DYNAMIC SOIL-STRUCTURE INTERACTION FOR LAYERED GROUND

By

HAO SHEN

A thesis in fulfillment of the requirements for the degree of Bachelor of Engineering

School of Civil and Environmental Engineering

The University of New South Wales

Sydney, Australia

October 2013

i

Abstract Taking into account dynamic soil-structure interaction in layered soil is quite

essential for some real cases, like earthquakes or machine-induced vibrations.

Both the out-of-plane and in-plane motion of a homogeneous, semi-infinite soil

layer under dynamic load is studied. For this thesis, the main purpose is to explain

and test two different methods of getting the acceleration unit-impulse response

matrix and finding out which method is more reliable and efficient.

First of all, the dynamic stiffness is required, which represents the

relationship between applied forces and the displacements. But, the key point is to

find out the acceleration unit-impulse response matrix . Based on the scaled

boundary finite element method, the scaled boundary finite element equation is

obtained, which can be used to deduce the values of in each time step. In

this thesis, there are mainly two methods to calculate : the constant scheme

and the linear scheme. For the constant scheme, the whole time period is

discretized into many time intervals with finer mesh. In addition, the unit-impulse

response coefficient is assumed to be constant within each time step. While

for the linear scheme, within each time interval, the values of change

linearly.

There are three numerical examples used to test these two schemes in a

Matlab program. For each example, the results will be compared with the

reference solutions both with respect to accuracy and time consumption (CPU

time). Then, we can find out which method is more efficient.

In conclusion, for the constant scheme, the results are usually in high

accuracy. But it costs lots of time. Although there are some errors, the linear

scheme usually requires less time. In addition, the errors can be diminished if we

choose the appropriate parameters.

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Acknowledgements

I would like to express my deep sense of gratitude to Dr. Carolin Birk, my

supervisor, for advising me on this research topic. She is an expert in the structural

dynamics and engineering mechanics fields. Dr Birk continuously and persuasively

inspired me throughout my study here. Without his timely support,

encouragement and advice, this thesis would not have been completed.

I would also like to thank Professor. Chongmin Song and John. P Wolf, who

established and developed the theory of scaled boundary finite element method,

and the basic knowledge about finite element method, and the Matlab codes

created by Dr Song, which helps me a lot on numerical example.

I am grateful to Dr. Wei Gao for exchanging research ideas and some basic

knowledge about material properties, and also Jason Zhao about helping me on

English grammar, format and editing in my thesis writing.

I wish to thank my father Peng Shen, my mother Xi Wang as well as other

members of my family for teaching, raising and supporting me.

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ORIGINALITY STATEMENT

‘I hereby declare that this submission is my own work and to the best

of my knowledge it contains no materials previously published or

written by another person, or substantial proportions of material which

have been accepted for the award of any other degree or diploma at

UNSW or any other educational institution, except where due

acknowledgement is made in the thesis. Any contribution made to the

research by others, with whom I have worked at UNSW or elsewhere,

is explicitly acknowledged in the thesis. I also declare that the

intellectual content of this thesis is the product of my own work, except

to the extent that assistance from others in the project's design and

conception or in style, presentation and linguistic expression is

acknowledged.’

Signed ……………………………………………..............

Date ……………………………………………..............

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Table of Contents

ABSTRACT ................................................................................................................................................ I

ACKNOWLEDGEMENTS ....................................................................................................................... II

ORIGINALITY STATEMENT .............................................................................................................. III

1 INTRODUCTION ................................................................................................................................. 1 1.1 STATEMENT OF PROBLEM ........................................................................................................................ 1 1.2 DIRECT METHOD ....................................................................................................................................... 2 1.3 SUBSTRUCTURE METHOD ......................................................................................................................... 3 1.4 THESIS OUTLINE ........................................................................................................................................ 6

2 LITERATURE REVIEW ...................................................................................................................... 8 2.1 INTRODUCTION .......................................................................................................................................... 8 2.2 FINITE ELEMENT METHOD (FEM) ........................................................................................................ 9 2.3 BOUNDARY ELEMENT METHOD .......................................................................................................... 10 2.4 INFINITE ELEMENTS .............................................................................................................................. 11 2.5 ABSORBING BOUNDARY CONDITIONS................................................................................................. 12 2.6 SCALED BOUNDARY FINITE ELEMENT METHOD ................................................................................ 14

3 THEORETICAL CONCEPTS ............................................................................................................17 3.1 INTRODUCTION OF HOMOGENEOUS, SEMI-INFINITE SOIL LAYER PROBLEM ................................. 17 3.2 TRANSFORMING CARTESIAN COORDINATES INTO LOCAL COORDINATES ..................................... 21 3.3 DERIVATION OF THE SCALED BOUNDARY FINITE ELEMENT EQUATION IN FREQUENCY DOMAIN

FOR OUT-OF-PLANE MOTION IN A 2D LAYER ............................................................................................ 26 3.4 DERIVATION OF SCALED BOUNDARY FINITE ELEMENT EQUATION IN TIME- DOMAIN .............. 32 3.5 CONSTANT SCHEME OF DERIVING THE ACCELERATION UNIT-IMPULSE RESPONSE MATRIX ... 37 3.6 LINEAR SCHEME OF THE OBTAINING ACCELERATION UNIT-IMPULSE RESPONSE MATRIX ....... 39 3.7 TIME-DOMAIN SOLUTION FOR ................................................................................................... 43

4 NUMERICAL EXAMPLES ................................................................................................................49 4.1 OUT-OF-PLANE MOTION OF HOMOGENEOUS-SOIL LAYER ............................................................... 49 4.2 IN-PLANE MOTION OF HOMOGENEOUS-SOIL LAYER ......................................................................... 67 4.3 HOMOGENEOUS SEMI-INFINITE SOIL LAYER WITH TRENCH ............................................................ 80

5 CONCLUSION AND FUTURE WORK ............................................................................................96 5.1 SUMMARY ................................................................................................................................................ 96 5.2 RECOMMENDATIONS FOR FUTURE RESEARCH................................................................................... 98

APPENDIX A ..........................................................................................................................................99

APPENDIX B ....................................................................................................................................... 100

APPENDIX C ....................................................................................................................................... 101

APPENDIX D ....................................................................................................................................... 102

BIBLIOGRAPHY ................................................................................................................................. 103

1

Chapter 1

Introduction

1.1 Statement of problem

To civil engineers, the analysis of structures on soils or fluids is one of the most

difficult technical problems need to be solved. For both the design and

construction, the bearing capacity of static loads of structures is one of the major

problems to be dealt with. At present, a lot of proven techniques can be used to

solve that problem, like ground improvement, foundation work and reinforced

concrete. Not just the static load, dynamic soil-structure interactions, like

earthquake, waves from moving train underground and the machine induced

vibration, are also major factors of the collapse of the buildings and structures.

Because of overlooking and lacking of techniques, a large amount of buildings are

damaged from dynamic loads. In the past few years, many experts and engineers

studied in this area and try to find a method to solve the problems.

In order to study how the dynamic action affects structures, we need to

determine the actual response of the structures and surrounding soil materials,

like the displacements. A so-called dynamic soil-structure interaction analysis

should be performed. Before that, a numerical model of structure and soil domains

needs to be established. In this chapter, some previous strategies of modeling

media of infinite extent and their shortages will be introduced.

According to Trinks (2005), there is a strong relationship between response

of structure and the condition of the underlying ground. Waves can propagate

through the soil and the structure interacts dynamically with the unbounded

medium. In that case, those numerical models should include both the bounded

structure and the unbounded medium. From Wolf and Song (1996), the most

difficult problem remained is how to model and analyze the unbounded soils.

2

For statics analysis, the displacement of unbounded domain tends to zero at

some distance from the loading position. Hence, a fixed boundary can be created

with specific boundary condition in the model. Thus, finite element method can be

used for statics load problems. But it is totally different for the dynamic analysis of

the unbounded domain.

Nowadays, there are mainly two methods to analyze the dynamic

unbounded medium-structure-interaction, Direct Method and Substructure Method.

Both of them are summarized in the next few paragraphs.

1.2 Direct method

In direct method, only a part of soil adjacent to structure will be analyzed together

with the structure. In order to satisfy the radiation conditions, an artificial

boundary need to be created in a suitable place to enclose the finite part. The finite

part or bounded domain is the combination of structure and near-field medium.

Finite Element Method can be used to analysis this coupled system. Finite

Element Method (FEM) is an efficient and well-developed numerical method to

obtain approximate solutions of vibration systems. But the problem of this method

is the reflecting of waves on artificial boundary, which may require additional

measures. So, finite element method is not suitable for the dynamic case here.

In order to solve that problem, some applicable strategies were created to

simulate the boundary condition. One of such methods is called Absorbing

Boundaries. The key point of Absorbing Boundaries Strategy is assuming that the

artificial boundaries can absorb the wave propagating through the interface to the

unbounded domain without any reflection. In that case, no additional measures

need to be done and this makes FEM possible here. But it is still unsuitable because

of the low level of accuracy and efficiency. These strategies are also referred to as

Transmitting Boundaries and non-reflecting boundaries. Comprehensive reviews

about those methods can be founded in the literature review part.

3

1.3 Substructure method

Compare to Direct Method, Substructure Method is more efficient and accurate for

dynamic interaction analysis. For this kind of method, the problem domain is

divided into two parts. First part is the combination of structure and the soil

surrounding structure, which is called bounded domain. The rest of the infinite

medium is the second part, called unbounded domain.

For the bounded domain only, an external load { } applies on the

structure. Along the artificial boundary, there is an internal load { }. In

addition, there are also nodal displacement responses to the dynamic load along

the boundary. The near field part is modeled by finite element method.

Figure 1.2. 1, Model of interaction problem in near-field part

Equation of motion in time-domain is shown below:

[ ] { } [ ] { } { } { } 1.2. 1

The purpose of this formulation is finding out the displacements { }. In

this equation, term { } is the static stiffness matrix of nodes for the bounded

domain, which can be calculated out. The amount of { }can be monitored from

equipment. The sum total mass of the structure and surrounded soil { }is also

given. In that case, the only unknown is the internal force { }.

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In order to find { } , we need another equation to analysis the

unbounded domain.

Figure 1.2. 2, Model of dynamic interaction problem in unbounded domain

For substructure method, the discretized boundary, which coincides with

the soil-structure interface, encloses the bounded domain. Only the nodes along

the discretized boundary will be analyzed.

There are several different numerical methods which are suitable to model

unbounded domain. Typically, all these methods assume that the internal force

and displacements are harmonic. The equations look like:

{ } { } 1.2. 2

{ } { } 1.2. 3

Equation for unbounded domain is in frequency domain at first,

{ } [ ] { } 1.2. 4

In equation 1.2.4, term { }is the displacement vector of the discretized

nodes along the boundary in frequency domain. The symbol [ ]is the dynamic

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stiffness matrix of the soil of the interface, and it is also in frequency domain. One

of the major problems of substructure method is to find out the values of [ ].

The matrix [ ] can be found using a numerical method, like Boundary

Element Method and Scaled Boundary Finite Element Method. Compare to the

Boundary Element Method, Scaled Boundary Finite Element Method (SBFEM) are

more suitable to model the far-field part. From Wolf and Song (1996), this method

combines advantages of finite element method and boundary element method. It is

also an accurate semi-analytical model of the radiation damping effect. The details

of this method will be introduced in literature review part.

We cannot just replace { }in equation 1.2.1 into equation 1.2.4, since

they are in different domain. So, first of all, equation 1.2.4 should be transferred

from frequency domain into time domain. A method called inverse Fourier

Transformation (James, 2011) is employed. The basic equations of Fourier

Transformation Method are:

∫

1.2. 5

∫

1.2. 6

Equation 1.2.5 is used to transfer the interaction force in frequency domain

into time domain. Equation 1.2.6 is the inverse form of 1.2.5. After transforming,

equation 1.2.4 becomes

{ } ∫ [ ] { }

1.2. 7

Here, the matrix term [ ] is called the displacement unit-impulse

response matrix. According to Prempramote (2011), there is a force applied at a

point on discretized boundary and displacements take place at all degree of

freedom along this boundary. The displacements in each degree of freedom not

just only depend on the present applied force, but also the previous applied force.

6

In that case, a convolution integral needs to be calculated at each time step, which

leads to a huge computational effort. Although this method is quite accurate,

because of the using of inverse Fourier transformation, it is still inefficient.

Recently, a more efficient method has been developed to calculate the unit-

impulse response matrix [ ] and evaluate the convolution integral (Radmanovic

& Katz, 2010). It is based on approximating the displacement unit-impulse

response matrix by a piece-wise linear function and truncating the convolution

integral after a certain point in time, when the displacement unit-impulse

responses close to zero. It is named as the new scheme or linear scheme to find out

the acceleration unit-impulse response matrix, which is employed in later study.

In this research, this idea will be extended to layered ground. I will focus on

the study about dynamic soil-structure interaction in a homogeneous semi-infinite

soil layer, which will be described later. The scaled boundary finite element

method will be used to find the dynamic stiffness matrix [ ]. The details about

SBFEM will be discussed in chapter 3 after the literature review.

