic simulations of

structures. The dynamic components of the analysis, which includes the mass and damping

components, are handled by this program and the restoring forces are obtained from structural

analysis applications or experimental specimen modules. Utilizing the alpha-operator splitting

0

50

100

150

200

250

300

350

400

0 0.002 0.004 0.006 0.008 0.01

Stre

ss [

MP

a]

Strain

Steel Stress-Strain Relationship

113

time integration scheme (Combescure & Pegon, 1997), the program predicts displacement at

each time steps and imposes these values on each substructure module monotonically. After

imposing these displacement and velocities, the modules determine the resulting forces and

return these values to UI-SimCor, where at that point the actual dynamic acceleration, velocities

and displacements are determined for the next time step.

For this example the structure and soil domains have been substructured with the restoring forces

of the structure being obtained from an OpenSees model and the restoring forces of the soil being

obtained from a new module that utilizes the Nakamura transform method. This Nakamura

model determines the restoring force based on the history of displacement and velocity and

instantaneous acceleration (Nakamura, 2006b). A non-causal impedance function is not expected

in this example so the partial data method introduced in Chapter 4 will not be used.

Since the Nakamura model conducts a convolution of the velocity histories in addition to the

displacement history convolution, UI-SimCor had to be modified to be able to send the predicted

velocities as well. It was unknown what would be the implications of using the history of

predicted displacement and velocities rather than the actual determined variables on the stability

of the Nakamura algorithm, but it was observed not to be an issue in this validation exercise.

A number of additional changes had to be made to the UI-SimCor program procedure in order to

conduct analyses using the Nakamura transform model. It was decided that the determination of

the coefficients of the force response function used in the convolution calculation should be

separated from the analysis algorithm. A pre-processor named REMUS was developed that

performs the coefficient formulation. The pre-processor creates files that contain the parameters

and coefficients required to run the Nakamura convolutions, which are read by the new module.

The instantaneous mass components determined by REMUS had to be manually inputted into the

UI-SimCor configuration files as mass at the foundational degree-of-freedom since mass

components were exclusively handled by the main program. However since UI-SimCor is only

capable of applying an acceleration to nodes with mass indiscriminately during the simulation,

these new masses had to be excluded from the force determining step in the time integration

scheme. The problem of the mass also complicated the issue of damping since UI-SimCor

currently is only capable of utilizing Rayleigh damping using the effective static stiffness of the

substructure modules and the mass terms that were inputted. To circumvent this problem UI-

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SimCor was revised to be able to accept a user defined damping matrix from a text file. In this

manner the damping of the structure in UI-SimCor was ensured to be identical to that of the

structure in the reference model using only OpenSees.

Once the coefficients have been determined by REMUS, the module program that determines the

restoring force for the soil domain will read these REMUS generated input files. The module

program is called NICA_M and it will be responsible for determining the restoring force using

the Nakamura convolution calculation and sending that value to the UI-SimCor program.

6.3.1. REMUS Pre-processor

REMUS was originally created with the ability to determine the coefficients for any of the three

transform methods investigated in Chapter 3 using any inputted impedance functions. The latest

version however is only operational using Nakamura’s method and it creates the aforementioned

data files that are read by the new module NICA_M. REMUS also allows the stability

assessment analysis, introduced in Chapter 3, to be conducted in case the stability of the

impedance transform and the given structural period is to be investigated.

The primary input file of the pre-processor is a text file that contains four input fields and an

example file is presented below in Figure 6.3. The first input field is the title of the analysis

which presently has little significance. The ‘Impedance File’ field should contain the filename of

the text files that contain the impedance functions corresponding to each degree-of-freedom that

is to be considered by the Nakamura transformation. These impedance files should contain three

columns of data: the frequency data, the real portion of the impedance function and the

imaginary portion. The ‘Interaction DOFs’ field expresses the degrees-of-freedom that the

corresponding impedance functions are associated with. For example, the ‘Sxr.txt’ file contains

the impedance function of the coupled first and third degrees-of-freedom as expressed by ‘1 3’. It

is important that these degrees-of-freedom correspond precisely to the degrees-of-freedom

present in the NICA_M module. The last input is that of the time step of the time integration

scheme. This variable is needed here since it is not communicated between UI-SimCor and the

module at any point.

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Figure 6.3 Input File for REMUS

The only input argument that REMUS requires is that of the name of the input file which in this

case is simply ‘input.txt’. Once the function is called the graphic user interface is started

displaying the first impedance function in the input text file. The GUI is presented below in

Figure 6.4. A number of variables are displayed immediately such as the number of data points

in the impedance function and the time step ratio which is the ratio between the chosen

convolution time step dependent on the maximum frequency of the impedance functions and the

inputted time integration scheme time step.

Figure 6.4 REMUS Graphic User Interface

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At this point the user may specify which model type is to be used to represent the impedance

function in the time domain. REMUS presents the three transform methods that were used in

Chapter 3 however, given the instability of two of the methods, only Nakamura’s method has

been fully implemented. If the Nakamura model is selected as the transform method further

options are presented to mitigate the formation of a transform with a negative mass, consisting of

forcing the mass to be zero and the Impedance Expansion Procedure introduced in Section 4.2.1.

If the extension option is selected, two sliders become visible which are used to determine which

two points on the impedance function are used for the parabolic extension of the real data.

Once the model options are selected the user is to press the Refresh Graph button at which point

the coefficients for the impedance function are ascertained. The actual impedance function

corresponding to the transform coefficients is determined numerically and presented visually on

the graph alongside the original input impedance data. The user may then evaluate the effective

impedance function’s accuracy and decide if it is appropriate and click the OK button to output

the coefficients. Figure 6.5 demonstrates how REMUS appears when the effective impedance is

being presented. If the model options are changed the user simply needs to click the Refresh

Graph button again to generate the new corresponding coefficients.

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Figure 6.5 Impedance Comparison in REMUS

Should the user want to execute the stability assessment analysis, described in Section 3.1.7, on

the given determined coefficients a period must be first input in the Structure Period field. The

analysis can only consider single degree-of-freedom structures so the user must use his best

judgment in choosing a period of the interested structural model here. An input value of one

second for the structural period will generate Figure 6.6 as the stability analysis curve for this

given impedance transformation.

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Figure 6.6 Stability Analysis Curve

Once the user agrees on the coefficients for the transformation model and has clicked on the OK

button this entire process will repeat three times more for the remaining impedance functions that

were specified in the input text file. Once all four impedances have been processed, REMUS

creates an output file containing all the stiffness, damping and mass terms and other necessary

variables that the module processor requires.

6.3.2. Algorithm Alterations

A number of alterations had to be made to both the UI-SimCor program and the convolution

procedure to facilitate the hybrid simulation. These changes are required because the alpha-OS

time integration scheme treats the instantaneous structural component differently than the

previously analyses that used the Newmark time integration scheme. This subsection details how

these discrepancies are overcome through a number of steps.

