Soil-structure interaction for traffic induced vibrations ...1237322/FULLTEXT01.pdf · EXAMENSARBETE INOM SAMHÄLLSBYGGNAD, AVANCERAD NIVÅ, 30 HP STOCKHOLM, SVERIGE 2018 Soil-structure

INOM EXAMENSARBETE SAMHÄLLSBYGGNAD,AVANCERAD NIVÅ, 30 HP

, STOCKHOLM SVERIGE 2018

Soil-structure interaction for traffic induced vibrations in buildings

MARCEL HOFSTETTER

NIMA PASHAI

KTHSKOLAN FÖR ARKITEKTUR OCH SAMHÄLLSBYGGNAD

KTH Royal Institute of TechnologySchool of Architecture and the Built Environment Department of Civil and Architectural Engineering Division of Structural Engineering and Bridges AF222X Degree Project in Structural Design and Bridges

TRITA-ABE-MBT-18361ISBN: 978-91-7729-889-2©Marcel Hofstetter and Nima Pashai

Abstract

Major cities in Sweden experience a population growth, demanding innovative solutionsregarding land exploitation for residential housing. One solution is to build closer to existingrailway tracks, however difficulties arise regarding determining traffic induced vibrationsfrom trains. This sometimes results in vibrations being too large in buildings regardingcomfort, resulting in expensive measures taken as to reduce the vibrations. The scope ofthis thesis is to investigate the soil-structure interaction caused by traffic induced vibrationsin buildings using ABAQUS FE software, where the aim is to partly investigate how astructure effects surrounding soil, partly to investigate which parameters of a structurehas largest favorable impact on foundation vibrations. Major results include that groundvibrations at 2-4 meters parallel to a structure relative to the vibration source remainconstant, independent on whether a house is present or not. Further results show thatincreasing the thickness of the foundation slab has a mitigating effect on the inducedvibrations. The main conclusions of this thesis include that quadratic elements are superiorto linear elements for dynamic analyses for soil, and that accelerometers should be placedat least 2-4 m next to an existing structure to obtain accurate measurements comparableto if no structure was present.

Keywords: Soil-Structure Interaction, Frequency Response Function, Traffic-InducedVibrations, Structural Dynamics, Signal Processing, Calibration

Sammanfattning

Större städer i Sverige upplever en befolkningstillväxt, vilket resulterar i att kreativa lös-ningar måste introduceras gällande markexploatering för bostadshus. En sådan lösningär att bygga närmre befintlig järnväg, dock resulterar detta i svårigheter gällande attkvantifiera magnituden av trafikinducerade vibrationer i byggnadsfundament orsakadeav tågtrafik. En konsekvens av detta är att vibrationsnivåerna i husen ibland blir förstora sett till komfortvibrationer, vilket resulterar i att dyra åtgärder måste tas för attminska vibrationerna. Denna avhandling syftar till att genom att använda ABAQUSFE-mjukvara utforska jord-strukturinverkan i hus orsakade av trafikvibrationer. Måletär delvis att undersöka hur byggnation påverkar omgivande markvibrationer, delvis attundersöka vilka parametrar som har störst gynnsam effekt gällande dämpning av trafikin-ducerade vibrationer. De viktigaste resultaten indikerar att markvibrationer 2-4 meterbredvid ett hus relativt vibrationskällan förblir oförändrade oberoende av om byggnationexisterar eller ej, samt att en ökning av tjockleken av grundplattan resulterar i minskadefundamentvibrationer. Slutsatserna som presenteras är flera, däribland att kvadratiskaelement är mer beräkningseffektiva än linjära element för dynamiska analyser för jord,samt att accelerometrar bör placeras minst 2-4 m bredvid ett befintligt hus för att erhållamätdata jämförbara med om ett hus inte skulle finnas på platsen.

Nyckelord: Jord-strukturinverkan, Frekvenssvarsfunktion, Trafikinducerade Vibrationer,Strukturdynamik, Signalbearbetning, Kalibrering

Preface

This master’s thesis was developed at the department of Civil and Architectural engineeringat KTH Royal Institute of Technology in cooperation with ELU Konsul AB.First, we would like to thank professor Jean-Marc Battini of KTH and professor CostinPacoste-Calmanovici of KTH and ELU Konsult AB for their invaluable guidance andsupervision throughout this thesis.We would also like to thank Ph.D student Johan Lind Östlund and Tekn. Lic AbbasZangeneh Kamali for their invaluable input and assistance.A big thank you to all our colleagues at KTH, we are truly grateful for the friendships thathave developed through the years together at campus.Finally, we would like to thank our friends, family and loved ones for their support duringthe 5 years of studying at KTH.

Table of contents

Nomenclature viii

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Purpose and Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Scope and Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Theoretical Background 52.1 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Time and frequency domains . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.4 Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.5 Frequency response functions . . . . . . . . . . . . . . . . . . . . . 162.1.6 Coherence functions . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Geodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.1 Soil Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Process of Transmission of Vibrations . . . . . . . . . . . . . . . . . 212.2.3 Sources of Train Induced Vibrations . . . . . . . . . . . . . . . . . . 222.2.4 Wave Propagation in Elastic Media - Analytical Solutions . . . . . 222.2.5 Wave Propagation in Elastic Media - Practical Implementations . . 252.2.6 Wave Propagation in Elastic Media - Numerical Considerations . . 26

2.3 Axisymmetric model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4 Optimization theory -

Introduction to Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . 33

Table of contents

3 Method 363.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 Axisymmetric model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.2 Physical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.3 Idealization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.4 FE-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.5 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.6 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2.7 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2.8 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2.9 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.3 3D Soil Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.3.2 Modeling approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4 3D Structure Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.4.2 Geometry of the structures . . . . . . . . . . . . . . . . . . . . . . . 713.4.3 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.4.4 Modeling of structure . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.5 3D Soil-Structure Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.5.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.5.3 Modeling of the force . . . . . . . . . . . . . . . . . . . . . . . . . . 763.5.4 Variation of parameters . . . . . . . . . . . . . . . . . . . . . . . . 763.5.5 Curves from literature . . . . . . . . . . . . . . . . . . . . . . . . . 77

4 Results 794.1 Comparison between free field and SSI . . . . . . . . . . . . . . . . . . . . 794.2 SSI - Ground Slab Acceleration . . . . . . . . . . . . . . . . . . . . . . . . 84