1.4 Thesis outline

The further outline of this thesis is as follows:

In Chapter 2, the literature review of the existing methods of solving

dynamic interaction problem in unbounded domain is summarized. Those

methods are divided into two groups, global procedures and local procedures. The

global procedures include boundary element method and scaled boundary finite

element method. And the local procedures include most absorbing boundary

methods and finite element method. Each method is introduced in details and their

advantages and disadvantage will also be discussed. Then, a suitable method is

chosen as the theoretical framework of this research.

In Chapter 3, the project about out-of-plane motion of a homogeneous,

semi- finite soil layer is introduced at first, which includes the major difficult

confronted, some figures and governing equations. Then, the governing equation is

7

derived, which is called the scaled boundary finite element method in this research.

The equation is in frequency-domain at first. A method of transforming the

equation into time-domain is also introduced in detail. The key step is to calculate

values of the acceleration unit-impulse response matrix. Two different methods

are introduced here.

In Chapter 4, studies about three different numerical examples are

represented. First one is about the out-of-plane motion of homogeneous semi-

infinite soil layer and the second example is the study about the in-plane motion of

the same soil layer. Two methods of obtaining the acceleration unit-impulse

response matrix are employed here, which has been talked about in chapter 3.

After comparing the results in terms of accuracy and time consumption, there is a

conclusion about which method is more efficiency and reliable. For the third

example, a trench is added in the same soil layer near the structure. There is a

study about how the trench influences the propagation of dynamic waves and

displacements of soils.

In Chapter 5, all the works in this research are summarized, and the

possible work in the future research are recommended.

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Chapter 2

Literature Review

In this chapter, some methods of solving the dynamic interaction problem in

unbounded domains will be introduced. Those methods can be clarified into two

groups, global procedures and local procedures. Global procedures include

boundary element method and scaled boundary finite element method, while most

absorbing boundary methods and finite elements belong to local procedures. For

each method, their principles and the process of developments will be talked about.

Their advantages and disadvantages will also be discussed in detail and then a

suitable method will be chosen as the theoretical framework in the soil-structure

interaction problems in layered soil.

2.1 Introduction

The laying of soils can importantly influence the dynamic behavior of the structure.

In this thesis, the problems of in-plane and out-of-plane motion of a homogeneous,

semi-infinite soil layer will be discussed in detail. In the past few decades, a lot of

researches had been undertaken to deal with this kind of problems and some

methods were also developed, for example, the Green’s functions, created by Luco,

Apsel (1983) and Bouchon (1981). But this kind of model is quite complicated and

expensive. Compared to Green’s function, Cone models are much simpler. Meek

and Wolf used this method to calculate the dynamic stiffness coefficients of a disc

of a single layer on rigid rock (Wolf & Meek, 1994). Lysmer and Waas derived the

thin-layer method to solve dynamic interaction problems in layered media

(Lysmer & Waas, 1975). It was developed between 1970s and 1990s. This method

has been considered as the most sophisticate one and mainly used in plane and

cylindrical problems. At recent, the scaled boundary finite element method

(SBFEM) has been widely used in the dynamic interaction problems. Wolf and

9

Song developed it in 1990s (1996). And it is usually in high efficiency and accuracy.

The basic theory of SBFEM is introduced in sub-section 2.6.

For the problem in this article is about the dynamic interaction in layered

soils. After discussion and comparing, scaled boundary finite element method

(SBFEM) is chosen to model the far field part. All the calculations and formula

derivations are based on this method, for example the obtaining of the governing

equation and the dynamic stiffness matrix.

2.2 Finite Element Method (FEM)

The finite element method is a numerical technique to solve some field problems,

like temperature distribution and displacement distribution in objects. Those

structures are modeled by small-interconnected elements, called finite elements,

and connect with each other by nodes, then solved by approximating solutions to

partial differential equations. In most case, it is in high efficiency with ideal

accuracy. This kind of analysis is a fairly recent discipline crossing the boundaries

of mathematics, physics, and engineering and computer science. It is widely used

in the structural, thermal and fluid analysis areas (Nikishkov, 2001).

There are mainly two methods of transforming the physical formulation of

the problem into finite element discrete analogue; they are Galerkin method and

vibrational formulation (Nikishkov, 2001). Galerkin method is suitable for

differential equation problem, while vibrational formulation can be used when

physical problem can be formulated as the minimization of a functional.

From the above introduce, it can be find that finite element method is a

widely used method in many fields. In that case, a lot of commercial software about

this method are developed and put into use. So, it is really convenient for us to

solve some questions. Another advantage is that this method can be used in the

arbitrary geometry problems, which means it is suitable for some actual issue with

irregular shaped domain. Furthermore, it can also be used in anisotropic materials

and nonlinear domain problems. However, there are also some limitations. This

method only offers approximate solution and the accuracy of solutions is usually

10

affected by the shape quality and the density of element. Moreover, it is not

suitable for dynamic problems in unbounded domain, which has been explained in

the introduction part. If there are some cracks in specified medium, a large amount

of fine meshes have to be created around those cracks, which may increase the

computational effort a lot and the accuracy of results can be much lower.

2.3 Boundary Element Method

Boundary element method is an important numerical method in the computational

solution of a number of physical problems. In the past few years, the work

presented by Green (1828)really contributes to the development of BEM.

The first step of boundary element method is representing the boundary

value problem in the form of integral formulation, but not the equations of motion

and boundary conditions (Trinks, 2005). There are two methods of doing that,

weighted residuals and Green’s reciprocal theorem. After that, there is a spatial

discretization on the boundary of specific domain and a boundary integral

equation is created. Then, for all nodes of the surface, a global system of algebraic

equations for displacements and forces’ values can be obtained. Thus, the solutions

of the nodes on that boundary can be obtained.

There are many advantages of using boundary element method. Among

them, the most important one is that the spatial dimension of problem can be

reduced by one. This is because the volume integrals of problem can be

transformed into surface integrals, or surface integrals can be transformed into

line integrals. Therefore, the computational process is much simpler. Furthermore,

this method can be used in unbounded medium. Not like finite element method,

boundary element method is applicable when some cracks appear in the medium.

Only some nodes along the edge of crack will be included in the computing.

But there are also some shortages. BEM is very inefficient for the transient

load problems (Trinks, 2005), like earthquake and waves created by travelling

subway. Moreover, an important feature of BEM is the using of fundamental

solution, which increases the inapplicable of this method. For medium with

11

anisotropic materials, the fundamental solutions are not easy to obtain. In order to

get a time-domain solution, the Fourier transformation method needed to be

utilized in, which leads to more computational effort. For time-dependent problem,

like dynamic interaction between structure and soil, the observation period also

needs to be discretized into time increment, which is quite expensive in terms of

numerical effort.

2.4 Infinite Elements

The infinite element method is developed from the finite element method and is

quite similar to it. The key step of infinite element is extending the shape function

of elements to infinite. Bettess (1977) did the original work of infinite elements.

The accuracy of the infinite element is controlled by the choice of the shape

function and the order of approximation (Prempramote, 2011). According to Trink

(2005), there are mainly three ways of extending the finite element domain to the

infinity: decay function, mapped infinite elements and wave envelope element.

In order to incorporate the correct decay in the infinite element, Bettess

(1977) and Zienkiewicz (Zienkiewicz, Taylor, & Zhu, 2005) developed the infinite

element with the help of a finite to infinite mapping. The formula of this method

looks like:

∑

∑

2.4. 1

The wave envelope concept was first used in finite elements. Then, this

method developed and transforms into the time-domain with the help of inverse

Fourier transformation.

There are also some disadvantages of infinite element. In order to improve

the accuracy of the results, the order of elements needs to be increase and this may

cause the ill-conditioning problems (Prempramote, 2011). Moreover, the geometry

12

of the infinite-element mesh has to be conformed to a separable coordinate system

for the wave equation.

2.5 Absorbing Boundary Conditions

From the problem statement in introduction part, the major problem of direct

method is the reflection of outgoing waves at artificial boundaries. In order to

avoid the errors caused by reflecting waves from artificial boundary, a special

boundary condition can be created here. In the literature, such method is called

absorbing boundary conditions.

From Trink PhD thesis (Trinks, 2005), absorbing boundaries can be divided

into two categories, local and nonlocal. The differential operators with respect to

time and space are used in local absorbing boundaries. But differential and integral

operators are both used in nonlocal-absorbing boundaries.

For nonlocal boundary conditions, the relationship between forces and

displacements [ ] at artificial boundaries needs to be founded to describe it.

The DtN method (Bikri, Guenther, & Thomann, 2010) applies here to find this

relationship in a formula form, which is:

{ } [ ]{ } 2.5. 1

Moreover, the force-displacements relationship can be obtained from

approximate solutions of wave transmitting problems in a semi-infinite far field.

The formulation about this relationship is usually derived from frequency domain.

In order to find out the formulation in time domain, the inverse Fourier

transformation need to be used. To prevent the big computational work, some

other strategies are founded to replace Fourier transform, for example

approximating the exact DtN map by a frequency-dependent rational function. A

least-squares method (Wolf J. , 1994) also can be used to calculate the coefficients

of a scalar rational dynamic stiffness approximation. Another method is called

13

lumped-parameter model, which were assigned to the corresponding partial

fractions (Wolf J. , 1994).

The major benefit of using global boundary conditions is the result is in

high accuracy and rigorous. For the formulation needs to be transformed from

frequency domain into time domain, the computational effort is very large and

time spending. Compared to the local boundaries condition, it will be more

expensive.

Some absorbing boundaries conditions are a kind of local procedures to

model the wave propagation in unbounded domain. Not like some global

procedures, this method is simpler and approximate. This method first appeared

on 1969, which is called “viscous boundary”, proposed by Lysmer and Kuhlemeyer

(Lysmer & Kuhlemeyer, 1969). For two-dimensional cases, the formulas of viscous

boundary condition are shown here:

2.5. 2

2.5. 3

where a and b are dimensionless parameters, is the mass density, is the

velocity of P-waves, is the velocity of S-waves, is a normal stress, is a shear

stress, is a normal velocity and is a tangential velocity. The ability of absorbing

the waves by viscous boundary depends on the ratio of the transmitting energy

from reflected waves and that from incident wave (Prempramote, 2011). After

research, Lysmer and Kuhlemeyer made a conclusion that the absorption

condition is good when a=b=1 (Lysmer & Kuhlemeyer, 1969).

From the above introduction, it can be easily found that local boundary

conditions is simple to implement and very efficient, when compared to nonlocal

boundaries conditions. The limitation of this method is the results are usually in

low accuracy. Accuracy can be improved by placing the boundary farther away,

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which means enlarge the near field. But this method may also increase the

numerical effort.

2.6 Scaled boundary finite element method

Scaled boundary finite element method (SBFEM) is a fundamental solution-less

boundary-element method. It is also a semi-analytical approach, which combines

the advantages of finite element method and boundary element method. This

method was initially developed by Song and Wolf (1997) and successfully utilized

in solving the elasto-dynamic and allied problems in civil engineering. For the

problems about dynamic interaction within unbounded medium, SBFEM will be

applied here as the theoretical framework.

At first, this method is called consistent infinitesimal finite-element cell

method, which developed for two-dimensional scalar waves in unbounded

domains. The term, called ‘scaled boundary finite element method’, first appeared

in 1997 created by Song and Wolf (1997). After extending this method, it can be

used to solve problems in diffusion (Song & Wolf, 1999), dynamic fluid-structure

interaction (Fan, Li, & Yu, 2005) and acoustics (Lehmann , Langer, & Clasen, 2006).

The scaled boundary finite element method can also apply to analyze problems

with stress singularity. Orders of singularity and stress intensity factors for multi-

material plates can be evaluated by it (Song & Wolf, 2002). Then, it is extended to

model the power-logarithmic singularities and to calculate the T-stresses, high

order terms, angular distributions of stresses under mechanical loading (Song C. ,

2005).

According to Wolf and Song (1999), there are two derivations of the scaled

boundary finite element equations in displacement and dynamic stiffness. One is

called the scaled-boundary-transformation-based derivation and another one is

called mechanically based derivation. For mechanically based derivation, the

finite-element cell is constructed between a similar fictitious boundary and an

artificial boundary. After limiting the cell width to zero, scaled boundary finite-

element equations can be obtained.

15

For transformation-based derivation, with the help of Galerkin weighted-

residual technique, scaled boundary finite-element equation in displacement can

be obtained from governing partial differential equation. Like boundary element

method, only the boundary needs to be discretized. So, the spatial dimension can

be reduced by one. In that case, the computational effort can be reduced, which is

much less than that of finite element method. Dislike the boundary element

method, the fundamental solution is not necessary (Prempramote, 2011). From

Wolf and Song (2005), another unique advantage of SBFEM is that the calculation

of stress intensity factors is in high efficiency and accuracy. Furthermore, for

unbounded domain, the radiation condition at infinite for this SBFEM is satisfied.