In the analyses that were conducted in previous Chapters, the instantaneous components of the

convolutions were incorporated into the structural matrices in order to consider their effects. In

the hybrid simulation framework however the restoring force of the substructures are determined

0 50 100 150 200 250 300 3500

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considering the current time step’s predicted displacement. That is to say that the UI-SimCor

program will transmit the current predicted time step to the NICA_M module program. NICA_M

will record and maintain the history of these predicted displacements in order to be used in the

convolution calculation of the restoring force. The instantaneous stiffness will be used in the

calculation since the current predicted displacement is being transmitted and not being handled

by the integration calculation.

A number of functions in UI-SimCor had to be altered in order to permit the transmission of

predicted velocities in addition to the predicted displacements. Only displacements were

necessary to be transmitted up to now since the pseudo-dynamic algorithm only required the

modules to apply the displacements monotonically to determine the resulting static restoring

force. The NICA_M program is in fact the first module that is not a purely static analysis on

account of the velocity’s contribution.

All mass elements in the simulation are to be specified by the UI-SimCor program since it

handles the dynamic analysis. The pre-processor will output a text file with the mass values

which need to then be manually input into the UI-SimCor configuration file at the foundation

node. Since the applied force on the model is based on the acceleration history of the Kobe

Earthquake, UI-SimCor would typically apply a load on the model that is equal to the negative

product of the masses present in the model and the acceleration history. The new input masses

from the Nakamura model however should not be included in the external force calculation since

it is desired that the external force only be applied on the structure. This will simulate the inertial

interaction portion of the soil-structure interaction that is being investigated in this study. The

UI-SimCor code therefore had to be edited in order to reflect this external force requirement.

Although the predicted velocities are now to be transmitted to the NICA_M module it was found

that the instantaneous velocity component was still needed to be managed by the UI-SimCor

algorithm rather than the module program. This is perhaps due to the fact that the damping

component was not intended to be used in the restoring force calculation but rather suppose to be

used in the determination of the pseudo-force in the alpha-OS procedure (Combescure & Pegon,

1997). UI-SimCor however presently is only capable of generating damping matrices by using

Rayleigh damping proportional to the global stiffness of all the modules combined and the mass

matrix inputted. The damping matrix that is desired however should best represent the isolated

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damping matrix of the structure in the OpenSees model used in the validation example. To do so

the additional masses and stiffness originating from the Nakamura model must not be included in

the damping formulation. The easiest procedure to accomplish this was found to be the manual

determination of the damping matrix and inputting it to UI-SimCor through a text file input.

6.3.3. Simulation Procedure

This subsection will detail every step in chronological order that is required in order to perform

the soil-structure analysis using the edited UI-SimCor program, the pre-processor REMUS and

the new module NICA_M. These steps reflect the changes and consideration listed in the above

section on the alterations that were required.

Having chosen the reference foundation and soil system that is to be modelled, the user will have

to determine the impedance functions of the soil model and discretize the data into three columns

containing the frequency, real data and imaginary data. The input text file to REMUS would then

need to be created in order for REMUS to determine the coefficients. The user would specify the

number of coefficients and the time step ratio in the pre-processor and REMUS would output

five .bin extension files and one text file which are described in Table 6.2.

Table 6.2 Description of the REMUS output files

File Name Description

Ki.bin Binary output file containing all the stiffness coefficients of all

impedance transforms

Ci.bin Binary output file containing all the damping coefficients of all

impedance transforms

dt.bin Binary output file containing the time step ratios for each impedance

transform

DOF.bin Binary output file containing the matrix that describes the interaction

between the degrees-of-freedom for each impedance function

N.bin Binary output file containing the number of data points and the number

of impedance functions

Mi.txt Text output file containing the instantaneous mass values that the user

needs to input into the UI-SimCor configuration file.

At this point it is important for the user to create the files necessary to run the modules that

contain the structures that are to be analyzed in the simulation. For this validation example this

would include the OpenSees .tcl extension file that contains substructure finite element structure

model that also includes the nonlinear and inelastic material properties. In addition to the

OpenSees input file, a configuration file for the NICA module is required that should be labelled

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‘NICA.cfg’. This file contains the necessary information required to communicate between both

the OpenSees software and the UI-SimCor program.

The user should manually input the instantaneous masses at the foundational degree-of-freedom

into the UI-SimCor configuration file. The damping matrix of the structural model is then to be

determined by the user algebraically and the instantaneous damping coefficients associated with

each of the impedance transforms need to be added to this matrix creating the global damping

matrix for the combined system. The damping matrix is to be inputted by the user into a text file

labelled ‘Global_C.txt’ so UI-SimCor can read it during the simulation.

UI-SimCor now contains two mass matrices, one possessing the instantaneous mass terms from

the transforms labelled the Total Mass Matrix. The other is used in the external force calculation

and only contains the masses present in the structure onwards called the Structural Mass Matrix.

The Total Mass Matrix is utilized everywhere else in the time integration scheme other than the

external force calculation.

All the preliminary steps have been accomplished and now the hybrid simulation may be

initiated. Upon commencing the NICA_M module, the program will load the data content of the

five binary REMUS output files which contain all the convolution calculation data. Binary data

files were used in order to allow the user to input variable number of impedance functions and

variable number of coefficients per impedance transform. Before commencing the hybrid

simulation, the UI-SimCor configuration file needs to be updated so that the module that

corresponds to the soil domain modelled using Nakamura’s transform method has a

‘SendTargetV’ variable value of one. This will inform UI-SimCor that it must send the predicted

velocities to that module in addition to the predicted displacements.

The UI-SimCor program determines the global static stiffness of the system by applying a static

displacement load to each module. The resultant restoring forces is determined in each module

and communicated back to UI-SimCor so it may assemble the global static stiffness matrix.

Upon starting the dynamic simulation the damping matrix is inputted from the text file

‘Global_C.txt’ and the pseudo-mass is determined by Eq. (6.1). The input acceleration history is

transformed into a force vector by multiplying the acceleration to the Structural Mass Matrix and

the influence vector.

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𝑀𝑝𝑠𝑒𝑢𝑑𝑜 = 𝑀𝑡𝑜𝑡𝑎𝑙 + 𝛾∆𝑡 1 + 𝛼 𝐶 + 𝛽∆𝑡2 1 + 𝛼 𝐾𝑠𝑡𝑎𝑡𝑖𝑐

At each time step UI-SimCor will predict displacement and velocity values based on previous

displacement, velocities and acceleration. These predicted values are sent to each module, which

in turn determine a restoring force value that is transmitted back to UI-SimCor. The pseudo-force

is then determined from all the returned values and the actual acceleration of the next time step is

determined as the product of the inverse of the pseudo-mass and the pseudo-force. Using the new

acceleration values the predicted displacement and velocities are corrected to obtain the actual

velocities and displacements of the current time step.