5 Discussion and Conclusions 895.1 Meshing in steady-state dynamics problems . . . . . . . . . . . . . . . . . 895.2 Considerations regarding soil calibration . . . . . . . . . . . . . . . . . . . 905.3 Effect of a structure on surrounding soil . . . . . . . . . . . . . . . . . . . 905.4 Important aspects of soil-structure interaction . . . . . . . . . . . . . . . . 91

vi

Table of contents

6 Recommendations 926.1 Propositions for consultation . . . . . . . . . . . . . . . . . . . . . . . . . . 926.2 Recommendations for future research . . . . . . . . . . . . . . . . . . . . . 93

References 94

Appendix A Convergence analysis for Axisymmetric model 96

Appendix B Sensitivity check for Axisymmetric model 108

vii

Nomenclature

Roman Symbols

u Acceleration

u Velocity

A Amplitude

a Acceleration

C Wave Velocity

d Damping coefficient

E Young’s modulus

f Frequency

G Power Spectrum

G Shear modulus

H Transfer function

h Mesh size

m Constant depicting mass matrix formulation

N Sample size

S Linear Spectrum

t Time

u Displacement

viii

Nomenclature

x Ratio between Rayleigh/Shear

a Fourier coefficients

b Fourier coefficients

c Fourier coefficients

Greek Symbols

α Dimensionless constant

α Mass proportional coefficient considering Rayleigh damping

α Ratio between Shear/Pressure

β Dimensionless constant

β Stiffness-proportional coefficient considering Rayleigh damping

γ Dimensionless constant

γ Shear strain

γ2 Coherence Function

λ Lame’s 2nd Constant

λ Wavelength

µ Shear Modulus

∇2 Laplacian Operator

ν Poisson ratio

ν Poisson’s ratio

ω Circular frequency

ρ Density

σ Stress

τ Shear Stress

ix

Nomenclature

θ Phase angle

ε Volumetric Strain

ε Strain

ξ Damping Ratio

Subscripts

θ Angle circular direction

L Love Wave

P Pressure Wave

r Radial distance (Cylindrical)

R Rayleigh Wave

S Shear Wave

x x-direction (Cartesian)

y y-direction (Cartesian)

r Axial direction

z z-direction (Cartesian and cylindrical)

Acronyms / Abbreviations

DEM Discrete Element Method

DFT Direct Fourier Transform

DoF Degree of freedom

FAAC Frequency Amplitude Assurance Criterion

FEA Finite Element Analysis

FEM Finite Element Method

FRAC Frequency Response Assurance Criterion

x

Nomenclature

GA Genetic Algorithm

ODE Ordinary Differential Equations

FFT Fast Fourier Transform

FRF Frequency Response Function

HHT Hilbert-Hughes Taylor

IFFT Inverse Fast Fourier Transform

MDoF Multi Degree of Freedom

PML Perfectly Matched Layers

SASW Spectral Analysis of Surface Waves

SDoF Single Degree of Freedom

SLL Stockholm County Council

SPT Standard Penetration Test

xi

Chapter 1

Introduction

1.1 BackgroundExpansion of metropolitan cities, due to high urbanization rate, has provided todays ur-ban planners and politicians with clear guidelines for future infrastructure projects. It isestimated that Stockholm County is growing with 35.000-40.000 people per year, and by2020 the total population will have reached 2.6 million. In order to meet the high demandof housing, an extension and branching of existing districts must be established in formsof housing, offices and essential social services in order to create sustainable communities(Svanström, 2015). One attractive solution, that has been proposed by the StockholmCounty Council (SLL), is to build nearby existing or future planned railway traffic to directlyachieve maximum accessibility for the citizens without making any major intervention interms of further expansion of public transport (Landsting, b). After years of negotiationsbetween four municipalities in the Stockholm region and the county of Stockholm, a newdevelopment plan for the subway system has been allocated. The planned expansion ofStockholm’s subway system is divided into six projects, which includes 20 kilometers of newtracks, 10 new stations and a entirely new metro line. These actions enable 78.000 newresidencies and approximately 100.000 new workplaces to be constructed in conjunctionwith this extension of the current subway system (Landsting, a).

1

1.1 Background

ARENASTADEN

ROPSTEN

MÖRBY CENTRUM

T-CENTRALEN

NORSBORG

BARKARBY STATION

NACKA CENTRUM

HJULSTA

GULLMARSPLAN

FARSTA STRAND

HÄSSELBY STRAND

HAGSÄTRA

ÅKESHOV

ALVIK

SKARPNÄCK

FRUÄNGEN

SLUSSEN

Hagas

tade

n

Stadion

Danderyds sjukhus

Bergshamra

Tekniska högskolan

Universitetet

Karlaplan

Bredäng

Skärholmen

Vårby gård

Masmo

Fittja

Vårberg

AlbyHallun

da

Sätra

Mälarhöjden

Östermalmstorg

Gamla stan

Gärdet

Zink

ensd

amm

Mar

iator

get

Horns

tull

Liljeh

olmen

Duvbo

Tensta

Rinkeby

Solna centrum

Hallonbergen

Husby

Bark

arby

stade

n

Akalla

Rissne

Kista centrum

Sofia

Hammar

by ka

nal

Järla

Sundbybergs centrum

Huvudsta

Solna strand

Västr

a sko

gen

Stad

shag

en

Rådhu

set

Näckrosen

Sick

la

Svedmyra

Ny station Slakthusomr.

Stureby

Bandhagen

Sockenplan

Joha

nnelu

nd

Hässe

lby gå

rd

Vällin

gby

Råcks

ta

Blac

kebe

rg

Islan

dsto

rget

Ängby

plan

Brom

maplan

Abrah

amsb

erg

Stor

a mos

sen

Fridh

emsp

lan

Thor

ildsp

lan

Kristin

eber

g

S:t E

riksp

lan

Skanstull

Medborgarplatsen

Hötor

get

Rådman

sgat

an

Skärmarbrink

Hökarängen

Sandsborg

Skogskyrkogården

Tallkrogen

Farsta

Hammarbyhöjden

Björkhagen

Kärrtorp

Bagarmossen

Blåsut

Gubbängen

Rågsved

Högdalen

Örnsb

erg

Axelsb

erg

Aspud

den

Kungsträdgården

Odenp

lan

Midsommarkransen

Telefonplan

Hägerstensåsen

Västertorp

Planerad utbyggnad

Planerad utbyggnad

Trafikering på befintligaspår mot Farsta strandeller Skarpnäck

The Metro map of the future

Planned development

Planned development

Traffic on existing lines to Farsta Strand or Skarpnäck

Fig. 1.1 Future planned map of Stockholm Metro System (Landsting, a)

When heavy traffic travels along a railway, vibrations are induced in the surrounding soil.These vibrations propagate throughout the soil and cause vibrations in buildings situatedin the vicinity of the railway track. This process is called the process of transmission ofvibrations, and can be seen in figure 1.2 below. These vibrations are not harmful for thestructural integrity of the building, however they may cause comfort problems for theresidents of the building.