At first, a scaling center O needs to be determined. The center should be

located in the problem domain and the whole boundary is visible from O. Then, the

boundary will be discretized into lots of nodes. In two dimensions problem,

curvilinear coordinates , are used to describe the locations of nodes. Symbol is

a dimensionless radial coordinate. It is also a scaling factor to represent the

location of scaling boundaries. For scaling center O, is equal to zero. Symbol

represents circumferential coordinate, which describes the location of nodes along

a specific boundary.

Figure 2.6. 1, Scaled boundary finite element method of spatial discretization of two-dimensional

unbounded domain

In figure 2.6.1, for any scaling boundary in unbounded domain, value of is

larger than one.

16

Figure 2.6. 2, Scaled boundary finite element method of three-node line element

For each node along the boundary, the Cartesian coordinates x and y can be

obtained from the shape functions of the element. A typical 3-node line element is

shown in Figure 2.6.2. The coordinates of two end points are known.

For in-plane and out-of-plane motion in semi-infinite soil layer problem, the

major challenge is to find out the displacement at a specific point. At first, the

governing differential equation of displacement, which represents relationship

between nodal displacements and interaction, was required to be found out. Then,

the governing equation should be solved numerically with the method of weighted

residuals. It also needs to be transferred into the local coordinates system. After

simplifying, a new equation called scaled boundary finite element equation in

displacement is obtained. At the same time, the dynamic stiffness [ ] also

needs to be obtained. After processing, dynamic stiffness matrix in frequency is

founded out from the “quadratic equation”. With the help of Fourier

transformation method, scaled boundary finite element equation can be

transferred into time-domain. In my thesis, the critical step is to find out the value

of acceleration unit-impulse response matrix. Finally, those unknowns about nodal

displacement can be calculated out in commercial software-Matlab. The scaled

boundary finite element equations for the out-of-plane motion of a homogeneous

soil layer are derived in detail in Chapter 3. After that, two methods about

calculating the acceleration unit-impulse response matrix will be introduced,

which are called constant scheme and linear scheme. The purpose of this research is

to find out which method is more efficiency.

17

Chapter 3

Theoretical Concepts

After choosing the scaled boundary finite element method, the methodology about

this project needs to be developed. Sub-section 3.1 is about introducing my project

and some basic equations. The governing equation called scaled boundary finite

element equation will also be found, which is in sub-section 3.2 and 3.3. In addition,

there are two schemes about finding the acceleration unit-impulse response

matrix. Those two schemes are constant scheme and linear scheme, which will be

introduced in details in 3.5 and 3.6. Finally, in sub-section 3.6, we will have a look

how to get the in-plane and out-of-plane displacements at a specific point in the

soil layer.

3.1 Introduction of homogeneous, semi-infinite soil layer problem

As mentioned before, the main challenge of this project is to find a suitable way to

solve the dynamic interaction problem in unbounded domain. In most cases of real

conditions, the far field medium we study is usually overlaying on some hard and

fixed soils layers without any nodal displacements, like bedrock. But the whole

layer is unlimited in the horizontal direction. This kind of medium is called

homogeneous semi-infinite soil layer, which is focused on in this thesis. In the first

18

step, a model of homogeneous semi-infinite soil layer needs to be created.

Figure 3.1. 1, Homogeneous, semi-infinite soil layer with bounded and unbounded domain

From figure 3.1.1, there is a structure with dynamic load applied attached to

the soil layer and the layer is unbounded in x direction on both sides. There is a

dynamic interaction between structure and bounded medium. The semi-infinite

soil layer is bounded by the fixed boundary at bottom, which the interaction waves

cannot transmit through and there is no in-plane and out-of-plane displacement

along it. Similar to the example in introduction part, the whole section is divided

into three parts, near field and far field part. For model shown in figure 3.1.1, the

near field part is section A, which includes structure and part of the medium near

the structure. The far field is section B and C. For section A, the finite element

method can be used to model and solve it. In this chapter, only the far field domain

will be discussed. For the soil is homogeneous, we only focus on section B. The far

field region is shown in figure 3.1.2.

Figure 3.1. 2, unbounded domain of the homogeneous, semi-infinite soil layer

19

For simplicity, the depth of thickness of the semi-infinite layer is constant, h.

The upper boundary is stress free boundary, so the stress on each node along the

boundary is zero. For the lower boundary is fixed, the displacement is zero. Along

the left side vertical boundary, where x=0, internal stresses apply on the boundary.

Those stresses are variable along the depth with coupled stresses applied on the

near field part in opposite direction. Thus, the boundary condition is:

1) At ,

2) At ,

3) At ,

Figure 3.1. 3, an infinitesimal element in the homogeneous, semi-infinite soil layer

An infinitesimal element is split out from the medium with dimension

and , shown in figure 3.1.3. Force analysis in z direction of this element will

be discussed and the governing equations will be found out. The only unknown is

the displacement in z direction which depends on the x, y coordinates and time.

Equilibrium of force in z direction is:

3.1. 1

20

Here, is the density of the soil. So, the mass of the infinitesimal element is

Symbol is the second derivative of displacement with respect to time of

this element. It can also be considered as the acceleration. The right hand side of

the equation is the inertia force of this element. After simplify, equation 3.1.1

becomes:

3.1. 2

According to the Hooke’s law:

3.1. 3

3.1. 4

For the strain can be derived as the first derivative of the displacement in each

direction. Thus, the equation 3.1.3 and 3.1.4 can be modified into:

3.1. 5

3.1. 6

In this formula, symbol G [ ] is the shear modulus of the soil. Symbols and

are the strains of elements in x and y directions. After combing equations 3.1.2,

3.1.5 and 3.1.6, the governing differential equation of displacement can be found,

which is:

3.1. 7

21

It can also be considered as the wave equation in far field domain. Soil property

can be derived as symbol c, which is called wave propagation velocity. The unit of c

is m/s and the equation is:

√

3.1. 8

Then, equation 3.1.7 can be transferred into a new equation:

( )

3.1. 9

or

3.1. 10

In equation 3.1.10, the symbol is the gradient operator, which can be derived

as {

}. Thus, equation 3.1.9 and 3.1.10 can be combined as:

{

} {

}

3.1. 11

Then, the governing equation can be obtained later.

3.2 Transforming Cartesian coordinates into local coordinates

In most case, Cartesian coordinates are used to find out the relationship

between nodal displacements and interaction. For this problem, the local

coordinates are more suitable. Before find out the governing equation, the way of

22

translating system from Cartesian coordinates into local coordinates should be

introduced. At first, a 2D far field model in Cartesian coordinates system is created:

Figure 3.2. 1, unbounded domain in Cartesian coordinates

In figure 3.2.1, the boundary in the left edge, which contacts with the near-

filed part, is in curve. This is because, in real condition, the boundaries usually are

not in perfect vertical. The same model in scaled boundary coordinate system is

created in figure 3.2.2.

Figure 3.2. 2, far field model in nodal coordinates system

As the similar physical model created in previous section, there are three

boundaries in this model, and . On boundary , the load { } applied varies

with time and depth. For (y=0) is a fixed boundary, there is no displacement

along it. The boundary is considered as a free surface with a constant depth h.

Thus, the stress on that boundary is zero. In this system, two new coordinates

and are defined. Symbol is called radial coordinate, which is used to represent

the location of boundary . For example, if this boundary is located in its initial

location in left side, is equal to zero. This boundary can shift right horizontally

with an increment , then all the nodes along the boundary shift right with the

same increment without any vertical displacement. As shown in figure 3.2.3:

23

Figure 3.2. 3, shift of the boundary along the horizontal direction

The symbol represents the geometry of the boundary, which is called

circumferential coordinate. The equations of scaled boundary transformation can

be derived as:

3.2. 1

3.2. 2

Here, is the value of Cartesian coordinate at boundary and y is the value

of Cartesian coordinate . These two equations represent the relationship between

the Cartesian coordinates of the nodes with theirs scaled boundary coordinates.

And they are called the scaled boundary transformation. For the boundary ,

there is only one element with two nodes at ends.

Figure 3.2. 4, circumferential coordinate of boundary

For these two end nodes, they both have their known Cartesian coordinates,

and , which can be represented in two vectors:

24

{ } { }

3.2. 3

{ } { }

3.2. 4

According to Finite Element Method for two nodes element, these two end

nodes can represent the coordinates of any other nodes in the boundary. Thus, the

equation can be written as:

[

] { }

3.2. 5

In equation 3.2.5, formulas

and

are the two shape function of

this boundary. They can also be written as and . So, the general

equations of the coordinates for each node along the boundary are:

[ ]{ } 3.2. 6

[ ]{ }

3.2. 7

The major challenge is to represent the equation 3.1.10 by the scaled

boundary coordinates. From equations 3.2.6 and 3.2.7, the relationships between

coordinates x, y and coordinates and are obtained, which are and .

It is also very easy to get the equations of displacements in x and y coordinates u(x,

y). Thus, the displacements of nodes along boundary are represented

as , . Then, we can use the chain rule to deal with that problem.

First of all, the first derivative of displacement with respect to needs to be

calculated out. The equations are represented below:

3.2. 8

25

3.2. 9

We can combine these two equations into a matrix form:

{

} (

) {

}

3.2. 10

In equation 3.2.10, the matrix term (

) is called Jacobian matrix[ ]. Moving

the Jacobian matrix to the left-hand side, the equation becomes:

{

} [ ]

{

}

3.2. 11

According to equations 3.2.1 and 3.2.2, the Jacobian matrix can be simplified into:

[ ] [

]

3.2. 12

From equation 3.2.12, it can be easily find that Jacobian matrix is not depending

on . In that case, this matrix can be symbolized as [J]. According to the method of

inverse function of the two by two matrixes, the inverse of [J] can be easily found,

which looks like:

[ ]

| |[

]

3.2. 13

where, term

| | is the determinant of [J]and[

]

equals to [

].The

gradient operator is defined as:

26

{

} 3.2. 14

Applying equation 3.2.14 to the displacement function gives:

| |{

}

| |{ }

3.2. 15

It can be simplified as:

{ }

{ }

3.2. 16

where:

{ }

| |{

}

3.2. 17

and:

{ }

| |{ }

3.2. 18

3.3 Derivation of the Scaled Boundary Finite Element equation in

frequency domain for out-of-plane motion in a 2D layer

After getting the operator equation in local coordinate, the governing equation

3.1.10 should be solved numerically. The method of weighted residual technique

will be used here. The term ‘residual’ means error in the approximate solution.

At first, equation 3.1.10 can be rewritten as:

3.3. 1

27

Where, is in Cartesian coordinates system with respect to time.

Evaluating equation 3.3.1 for an approximate solution and integrating yields:

∫

∫

3.3. 2

In this equation, the term ∫ means the accumulation of the

residual values in Cartesian coordinates with respect to time. To derive a finite-

element approximation, the weighted-residual technique is applied to the

equilibrium equation. The general formula of this method is shown:

∫

3.3. 3

The symbol is the domain of this researched system. According to this general

formula, equation 3.3.1 is transferred into:

∫

3.3. 4

In equation 3.3.4, term represents displacement of soil node in Cartesian

coordinates as a function of time . Assumed in time-harmonic behavior,

the displacements can be derived as:

3.3. 5

The second derivative of displacement (acceleration) is:

3.3. 6

28

Replacing the term in equation 3.3.4 with equation3.3.6 gives:

∫ (

)

3.3. 7

For this equation, it is really hard to integrate the term . So, the method of

integration by parts is employed here. The general formula for 1D integration of

this method looks like:

∫ ∫

3.3. 8

and

∫ [ ] ∫

3.3. 9

In equation 3.3.9, term [ ]

is the boundary term. Employing this method in

equation 3.3.7, integrating by parts for the first term gives:

∫

∫

∫

3.3. 10

Term∫

is the boundary term and .

Figure 3.3. 1, the far-field domain in local coordinates system with boundary condition

29

Substituting equation 3.3.10 into equation 3.3.7 gives:

∫

∫

∫

3.3. 11

The boundary is discretized into a number of elements, and then the

displacements and weighting functions of boundary can be derived as:

[ ]{ }

3.3. 12

[ ]{ } 3.3. 13

The transpose of the weighting function is:

{ } [ ] 3.3. 14

Substituting equation 3.3.12 into equation 3.2.16 gives a new equation for the

gradient operator:

{ }[ ]{ } { }[ ] { } 3.3. 15

The following matrices are introduced as:

[ ] { }[ ] 3.3. 16

[ ] { }[ ] 3.3. 17

Substituting equation 3.3.16 and 3.3.17 in equation 3.3.15 yields:

30

[ ]{ } [ ]{ } 3.3. 18

For the gradient operator of weighting function:

[ ]{ } [ ]{ } 3.3. 19

Gradient operator of the transpose weighting function looks like:

{ } [ ]

{ } [ ] 3.3. 20

Equation 3.3.11 can be rewritten as:

∫

{ } [ ]

{ } [ ] ([ ]{ } [ ]{ })

∫ { } [ ] [ ]

{ } ∫

3.3. 21

where, , which can be transferred into local coordinates system:

| | . 3.3. 22

The term ∫

in equation 3.3.21is the boundary condition corresponds to

applied forces, which can be neglected at the moment. After substituting equation

3.3.22 into 3.3.21 and

, multiply the whole equation by G yields:

∫ { }

∫ [ ]

[ ]| | { } ∫ { }

∫ [ ]

[ ][ ] { }

∫ { } ∫ [ ] [ ]

| |

{ } ∫ { } ∫ [ ] [ ]| |

{ }

∫ { } ∫ [ ] [ ]

| | { }

3.3. 23

31

In equation 3.3.23, the integration of the terms and can be evaluated. The

coefficient matrices are defined as:

∫ [ ] [ ]| |

[ ] 3.3. 24(a)

∫ [ ] [ ]

| | [ ] 3.3. 24(b)

∫ [ ] [ ]| |

[ ] 3.3. 24(c)

∫ [ ] [ ]

| | [ ]

3.3. 24(d)

Matrices [ ] [ ] [ ] are the stiffness matrices in a finite element model.