During the simulation the UI-SimCor algorithm will send the NICA_M module the current time

step’s predicted displacement and velocities and the module program will compile a history of

these values. These histories will be utilized to perform the convolution calculation needed to

determine the restoring force the soil imparts. The value for the restoring forces is then

communicated back to UI-SimCor who uses it to determine the actual displacement and velocity

of the system for that time step which is used to determine the next time step’s predicted

displacement and velocity.

At this point the UI-SimCor algorithm will be running for however many time steps necessary

required by the time-history analysis and it will be continuously communicating with both the

NICA_M module and the OpenSees module that contains the structure. Once the analysis is

complete all the modules and the UI-SimCor program will be terminated and the resultant

dynamic displacements of the nodes and degrees-of-freedom of interest are outputted in the

‘NodeDisp.txt’ file. From the displacements exported by UI-SimCor the remaining variables can

be derived if needed.

6.4. Structure Response Comparison

The displacements recorded from the hybrid simulation were compared to that of the same

system modelled entirely in OpenSees. Should the responses be in agreement then it would be

indicative that Nakamura’s transform is capable of modelling the soil domain when used in a

hybrid simulation framework for this example. Figure 6.7 through Figure 6.12 present visually

the comparison between the two nonlinear analyses, and in addition the linear-elastic response of

the total system is also presented.

(6.1)

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This linear-elastic model is labelled in the graphs simply as ‘Linear-Elastic’ and the nonlinear-

inelastic structure and soil system modelled entirely in OpenSees as the reference model is

termed ‘NonLin OpenSees’. The displacement response of the nonlinear system modelled in a

Hybrid framework using the Nakamura transform to model the soil domain is labelled ‘NonLin

Hybrid’.

Figure 6.7 Total Displacement Response of Topmost Node in Horizontal DOF

Figure 6.7 displays the horizontal response of the topmost node whose response is dominated by

the soft response of the yielding structural element. It is evident from that graph that the long

term structural response possesses a lasting plastic deformation since it deviates consistently

from the Linear-Elastic response. The linear-elastic response experiences greater peak

displacement amplitude at around 8 seconds perhaps for a few reasons. The displacement peak at

6 seconds likely induced plastic deformations in the column that required the system to reverse

this stored deformation which ultimately diminished the next cycle’s total amplitude.

Furthermore the hysteric behaviour the structure possesses develops an additional damping

mechanism which perhaps diminishes the amplification of the system.

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Figure 6.8 Stress-Strain Curve of the Yielding Column at Topmost Fibre during

Simulation

The stress-strain relationship presented above in Figure 6.8 demonstrates the hysteric response of

the structural yielding column being tested. As the horizontal load of the earthquake is applied to

the structure the beam-column’s flanges experience the greatest stresses and yield at certain

points during the simulation. Due to the inelasticity of the model the structure exhibits a

hysteretic response and dissipates some energy in this manner.

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-4

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Figure 6.9 Total Displacement Response of the Foundation in Horizontal DOF

The remaining graphs display the horizontal deformation of the interface node where the

foundation is located and the rotational deformation of the two nodes. It is important to notice

that the deformations of the foundation in Figure 6.9 are much lower in amplitude than that of

the structure’s response. The comparatively small response of the soil is perhaps indicative that

the system possessed little significant soil structure interaction. Insignificant interactivity would

suggest that the response of the combined system would be identical to that of a fixed base mode.

Figure 6.10 displays the horizontal response of the middle node in both the Fixed Case model

response and that of the model considering soil-structure interaction. The response presented in

the figure is between 4 and 14 seconds where the greatest seismic activity transpires and the

horizontal response presented is relative to the surface response. The Figure demonstrates the

error produced if the fixed base assumption is used, which demonstrates the importance of soil-

structure interaction since it generated greater inter-storey drift in the structure.

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Figure 6.10 Relative Displacement Response of the Fixed Base Model and the Hybrid

Simulation including SSI

The remaining Figures demonstrate the consistency between the reference model in OpenSees

and that of hybrid model using Nakamura’s method in the rotational degrees-of-freedom. The

identical behaviour is indicative that the coupled soil behaviour is being captured adequately by

the transform method.

Figure 6.11 Total Displacement Response of Structure in Rotational Degree-of-Freedom

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Figure 6.12 Total Displacement Response of Foundation in Rotational Degree-of-Freedom

These graphs ultimately demonstrate that the valid OpenSees model and the hybrid simulation

using Nakamura’s transform method produce identical results. This validates that Nakamura’s

method is a suitable method to capture the soil domain interaction when performing nonlinear

structural hybrid simulations. This type of analysis is an improvement on current industry

practices industry which relies on approximate linear soil and linear-elastic structure analyzed in

the frequency domain (Tyapin, 2007). Though nonlinear finite element modelling can analyze

soil nonlinearity in addition to structural nonlinear behaviour, which the investigated hybrid

simulations are incapable of, this method provides far greater computational efficiency when

modelling linear-elastic soil systems.

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Chapter 7 Conclusion

7.1. Summary

This study endeavoured to determine the capabilities of various transform methods to capture the

soil-domain behaviour in the time domain by utilizing impedance functions. It was demonstrated

that these methods allow for efficient time-domain analysis to be conducted on systems that

consider only the inertial soil-structure interaction of foundation systems. The use of these

transform methods provides an accurate means to account for soil-structure interaction that is

often computationally more efficient than the use of combined finite element models of the soil

and structure systems.

Through a number of validation examples it was shown that the Nakamura transform method

was by far the most robust of the three investigated transform methods. It successfully

reproduced response of structures with both single and multiple degrees-of-freedom resting on

soil and foundation systems that were modelled in reference finite element models. Lastly the

Nakamura model showed great potential in the last validation example where it was

demonstrated that the transform method can accurately model a linear elastic soil domain when

modelled together with a nonlinear inelastic structure in a hybrid simulation.

Though these models showed great promise in capturing the soil’s influence in simulations

involving soil-structure interactions, the transform models proved to be conditionally stable.

Known issues associated with non-causal impedance functions were explored and solutions

developed by other researchers were presented in this study. In addition to this, a solution was

developed to overcome the instability that arises in the Nakamura model when a negative mass

coefficient is formulated. Besides these known stability concerns, the transform methods have no

known general stability criteria and as a result each developed transform’s stability must be

evaluated. To facilitate this, a stability assessment procedure was developed that allows one to

discern the stability of a given transform prior to executing any time-history analysis.

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7.2. Contribution

Though all three methods are not original to this study, there was an evident lack of in-depth

evaluation of these methods when utilized in dynamic time-history analysis. A number of

validation examples were conducted in order to evaluate the various methods’ ability to model

increasingly more complicated soil-structure interaction systems and to demonstrate how each

method is conducted and integrated into the time-history analysis.