Fig. 1.2 Process of transmission for vibrations.

2

1.1 Background

Today, measures are conducted at the site for a planned building, and the expected mag-nitude of transferred vibrations into the foundation of the planned building are determined.This expected value of vibrations then serves as a basis regarding making the decisionwhether a building can be constructed at the site or not. The major problem today is thatthese expected values of the transferred vibrations contain a large amount of uncertainties,where one important uncertainty is the transfer of vibrations between soil and buildingfoundation. Due to these uncertainties, erratic decisions are occasionally made, resultingin buildings not being constructed where they can be, and buildings being constructedwhere they shouldn’t be, where the latter situation results in expensive measures taken asto reduce the comfort disturbing vibrations in the structure.Mainly two different approaches are used today in order to quantify the vibrations of thebuilding foundation, the first assuming that the vibrations in the building foundation areequal to the vibrations in the surrounding soil, however this proves too conservative inmost cases. The second approach is to use transfer functions from literature, as presentedin figure 1.3. A problem arises when using these transfer functions, mainly that differenttransfer functions exist, and the variation between different transfer functions are large,meaning there is a large risk that the expected vibrations are incorrect. Another problemis that many transfer functions compare vibrations between a building foundation and soilwhere no buildings are present, meaning difficulties arise in cases where a building is firstdemolished before the new building is constructed.

Fig. 1.3 Experimental curves for the transfer rate between soil vibrations and buildingvibrations.

3

1.2 Purpose and Aim

1.2 Purpose and AimThe aim and purpose of this thesis is twofold, as will be described below:

1. To analyze how vibrations in the soil propagate when buildings are present and absent,in order to gain deeper understanding of how the accuracy of measurements are affectedwhen structures are present at the site. This is done through comparing numerical solutionsperformed in ABAQUS for a solid soil medium, and compare with a soil medium with anadded structure, both subjected to a dynamic force.

2. Through calculations in ABAQUS and data processing in MATLAB, the second aimis to analyze which parameters of a structure has the largest impact on the transferredvibrations from the soil to the foundation of the structure, be it mass, stiffness, etc.In order to achieve an accurate representation of the soil profile in the numerical simulations,the soil model was calibrated using measurement results from a test site in Portugal.

1.3 Scope and LimitationsThe scope of the work done for this thesis is to analyze vibrations in selected points ofa structure, when a part of the surrounding soil is subjected to dynamic loading. Thethesis is limited to strictly linearly-elastic relationships, the analyzed structures are verysimple frame models, and the soil is assumed to be a layered medium, where each layer iscompletely homogeneous. Furthermore, the final frequency range that has been analyzedwithin this thesis are frequencies between 1 − 30 Hz.

1.4 Previous workAs the work done in this thesis includes calibration of a soil model to measurement data,it is important to consider the performed measurements. The measurements have beenperformed on a stretch of the Portuguese railway network, at a private property closelysituated to Carregado (dos Santos et al. (2016)). The measurements analyzed within thisthesis are SASW (Spectral Analysis of Surface Waves) tests, which are used to calculate asoil profile based on accelerometer results from impulse loading by a hammer mechanism.At the test site, SASW tests were performed and analyzed in order to obtain an accuraterepresentation of the layered soil profile.

4

Chapter 2

Theoretical Background

2.1 Signal processing

2.1.1 Overview

In this section the concepts regarding signal processing, including a minor introductionto time- and frequency domain and its corresponding inherent theories, will be presented.This will be followed by a description about frequency response- and coherence functions.

2.1.2 Time and frequency domains

The time domain

The most conventional way of analyzing signals is to observe them in the time domain.Basically the time domain includes a ledger of what occurred in a system and its corre-sponding parameters against time, (Hewlett-Packard (1994)), as shown as an example infigure 2.1.

Fig. 2.1 Representation of a random signal in time domain for the theoretical purpose

5


Numerical methods for ordinary differential equations (ODEs)

Solving the equation of motion for a multi-degree-of-freedom system magnitude analyticallyis almost impossible when a field acceleration u(t) varies swiftly over time (Chopra (2014)).To deal with these difficulties, this section will present time-stepping methods that havebeen implemented in the modeling procedure.

Newmark’s Method - Implicit Direct Integration

On of the most widely used iterative time-stepping methods is the Newmark’s method. Itwas developed by professor Nathan M. Newmark in 1959 for use in structural dynamics.It’s a unconditionally stable method, which implies that if very large values of ∆t are usedit will lead to inaccurate but not "absurd" values. It’s established by following relations(Chopra (2014)):

ui+1 = ui + [(1 − γ)∆t]ui + (γ∆t)ui+1 (2.1)

ui+1 = ui + (∆t)ui + [(0.5 − β)(∆t)2]ui + [β(∆t)2]ui+1 (2.2)

Which are combined with the equation of motion at the end of the time step

Mui+1 + Cui+1 +Kui+1 = F (t) (2.3)

By knowing the initial values, typically ui = u(0) and ui = u(0), ui+1 and ui+1 can besolved through

ui+1 = ∆ui

β∆t2 − ui

β∆t + ui1 − 2β2β (2.4)

The expression is then substituted into 2.1 in order to attain ui+1. Parameters β andγ indicate the variation in acceleration over a time-step and determine its stability andcertainty of the method and are usually set to

γ = 12 (2.5)

16 < β <

14 (2.6)

Hilber-Hughes-Taylor method (HHT)

For this project, due to the finite element analysis software ABAQUS, an extension ofNewmark’s method has been applied, namely the Hilber-Hughes-Taylor method, whichprovides a possibility to apply numerical dissipation without degrading the order of accuracy.