The matrix term [ ] is the mass matrix.

Substituting all those stiffness matrixes and mass matrix into equation 3.3.23 gives:

∫ { }

([ ]{ } [ ] { }) ∫ { }

([ ]{ } [ ]{ }

[ ]{ })

3.3. 25

Integrating this equation by parts once more yields:

∫ { } [ ]{ } [ ] { } [ ]{ } [ ]

{ }

[ ]{ }

3.3. 26

As said before, this equation represents residual value. In order to get the

appropriate solution, value of residual must be zero. However, the transpose

weighting function { } cannot be zero. So, equation 3.3.26 can only satisfied for

32

arbitrary weighting function if the term in brackets is zero. Then, the scaled

boundary finite element equation in displacement can be obtained:

[ ]{ } [ ] { } [ ]{ } [ ] [ ] { } 3.3. 27

3.4 Derivation of Scaled Boundary Finite Element Equation in time-

domain

After getting the scaled boundary finite element equation in displacement { },

the values of dynamic stiffness [ ] needs to be obtained too. First of all, internal

nodal forces { }should be defined. In John P. Wolf’s book (1994), a method of

equating the virtual work of the internal nodal forces { } to the virtual work of

the surface tractions is discussed, which gives:

{ } [ ]{ } [ ] { } 3.4. 1

For unbounded domain, external nodal force { { }} applied, which is equal to

{ } with opposite direction. So, the relation between these two forces is:

{ { }} { } 3.4. 2

In introduction part, the equation 2.5.1 can be modified into local coordinates,

which is:

{ { }} [ ]{ } { } 3.4. 3

This equation is in frequency domain. In our case, the term { } in equation 3.4.3,

which relates to body loads or surface tractions, can be neglected. Substituting

equation 3.4.2 into 3.4.3 yields:

33

{ } [ ]{ } 3.4. 4

After substituting equation 3.4.1 into 3.4.4, a new equation can be obtained:

[ ]{ } [ ] { } [ ]{ } 3.4. 5

Then, differentiate equation 3.4.5 with respect to yields:

[ ]{ } [ ] { } [ ] { } [ ]{ } 3.4. 6

It can be modified into:

[ ] { } [ ]{ } [ ]{ } [ ] { } 3.4. 7

Substituting scaled boundary finite element equation in displacement 3.3.27 into

equation3.4.7 gives:

[ ] { } [ ]{ } [ ] { } [ ]{ } [ ]{ } 3.4. 8

Equation 3.4.5 can be solved to obtain the value of { } . The equation of { } is

shown below:

{ } [ ] [ ]{ } [ ] [ ] { } 3.4. 9

And then, equation 3.4.8 can be modified as:

[ ] { } [ ][ ] [ ] [ ] { }

[ ] [ ] [ ] [ ] { } [ ]{ } [ ]{ }

3.4. 10

34

For the soil layer has a constant depth with identical material coefficients,

the coefficient term [ ] does not depend on the value of . So, the value of

[ ] is zero. And equation 3.4.10 can be simplified as:

[ ] [ ] { }[ ] [ ] [ ] { }

[ ]{ } [ ]{ }

3.4. 11

All the terms { } in equation 3.4.11 are cancelled, so the equation becomes:

[ ] [ ] [ ] [ ] [ ] [ ] [ ] 3.4. 12

Equation 3.4.12 is called “quadratic equation” for dynamic stiffness matrix[ ].

It is referred to as an algebraic Riccati equation. This equation is also the scaled

boundary finite element equation in frequency domain.

At this stage, the scaled boundary finite element equation in time-domain

can be obtained. At first, the displacement-unit-impulse response [ ] is defined

as:

∫ [ ]{ }

3.4. 13

And the acceleration-unit-impulse response matrix [ ] is defined:

∫ [ ]{ }

3.4. 14

There is a relationship between acceleration dynamic stiffness [ ] and the

dynamic stiffness [ ] (Wolf & Song, 1996), which is shown below:

[ ] [ ]

3.4. 15

Equation 3.4.15 can be modified into:

35

[ ] [ ] 3.4. 16

In this equation, symbol “ ” is imaginary unit. Substituting equation 3.4.16 into

equation 3.4.12 yields:

[ ] [ ] [ ] [ ] [ ]

[ ] [ ]

3.4. 17

For equation 3.4.17, dividing the whole equation by gives:

[ ][ ] [ ]

[ ][ ] [ ] [ ][ ] [ ]

[ ][ ] [ ]

[ ]

[ ]

3.4. 18

In order to transfer this equation into time domain, the Fourier

transformation method needs to be employed. Transforming rules are shown

below:

3.4. 19(a)

3.4. 19(b)

With the help of convolution theorem and integration by parts, scaled boundary

finite element equation in time domain is obtained, which looks like:

∫ [ ]

[ ] [ ]

[ ][ ] ∫ ∫ [ ]

∫ ∫ [ ] [ ] [ ]

3.4. 20

36

[ ] [ ][ ] [ ] [ ]

In equation 3.4.20, the matrix term [ ] is the acceleration unit-impulse

response matrix.

To simplify this equation, Cholesky decomposition of [ ] is utilized.

Equation of [ ] is:

[ ] [ ] [ ] 3.4. 21

Then, inverse matrix of is shown below:

[ ] [ ] [ ]

3.4. 22

Equation 3.4.22 is substituted into equation 3.4.20. At the same time, all terms are

pre-multiplied by [ ] and post-multiplied by [ ] . A new equation is obtained:

For simplify, some multiplying of matrices can be replaced by one term only, for

example:

[ ] [ ][ ] [ ] 3.4. 24(a)

[ ] [ ][ ] [ ]

3.4. 24(b)

[ ] [ ] [ ][ ] [ ] [ ] [ ] 3.4. 24(c)

[ ] ∫ [ ]

[ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ] ∫ ∫ [ ]

[ ]

[ ] ∫ ∫ [ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ]

[ ] [ ][ ] [ ] [ ] [ ] [ ] [ ][ ]

3.4. 23

37

[ ] [ ][ ] [ ] 3.4. 24(d)

Finally, this equation looks like:

∫ [ ][ ] [ ] ∫ ∫ [ ]

∫ ∫ [ ]

[ ]

[ ] [ ]

3.4. 25

where the values of [ ], [ ] and [ ] have been given. The only unknown in this

equation is the term [ ], which is called acceleration unit-impulse response

matrix.

3.5 Constant scheme of deriving the Acceleration Unit-Impulse Response Matrix

In the scaled boundary finite element equation in time-domain, the term [ ]

is unknown. The major problem in my project is how to find the values of [ ].

Response matrix [ ] can be solved numerically by splitting the whole time range

into many small time steps. There are two methods of obtaining the values

of [ ]. In this sub-section, the constant scheme is introduced first.

Figure 3.5. 1, constant acceleration unit-impulse response coefficient in each time step

Figure 3.5.1 shows the change of [ ] with respect to time. The constant

time step size equals to . For the original scaled boundary finite element method,

38

term is assumed to be constant in each time interval. The integration of

can be expressed as:

∫ [ ]

∫ [ ]

∫ [ ]

∫ [ ]

3.5. 1

Then, this equation can be simplified into:

∫

∑

3.5. 2

In equation 3.5.2, the values of for each time step are in constant. All other

terms are discretized analogously. First of all, the scaled boundary finite element

equation for the first time interval can be obtained by the initial condition. The

equation is shown below:

[ ] [ ]

[ ] [ ]

[ ]

[ ] [ ]

3.5. 3

For equation 3.4.25, the term t is replaced by

. Dividing the whole equation by

yields:

[ ] [ ]

[ ] [ ]

[ ]

[ ] [ ]

3.5. 4

This quadratic equation for matrices is called Riccati equation. In commercial

software Matlab, the function code called “care.m” can be used to obtain the value

of [ ] .

Now, let’s look at steps that . Equation 3.4.25 is discretized into:

∑ [ ] [ ] [ ] ([ ] [ ]

[ ] )

3.5. 5

39

([ ] [ ]

[ ] ) [ ]

[ ] [ ]

Modifying the equation 3.5.5 yields:

[ ][ ] [ ] [

]

[ ][

]

[ ] [

]

∑ [ ] [ ] ([ ]

[ ] ) [ ]

[ ] [ ]

3.5. 6

This method will be used frequently in numerical examples part. Although

it can give us the results in high accuracy, the calculating process is time

consuming and complicated. Those equations related to constant scheme will be

transferred into Matlab codes for computation. The results obtained will also be

compared with reference solution and the results obtained from Linear Scheme

Method in terms of accuracy and efficiency. The linear scheme is the second

method of getting the acceleration unit-impulse response matrix. Details about

Linear Scheme Method will be introduced in the next sub-section.

3.6 Linear scheme of the obtaining Acceleration Unit-Impulse Response

Matrix

In the method above, scaled boundary finite element method (SBFEM) in time

domain is used for the computation of acceleration unit-impulse response matrix,

which is first developed by Wolf and Song (1996) and extended to non-

homogeneous medium by Bazyar & Song (2006). In order to capture main

characteristics of the wave propagation, the whole time period is discretized into

many time intervals with finer mesh. In addition, the unit-impulse response

coefficient is assumed to be constant within each time step. In that case, the

computational effort is extremely large for this method (Radmanovic & Katz, 2010).

A new integration scheme for the solution of the acceleration unit impulse

response matrix and the evaluation of the soil-structure interaction vector is

developed. There are two characteristics of this method. Firstly, within each time

40

interval, the acceleration unit-impulse response matrix changes linearly.

Furthermore, the extrapolation parameter is used to increase the stability of the

solution, which allows the using of large time step size. Compared to the previous

method, the computational effort decreases significantly.

The transformed scaled boundary finite element equation in acceleration

unit-impulse response for a two-dimensional layered medium is represented as:

∫ [ ][ ]

[ ] ∫ ∫ [ ] ∫ ∫ [ ]

[ ]

[ ]

[ ]

3.6. 1

In equation3.6.1, the value of [ ] [ ] [ ] [ ] are already expressed in

equations 3.4.24. This equation is definitely from the equation (5) in Radmanovic

& Katz’s report (2010). The only difference is the term ∫ [ ]

is missing.

For the first time step m=1, algebraic Riccati matrix equation can be used

for solving[ ] :

[ ] [ ][ ] [ ][

] [ ] [ ] [ ] 3.6. 2

In this equation, terms [ ] and [ ] are represented as:

[ ] [ ] [ ] 3.6. 3

[ ] [ ] [ ][ ] [ ] [

] [ ]

[ ]

3.6. 4

For second time step m=2, the algebraic Riccati equation 3.1.7 is solved for[ ] ,

[ ] [ ][ ] [ ] [ ] [ ] [ ]

[ ] 3.6. 5

with

41

[ ] [ ] [ ] [ ] 3.6. 6

[ ] [ ][ [ ] [ ] ]

[ [ ] [ ] ][ ]

[ [ ] [ ]] [[ ] [ ] [ ] [

] ]

[ ]

3.6. 7

For next few time intervals m 3, the Lyapunov equation 3.6.7 is solved for each

time step,

[ ] [ ] [ ] [ ] [ ] 3.6. 8

with

[ ] [ ] [ ] [ ]

[ ]

3.6. 9

[ ] ∑ [ ] [ [ ] ]

[ ] [ [ ]

[ ] [ ] ] [

[ ]

[ ]

[ ] ] [ ] [[ ] [ ]

[ ] ][ ] [ [ ] [ ]]

[ ] [[ ] [ ] ]

[ ]

3.6. 10

Furthermore, a new and very efficient recursive integration scheme for the

evaluation of the soil-structure interaction vector described by convolution

integral has been developed in Radmanovic & Katz’s report (2010), which is based

on integration by parts.

The soil-structure integration force vector is given by the convolution

integral:

42

∫

3.6. 11

With the help of integration by parts, we can derive efficient integration scheme

for evaluation of the soil-structure interaction vector as follows:

[

] 3.6. 12

The derivation of equation 3.6.12 and definition of the terms

and are described in detail in Radmanovic & Katz’s

reference(2010). The time-domain integration scheme for the layered soils is the

same as for half space. Moreover, integration by parts and linearization allow

truncating convolution integral.