From even the simplest validation exercise conducted in this thesis it became evident that the

proposed transform methods possessed obvious stability concerns when incorporated into a time

integration scheme which had been little discussed in literature. The stability of the transform

dependent greatly on the characteristics of the impedance functions in the frequency domain. The

data content is difficult to judge but certain user defined impedance function parameters could be

manipulated so as to obtain a stable simulation. These parameters include but were not limited to

the maximum considered frequency and the frequency step of the impedance function. These

values determine the number of data points in the force response function and the time step of the

convolution calculation.

Defining a clear and general stability criterion for any of the proposed transform methods proved

to be unattainable given the uniqueness of each set of coefficients to each impedance function.

As an alternative a stability assessment procedure was developed to evaluate the stability of a

given set of transform coefficients when modelled in conjunction with a single degree-of-

freedom structure in a Newmark time integration scheme. The stability assessment evaluates the

system’s capacity to dissipate free vibration motion as simulation iterations are executed since in

unstable systems these free vibrations instead of dissipating, grow exponentially. The assessment

was demonstrated to be accurate since its evaluations of stability corresponded to the stability

witnessed in the executed simulations. Since the evaluation only models single impedance

transform in conjunction with a structure of a specified period, the assessment cannot determine

the precise stability of systems composed of multiple degrees-of-freedom and multiple

impedance transforms utilized simultaneously. It does however provide a means to estimate

individual impedance’s accuracy which may affect the overall stability of the global system.

The first validation example, that modelled a single storey structure with a single degree-of-

freedom interface with a simple soil domain, provided significant insight into the robustness of

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the transform models. It was shown that the Inverse Fourier transform and Şafak’s transform

models were unable to produce a stable numerical integration algorithm and thus failed to

produce a response for the reference system. This did not discredit the model’s ability to

transform other impedance functions, but it did demonstrate that these models lack general

applicability. Nakamura’s model was the only one able to reproduce the response of the system

and thus proved to be most robust model of those investigated.

The second validation exercise further demonstrated Nakamura’s transform model’s aptness in

time domain analysis. The reference system contained a more realistic soil domain modelled as a

two dimensional finite element model in OpenSees. It contained a multiple degree-of-freedom

interface which consequently produced a number of impedance functions that were necessary for

transform method. This validation illustrated the model’s capability of directly substituting a

complex finite element model of the soil domain with a greater number of interface degrees-of-

freedom. This demonstrated the suitability of Nakamura’s transform to model soil-structure

interaction systems that utilized more realistic and complex soil systems, provided that an

impedance function can be generated.

The final validation example was performed in order to demonstrate the potential of these

transform functions in modelling soil-structure interaction problems with consideration of the

non-linearity and inelasticity of the superstructure. In these transform models however the soil

domain must remain linear-elastic for the duration of the simulation. Nevertheless this

development is an advancement of most contemporary analyses in industry that rely on linear-

elastic frequency domain analysis since few engineers are willing to pursue the arduous task of

developing a non-linear finite element model. Furthermore there exists readily available software

that conducts frequency domain analysis considering soil-structure interaction (Tyapin, 2007).

This demonstrated that the transform methods present a convenient and efficient means of

modelling structure nonlinearity considering the interaction of the underlying soil and structure.

A number of stability issues and mitigation strategies, either from literature and original to this

study, were presented in this thesis. Non-causal impedance functions pose a serious stability

concern for the transform functions and a number of researchers have developed methods that try

eliminating or at least diminishing these obstacles. The presented strategies in exchange for

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greater stability compromise the accuracy of the transform methods when used in time-history

analysis. These inaccuracies are at times justified if a stable simulation is impossible otherwise.

A procedure was developed for the Nakamura’s transform model that prevents the transform

model from obtaining a negative instantaneous mass, which would surely produce instability in

the time integration schemes. The procedure forces the impedance to have certain characteristics

in the frequency range beyond where it was originally defined. Since this region is by definition

undefined and arbitrary in the analysis, defining it during the procedure should not affect the

accuracy substantially. Forcing the extended impedance to have a negative parabolic relationship

with frequency in the read data portion of the extended frequency range will force the mass to be

positive. This also consequently halves the time step and doubles the number of data points. This

method was shown to be accurate and frequency implemented in the latter two validation

examples.

Ultimately this investigation demonstrates the great potential these transform methods have in

the efficient time domain modelling of soil-structure interaction system. Potential stability issues

do exist in these methods, which have been discussed at length in this model, there do exist a

number of mitigation and stability assessment strategies to overcome these difficulties. These

stability concerns however are outweighed by the considerable capability that the transform

models offer in the modelling nonlinear structures considering the soil-structure interaction in the

time domain.

7.3. Limitations

Though this thesis has demonstrated the potential of using time-domain representations of

impedance functions to capture soil-structure interaction problems, they do possess some

obvious limitations. Most of these limitations have been addressed among the preceding chapters

and some are just inherent to the transformation methods in general.

In the analysis of the single degree-of-freedom structure it became apparent that stability of time

integration scheme can become imperilled by the introduction of the transform methods. Though

Nakamura’s method did prove to be capable of reproducing the validation examples, the method

still was very sensitive at times to the selection of the impedance parameters. Often times a great

deal of trial and error is needed to discern the appropriate parameter values. Though the stability

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assessment tool developed here is intended to ascertain a specific transformation’s stability, it

only does little to illuminate the source of the instability.

It was shown that non-causal impedance functions tend to produce force response functions that

are not impulses which in turn may negatively affect the stability of the combined algorithm.

This had been investigated by researchers; however it does not explain the instability that can

arise in causal impedance function transformations. Ultimately the lack of a clear and precise

stability criteria or a transform method that is unconditionally stable limits the general

applicability of these methods to some degree.

The transform methods are also inherently limited by their dependence on impedance functions.

All these methods are predicated on the fact that a suitable impedance function exists for the

given soil conditions and foundation configuration and without an impedance function these

methods are unusable. Furthermore any errors or assumptions found in the impedance functions

will exist in the time domain analyses using transformations of these functions. Nevertheless

impedance functions are utilized in industry for nuclear reactor design, albeit in frequency

domain analysis, using such computer tools as Dyna6 (Elkasabgy & El Naggar, 2013) or SASSI

(Tyapin, 2007). These tools can likewise be easily used in conjunction with transformation

methods investigated in this thesis to run time-domain analysis of non-linear structures

considering soil-structure analysis.

It is important to recall that all the analyses conducted in this thesis were strictly considering

only the inertial interaction portion of soil-structure interaction. The force that was utilized in

these analyses was that of negative product of the mass and acceleration history which is the

external force used in fixed base condition analysis. This external force allows one to assert that

the ground surface accelerates according to the input acceleration history. In these analyses

though the ground was not fixed and consequently neither the rigid base nor the foundation node

possessed an acceleration history similar to the inputted acceleration history.