6


It’s endowment came as an improvement to maintain the A-stability and numerical dampingproperties of a system, but still achieving second order accuracy when used in conjunctionwith the second order linear ODE problem (Negrut et al. (2006)).The differences between these methods do not include Newmark integration formulas,instead the form of the discretized equations of motion in (2.3) are adjusted to

Mui+1 + (1 + α)Cui+1 − αCui + (1 + α)Kui+1 − αKui = F (ti+1) (2.7)

where the time step is defined as

ti+1 = ti + (1 + α)h (2.8)

The HHT method will seize the proclaimed stability and order properties provided byαϵ[−1

3 , 0] where α is defined as

γ = 1 − 2α2 β = (1 − α)2

4 (2.9)

With these conditions, as well as a smaller value of alpha, the damping becomes larger inthe model. In ABAQUS, the presets that have been applied for a transient fidelity actionwhich corresponds to:

α −0.05β 0.275625γ 0.55

The frequency domain

An alternative approach of studying signals is by analyzing it through the frequency domainview. As is described in section 2.1.3, it is possible to represent all signals from the realworld by adding up sine waves with various frequencies, phases and amplitudes in order toproduce the desired signal. However, only one combination of sine waves can represent thesignal, (Hewlett-Packard (1994)). In the figures below, the transformation and relationshipbetween time, frequency and amplitude are illustrated.

7


↘

Fig. 2.2 Signal in time domain view

↗

Fig. 2.3 Signal in frequency domain view

Fig. 2.4 Three dimensional figure of signalwith time, frequency and amplitude as coordi-nates. One should note, both real and imagi-nary parts of the Fourier transformation areincluded in this case.

2.1.3 Fourier series

In 1822, Jean-Baptiste Joseph Fourier presented a method of writing periodic functions asa sum of sine and cosine terms. Today, his theory is widely used in signal processing as atool to analyze measurements through transformation from the time domain of the signalinto the frequency domain. In this section, the relevant aspects of the Fourier series arepresented. For proof of the concepts introduced in this section, the reader is referred torelevant literature treating the subject.Leonhard Euler discovered that the natural number e could be connected with sine andcosine by using Maclaurin series. First, consider the Maclaurin series of the complexexpression eiy, as shown in equation 2.10:

eiy = 1 + iy + (iy)2

2! + (iy)3

3! + (iy)4

4! + (iy)5

5! + (iy)6

6! + (iy)7

7! + (iy)8

8! + ... (2.10)

8


By using the identities i2 = −1, i3 = −i, and i4 = 1, the expression can be written asshown in equation 2.11.

eiy =(

1 − y2

2! + y4

4! − y6

6! + y8

8! − ...)

+ i(y − y3

3! + y5

5! − y7

7! + ...)

(2.11)

The discovery was made when the series expansion shown in 2.11 was compared to theMaclaurin series of cos y and sin y, which are shown in equations 2.12 and 2.13 below.

cos y = 1 − y2

2! + y4

4! − y6

6! + y8

8! − ... (2.12)

sin y = y − y3

3! + y5

5! − y7

7! + ... (2.13)

It is then evident that the natural number e can be connected to sine and cosine terms, asshown in 2.14 below.

eiy = cos y + i sin y (2.14)

Let f(t) be a periodic function of time t with period 2π, it can then be expressed as theFourier series shown in equation 2.15 below (Vretblad (2003)),

f(t) =∞∑

n=−∞cne

int (2.15)

where cn are complex numbers calculated through equation 2.16.

cn = 12π

∫ π

−πf(t)e−int dt (2.16)

By utilizing the Euler identity presented in equation 2.14, the series can be presented asthe trigonometric series shown in equation 2.17,

f(t) = a0

2 +∞∑

n=1(an cosnt+ bn sinnt) (2.17)

where the constants a0, an, and bn are calculated through equations 2.18, 2.19, and 2.20respectively.

a0 = 1π

∫ π

−πf(t) dt (2.18)

an = 12π

∫ π

−πf(t) cosnt dt (2.19)

bn = 12π

∫ π

−πf(t) sinnt dt (2.20)

9


By manipulation using trigonometric rules, the Fourier series representation presentedabove can be adapted for signal analysis, as shown in equation 2.21.

f(t) =∞∑

n=0An cos(nt− ϕn) (2.21)

In equation 2.21, the term An is the amplitude and ϕn is the phase of the signal. Theexpressions for An and ϕn are presented in equations 2.22 and 2.23 below.

An =√a2

n + b2n (2.22)

ϕn = arctan(bn

an

)(2.23)

Example - Fourier SeriesLet f(t) = t2 be defined for t ∈ [−π, π]. The defined function can be seen in thefigure below.

The Fourier coefficients for this expression are calculated as follows:

a0 = 1π

∫ π

−πt2dt = ... = 2π2

3

an = 12π

∫ π

−πt2 cosnt dt = ... = (−1)n 4

n2

bn = 0, n = 1, 2, 3, ...

10


Example - Fourier Series cont.Thus, for −π ≤ t ≤ π, the Fourier series become

f(t) = π2

3 +∞∑

n=1(−1)n 4

n2 cosnt

Presented graphically below, the Fourier series are visualized using 1, 2, 5, and 10terms. It can here be seen that the Fourier series of the expression corresponds verywell to the initial function for 10 terms.

11


2.1.4 Fourier transforms

The tool for conversion of a signal from the time domain to the frequency domain is theFourier transform, which transforms a signal f(t) of time into a signal F (ω) of frequency,without adding or losing information (Vretblad (2003)).

Continuous Fourier transform

For continuous functions of time, the continuous Fourier transform is utilized, which isbased on integration of the time signal.Let f(t) be a piecewise continuous function of time over [−P, P ], and let it be periodic withperiod 2P . Next, introduce ω = nπ

Pand let P → ∞. The continuous Fourier transform is

then defined as shown in equation 2.24,

F (ω) =∫ ∞

−∞f(t)e−iωt dt (2.24)

and the inverse Fourier transform (IFFT) is defined as presented in equation 2.25.

f(t) = 12π

∫ ∞

−∞F (ω)eiωt dω (2.25)

Discrete Fourier transform

Due to the fact that measurements are not intrinsic continuous, a discrete method ofcomputing the Fourier transform is needed. Let f(t) be a continuous function of time, aspresented graphically in figure 2.5 below.

Fig. 2.5 A continuous function f(t)

12


This signal, subjected to a discrete division of 20 evenly spaced points, takes theappearance of figure 2.6.