If there is no truncation, the value of acceleration unit-impulse matrix

in equation 3.6.11 grows with time and may cause severe numerical

problem. For the original discretization scheme with n time steps, the matrix-

vector multiplication and vector summations should be carried out at each time

station. If the time step n is huge, large amounts of calculations should be operated,

which leads lots of CPU time consumption. So, it is not efficient. In order to reduce

the computational effort, the truncation time is employed. We may only deal with

the data in the first M time steps, which are only a small portion of the total

number of time steps. After that, the acceleration unit-impulse response matrix is

assumed to change linearly.

Linear scheme with those improvements will be employed in numerical

examples in chapter 4 to compare with the original scheme. In order to show how

the efficiency and accuracy they may achieve, some data analyses will also be

carried out.

43

3.7 Time-domain solution for

After getting the acceleration unit-impulse response matrix , the value of

displacement in each time point can be obtained.

Figure 3.7. 1, the internal reaction forces between near-field and far-field domain

From figure 3.7.1, we can find that the internal reaction forces, which apply on the

boundary of far-field and near-field domain, are in the same value with opposite

direction. The equation about the force-displacement relationship is represented

below:

∫

3.7. 1

In equation 3.7.1, internal reaction forces are given. Term t equals . The

only unknown is the value of acceleration in each time step.

44

Figure 3.7. 2, acceleration unit-impulse response coefficient in each time step

In order to simplify the calculation, the entire period is divided into n time

steps with size equals to . For each time step, the value of acceleration unit-

impulse response coefficient is constant.

During each time step, displacement of any node can be represented by the

displacements of that node at two end points. For example, in the first time step,

displacements of one node at two end points are and . Then, the value in

other time is:

(

)

3.7. 2

According to figure 3.7.2, the internal reaction force can be calculated from the

sum of the integration value in each time step. Then, equation 3.7.1 can be

modified into:

∫ ∫

∫

3.7. 3

with . All the terms t in equation 3.7.3 can be replaced by , then the

equation becomes:

45

∫ ∫

∫

3.7. 4

Equation 3.7.4 can be rewritten as:

∫ ∫

∫

3.7. 5

For each time step, coefficient terms can be split out from the integral sign. A new

equation is shown below:

∫

∫ ∫

3.7. 6

Then, equation 3.7.6 can be simplified as:

∑ ∫

∑

|

3.7. 7

Finally, an equation about the internal reaction forces, with respects to

acceleration unit-impulse response coefficients and the velocities in each time step,

can be obtained.

∑

3.7. 8

In equation 3.7.8, the value of is given or obtained from last time step. The

only unknown is the velocity in the next time step , which need to be calculated

out. In first step, if the term n equals to 1, equation 3.7.8 becomes:

3.7. 9

The initial velocity has been given and equals to zero in this case. Then, the

formula of expressed as:

46

[ ] 3.7. 10

In the second step, the term n equals to 2 and equations 3.7.8 transferred into:

3.7. 11

In equation 3.7.11, term has been obtained from last step and the only unknown

is term . So, the expression of is:

[ ] 3.7. 12

In the case of n=2, terms and the term can be replaced by the coefficients

and respectively. So, equation 3.7.12 can be modified into:

[ ] 3.7. 13

Thus, the general equation of the displacement in the nth time step can be

obtained:

∑

3.7. 14

Now, we can calculate the displacement.

(

)

3.7. 15

Equation 3.7.15 represents the displacement of nodes in each time step. Then, the

velocity in each time step can also be derived as:

(

)

3.7. 16

47

∫

3.7. 17

In equation 3.7.17, term equals to the value of the term . Then, the value of

velocity in each time step can be derived as:

3.7. 18

Substituting equations 3.7.16 and 3.7.18 into equation 3.7.17 yields:

∫ (

(

)

)

3.7. 19

After integral calculation, equation 3.7.19 modifies into:

3.7. 20

In equation 3.7.20, term represents the initial displacement, which is usually

given. In that case, the value of the term can be obtained. The equation of is

shown below:

3.7. 21

Then, substituting equation 3.7.21 into 3.7.15, the displacement for each time step

is obtained.

All those theories and equations will be used in numerical examples in the

next chapter to study about the dynamic interaction problems. Among them, the

two essential steps are the calculation of acceleration unit-impulse response

matrix and in-plane or out-of-plane motions of the soil layer. For simplicity, those

48

equations and theories will be translated into Matlab codes to have a computation.

In next chapter, three different numerical examples will be studied and discussed.

After comparing the results obtained with the reference solutions, there will be a

conclusion about which scheme, constant scheme or linear scheme, is more

suitable for calculating the acceleration unit-impulse response matrix.

49

Chapter 4

Numerical Examples

There are three numerical examples in my thesis. In the first example, there is a

study about the out-of-plane motions, which are caused by dynamic load, in a

homogeneous-soil layer. For the second example, we will focus on the in-plane

motion. In those two examples, constant scheme and linear scheme will be tested

and find out which one is more efficiency. In the last example, a trench is added in

the soil layer near the structure. Then, a study about how the trench influences the

displacements in soil layer will be carried out.

4.1 Out-of-plane motion of homogeneous-soil layer

At present, a new model of homogeneous, semi-infinite soil layer is created. It will

be analyzed and solved by Matlab. Then, the results of acceleration unit-impulse

response matrix will be compared with the standard solution, which is

represented in Wolf and Song’s book (1996).

In this model, the thickness of the layer is 2 meters. The vertical boundary

is discretized into two elements with two nodes each. For the soil properties, the

density is assumed to be 2000kg/m^3, Young’s modulus is 1.25e8 N/m^2 and the

Passion’s ratio is 0.25. The soil is considered as isotropic. The dimensionless end

time (tend) calculates up to 4 and the dimensionless time step size is 0.01.

The value of the stiffness and mass matrices in this example are already given

below:

[ ] [

]

4.1. 1

50

[ ] [

] 4.1. 2

[ ] [

]

4.1. 3

[ ] [

]

4.1. 4

The relationship between the dimensionless time and the real time t in [ ] is as

follows:

4.1. 5

In equation 4.1.5, term represents the time in units, means the dimensionless

time, h is the thickness of the layer and term denotes the shear wave velocity in

soil.

First of all, an upper triangular matrix [ ] is obtained through the Cholesky

decomposition of [ ], which has been explained in equations 3.4.21 and 3.4.22. In

Matlab, the code of Cholesky factorization is:

[ ] 4.1. 6

The value of matrix u is:

[ ] [

]

4.1. 7

According to equations 3.4.24, 3.4.25, 3.4.26 and 3.4.27, those simplified stiffness

and mass matrices, like [ ] [ ] and [ ] can be obtained:

51

[ ] [

]

4.1. 8

[ ] [

] 4.1. 9

[ ] [

]

4.1. 10

With the help of a function called “care”, equations 3.5.4 and 3.5.5 can be

solved in Matlab. Then, the values of for each time step can be obtained. All

these values need to be transferred into acceleration unit-impulse response

matrices[ ]. The equation of transformation is represented as:

[ ] [ ] [ ][ ] 4.1. 11

In most case, discretization of structure-medium interface can be much

finer, like dozens or even hundreds of nodes in each element. In order to simplify

the calculation, a method called impose quadratic variation will be used to transfer

the enormous original acceleration unit-impulse response matrices into a very

simple two by two matrix. For example, the structure medium interface consists of

2 elements, with 5 nodes in each. The graph is shown below:

Figure 4.1.1, structure medium interface with 9 nodes

52

In figure 4.1.1, within those nine nodes, only the displacements of node 5

and 9 are known, which can represent the displacements of other seven nodes. For

example, for the displacement of node 2, the equation looks like:

[

(

)

] {

}

4.1. 12

From the equation 4.1.12, the term [

(

)

] is the shape

function of the displacement of node 2. The equation of the displacements of these

nine nodes can be represented as:

{

}

[ ] {

}

4.1. 13

The term [ ] denotes the shape functions of nine nodes’ displacements, which is a

nine by two matrix. Then, acceleration unit-impulse response matrices [ ]

can be transferred into a 2 by 2 matrix from multiplying [ ] forward and [ ]

backward respectively. The equation looks like:

[ ] [ ] [ ] [ ] 4.1. 14

with

[ ] [

]

4.1. 15

After getting the acceleration unit-impulse response matrices, for the different

unit-impulse response coefficients ,

and , three figures are created to

53

show the change of each coefficient’s value with respect to dimensionless time.

Before plotting, real time in units should be changed back into dimensionless time

again.

54

Figure 4.1.2, acceleration unit-impulse response matrix of out-of-plane motion of semi-infinite layer of

constant depth discretized with quadratic finite element

In figure 4.1.2, the acceleration unit-impulse response coefficients in y-axes

also need to be changed into dimensionless. The tendency of each curve is the

same as the reference solutions. This references solution of acceleration unit-

impulse response coefficients are from figure A-9 in Wolf & Song’s reference

(1996)(see in Appendix A). But there are still some errors here, for example the

value of some key points. With the purpose of diminishing the errors, the

structure-medium interface will be discretized into a finer mesh. In this case, there

are five nodes in each element, but the number of elements along the boundary

keeps the same.

0

1

2

3

4

5

6

0 1 2 3 4 5

2 nodes

5 nodes

U

nit

-Im

pu

lse

Res

po

nse

Co

effi

cien

t

Dimensionless Time

55

Figure 4.1.3, curves of diagonal coefficients for 2 nodes mesh VS 5 nodes mesh

Figure 4.1.3 show the curves of diagonal coefficients for 2 and 5 nodes

meshes. As can be seen, for the curve of the coefficient , the difference between

two cases is quite small. But there is a big distinction for the curves of

coefficients

. The curves for the case of 5 nodes per element are much

closer to the reference solution. In next step, we will try the finer the mesh with 10

nodes per element for constant scheme. The comparison of the acceleration unit-

impulse response coefficients within three different meshes are shown in the

figures below:

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0 1 2 3 4 5

2 nodes

5 nodes

U

nit

-Im

pu

lse

Res

po

nse

Co

effi

cien

t

Dimensionless Time

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5

2 nodes

5 nodes

U

nit

-Im

pu

lse

Res

po

nse

Co

effi

cien

t

Dimensionless Time

56

Figure 4.1.4, curves of diagonal coefficients with respect to time in 2, 5 and 10 nodes meshes

0

1

2

3

4

5

6

0 1 2 3 4 5

2 nodes

5 nodes

10 nodes

U

nit

-Im

pu

lse

Res

po

nse

Co

effi

cien

t

Dimensionless Time

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0 1 2 3 4 5

2 nodes

5 nodes

10 nodes

U

nit

-Im

pu

lse

Res

po

nse

Co

effi

cien

t

Dimensionless Time

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5

2 nodes

5 nodes

10 nodes

U

nit

-Im

pu

lse

Res

po

nse

Co

effi

cien

t

Dimensionless Time

57

From these three figures, it can be easily found that the curves of the 5

nodes and 10 nodes converge. As the mesh finer, the results are much closer to

reference solutions. In addition, 5 nodes per element are fine enough.

At present, a model with 5 nodes per element is created in Matlab to

compare the stability of the linear and constant scheme. All related equations are

transferred into Matlab code to compute the acceleration unit-impulse response

matrix. In Matlab code, the linear scheme is named as intTyp=1, while the constant

scheme is defined as intTyp=0.

Figure 4.1 5, structure medium interface with 5 nodes per element

The code of constant scheme is expected to run with two different

dimensionless time steps 0.1 and 0.13. The linear scheme will also run in two

different conditions, time step is 0.1, theta is 1.0 and time step is 0.5, theta is 1.7.

Then, those four curves, which show the change of coefficient M11 respect to time

in four different cases, can be displayed in the same graph to make a comparison in

terms of stability. The figure is shown below:

58

Figure 4.1.6, comparison of the stability of the linear and constant scheme for determination of

acceleration unit-impulse response matrix of 5 nodes element model

As can be seen from figure 4.1.6, for the constant scheme, the result is stable

in 0.1 dimensionless time step size. But the curve is unstable when the size

increases to 0.13. If we use the linear scheme with the step size equals to 0.1, the

result is unstable quite early and becomes even worse when theta equals to 1.0. If

the theta increases to 1.7, the result will be stable again even for larger time step

size, like 0.5. In conclusion, the linear scheme can be used efficiently for a larger

time step, if we just increase the value of theta.