These analyses are equivalent to analyzing the response of the soil and structure system to

vibrations originating from the structure rather than the movement of the ground. An accurate

soil-structure interaction analysis would require the modelling of the kinematic interaction in the

analysis procedure. The kinematic interaction analysis would involve the consideration of the

wave propagation of shear and pressure waves from the bedrock to the foundation. These waves

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would generate interaction forces at the interface which would then need to be utilized in the

inertial interaction analysis. Complete kinematic and inertial interaction analysis using

impedance functions transform methods have not been proposed or discussed.

7.4. Future Studies

This thesis demonstrated through increasingly complex examples the effectiveness of impedance

function transform methods, especially Nakamura’s method (2006b), to capture the inertial

interaction caused by combined soil and structure systems. What would be beneficial for the

progress of these methods is the larger scale verification analysis of three-dimensional model of

a realistic foundation, soil and structure system. Though multiple degree-of-freedom interface

foundation had been analyzed here, a three-dimensional analysis has yet to be performed and it

would be a significant validation in addition to those established here.

Furthermore it would be valuable to validate the transform methods against a realistic test subject

with recorded seismic data. One such large scale seismic test was conducted in Hualien, Taiwan

using a ¼-scale model of a nuclear power containment building (Choi, Yun, & Kim, 2001). The

experiment coordinators have made available acceleration history data of the structure and soil

when the test site was experiencing real seismic events. Modelling such a soil-structure system

experiment, using the methods investigated in this thesis, and comparing the results with field

measurements would demonstrate exactly how accurate these models and assumptions of linear-

elasticity are in comparison to a physical specimen.

It was discussed in the previous section that these methods are currently only capable of

capturing the inertial interaction portion of soil-structure problem. Any analysis attempting to

model realistic seismic soil-structure interaction systems using these transform methods will

require the consideration of the kinematic interaction. A logical procedure needs to be developed

and validated that allows the two interactions to be accounted for. Should the kinematic

interaction portion of soil-structure interaction analysis be incorporated into these investigated

transform methods, it would facilitate the computationally efficient consideration of complete

soil-structure analysis into non-linear inelastic structure modelling.

Such a development would allow other researchers to investigate the effect soil-structure

interaction has on structural response without the need of laborious and time-consuming finite

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element models of soil-structure systems. The transform methods allow for the relatively

efficient reuse of soil models since the speed of the convolution calculation is independent of the

complexity of the soil geometry or material. The ability to run quick successive simulations

would facilitate the analysis of the influence a particular soil system has on vast array of

structures with varying fundamental periods, higher mode excitations and perhaps using various

earthquake records.

Though the analysis conducted here always utilized a linear-elastic soil model in the simulations,

there exist a number of potential strategies that allow for the modelling of nonlinear soil

behaviour when using the investigated transform methods. One means of accounting for

nonlinear soil behaviour would be to utilize impedance functions generated from soil models

with properties adjusted iteratively to account for the soil’s degradation during an earthquake

event. This approximate technique is known as ‘Equivalent Linearization’ and it used often in

the Nuclear industry where soil nonlinearity is especially of interest (Lysmer et al., 1981). Given

the familiarity engineers have with this approximate method it would be worthwhile to

investigate how effective this approximation would be in conjunction with the transform

methods to conduct time-history analysis of a soil-structure interaction system where both

domains behave nonlinearly.

A more accurate means of capturing the soil’s nonlinear behaviour is to incorporate the portion

of the soil that will behave nonlinearly into the finite element model of the structure. The rest of

the soil can be modelled using the transform methods discussed in this thesis. In this way the

nonlinearity of the soil near the structure and the infinite medium behaviour of the soil further

away from the structure is captured adequately using modelling tools best suited for that

behaviour. The division between the far and near field soil is presented in Figure 6.13. In this

type of model each of the nodes at the interface of the two fields would contain impedance

functions that would need to be transformed and incorporated into the time-history analysis. The

impedance function generation may perhaps prove to be rather complicated; however the finite

element model that would be used in the near field modelling would still contain a reduced

number of elements compared to a finite element model that attempts to capture the complete

soil domain. Consequently this type of analysis may prove to be less computationally intense

compared to conventional finite element modelling.

135

Figure 6.13 Near Field and Far Field Modelling (Tzong & Penzien, 1986)

The hybrid simulations performed in Chapter 6 provided the initial framework of conducting

combined soil-structure analysis and nonlinear inelastic structural modelling however there are

still improvements to be made. Greater integration could be made between UI-SimCor and the

new NICA_M module that would eliminate the need for the user to change the code or transfer

data from the REMUS output to the UI-SimCor configuration input text file. Ultimately a

standalone user-friendly program could be developed to interact with the hybrid simulation

framework.

136

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141

Appendix A

This appendix defines explicitly variables that were introduced in the stability assessment

sections of Chapter 3. Variable definitions are unique to each of the transformation methods.

A.1 Inverse Fourier Transformation

The equation that defines all state variables in terms of previous state variables was described in

Eq. (3.27)

but is repeated once more here.

𝑈𝑖+1 = 𝐾0 × 𝑈𝑖 + 𝑎2 ∙ 𝐴0 × 𝑈𝑖−1 + 𝑎3 ∙ 𝐴0 × 𝑈𝑖−2 + ⋯

This equation is used to formulate the compounded iteration matrix. The 𝐾0 matrix is composed

of three matrix variables that were previously introduced.

𝐾0 =

𝕊𝑢𝕊𝑣𝕊𝑎

∙ 𝜌

Where:

𝜌 =1

𝑚 ∙ 𝑕𝑘𝛽 + 𝑐𝛾 + 𝑎0 ∙ 𝑕𝛽 ∙ 𝑚 + 𝑕 ∙ 𝑕𝑘𝛽 + 𝑐𝛾

The definitions of the variables in Eq. (A.1) are as follows:

𝕊𝑢 = 𝑚𝜃1 + 𝜑0𝜃2 −𝑕2𝛽 𝑎0𝑐𝛾 + 𝑎1𝜃1 𝑕 𝑚𝜃1 + 𝜑0 𝜃2 − 𝛽𝑐𝑕 𝜑0𝑐𝑕𝜃3

𝑕𝑘𝑚𝛽 𝑐𝑚𝛾 − 𝑎1𝑕𝛽 𝑚 + 𝑕𝜃1 𝑕𝑚𝛽𝜃4 −𝑐𝑚𝜃3

…

…−

1

2𝑕2 𝜑0 2𝛽 𝑚 + 𝑐𝑕 − 𝜃2 + 𝑚𝜃1 2𝛽 − 1

1

2𝑎0𝑐𝑕

3𝛽𝜃5

−1

2𝑕2𝑚𝛽 2𝜃1 − 𝑕𝑘 + 2𝑐 −

1

2𝑐𝑕𝑚𝜃5

𝕊𝑣 = −𝜑0𝑕𝑘𝛾 −𝑕𝛾(𝑎1𝜃1 + 𝑎0𝑐𝛾) 𝜑0(𝑘𝜃3𝑕 + 𝑚) + 𝑚𝜃1 𝑎0𝑐𝑕𝛾𝜃3

𝜑1 𝜑1 − 𝛾𝑎 (𝑚 + 𝑕𝜃1) 𝑚𝛾𝜃4 𝜃3 𝑘𝑚 + 𝑎0 𝑚 + 𝑕𝜃1 …

(A.1)