Fig. 2.6 Discrete representation of f(t)

In order to integrate this discrete function, integration can only occur at the discretepoints shown in figure 2.6, meaning the value of at integration points is assumed to beconstant between the integration points. This is illustrated graphically in figure 2.7, wherethe area under the continuous graph represents the continuous integration of f(t) and theareas of the rectangles represent the discrete integration of f(t) using 20 equidistant points.

Fig. 2.7 Comparison between the discrete and continuous representation of f(t)

It is evident from figure 2.7 that if the number of points increase (and thus the distancebetween points decrease), the discrete representation will converge towards the continuousrepresentation. if an infinite amount of discrete points are included, the representationswill be identical. However, when conducting measurements, it is not possible to measure

13


at an infitesimal time interval, resulting in the need for a discrete procedure in order tocompute the Fourier transform of the signal.Let f(t) be a discrete function defined at N discrete points. Let the function values off(t) be considered as the areas described above, with values f [0], f [1], f [2], ..., f [N − 1]. Byassuming that f(t) is a periodic function with period N , defined over the interval [0, N − 1],the Discrete Fourier transform (DFT) is then defined as (Vretblad (2003)):

F (ω) =N−1∑k=0

f [k]e−i2πkn/N (2.26)

The value of n in equation 2.26 is equal to the distance between the discrete points.

Fast Fourier transform

A drawback of the Discrete Fourier transform is that it is computationally heavy, as eachDFT requires N2 operations, where an operation is defined as a complex multiplicationfollowed by a complex addition, and N is the number of samples subjected to the DFT. In1965, Cooley and Tukey found an algorithm for efficient calculation of the DFT, given thecondition that the sample size was 2m, m ∈ Z, called the Fast Fourier transform (FFT).This algorithm will be presented generally in this section.First, introduce the substitution as shown in equation 2.27.

ω−nk = e−i2πkn/N (2.27)

The FFT works under the assumption that the Fourier transform is a polynomial in thevariable ω−n, i.e. uses the relation ω−nk = (ω−n)k. The Fourier transform is then calculatedusing Horner’s method, which is presented below.By working under the assumption that F (ω) is a polynomial, it is then rewritten accordingto equation 2.28

F (ω) =N−1∑k=0

f [k](ω−n)k = f [0] + f [1]ω−n + f [2](ω−n)2 + ...+ f [N − 1](ω−n)N−1 (2.28)

To evaluate equation 2.28 at an arbitrary value of ω, say ω0, new constants ψk are introducedwith the following values:

ψN−1 = f [N − 1]ψN−2 = f [N − 2] + ψN−1ω0

...

14


ψ1 = f [1] + ψ2ω0

ψ0 = f [0] + ψ1ω0

Where ψ0 is the value of F (ω0). The time gain of using FFT instead of DFT is that thenumber of operations decrease from N2 as is the case with DFT to 2N log2 N for the FFT.The time gain is illustrated graphically in figure 2.8 below.

Fig. 2.8 Comparison of number of operations needed between the Discrete Fourier transformand the Fast Fourier transform for a signal with 128 samples.

15


2.1.5 Frequency response functions

A frequency response function (FRF) is a function that describes how a model reacts whensubjected to dynamic loading. The FRF presents the response of the structure in thefrequency, where the response of the structure is normalized with respect to the input force(Bodén and Carlsson (2014)). The meaning of this is that the FRF presents how a structurewould react under perfect linear elastic conditions when subjected to a perfect impulseof unit amplitude. These characteristics make FRF’s very powerful regarding analysis ofresonance peaks and material damping inherent to the structure, as the resonance peaksare clearly visible in the FRF. The width of the peaks reveal the damping, should the FRFbe subjected to analysis of material damping using the half-power bandwidth method.Calculation of a FRF starts with the calculation of the linear spectrum of the input forceand the linear spectrum of the response of the system. These linear spectra are calculatedby subjecting the time histories of the input and the output to Fourier transformation, asshown in equation 2.29 and 2.30.

Sxx(f) = F (p(t)) (2.29)

Syy(f) = F (a(t)) (2.30)

Sxx is the linear spectrum of the input, p(t) is the input force, Syy is the linear spectrum ofthe output, a(t) is the output, and the notation F is used to denote Fourier transformation.As these spectra are complex valued with a real and an imaginary part, it is sometimesnecessary to display the magnitudes of these complex-valued functions. For this reason, theautopower spectrum of the input, Gxx, and the autopower spectrum of the output, Gyy,are defined as shown in equations 2.31 and 2.32 respectively.

Gxx(f) = Sxx(f)S∗xx(f) (2.31)

Gyy(f) = Syy(f)S∗yy(f) (2.32)

The asterix ∗ denotes the complex conjugate of the function. By cross correlating the twolinear spectra in equations 2.29 and 2.30, the cross power spectrum is defined as shown inequation 2.33.

Gyx(f) = Syy(f)S∗xx(f) (2.33)

16


The frequency response function is now given as presented in equation 2.34 below.

H(f) = Gyx(f)Gxx(f) (2.34)

2.1.6 Coherence functions

For perfect linear systems in an isolated environment where no ambient noise can affect theresponse, it is safe to assume that 100% of the response is caused by the input. This ishowever not the case in practice, as measurements conducted always have some disturbingelements present, stemming from different sources. The theoretically perfect system isillustrated graphically in figure 2.9, where p(t) is the input and a(t) is the output.

Fig. 2.9 Perfect linear system

For real systems, as seen in figure 2.10, the input p(t) is subjected to noise at theinput, denoted x(t), example of such noise could be car traffic from a nearby road whenmeasuring the response from train passages. The effect of this noise, coupled with theinput, produces a noisy input signal, denoted p∗(t). This signal produces an output signaldisturbed by noise at input, denoted a∗(t). Since there is also noise present at output, e.g.from eigenfrequencies in the accelerometer used for the measurements, denoted y(t), theoutput signal is also disturbed by noise at output. This final signal, the measured response,is denoted a∗∗(t).

Fig. 2.10 Real system

17


In order to deal with this noise, coherence functions have been developed. The coherencefunction present, on an absolute scale, a value that reveals how much of the output iscaused by the input. The coherence function is presented in equation 2.35.

γ2(f) =Gyx(f)G∗

yx(f)Gxx(f)Gyy(f) (2.35)

A coherence function value of 1 means that 100 % of the output is caused by the input,as is the case with the system shown in 2.9. A coherence function value of 0 means thatnothing of the output is caused by the input.Normally, a coherence function value of γ2(f) > 0.95 signifies a signal of good quality,whereas signals with coherence below 0.95 could be considered signals of low quality(another explanation is that the real system cannot be assumed to be linear). Coherencefunctions should be used on averaged values of the power spectra, in the case that severalmeasurements have been performed.