Figure 4.1.7, effect of the extrapolation parameter on the stability of the linear integration scheme

For linear scheme only, the program runs four times with the same time

step 0.5, but in different value of theta. From figure 13, we can find that the result

is unstable for 1.0 and 1.4, but it becomes stable when the theta increases to 1.7

0

5

10

15

20

25

30

35

40

45

0 10 20 30 40

const, timestep=0.1

const, timestep=0.13

theta=1.0,time step=0.1

theta=1.7,time step=0.5

Un

it I

mp

uls

e R

esp

on

se C

oef

fici

ent

Dimensionless TIme

0

5

10

15

20

25

30

35

40

45

50

0 10 20 30 40

theta=1.0

theta=1.4

theta=1.7

theta=1.8

U

nit

- Im

pu

lse

Res

po

nse

Co

effi

cien

t

Dimensionless Time

59

and 1.8. In addition, compare to the curve with 1.7 theta value, the curve with theta

equals to 1.0 becomes unstable quite early. Thus, as the value of theta increase, the

results are more stable.

Now, the same research is carried out for a finer mesh. All the coefficient

values keep the same. For this new model, each element consists of 9 nodes in the

structure medium interface, which is represented in the graph below:

Figure 4.1.8, structure medium interface model with 9 nodes per element

Figure 4.1.9, comparison of the stability of the linear and constant scheme for determination of

acceleration unit-impulse response matrix for finer mesh

0

5

10

15

20

25

30

35

40

45

0 10 20 30 40

const, timestep=0.02

theta=1.0, timestep=0.02

const, timestep=0.0375"

theta=1.7, timestep=0.5

Dimensionless Time t

Un

it-I

mp

uls

e R

eso

np

on

se C

oef

fici

ent

60

Figure 4.1.10, zoom in of the curves of coefficient for two stable cases in the initial steps.

In this figure, the curve for the larger time step (0.5) is quite rough. While

for smaller time step size 0.02, the curve looks smoother. Thus, the results will be

more accurate, if the time step size decreases.

Figure 4.1.11, effect of the extrapolation parameter on the stability of the linear integration scheme for

finer mesh

From figure 4.1.9 and 4.1.11, for the finer mesh, we can find that the

conclusion will be the same as the previous study. In addition, for both constant

and linear scheme, the coefficient values will be stable with smaller critical time

step sizes in a finer mesh. For the new scheme only, the critical time step sizes of

the stabilization of coefficient values does not depend on the theta value.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5

const,timestep=0.02

theta=1.7,timestep=0.5

Dimensionless Time

Un

it I

mp

uls

e R

esp

on

se C

oef

fici

ent

-20

-10

0

10

20

30

40

50

60

70

0 10 20 30 40

theta=1.0

theta=1.4

theta=1.6

theta=1.7

U

nit

-Im

pu

lse

Res

po

nse

Co

effi

cien

t

DImensionless Time

61

The accuracy and efficiency of the new procedure will be tested in a similar

numerical example. In order to compare the accuracy, the curves of out-of-plane

displacements with respect to time in each case need to be obtained at first. For

the same homogeneous, semi-infinite soil layer, there is a uniformly distributed

load applied on the boundary. The figure of the model looks like:

Figure 4.1.12, time-dependent uniformly distributed load applied on the boundary of the

homogeneous, semi-infinite soil layer.

The load is a time dependent cosine function.

4.1. 16

with

4.1. 17

In equation 4.1.17, term is the first natural frequency of homogeneous layer.

After applying the dynamic load, the out-of-plane displacements for each soil

element are generated. The equation below represents the getting of the out-of-

plane displacements, which is from the honor thesis of Truong (2012).

∑ [ (

)]

√

√

4.1. 18

62

with

(Where j=0, 1, 2…. etc.)

4.1. 19

where is the eigenvalue.

In this example, a homogeneous, semi-infinite soil layer model with 2

meters thickness is still employed. But the vertical boundary is discretized into

two elements with 5 nodes each. The material properties keep the same and

dimensionless time step size for time-integration is 0.01. The dimensionless

finishing time is 30. At first, the curve of displacements for constant scheme is

created, which compared with the reference solution. The figure is shown below:

Figure 4.1.13, out-of-plane motion of numerical result VS analytical result

From figure 4.1.13, we can find that the numerical result from constant

scheme is quite close to the exact solutions, except for some small errors. There

are two kinds of errors here, the amplitude errors and phase errors. For the case

that creates the exact solution, there is only force vibration response. But for the

constant scheme, free vibration response also needs to be considered, which leads

to the errors. But the free vibration response can be diminished after at certain of

time.

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 10 20 30 40

exactsolutions

const

Dimensionless Time

Dis

pla

cem

ent

𝑣

63

Figure 4.1.14, numerical result VS analytical result over longer time

As can be seen from this figure, two curves are slightly different on the

amplitude values at the beginning. Then, they converge together. In the next few

studies, the numerical results from constant scheme can be used directly as the

reference solutions.

For the new scheme, the value of theta keeps constant at 1.7 in different

cases. Time step size for the computation of the displacements is taken as

0.01, which is the same as original scheme. The time step size for the

derivation of is , which is N time larger than the time step size of

displacements. For the linear scheme only, the Matlab program runs several time

with different value of N, but there is no truncation time. Through observing the

results, we can find that if the value of N is small, the displacements curve is

usually unstable. It becomes stable until N increases to around 50. But the results

are still inaccurate when compared with the reference solution. If the truncation

time is employed, the curve looks stable even for a smaller N.

At first, there is no truncation and a big value of is chosen to investigate

the error. In this example, the value of N is 50, so equals 0.5. Other parameters

keep the same.

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 20 40 60 80 100 120

const

exactsolution

Dimensionless Time 𝑡 /h

Dis

pla

cem

ent 𝑣

64

Figure 4.1.15, curve of new scheme case with N=50 and no truncation employed.

From this figure, there is almost no difference between the reference

solution and the new scheme, no truncation results in the first half period. As time

progress, the new scheme curve looks increasingly inaccurate. Those errors

include both the amplitude errors and the shift errors, which are mainly from the

choosing of the large time step size.

Now, truncation time is employed and we would like to know how it

influences the results. In this case, the truncation time is varied, while the value

of N keeps constant.

Figure 4.1.16, influence of the truncation time on results

-12

-10

-8

-6

-4

-2

0

2

4

6

8

0 10 20 30 40

const

N=50, notruncation

Dimensionless Time𝑡

Dis

pla

cem

ent 𝑣

-6

-4

-2

0

2

4

6

0 5 10 15 20 25 30 35

const

N=30,tc=3

N=30,tc=2

N-30,tc=1

Dimensionless Time𝑡

Dis

pla

cem

ent 𝑣

65

In figure 4.1.16, for the new scheme cases with later truncation time, like

or , there are some amplitude and shift errors. So, with the increasing of

truncation time , the results become more accurate. Compared with the no

truncation, larger time step size case in figure 4.1.15, the results of case N=30 and

will be more accurate. In the next part of study, the truncation time keeps

constant and we would like to find out how the value of N affects the results.

Figure 4.1.17, influence of the value of N on results

From this figure, large value of N means large time step size of derivation of

coefficients, which leads to inaccurate results.

At present, we would like to look at CPU time consumption of some new

scheme cases, with different values of N and truncation time.

-6

-4

-2

0

2

4

6

0 5 10 15 20 25 30 35

const

N=30, tc=1

N=40, tc=1

N=50, tc=1

Dimensionless Time 𝑡

Dis

pla

cem

ent 𝑣

66

Figure 4.1.18, vertical displacements vs. time in original and new schemes

Figure 4.1.19, CPU time required for the computation

As can be seen from the figure 4.1.18 and 4.1.19, although there are some

errors for linear scheme, the computational CPU time is quite small. From the bar

chart in figure 4.1.19, no matter how much the N value and truncation time are,

time spending of new scheme cases are all less than 5% of that of the constant

scheme. Moreover, there is no big difference within each other. Thus, in order to

find out which method is more efficiency, we will look at the accuracy of the result

more, while not taking its CPU time consumption into a serious consideration. In

-6

-4

-2

0

2

4

6

0 5 10 15 20 25 30 35

const

N=30,tc=1.0

N=40,tc=3.0

N=30,tc=3.0

Dimensionless Time 𝑡

Dis

pla

cem

ent 𝑣

0

0.2

0.4

0.6

0.8

1

1.2

const

N=30, tc=1

N=30, tc=3.

N=40, tc=3

% C

PU

tim

e o

f co

nst

ant

and

new

sch

eme

67

this example, the case with N=30 and will be banned, for its lower accuracy.

Although the CUP time consumption of N=30, is slightly longer than that of

N=40, , the former one is chosen as the best scheme since its results are more

accurate.

In conclusion, for first example, linear scheme is more suitable. The errors

can be diminished if we choose the appropriate values of parameters, like N,

truncation time, and theta value. In the next sub-section, the in-plane-motion

responses to dynamic load in a homogeneous semi-infinite soil layer will be

studied and discussed. And then, find out which scheme, linear scheme or linear

scheme, are more suitable.

4.2 In-plane motion of homogeneous-soil layer

In this example, a model of in-plane motion of a homogeneous-soil layer is created,

which is quite similar to the soil layer model in the last sub-section. But some

parameters and dimensions have been changed. The thickness of the layer is

shorted into 1 meter. In horizontal direction, the layer is still extended to infinite.

However, it is impossible to get the analytical solutions. In that case, the reference

solutions are needed. The model for reference solution will be created in ANSYS

with 50 meters in length, which will be discussed later. The soil properties, like

young’s modulus, density and Passion’s ratio, keep the same as those in example 1.

Figure 4.2. 1, model of a homogeneous soil layer

From this graph, the soil layer is overlaying on hard rock layer, which

means there is no displacement in both horizontal and vertical directions along the

bottom surface. On the boundary in the left end, a time-dependent surface traction

68

applied and it is described as a triangular function. The plotting of this

function load is shown in figure 4.2.2.

Figure 4.2. 2, time history of the triangular function surface traction

In figure 4.2.2, the time in x-axis is in dimensionless. The amplitude of the

triangular force is 10,000 N.

First of all, this kind of model is created in commercial software-ANSYS (see

Appendix B). The solutions, which generated from ANSYS model, can be used as

reference solution in later studies. The homogeneous layer is in the dimension of

1meter height and 50 meters length. The whole layer is discretized into 8 elements

in y-direction and 400 elements in x-direction with 4 nodes in each element. In this

example, we will only look at the horizontal and vertical displacements responses

at the upper-left corner point of this layer. In addition, the soil layer in ANSYS is

recreated into finer meshes, which are 16 elements in y- direction and 800

elements in x-direction. The displacement responses of this point to the horizontal

surface traction under two different meshes are represented in the figures below:

69

(a)

(b)

Figure 4.2. 3, Displacement responses to horizontal surface traction at the upper-left corner point for a

finer mesh model and a coarse mesh model computed using ANSYS: (a) horizontal displacement and (b)

vertical displacement

As can be seen, the curves for finer and coarser meshes converge and match

perfectly in both horizontal and vertical displacements. In that case, the results,

which obtained from coarse mesh model, are accurate enough as the reference

solutions.

In Matlab, the same model is also created. Similar to the example 1 in last

sub-section, two methods will be tested here, constant scheme and linear scheme,

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40

Finermesh

Coarsemesh

Horizontal Displacement

Dimensionless Time

Dis

pla

cem

ent

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 10 20 30 40

Finermesh

Coarsemesh

Verical Displacement

Dimension less Time

D

isp

lace

men

t

70

which has been explained in details in chapter 3. In order to get the stable results,

the biggest dimensionless time step size for constant scheme is 0.07, which is

obtained from running the program many times with different values of . In next

step, I am trying to find out which scheme is more suitable for this example. First

of all, the constant scheme is used with dimensionless time step size equals to 0.07

and finished at 30.Then, the results will be compared with the reference solutions

from ANSYS.

(a)

(b)

Figure 4.2. 4, Displacement responses at the upper-left corner point in Matlab model with constant

scheme VS reference solutions: (a) horizontal displacement and (b) vertical displacement

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 20 25 30 35

ReferenceSolution

Dimensionless Time

Dis

pla

cem

ent

Horizontal Displacement

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 10 20 30 40

ReferenceSolution

ConstantScheme,timestep=0.07

Dimensionless Time

D

isp

lace

men

t

Vertical Displacement

71

From figure 4.2.4, it can be seen that the curves of constant scheme perfectly

match the curves of reference solution. The high accuracy of the results is caused

by the small time step size 0.07.

At this stage, the linear scheme with no truncation is employed here. After

running the Matlab program with different time step size, we can find that the

critical dimensionless time step size for stable and accurate displacement curves is

0.008. When enlarge the time step size, the curves are unstable at first. It becomes

relatively stable again until time step equals to 0.5 (Nint=50). But the results are

very inaccurate.

Figure 4.2. 5, horizontal displacement curve for the linear scheme case with critical dimensionless

time step 0.008, no truncation VS reference solution

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 20 25 30 35

Linearscheme,0.008, notruncation

ReferenceSolution

Horizontal Displacement

Dimensionless Time

D

isp

lace

men

t

72

Figure 4.2. 6, horizontal displacement curve for linear scheme case with time step size equals to 0.5, no

truncation VS reference solution

Figure 4.2. 7, zoom in the first two periods of figure 4.2 7

From the figure 4.2.6 and 4.2.7, the curve is only stable up to 5 in the first

period in a low accuracy. The main reason of that is the time step size is too big.