142

⋯

1

2𝑕 𝜑0 𝑕𝑘𝜃5 − 2𝑚 𝛾 − 1 𝜑0 + 𝜃1

1

2𝑎0𝑐𝑕

2𝜃5𝛾

1

2𝑕𝑚𝛾 𝑕𝑘 + 2𝑐 − 2𝜃1

1

2𝑕𝜃5 𝑘𝑚 + 𝑎0 𝑚 + 𝑕𝜃1

𝕊𝑎 =

−𝜑0𝑘 −𝑎1𝜃1 − 𝑎0𝑐𝛾 −𝜑0𝜃4 𝑐𝑎0𝜃3

𝑘𝑚

𝑕−𝑘𝑚 + 𝑎 𝑚 + 𝑕𝜃1

𝑕

𝑚𝜃4

𝑕−𝑎0𝑕

2𝜃1 + 𝑚𝑕 𝑎0 + 𝑘 + 𝑐𝑚

𝑕

…

⋯

1

2𝑕𝜑0 2𝜃1 − 𝜃4 − 𝑐

1

2𝑎0𝑐𝑕

2(2𝛽 − 𝛾)

−1

2𝑚 2𝜃1 − 𝜃4 − 𝑐

1

2(𝜃6(𝑘𝑚 + 𝑎0(𝑘𝛽𝑕2 + 𝑚)) + 𝑐(𝑎0𝜃6𝛾𝑕 + 2𝑚(𝛾 − 1)))

Where

𝑎 = 𝑎0 + 𝑎1 𝜃1 = 𝑕𝑘𝛽 + 𝑐𝛾 𝜃2 = 𝑚 + 𝑐𝑕𝛾

𝜃3 = 𝑕 𝛽 − 𝛾 𝜃4 = 𝑐 + 𝑕𝑘 𝜃5 = 𝑕(2𝛽 − 𝛾)

𝜃6 = 𝑕(2𝛽 − 1) 𝜑0 = 𝑎0𝑕𝛽 𝜑1 = 𝑘𝑚𝛾

The 𝐴0 matrix is defined as follows:

𝐴0 =

0 −𝑕2𝛽𝜃1 0 0 0 0

0 −𝑕𝛽 𝑚 + 𝑕𝜃1 0 0 0 00 −𝑕𝛾𝜃1 0 0 0 00 −𝛾(𝑚 + 𝑕𝜃1) 0 0 0 00 −𝜃1 0 0 0 0

0 −𝑚 + 𝑕𝜃1

𝑕0 0 0 0

∙ 𝜌

A.2 Nakamura Method (2006b)

The equation that is used to define the compound iteration matrix 𝐾𝐶, is presented below in Eq.

(A.3). It contains two convolutions which correspond to the convolution dependent on the

displacement and velocity history.

𝑈𝑖+1 = 𝐾0 × 𝑈𝑖 + (𝑘2 ∙ 𝐷0 + 𝑐2 ∙ 𝑉0) × 𝑈𝑖−1 + (𝑘3 ∙ 𝐷0 + 𝑐3 ∙ 𝑉0) × 𝑈𝑖−2 + ⋯

(A.2)

(A.3)

143

The 𝐾0 matrix can be formulated by determining the following matrices.

𝐾0 =

𝕊𝑢𝕊𝑣𝕊𝑎

∙ 𝜌

Where:

𝕊𝑢 = 𝑐𝑕𝛾𝜂0 + 𝑚𝜂 −𝑕2𝛽 𝑐𝑕𝑘 𝛾 − 𝑘 𝑚0 − 𝑕𝜖1 𝑕 𝑚𝜂 − 𝑐𝜃3𝜂0

𝑕2𝑘𝑚𝛽 𝑚 𝑚0 + 𝑕𝜖2 − 𝑕𝜃1 𝑕𝜖1 −𝑚0 𝑕2𝜃4𝑚𝛽

…

… 𝑕2𝛽 𝑐 𝑚0 + 𝑕 𝑘0𝜃3 − 𝑐1𝛾 + 𝑕𝑘𝜖3 −

1

2𝑕 𝑕𝑐𝜂0𝜃5 + 𝑚𝜂 𝜃6

𝑕 𝑕𝜃1𝜖3 −𝑚 𝑐 𝜃3 + 𝑕𝑐1𝛽 + 𝑚0 −1

2𝑕3𝑚𝛽 2𝜃1 − 𝜃4 − 𝑐

….

…

1

2𝑕3𝛽 𝑐 𝑕𝑘0𝜃5 − 2𝑚0 𝛾 − 1 − 𝑘𝜖4

−1

2 𝑚𝑕𝜖5 + 𝑕2𝜃1𝜖4

𝕊𝑣 = −𝑕𝑘𝛾𝜂0 −𝑕𝛾 𝑐𝑕𝑘 𝛾 − 𝑘 𝑚0 − 𝑕𝜖1 𝑘 𝛽 − 𝛾 𝜂0 + 𝑚𝜂

𝑕𝜑1 −𝑕𝛾 𝑘 𝑘 𝛽𝑕2 + 𝑚 + 𝑘 𝜃2 𝑕𝜃4𝑚𝛾…

… 𝑕𝛾 𝑐 𝑚0 + 𝑕 𝑘0𝜃3 − 𝑐1𝛾 + 𝑕𝑘 𝑚0 − 𝜖3

1

2𝑕 𝑕𝑘𝜂0𝜃5 − 2𝑚 𝛾 − 1 𝜂

𝑚0 − 𝑐1𝛾 𝑚 + 𝑕𝜃1 + 𝑕𝜃3 𝑘 𝑚 + 𝑘0𝑕𝜃1 1

2𝑕2𝑚𝛾 −𝑘𝜃6 − 2𝑐 𝛾 − 1

…

…

1

2𝑕3𝛽(𝑐 𝑕𝑘0𝜃5 − 2𝑚0 𝛾 − 1 − 𝑘𝜖4)

−1

2𝑕 𝑚𝜖5 + 𝑕𝜖4𝜃1

𝕊𝑎 = −𝑘𝜂0 𝑘 𝑚0 − 𝑕𝜖1 − 𝑐𝑕𝑘 𝛾 −𝜂0𝜃4 𝑐1𝑕 c𝛾 − 𝑕𝑘𝛽 + 𝑚0𝜃4 − 𝑕𝜅𝜃3

𝑘𝑚 −𝑘 𝑘 𝛽𝑕2 + 𝑚 − 𝑘 𝜃2 𝑚𝜃4 − 𝑐 + 𝑕𝑘0 𝑕𝜃1 −𝑚 𝑐1 + c + 𝑘 𝑕 …

…

1

2𝑕𝜂0𝜖6

1

2𝑕2𝜃5𝜅 −

1

2𝑕𝑚0𝜖6

−1

2𝑕𝑚𝜖6

1

2𝑕𝜃6 𝑘 𝑚 + 𝑘0𝑕𝜃1 + 𝑕 𝑐 𝑚 + 𝑐0𝑕𝜃1 𝛾 − 1

144

The newly introduced variables are defined below.