Example - Frequency Response Function and Coherence FunctionFrom Portugal, tests have been performed by applying an impulse load and measuringthe response at a point 24 m from the impact. The impulse load and its responsecan be seen in the figure below.

From the response and the force, the frequency response function is calculated, andis presented in the figure below.

18


Example cont. Frequency Response Function and Coherence Function

The coherence function for the response is then calculated, which can be seen below.

The shaded areas of the coherence function signifies where the coherence functionvalue is below 0.95, i.e. where too much ambient noise is present. It is clearly visiblehere that a disturbance occurs for frequencies below 7 Hz, most probably becauseof eigenfrequencies of the accelerometer, and for some narrow frequency bands in the30 Hz range, stemming from external disturbance of the system.

19

2.2 Geodynamics

2.2 GeodynamicsIn this section, the most important aspects of geodynamics regarding traffic inducedvibrations are introduced. First, in section 2.2.1, the process of transmission from thevibration source to the receiver is described. After the process of transmission is described,the different sources of vibrations from railway traffic is introduced in section 2.2.3. Insection 2.2.4, the different types of waves propagating in elastic media are described.

2.2.1 Soil Models

When considering soil, the material exhibits highly non-linear behavior, as the governingparameters of the soil (stiffness, damping) are dependent partly on the rate of the applicationof the load, partly on the magnitude of the load (SGF (2013)). However, for small strainsin the soil, γ ≤ 10−4, the material model can be assumed to be linear elastic due to theinsignificance of non-linear behavior. In theory, this linear model exhibits zero energydissipation, but it is shown (Bodare (1997)) that the material damping still has to beincorporated in the model, as some damping still exist.Further material models are utilized in geodynamics, namely elasto-plastic material modelsfor medium strains (10−4 ≤ γ ≤ 10−2) and non-linear models for large strains (10−2 ≤ γ).However, due to the small strains intrinsic to vibration problems, the linear elastic model ismotivated (SGF (2013)). These threshold values are visualized in figure 2.11 below.

Fig. 2.11 Suggested material models for different magnitudes of shear strain.

20

2.2 Geodynamics

2.2.2 Process of Transmission of Vibrations

When a train travels, it acts as a vibration source, displacing the surrounding ground, asdepicted in figure 2.12. The induced displacements propagate through the soil, knownas wave propagation. This propagation of waves is divided up into two main categories:surface waves, propagating along the surface of the medium, and body waves, propagatingthroughout the body of the medium (SGF (2013)).

Fig. 2.12 Process of transmission of vibrations for traffic-induced vibrations in buildings.

The body waves are subjected to phenomena such as refraction and reflection, the formeroccurs at the interface between two soil layers, and the latter occurs at both interfacesbetween soil layers and at the interface between soil and bedrock. The waves propagatingin the soil eventually reach the building foundation in figure 2.12, causing the foundationto vibrate. These vibrations in turn cause the building to vibrate, providing phenomenasuch as comfort problems for residents of the structure.

21

2.2 Geodynamics

2.2.3 Sources of Train Induced Vibrations

The schematic illustration of the process of transmission in figure 2.12 depicts the train asthe source of vibration. There are many factors that contribute to vibrations caused bytrains, which will be described in this section.It is important to distinguish between two main sources of vibrations from trains: the firstsource stems from ground displacements caused by point loads from the wheels, whichgives rise to ground displacements with a low frequency and high amplitude. The secondsource originates from track unevenness etc., and induced vibrations of higher frequencyand lower amplitude.

Some of the sources found in the literature (SGF (2013)) resulting in traffic inducedvibrations are:

1. Deformation of soil due to loading from wheels. These vibrations are dependent on themass of the train, the bogey distance, the axle length and the velocity of the train.

2. Sources from the train-track interaction. Example of sources from train-track interactionare wheel imperfections and damage, and acceleration/retardation of the train.

3. Sources from rail unevenness. Example of train track unevenness is both horizontaland vertical unevenness of the track. Further sources are due to curves in the track, givingrise to centripetal forces in the system.

4. Vibrations induced by uneven support from the embankment. These vibrations stemfrom geometric variability of the embankment, geometry of the ballast and the stiffness ofthe both.

2.2.4 Wave Propagation in Elastic Media - Analytical Solutions

When discussing wave propagation of soils, it is important to distinguish between differentwaves stemming from a vibration source. Namely, there are 4 different types of wavespropagating through an elastic medium: Pressure waves (or P-waves), shear waves (orS-waves), Rayleigh waves (R-waves) and Love waves (L-waves). The two former waves areso-called body waves, meaning they propagate through the body of a medium, whereas thelatter two are surface waves, meaning they propagate at the surface. The different kinds ofwaves are illustrated visually in figure 2.13 below.

22

2.2 Geodynamics

Fig. 2.13 Wave types propagating through an elastic medium. Depicted top is the pressurewave, middle is the shear wave and bottom is the Rayleigh wave.In geodynamics analysis, it is important to characterize these waves with their correspondingwave velocities, as the waves propagate at different speed. This section is devoted to thecharacterization of velocity of the P-, S-, and R-waves (as they can be determined throughin-plane analysis). The theory regarding the L-waves is not presented here, as they playminimal role in the analyses performed within the scope of this thesis.First, consider the equations of motion for an element in a 3D-space, as shown in equations2.36 through 2.38 below.