But for the displacement curve with 0.008-dimensionless time step size, it is more

stable and accurate. However, it still becomes unstable after dimensionless time 20.

As discussed before, the values of acceleration unit-impulse response

matrix may also influence the values of displacement. For the constant scheme, the

-3

-2

-1

0

1

2

3

4

0 10 20 30 40

ReferenceSolution

Linear Scheme,dt=0.5, notruncation

Dimensionless Time

Dis

pla

cem

ent

Horizontal Displacement

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 2 4 6 8 10 12

LinearScheme,0.5, notruncationReference solution

Dimensionless Time

D

isp

lace

men

t

Horziontal Displacement

73

dimensionless values of coefficients , and at the dimensionless time

point 1.0 are 2.9846, -0.20701and 3.0505 respectively. But those values for the

linear scheme with 1.0 time step size equal to 2.9056, -0.20739 and 2.9953

respectively.

Figure 4.2. 8, curves of acceleration unit-impulse response coefficient in constant scheme case and

linear scheme case with 0.5 dimensionless time step size

From this figure, we can find that those two curves are not converging.

Compared to the linear scheme, the curve of constant scheme is smoother since the

smaller time step size. That is why the displacement curves of constant scheme are

much closer to the reference solutions.

At present, the truncation will be employed and see how it affects the

results. Firstly, a linear scheme case with a large time step 0.5 is tested. If no

truncation, the horizontal and vertical curves are relatively stable, which are

represented below:

0

0.5

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1 1.2

ConstantScheme,dt=0.07

LinearScheme,dt=0.5

DImensionless TIme

U

nit

-Im

pu

lse

Res

po

nse

Co

effi

cien

t

𝑀

∞𝑡′

density

𝑠

74

(a)

(b)

Figure 4.2. 9, Displacement response of upper-left corner point to horizontal surface traction in linear

scheme with , no truncation VS reference solution: (a) horizontal displacement and (b) vertical

displacement

As can be seen, if no truncation, the displacement curves are only stable in

the first two periods in very low accuracy. The results obtained contain lots of

errors and we can consider that this method is unsuitable. Now, the model is

-3

-2

-1

0

1

2

3

4

0 10 20 30 40

ReferenceSolution

LinearScheme,dt=0.5, notruncation

Dimensionless Time

Dis

pla

cem

ent

Horizontal Displacement

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 10 20 30 40

ReferenceSolution

LinearScheme,dt=0.5, notruncation

Dimensionless Time

D

isp

lace

men

t

Vertical Displacement

75

truncated at dimensionless time 5. The horizontal and vertical displacements

curves look like:

(a)

(b)

Figure 4.2. 10, Displacement responses of upper-left corner point to horizontal surface traction in

linear scheme with , truncate at 5 VS reference solution: (a) horizontal displacement and (b)

vertical displacement

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 5 10 15 20 25 30 35

Referencesolution

Lienarscheme,dt=0.5,truncate at 5

Dimensionless Time

Dis

pla

cem

ent

Horizontal Displacement

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 5 10 15 20 25 30 35

"Reference solution

Linearscheme,dt=0.5,truncateat 5

Dimensionless Time

D

isp

lace

men

t

Vertical Displacement

76

From figure 4.2 10, with the help of truncation, the curves can keep stable

over longer time. But results still have certain amount of errors. In that case, since

the low accurate results, the linear schemes with large time step size are not

workable whenever truncation or not.

Then, we will look at the linear scheme case with a smaller time step size

0.008 and discuss how the truncation influences the results.

(a)

(b)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40

Linearscheme,0.008, notruncation

Referencesolution

Dimensionless Time

D

isp

lace

men

t

Horizontal Displacement

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 10 20 30 40

Linearscheme,0.008, notruncationReferencesolution

Dimensionless Time

D

isp

lace

men

t

Vertical Displacement

77

Figure 4.2. 11, Displacement responses of upper-left corner point to horizontal traction in linear

scheme with and no truncation VS reference solution: (a) horizontal displacement and (b)

vertical displacement

As can be seen, because of the smaller time step size, the curves are more

accurate and quite close to the reference solution. They become unstable only after

dimensionless time point 25. Moreover, amplitude values of two curves are also

identical in the first five periods. Then, we still try to truncate it at dimensionless

time 5. The displacements curves are represented below:

(a)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40

Lienarscheme,0.008,truncateat 5Referencesolution

Horziontal Displacement

Dimensionless Time

D

isp

lace

men

t

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 10 20 30 40

Linearscheme,0.008,truncateat 5Referencesolution

Dimensionless Time

D

Isp

lace

men

t

Vertical Displacement

78

(b)

Figure 4.2. 12, Displacement responses of upper-left corner point to horizontal surface traction in

linear scheme with and truncate at 5 VS reference solution: (a) horizontal displacement

and (b) vertical displacement

From figure 4.2 12, we can find that the curves of linear scheme case can be

stable for a long time. But, for both the horizontal and vertical curves, the accuracy

of amplitude values incline after a certain time. It seems that there is a slight over

damping in this system. In conclusion, for a smaller time step size, the truncation

causes some error after a certain time. There are mainly two methods to improve

the accuracy: making time step smaller or truncating at a later time. Now, we will

have a look at how these two methods influence the results.

At first, for the same truncation time 5, different time step size will be

tested. Those chosen dimensionless time step sizes are 0.008, 0.006 and 0.002,

whose results will be compared with the reference solution. The curves are

represented below:

Figure 4.2. 13, horizontal displacement responses of upper-left corner point to horizontal surface

traction in linear scheme with different time step size, truncate at 5 VS reference solution

From figure 4.2.13, those curves for different linear schemes are identical

and the errors are not diminished. So, for the linear scheme under truncation,

decreasing of time step size does not improve the accuracy. For the second method,

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40

ReferenceSolution

t=0.008

t=0.006

t=0.002

Dimensionless Time

Dis

pla

cem

ent

Horizontal Displacement

79

the dimensionless time step size keeps constant as 0.008, while the truncation

time increases to 10 and 15. The curves look like:

Figure 4.2. 14, horizontal displacement responses of upper-left corner point to horizontal surface

traction in linear scheme with , truncate at 5, 10 and 15 VS reference solution

From figure 4.2.14, as the truncation time increased to 10, the horizontal

displacement curve converges to the reference solution’s curve. But it gets worse

when the truncation time increases to 15. There is some instability of the curves.

Then, it is hard to say that increasing the truncation time can always improve the

accuracy of results. But the optimal results can be obtained by employing a critical

value of truncation time.

In summary, the idea of using a big time step ( ) is not suitable here,

because of the loss of accuracy in both truncation and non-truncation conditions.

But for small time step size condition, which is smaller than or equal to 0.008, the

results is much more accurate, no matter truncate it or not. The accuracy can be

optimized if the appropriate truncation time is chosen.

Finally, we will look at the CPU time consuming of four different cases.

Those cases are constant scheme with time step size equals to 0.07, linear scheme

with 0.008 time step size and truncate at 5, 10 and 15 respectively.

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 10 20 30 40

Reference Solution

t=0.008, truncate at 5

t=0.008, truncate at 10

t=0.008, truncate at 15

Dimensionless Time

Dis

pla

cem

ent

Horizontal Displacement

80

Figure 4.2. 15, CPU time required for the computation

As can be seen from figure 4.2.15, the time consumptions for linear schemes

are much larger than that of constant scheme with 0.07 dimensionless time step

size. With the increasing of truncation time, the costs become larger. In this case,

we can make a conclusion that the constant scheme is the best choice since its high

accuracy and less time consumption. In the future, truncation may be employed in

the constant scheme. That may lead to a higher efficient method.

4.3 Homogeneous semi-infinite soil layer with trench

In this example, there will be a study about a new homogeneous soil layer, with a

building foundation laying on the top and a trench nearby. The main purpose of

this example is to find out how the trench influences the dynamic wave

propagation and the displacement responses to dynamic load at some specific

points. The layer is constant in depth and extends to infinite on both sides in

horizontal direction. It is overlaying on a hard concrete layer and no displacement

along the lower boundary. The simple sketching of the layer looks like:

0

10

20

30

40

50

60

70

Constant Scheme,t=0.07

Linear Scheme,t=0.008, truncateat 5Linear Scheme,t=0.008, truncateat 10Linear Scheme,t=0.008, truncateat 15

CP

U t

ime

of

con

stan

t an

d n

ew s

chem

e

81

Figure 4.3. 1, sketch of the homogeneous soil layer with a building foundation and a trench

For the construction in site, there is always some dynamic load applied on

the building foundation, which may damage the structures nearby. So, the function

of the trench here is to weak the propagation of the dynamic loading waves.

The whole layer will be divided into three parts, a bounded domain and two

unbounded domains. The bounded domain includes the foundation, the trench and

surrounding soils. Those three parts independently looks like：

Figure 4.3. 2, independent bounded and unbounded parts

At present, we will only study the bounded domain. In numerical study, the

trench can be considered as a crack in the layer. For traditional finite element

method or scaled boundary finite element method, the mesh around the crack is

much finer than other locations, which indeed increases the computer resource

and human effort. In order to overcome that problem, the super-elements method

will be employed. Super-elements method is usually suitable for some huge models

with different material properties or loading conditions in different parts. The big

model can be divided into several small sections and each section will be meshed

according to specific method, like finite element method and scaled boundary

82

finite element method. The figure of super-elements of the bounded domain is

represented below:

Figure 4.3. 3, super-elements of bounded domain

In our case, the dynamic stress intensity factors can be determined based

on scaled boundary finite-elements formulation in high efficiency with the help of

super-elements. According to Song, although the advantages of the scaled

boundary finite element method in modeling stress singularities are retained, the

size of the super-elements is limited by the highest frequency of interest. In order

to simulate the response at high frequencies, the problem domain should be

divided into smaller elements, which increases complexity and effort (Song C. ,

2008).

Recently, a continued fraction solution of the scaled boundary finite

element equation in the dynamic stiffness of bounded domain is employed, which

can greatly improve the efficiency. This method is developed from the derivation of

dynamic stiffness matrix of unbounded domain. As Song said, with applying this

continued fraction solution, the force-displacement relationship on the boundary

is formulated as an equation of motion expressed by symmetric, spare, high-order

static stiffness and mass matrices (2008). The derivation of continued fraction

solution is introduced here briefly.

First of all, the scaled boundary finite element equation in dynamic stiffness

is derived and the details are shown in Song’s thesis (2008). The equation looks

like:

[ ] [ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ]

4.3. 1

83

In this equation, the terms [ ], [ ], [ ]and [ ]are coefficient matrices,

which obtained by assembling the element coefficient matrices. They are derived

from the equation 3.3.24. Term [ ] is the dynamic stiffness matrix of a bounded

domain. In Song’s thesis (2008), based on the equation 4.3.1, the derivation of the

continued fraction solution is illustrated step-by-step by using a simple example.

Then, the equation of the continued fraction is expressed as:

4.3. 2

with the coefficients K, M,

and

( =1,2… ). Term is the order of the

continued fraction. Term equals to (

𝑠)

. According to Song, the accuracy of the

continued fraction solution at high frequency can be improved, as the order of

continued fraction increases (Song C. , 2008). In addition, a continued fraction

solution for the dynamic stiffness matrix of a bounded domain of arbitrary

geometry can be obtained. For simplicity in notation, the scaled boundary finite

element equation in dynamic stiffness 4.3.1 can be rewritten as:

[ ] [ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ]

4.3. 3

where

4.3. 4

Equation 4.3.2 can also be rewritten as the matrix equation, which looks like:

84

[ ] [ ] [ ] ([

] [

] ([

] [

]

([

( )] [ ( )])

)

)

4.3. 5

From Song’s thesis, after getting the continued fraction for a given order , the

dynamic stiffness matrix at specific frequency can be obtained by substituting

equation 4.3.4 into equation 4.3.5 (Song C. , 2008).

From continued fraction solution for the dynamic stiffness matrix, the force

displacement relationship of a bounded domain can be expressed as:

[ ] [ ] { } { } 4.3. 6

with coefficient matrices

[ ] ([ ] [

] [

] [

]) 4.3. 7(a)

[ ]

[ [ ] [ ]

[ ] [ ] [ ]

[ ]

[ ]

[

]]

4.3. 7(b)

{ }

{

{ }

{ }

{ }

{ }}

, { }

{

{ } }

4.3.7(c)

85

Equation 4.3.6 can be transferred into time domain, which looks like:

[ ]{ } [ ]{ } { } 4.3. 8

According to Song, the coefficient matrices of continued fraction solution

are determined for each super-element. Through assembling the equations of

motion of individual super-element, the global equation of motion is obtained,

which is represented below:

[ ]{ } [ ]{ } { } 4.3. 9

In equation 4.3.9, matrices terms [ ] and [ ] are the global static stiffness and

mass matrices respectively. Term { } is the displacement vector, includes all the

auxiliary variables. Term { } is the external fore vector (Song C. , 2008).