𝜂0 = 𝑚0 + 𝑕(𝑕𝑘0𝛽 + 𝑐0𝛾) 𝜂 = 𝑚0 + 𝑕(𝑕𝑘 𝛽 + 𝑐 𝛾) 𝜌 = 𝑕𝜂0(𝑕𝑘𝛽 + 𝑐𝛾) + 𝑚𝜂

𝜖1 = 𝑕𝑘1𝛽 − 𝑐0𝛾 𝜖2 = 𝑐 𝛾 − 𝑕𝑘1𝛽 𝜖3 = 𝑚0 − 𝑐1𝛽𝑕 − 𝑐0𝜃3

𝜖4 = 𝑚0𝜃6 + 𝑕𝑐0𝜃5 𝜖5 = 𝑚0𝜃6 + 𝑕𝑐 𝜃5 𝜖6 = 𝑘𝜃6 + 2𝑐 𝛾 − 1

𝑘 = 𝑘0 + 𝑘1 𝑐 = 𝑐0 + 𝑐1 𝜅 = 𝑘0𝑐 − 𝑘𝑐0

The 𝐷0 and 𝑉0 matrix that are used in the convolution portion of Eq. (A.3) are defined here.

𝐷0 =

0 −𝑕2𝛽𝜃1 0 0 0 0

0 −𝑕𝛽 𝑚 + 𝑕𝜃1 0 0 0 00 −𝑕𝛾𝜃1 0 0 0 00 −𝛾(𝑚 + 𝑕𝜃1) 0 0 0 00 −𝜃1 0 0 0 0

0 −𝑚 + 𝑕𝜃1

𝑕0 0 0 0

∙ 𝜌

𝑉0 =

0 −𝑕3𝛽𝜃1 0 0 0 0

0 −𝑕2𝛽 𝑚 + 𝑕𝜃1 0 0 0 0

0 −𝑕2𝛾𝜃1 0 0 0 00 −𝑕𝛾(𝑚 + 𝑕𝜃1) 0 0 0 00 −𝑕𝜃1 0 0 0 00 −𝑚 − 𝑕𝜃1 0 0 0 0

∙ 𝜌

(A.4)

(A.5)

145

Appendix B

This appendix will outline the various functions and scripts that were utilized in the examples

throughout this thesis. The main reason for this section is so that anyone in the future interested

in the precise code and algorithms that were used in this study may examine them.

B.1 Single degree-of-freedom Interface Example

The first validation example consisted of a simple structure and soil domain that were entirely

constructed from spring, damper and mass elements in series. The problem statement is available

in Section 3.1.2 of Chapter 3. The soil domain was modelled using the three proposed transform

methods: inverse Fourier transform, Nakamura’s transform, and Şafak’s transform models. The

first scripts created utilized Newmark time integration scheme which had the convolution

calculation embedded within the algorithm of the numerical integration. Later program iterations

would divorce these two functions so that the restoring force would be the output of the

convolution function and utilize the alpha-OS integration scheme.

Below the functions and important variables utilized in the version of the scripts that use a

combined convolution and time integration algorithm are presented. The script is found in the

folder directory Final Program Versions\SDOF Example and the main script file is

SDOF_Example.m. The configuration variables that control various aspects of the simulation

are presented in Table B.1 while the functions that are used in the script are found in Table B.2.

Table B.1 Configuration Variables for SDOF_example.m

Section Variable Description

Impedance

Model

Parameters

dtRatio Ratio between the time step used in the numerical time

integration and that used in the convolution calculation.

nFrq Number of frequencies in the impedance data that later dictates

the number of coefficients in the transform model

Structural

System

Parameters

k Structural Stiffness

m Structural Mass

xi Damping ratio

nDOF Number of degrees-of-freedom in the structural model

Model Scenario

SoilModel Type of Soil

[1: Stiff soil case, 2: Soft soil case]

AppliedLoad Type of External Load

[1: Sinusoidal, 2: Sinesweep, 3: Kobe Earthquake]

146

Table B.2 Functions Used in SDOF_Example.m

Function Name Arguments Outputs

ReproduceDynaStiff()

[Determine the impedance

function numerically from

the coefficients and plot

against the original

impedance functions]

Ki Stiffness Coefficients Void

Ci Damping

Coefficients

Mi Instantaneous mass

F_in Frequency data

S_in Impedance data

dt Time step

dt_Nak Convolution time

step

ReproduceZ()

[Determine the impedance

function from the using Z-

transform method and plot

the results]

Ki Stiffness Coefficients Void

Ci Damping

Coefficients


Frq Frequency data

S Impedance data

dt Time step

NakamuraCoef()

[Determine the Coefficients

for Nakamura the transform]

f Frequency data Kimp Stiffness Coefficients

S Impedance data Cimp Damping Coefficients

plotOut

put

Whether or not to

plot impedance data

Mimp Instantaneous Mass

Impeded_NewmarkMDOF()

[Time-history analysis of

MDOF structure and

Nakamura transform model

involving multiple

impedance function acting on

multiple degree-of-freedom

interface]

Mo Struct. Mass Matrix t Time history

Co Struct. Damping

Matrix

a Acceleration history

Ko Struct. Stiffness

Matrix

v Velocity history

Fo External Force u Displacement history

dt Time step T Fundamental Periods

Ki Stiffness Coefficients

Ci Damping

Coefficients


dt_Nak Convolution time

step

iDOF Impeded degree-of-

freedom matrix

convCoefficients()

[Determines Coefficients for

the inverse Fourier transform

method. Conv stands for

convolution of displacement]

Frq Frequency data t Time History

S_anal Impedance data St Convolution

Coefficients

Convoluted_Newmark2()


SDOF structure and inverse

Fourier transform model


Co Struct. Damp. Matrix a Acceleration history


Matrix

v Velocity history


147

using only one impedance] dt Time step T Fundamental Periods

dt_conv Time step of

Convoltuion

St Force Impulse

Coefficients

SafakCoef()

[Determine coefficients for

the Safak method]

Frq Frequency data Ai Previous Force

Convolution

Coefficients

S_anal Impedance data Bi Displacement

Convolution

Coefficients NumSaf

akCoef

Number of

Coefficients to be

determined

dt_Safak Time step of

convolution

frqCut Portion of impedance

to be considered in

optimization scheme

Safak_Newmark()


SDOF structure and Safak

transform model using only

one impedance]


Co Struct. Damping

Matrix

a Acceleration history


Matrix

v Velocity history


dt Time step T Fundamental Periods

dt_Saf Time step of

Convoltuion

ai Force Convolution

Coefficients

bi Displacement

Convolution

Coefficients

B.2 Stability Analysis Code

The stability analyses that were conducted in Chapter 3 are completely contained in three

functions corresponding to each of the three transform methods and they presented in Table B.3.