ρ∂2u

∂t2= ∂σxx

∂x+ ∂τxy

∂y+ ∂τxz

∂z(2.36)

23

2.2 Geodynamics

ρ∂2v

∂t2= ∂τyx

∂x+ ∂σyy

∂y+ ∂τyz

∂z(2.37)

ρ∂2w

∂t2= ∂τzx

∂x+ ∂τzy

∂y+ ∂σzz

∂z(2.38)

Furthermore, introduce the Lamé constants as presented in equations below

µ = E

2(1 + ν) (2.39)

λ = νE

(1 + ν)(1 − 2ν) (2.40)

Here, µ is the shear modulus and λ is Lamé’s second constant, both dependent on the Poissonratio ν and the Young’s modulus E. Using these constants, the applicable stress-strainrelationships become as shown in equation

σxx = λε+ 2µεxx σyy = λε+ 2µεyy σzz = λε+ 2µεzz (2.41)

τxy = τyx = µγxy τyz = τzy = µγyz τxz = τzx = µγxz (2.42)

With ε = εxx + εyy + εzz.By incorporating the Lamé constants in the stress-strain relationships for the stresses in theequations of motion above, the Navier wave equations are given as presented in equationsbelow:

ρ∂2u

∂t2= (λ+ µ)∂ε

∂x+ µ∇2u (2.43)

ρ∂2v

∂t2= (λ+ µ)∂ε

∂y+ µ∇2v (2.44)

ρ∂2w

∂t2= (λ+ µ)∂ε

∂z+ µ∇2w (2.45)

Where ∇2 is the Laplacian operator, defined as ∇2 = ∂2

∂x2 + ∂2

∂y2 + ∂2

∂z2 . These equations arethen solved for the P-, S-, and R-wave velocities. Solving for the P-wave velocity yields

CP =√λ+ 2µρ

(2.46)

24

2.2 Geodynamics

Solving for the S-wave velocity yields

CS =√µ

ρ(2.47)

The complete derivation for the solution of the Navier flow equations with regard to theR-wave velocity is long and laborious, hence it is not presented here. The interested reader ishowever referred to literature (Tschebotarioff (1962)) treating the subject for this derivation.For the R-waves, the solution arrives at the characteristic Rayleigh wave equation, aspresented in equation:

x3 − 8x2 + (24 − 16α2)x+ 16(α2 − 1) = 0 (2.48)

Where α = CS

CP=

√1−2ν2−2ν

and x = CR

CS. It can be shown mathematically that for real material,

i.e. 0 ≤ ν ≤ 0.5, the relationship CR ≤ CS ≤ CP holds true. By utilizing this, thecharacteristic Rayleigh wave equation is left with one root, with a solution dependent on ν,which can be written approximately as

CR = 0.87 + 1.12ν1 + ν

(2.49)

2.2.5 Wave Propagation in Elastic Media - Practical Implemen-tations

The Rayleigh waves possess a characteristic that they are dispersive - the Rayleigh wavevelocity is dependent on the wavelength of the Rayleigh wave. This dispersion relationship isutilized in the Spectral Analysis of Surface Wave (SASW) testing procedure. The dispersionrelationship implies that for a propagated Rayleigh wave consisting of several waves ofdifferent velocity and wavelength, a unique dispersion curve can be obtained, which servesas the base for determining the soil profile and its relevant parameters.By performing an impulse load test on the surface of a soil, accelerations are measuredat different distances from the impact. These accelerations are then subjected to signalanalysis, and from the signal analysis the dispersion curve is obtained. This dispersioncurve is commonly referred to as the experimental dispersion curve.Now, an inverse procedure is performed, where the soil profile is assumed, and from this soilprofile, an eigenvalue problem is solved in order to obtain the so-called theoretical dispersioncurve. This procedure is iterated using some optimization algorithm until correspondencebetween the theoretical and experimental dispersion curves are achieved.

25

2.2 Geodynamics

From the inversion procedure, the soil profile is obtained, as the governing parameters arethe body wave velocities along with the layer thickness. Furthermore, that damping isevaluated using e.g. the half-power bandwidth method for the signal, performed on thesignal in the frequency domain.

2.2.6 Wave Propagation in Elastic Media - Numerical Consider-ations

Regarding numerical calculations of the described wave propagation mechanisms, extracaution has to be considered regarding the calculation of wave propagation in e.g. FEA. Oneimportant aspect to considered are spurious waves, which are non-physical waves caused bylogical errors implemented into the analyzer. Examples of spurious wave propagation innumerical modeling are spurious wave reflection at boundaries due to incorrect modeling ofthe boundaries, and spurious wave reflection and refraction within the body of the medium,also due to incorrect modeling.The modeling of the boundaries play an important role, as normal rollers, pins or clamps act,according to elementary beam theory, as springs of infinite stiffness in different directionsdepending on the type of boundary. These infinitely stiff springs are physically interpretedas an infinitely stiff material (e.g. bedrock) when considering wave propagation, meaningfull reflection of any incident waves. In order to cope with this, several different methodshave been formulated in order to minimize the effect of spurious reflection of waves atboundaries. To name a few,

1. Absorbing boundaries, which includes added layers of material at the boundaries, withincreasing value of material damping, in order to dampen the incident wave without causingreflection. In order for this method to fully work, careful consideration has to be taken asto the determination of the damping values inserted into the FEA. The gain of using thismethod is that it is not subjected to the Sommerfeld radiation condition, however it resultsin a larger model due to the extra elements that need to be added, resulting in a drawbackregarding computational efficiency.

2. Perfectly Matched Layers (PML), which include damping not subjected to the Sommer-feld radiation condition. This method however is not inherently implemented in ABAQUS(however it is in COMSOL).

26

2.2 Geodynamics

3. Lysmer-Kuhlemeyer elements (or infinite elements), which are described in greaterdetail below. These elements are subjected to the Sommerfeld radiation condition (i.e. thatin order for the elements to work properly, the incident wave has to have an incident angleperpendicular to the element interface). The gain of these elements are that they are easilyimplemented, and that they result in a computationally efficient model.

The infinite elements named above act as dashpots at the nodes, with a dashpotcoefficient for the incident P-wave and one for the S-wave. The values of these dashpotcoefficients are shown in the expressions below.

dp = ρcp (2.50)

ds = ρcs (2.51)

Where ρ is the density of the material.As stated above, the Lysmer-Kuhlemeyer elements work optimally for cases where theincident wave is perpendicular to the element interface, meaning that the optimal modelingapproach when using these is a half-space model, as this implies that the probability of theradiation condition being fulfilled increases.As stated above, spurious reflection/refraction within the model is also a potential problem,as the selection of element size and continuity of element size, along with mass matrixformulation plays an important role. Zdeněk P. Bažant showed in 1978 and 1981 that, inorder to achieve correspondence between analytical and numerical solutions for the wavepropagation velocity for an elastic medium, the element size has to be selected based on thewavelength of the wave of the highest frequency taken into consideration. The wavelengthof a wave propagating through an elastic medium is calculated from λR,S,P = CR,S,P

ffor

R-, S-, and P-waves respectively. In figure 2.14, the relative wave velocity from numericalcalculations at the left and right side of an element interface is plotted, showing that forlinear elements, an element size of about 1/8 of the wavelength is required in order for thewave to be properly considered. Figure 2.15 shows the same, but for quadratic elements,and figure 2.16 shows a comparison between the two. The parameter m is called the massdistribution parameter, and denotes how the mass is distributed between the nodes andthe elements. The mass distributed over the element is mρh, where ρ is the density and h

is the mesh size. The mass lumped at the node is (1 − m)ρh. A value of m = 0 denoteslumped mass matrix formulation, and m = 1 denotes consistent mass matrix formulation.In the figures, v signifies the analytical value of the wave velocity, v0 is the numerical valueof the wave velocity, and λ is the wavelength considered.