Same as the last two examples, commercial software- Matlab will be used

here to establish the model and then find out the results. First of all, a simple

homogeneous layer, without trench and the building foundation, is created, which

is same as last example. It is still overlaying on a hard rock with a time dependent

surface traction applying on the left boundary surface. In addition, it extends to

infinite in horizontal direction and has a constant depth. All the other properties

will be the same as the last example, like dimensions, loading conditions and soil

properties. The way of meshing will be different.

Figure 4.3. 4, a homogeneous layer divided into two sub-domains

86

From figure 4.3.4, the whole layer is divided into two sub-domains, a

bounded domain and an unbounded domain. For bounded domain, the scaled

boundary finite element method is employed here with a scaling center . The

boundary of bounded domain is discretized into 8 elements with 5 nodes in each.

For unbounded domain, only the left boundary is discretized into two elements.

The values of horizontal and vertical displacements response at the upper-left

corner point of the bounded domain will be calculated out and then compared with

the reference solution obtained in last sub-section.

(a)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 20 25 30 35

Continuedfractionsolution,dt=0.01

Constantscheme,dt=0.01

Dimensionless Time

Dis

pla

cem

ent

Horizontal Displacement

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20 25 30 35

Constantscheme,dt=0.01

Continuted fractionsolution,dt=0.01

Dimensionless Time Dis

pla

cem

ent

Vertical Displacement

87

(b)

Figure 4.3. 5, Displacement responses to horizontal surface traction at the upper-left corner point of

the continued fraction solution VS constant scheme: (a) horizontal displacement and (b) vertical

displacement

From figure 4.3.5, we can find that curves of constant scheme and continued

fraction solution are completely identical for both horizontal and vertical

displacements. The new results are in high accuracy. Thus, the new discretizing

method is practicable and can be used in this example.

The particular homogeneous soil layer for example 3 and its dimensions are

represented below:

Figure 4.3. 6, sketch of the homogeneous soil layer with a building foundation and a trench for example

3

In order to find out the effects of the trench, the in-plane motion responses

will be monitored in point A and B, which are shown in figure 4.3.6. Point A locates

in front of the trench with distance of 0.5 meter, while point B is 0.8 meter from

another side of the trench. With the employment of super-elements, the whole

section is divided into several parts according to different loading and structure

conditions. After subdividing, the whole layer looks like:

88

Figure 4.3. 7, the super-elements mesh plot of the homogeneous soil layer with a trench

As can be seen from 4.3.7, the bounded domain is divided into eight sub-

domains. The sub-domain 9 is the unbounded domain. For each sub-domain, the

scaled boundary finite element method is used and the scaling center is located at

the center point. Each element contains 5 nodes. The similar mesh plot is also

created by Matlab (see in Appendix C). Before finding the displacement values, we

need to check whether this mesh is fine enough.

A time-dependent triangular function load, which is the same as the load in

example 2, applies on the building foundation in vertical direction. In

Prempramote’s PHD thesis, the same load is also used in his second example,

which is called semi-infinite layer subjected to horizontal surface traction. The

figures of time history and Fourier transform is shown in figure 7.7.9

(Prempramote, 2011)(see in Appendix D). From the Fourier transform figure, the

maximum value of dimensionless frequency is 5. Then, the minimum

requirement value of the wavelength can be obtained for this example. The

equation of the dimensionless frequency looks like:

4.3. 10

Where, term c is the wave velocity, term h is the depth of the layer. The equation of

traveling wave velocity is also represented below:

89

4.3. 11

In equation 4.3.11, term is the wave length and term f is the frequency of the

wave. Combination of equations 4.3.10 and 4.3.11 gives us the minimum value of

wavelength:

4.3. 12

After substituting in the values of and h into equation 4.3.12, the value

of is around 2.51m. In that case, the mesh of the subdomains is fine enough. In

addition, the longest radial distance contained in each subdomain also meets the

requirement. Thus, the meshing of the domain is fine enough to get the accurate

results.

With running the Matlab program, the in-plane motion at a specific point in

this soil layer can be obtained. In order to find out the effects of the trench, the

data of different two points, which locate in front and at the back of the trench

respectively, will be collected. In addition, when the trench has been removed, the

data will also be collected at the same pointsand then compared with the previous

data.

At present, we will look at how the trench influences the propagation of

dynamic waves. In the first case, the trench is kept and the values of horizontal and

vertical displacements of point A and B are collected. The curves look like:

90

(a)

(b)

Figure 4.3. 8, displacement responses to the dynamic load point A and B when the trench exists: (a)

horizontal displacement and (b) vertical displacement

From the figure 4.3.8, for both horizontal and vertical displacements, the

amplitude values of the curves of the point B are much smaller than those of point

A. So, the trench indeed weakens the propagation of the dynamic waves in the soil

layer. Moreover, reduction in amplitude values for horizontal displacement curves

is more pronounced.

In the second case, the trench is removed. The mesh plot is shown below:

-150

-100

-50

0

50

100

150

200

250

0 5 10 15 20 25 30 35

Horizontal Displacements

Point A

Point B

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Vertical Displacements

Point A

Point B

91

Figure 4.3. 9, the super-elements mesh plot of the homogeneous soil layer without trench

Since the trench does not exist, there are only five subdomains in this soil layer,

four bounded subdomains and one unbounded subdomain. This kind of mesh is

also fine enough to give us the accurate results. In this mesh plot, node points 9

and 10 have the same locations as the point A and B respectively. The horizontal

and vertical displacement curves of these two points are represented in the figure

4.3.10 below:

(a)

-40

-20

0

20

40

60

80

100

120

0 5 10 15 20 25 30 35

Horizontal Displacements

Point A

Point B

92

(b)

Figure 4.3. 10, displacement responses to the dynamic load at point A and B when the trench is

removed: (a) horizontal displacement and (b) vertical displacement

As can be seen, for the horizontal and vertical displacements curves, the

differences in the amplitude values are quite small. Two curves are not identical at

the beginning. But they converge as time goes. In that case, if there is no trench, the

wave can propagate freely. The dynamic waves can affect the structures or

buildings, which is far away from the loading point.

Furthermore, a study about how trench influences the displacement

responses at two points individually is also carried out:

-40

-20

0

20

40

60

80

100

120

0 5 10 15 20 25 30 35

Vertical Displacements

Point A

Point B

93

(a)

(b)

Figure 4.3. 11, displacement responses to the dynamic load at the point A with and without trench: (a)

horizontal displacement and (b) vertical displacement

From figure 4.3.11, for both the horizontal and vertical displacements at point A,

there are some differences of the amplitude values. The using of the trench leads to

larger displacement responses of soil layer, which is caused by the wave reflection

from the boundary and trench. For there is no material damping, the dynamic

waves can be trapped.

-150

-100

-50

0

50

100

150

200

250

0 5 10 15 20 25 30 35

With trench

Withouttrench

Horizontal displacement of point A

-80

-60

-40

-20

0

20

40

60

80

100

120

0 5 10 15 20 25 30 35

With trench

Withouttrench

Vertical displacement of point A

94

(a)

(b)

Figure 4.3. 12, displacement responses to the dynamic load at point B with and without trench: (a)

horizontal displacement and (b) vertical displacement

For the point at the back of trench, there is no big difference in the

amplitude values of displacements in both horizontal and vertical directions. For

horizontal displacement only, there are some shifts of the curves. So, if trench is

-30

-20

-10

0

10

20

30

40

50

0 5 10 15 20 25 30 35

Horizontal displacement of point B

Withtrench

Withouttrench

-20

-15

-10

-5

0

5

10

15

20

0 5 10 15 20 25 30 35

Vertical displacement of point B

Withtrench

Withouttrench

95

removed, the horizontal displacement responses may happen earlier at point B. So,

the existing of trench can prevent the propagation of waves and delay the

responses of soil in horizontal direction. But for vertical displacements, there is

almost no influence and two displacements curves are almost identical.

In conclusion, the using of trench can lead to a reduction of the

displacement amplitudes. Compared to the vertical displacements, the horizontal

displacement responses of the soil are more easily to be influenced at any place in

the soil layer.

96

Chapter 5

Conclusion and Future Work

5.1 Summary

The main objective of this research is to develop a reliable and efficient method to

obtain the acceleration unit-impulse response matrix in the dynamic soil-structure

interaction problems in homogeneous, semi-infinite soil layer. According to the

literature review part, the scaled boundary finite element method is chosen as the

theoretical framework of this study. The major advantage of using SBFEM is that it

can be used to model the unbounded domain soil layer. In addition, the

fundamental solution is not required. Only the boundary of the domain needs to

be discretized. So, the spatial dimension reduced by one, which significantly

decreases the computational effort. Other advantages include the using in a soil

layer with anisotropic materials and cracks.

This research focuses on the homogeneous, semi-infinite soil layer. Based

on the scaled boundary finite element method, the scaled boundary finite element

equation in frequency-domain of this system is derived. There is a challenge to get

the scaled boundary finite element equation in time-domain. In my thesis, the key

step is to find out the acceleration unit-impulse response matrix. According to the

previous studies (Radmanovic & Katz, 2010), two different technologies have been

developed, constant scheme and linear scheme. These two schemes are tested in a

homogeneous, semi-infinite soil layer for both in-plane and out-of-plane motion

responses to dynamic load. This research is also extended to a soil layer with a

trench near a foundation, whose purpose is to find out how the trench influences

of the displacements of the soil layer. A summary of conclusion for each chapter is

represented here.

In Chapter 1, the introduction of this thesis was presented. The background

and statement of problem are also introduced. Two different kinds of methods to

97

analyze the unbounded domain are also talked about, which are direct method and

substructure method.

In Chapter 2, a detailed literature of existing approach for soil-structure

interaction problems in unbounded domains was represented. Those approaches

are divided into two groups, global procedures and local procedures. Global

procedures include boundary element method and scaled boundary finite element

method. Local procedures include most absorbing boundary methods and finite

elements method. After discussing the advantages and disadvantages of each

method, scaled boundary finite element is chosen as the theoretical framework of

this research.

In Chapter 3, a detailed about derivation of the theoretical concepts of this

research was represented, which includes some essential equations and important

schemes. First of all, properties of homogeneous, semi-infinite soil layer were

introduced and the basic equation called operator equation was obtained. Then, a

method of transforming Cartesian coordinates into local coordinates was

described, which was used in obtaining the operator equation in local coordinates.

With the help of weighted residuals technique and method of integration by parts,

the scaled boundary finite element equation of displacement in frequency domain

was obtained. By using the Fourier transformation method, the scaled boundary

finite element equation in time domain was derived. Within this equation, the only

unknown is the acceleration unit-impulse response matrix [ ]. There were

two methods introduced to get the values of [ ], which are constant scheme

and linear scheme. For constant scheme, the coefficient value is assumed to be

constant. But for linear scheme, the coefficient value changes linearly within each

time interval. The main purpose of this research was to find out which scheme is

more efficiency and reliable, which had been done in chapter 4. Finally, the way of

calculating the in-plane and out-of-plane motions of soil layer under dynamic loads

were also represented.

In Chapter 4, three numerical examples were studied. First two examples

were about the out-of-plane and in-plane motion of homogeneous semi-infinite

soil layer problem. After comparing the results and the CPU time consumption of

running program, there is a conclusion that the constant scheme can always give

98

us the high accurate results. But it was usually time consuming. For linear scheme,

it might safe some time with using truncation, but the results contained lots of

errors. In order to get the results in high accuracy for linear scheme, very small

time step size can also be chosen. But it may cost more. So, linear scheme may still

not that efficient for certain problems. In addition, a trench was added in the same

soil layer nearby the structure, which was the third example. In-plane motion at

two separate points, which were in front and at the back of the trench respectively,

were collected and studied. Then, it could be easily found that the trench indeed

affects the propagation of dynamic waves in soil layer and the values of

displacement amplitudes were reduced significantly. Moreover, horizontal

displacements of soil were more easily to be influenced.

5.2 Recommendations for future research

In Chapter 3, the Fourier transformation method was employed to transfer

the scaled boundary finite element method from frequency domain into

time domain, which increased the computational efforts a lot. In future

work, a more reliable and efficiency method can be used to replace the

Fourier transformation method, for example approximating the exact DtN

map by a frequency-dependent rational function.

For the second numerical example in chapter 4, the linear scheme was not

that efficiency. The using of truncation always leads to some errors. Future

work of this research might involve the employment of truncation time for

constant scheme. In that case, the cost of using constant scheme may be

reduced. At the same time, the high level of accuracy can still be kept.

For the third example in chapter 4, a reference solution of that problem can

be created by commercial software. With the help of reference solutions, it

will be much easier to find out how the trench influences the displacements

of soil layer.

99

Appendix A

Reference solution of references solution of acceleration unit-impulse

response coefficients (Wolf & Song, 1996)

100

Appendix B

B.1 Mesh plot of the model of numerical example 2 from ANSYS

B.2 Zoom in of the mesh plot of the model of numerical example 2

101

Appendix C

Mesh plot of Homogeneous semi-infinite soil layer with trench

102

Appendix D

The figures of time history and Fourier transform (Prempramote, 2011)

103

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