Prior to running these functions the inputted impedance functions have to be transformed and the

convolution coefficients need to be determined. The stability analyses also require that the user

input the properties of the single degree-of-freedom structure that is to be considered.

Table B.3 Stability Analyses Functions Function Name Arguments Outputs

Stable_Conv()

[perform stability analysis on

given inverse Fourier

transform and plot

eigenvalues, cannot consider

unequal timesteps]

St Displacement Convolution

Coefficients

Void

dt Time step

m Struct. Mass

c Struct. Damping

k Struct. Stiffness

148

Stability_Nak()


given Nakamura transform and

plot eigenvalues]

Knak Stiffness Coefficients Void

Cnak Damping Coefficients

Mnak Instantaneous mass

dt Time step

dtNak Time step of Convoltuion

m Struct. Mass

c Struct. Damping

k Struct. Stiffness

Stability_Saf()


given Safak transform and plot

eigenvalues]

A Force Convolution

Coefficients

Void

B Displacement Convolution

Coefficients

dt Time step

dtNak Time step of Convoltuion

m Struct. Mass

c Struct. Damping

k Struct. Stiffness

B.3 Multiple Degree-of-freedom Interface Example

Chapter 5 focused on the modelling of a reference structure with an interface that contained

multiple degrees-of-freedom which required a number of impedance functions, corresponding to

each degree-of-freedom, to be transformed. This validation example no longer considered all

three transform methods but only focused on the Nakamura model and compared the results of

the simulation using that transform with that of the total system modelled in OpenSees.

The impedance functions were determined from an OpenSees model by applying a sinusoidal

force at the interface and determining the response of the system for every frequency of interest.

The script genTotalImpedance.m manipulated the OpenSees input files and ran this

impedance generation analysis for twenty five frequencies and it may be found in the directory

Open\Impedance Generation. The OpenSees file that was utilized in this analysis is in the same

folder and it is labelled as Main.tcl.

The results of this script are found in four text files that contain the numerically generated

impedance functions. They have been transferred to the folder Final Program

Versions\MDOF Example where the analysis scripts for this validation example are found.

The impedance files are Sxx.txt, Sxr.txt, Srr.txt, and Syy.txt. The analysis script is

MDOF_example.m and it utilizes the same functions corresponding to the Nakamura

149

transformation that are found in Table B.2. The analysis configuration variables are different

than those presented in Section B.1 and they are presented below in Table B.4.

Table B.4 Configuration Variables for the MDOF_example.m Script

Variable Description

AppliedLoad Type of External Load

[1: Sinusoidal, 2: Sinesweep, 3: Kobe Earthquake]

DampingType Type of Damping present in the Structure

[1: Rayleigh Damping, else: specific damping]

dt_Ratio Ratio between the time step used in the numerical time integration and that used in

the convolution calculation.

frqMult If larger than 1 than the number of data points in the impedance function will be

greater than the number of data points in the input impedance function. Values for

new frequency data determined through interpolation.

zeroMass Remove mass if the instantaneous mass is less than zero

extendImp If this value is one, the impedance expansion procedure specified in Section 4.2.1

will be performed on the input impedance functions

genFreqAnalysis If this value is one, frequency domain analysis will be performed on the system

and plotted against the result of the time-history analysis uses the transform

method.

openseesData If this value is one, the OpenSees data will be inputted from a text file and plotted

against the result of the time-history analysis that uses the transform method.

B.4 α-Operator Splitting time Integration and REMUS

Before conducting the hybrid simulations presented in Chapter 6 the above examples were

conducted once more with a new time integration scheme. Furthermore the convolution

calculation was made into a distinct function in order to separate the force calculation from the

time integration which is the case in the hybrid simulation procedure found in UI-SimCor.

In addition to these changes a pre-processor was developed that simplifies the creation of these

transform models and combines the several functions and tools that had been developed in this

thesis. These tools include the numerical generation of the actual impedance function the

convolution coefficients represent and conducting the stability analysis previously mentioned.

The pre-processor was named REMUS and a description of how to operate it was given in

Section 6.3.1. The version of REMUS described here is an earlier version that was not altered to

be used in the UI-SimCor framework so it may be used for other integration schemes and

purposes.

150

For this version of analyses the time integration scheme no longer requires that the stiffness,

damping coefficients and mass to be transferred individually as arguments. Instead a new class

called Coefficients was introduced that contains most of the data required for the

Nakamura transform. In fact the structure was constructed with the ability to store the data for

any of the three transform models and its property variables and class functions are presented in

Table B.5.

Table B.5 Elements for Class Coefficients

Properties

Variable Description

modelType Determines what transform method is to be used

[1: inverse Fourier transform, 2: Nakamura model, 3: Safak model]

Ki Stiffness Coefficients

Ci Damping Coefficients


Ai Force Convolution Coefficients

Bi Displacement Convolution Coefficients

dt_Ratio Convolution time step (array)

N Number of Coefficients

Methods

Functions Description

Coefficients() Initialization

clearData() Empties all variables

With the definition of this class established the functions utilized by the latest iteration of

functions can be described. These files are found in the directory Final Program

Versions\Alpha-OS Code and the functions are described in Table B.6.

Table B.6 Functions for α-OS analysis using REMUS

Function Arguments Outputs

SIM_Init()

[starts REMUS and

determines coefficients]

inputFile String input of the name

and extension of the file

input

Coeff Coefficients object that

contains the data

necessary for transform

method

dt Time step of analysis iDOF Impeded degree-of-

freedom matrix

AlphaOS2()

[Time-history analysis

using α-OS time


Co Struct. Damping Matrix a Acceleration history

Ko Struct. Stiffness Matrix v Velocity history

F External Force u Displacement history

151

integration and

Nakamura’s method]

trnsf_Coeffs Coefficients object that

contains the transform

method data

T Fundamental Periods

iDOF Impeded degree-of-

freedom matrix

Documents

Incorporating Time Domain Representation of Impedance