27

2.2 Geodynamics

1 2 3 4 5 6 7 8 9 10

/h

0

0.5

1

1.5

v/v

0Relative wave velocity with relative wave length, linear elements

m = 0

m = 0.25

m = 0.5

m = 0.75

m = 1

m = 1.5

Reference

Fig. 2.14 Relative wave velocity plotted against element size for linear elements. It can herebe seen that a mesh size of about 1/8 of the wavelength is required in order to achieve goodcorrespondence between the analytical and numerical solution for the wave velocity.

28

2.2 Geodynamics

1 1.5 2 2.5 3 3.5 4 4.5 5

/h

0

0.5

1

1.5v/v

0

Relative wave velocity with relative wave length, quadratic elements

m = 0

m = 0.25

m = 0.5

m = 0.75

m = 1

m = 1.5

Reference

Fig. 2.15 Relative wave velocity plotted against element size for quadratic elements. It canhere be seen that the wave velocity converges quicker toward the analytical solution comparedto linear elements.

29

2.2 Geodynamics

0 2 4 6 8 10 12 14 16

/h

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1v/v

0

Relative wave velocity with relative wave length, comparison

Linear

Quadratic

Fig. 2.16 Comparison between linear elements and quadratic elements regarding conver-gence towards wave velocity, using lumped mass matrix formulation, which is the preset inABAQUS.

30

2.3 Axisymmetric model

2.3 Axisymmetric modelTo grasp the soil as a three-dimensional solid, the understanding of small interactionsbetween the various soil layers and applied dynamic load are required. Essentially, this leadto construct an axisymmetric FE-model of soil in order to interpret how well the modelrelates to the load and given boundary conditions. This strategy will result in reduction ofcomputation time due to fact that an axisymmetric model represents a cross section of thesolid three dimensional earth.

Geometrical Axisymmetry

The basic concepts for an axisymmetric geometry is that it can be sealed around its ownsymmetrical axis into a cylinder by rotating a flat surface around it. However, from amathematical and physical point of view, an axisymmetric model should be treated astwo-dimensional problem. An illustration of an axisymmetric geometry for a cylinder areshown below:

θ

θ

AXIS OF SYMMETRY

Fig. 2.17 A cylinder conceived by revolving a plane surface around its axis of symmetry

31


Pressure load implemented on Axisymmetric model

Furthermore, an axisymmetric model have an intrinsic application regarding pressure loads,due to its integration approach. An axisymmetric geometry integrates the pressure loadover a quarter circle (see fig 2.18), then the problem is assumed to be double symmetric,resulting in the cylindrical representation shown in figure 2.17

Fig. 2.18 Conceptual illustration of a quarter circle of the soil when exposed to a pressureload

32

2.4 Optimization theory -Introduction to Genetic Algorithm


This section will describe the basic concepts regarding optimization theory and GeneticAlgorithm (GA), which has been implemented in order to optimize essential parameters toobtain a more realistic and accurate model of the soil in Portugal. First a briefing aboutoptimization theory will be presented, and after an introduction to GA will be discussed.

Overview - Optimization theory

The primary purpose of an optimization is to identify the best solution regarding a particularproblem that consists of a compilation of numerous appropriate solutions while fulfillingall constraints to which the optimization problem is subjected to. Initial phase in theoptimization process is to establish a suitable model, which in mathematical terms implies toidentify and express the objective function, the design variables and the constraints,Rao (2009). These are the three essential components to proceed an optimization, and aredefined as follows:

• Design variables: Are unknown parameters of the system where the aim is todetermine these values. Through the optimization scheme, it tries to discover valueswhich generates the most suitable solution.

• Objective function: Is a quantitative figure which the optimization aim to maximizeor minimize. Design variables and parameters of interest are deciding the shape andform of the objective function. For instance in production the goal is to minimize costsand maximize profits or in research to fit experimental data to a model representingthe objective.

• Constraints: Functions which explain the relationship between allowable values fordesign variables and their predicted values.

An optimization problem can be mathematical defined as following:

Design V ariables :E = [E1, E2, ..., En] ∈ Rn

ξ = [ξ1, ξ2, ..., ξn, ] ∈ Rn

(2.52)

33


Constraints :Plb = 0.5, 0.5, ..., 0.5 (Lower boundary)Pub = 2.0, 2.0, ..., 1.0 (Upper boundary)

(2.53)

Objective Function :Minimize SC(x)

(2.54)

Next step in a optimization process involves identifying a appropriate optimization algorithm.This mainly depends on which type of optimization problem fits the concerned issue.Usually a optimization problem is solved by combining various different categories, suchas constrained- or unconstrained optimization and Single- or multi-objective optimization,Rao (2009).Today there are numerous types of applicable optimization algorithms, which are distributedinto two major categories, (Mathworks (2018)):

• Minimum-seeking algorithms: Traditional gradient based methods, such asNewton’s method. They are swiftly to implement, but commonly stuck in localminimum.

• Metaheuristic methods: Applies probabilistic calculations, non-deterministic andapproximate methods. Gradual process, but more fortunate at identifying globalminimum.

For this specific case the Genetic Algorithm (GA) has been implemented, which is a heuristicsearch and optimization technique inspired by natural evolution.

Genetic Algorithm

GA is today one of the most sophisticated and well applied optimization algorithms. Itwas firstly introduced by John Holland, a researcher at the University of Michigan during1960s, in order to find suitable solutions to problems that were computationally intractable.Although, due to unpractical implementation during that time, it didn’t spread widely until1990s. Holland (1992) suggested GA in order to improve the coherence and robustness ofexisting optimization methods.

GA is a approach for determining both constrained and unconstrained optimization problems.It is a heuristic optimization technique, based on Darwinian principle of evolution andthrough genetic selection process, it reflects biological evolution with a higher level ofperception. It basically involves step like inheritance, selection, mutation and crossover. A

34


GA procedure launches by producing a randomly selected initial population of artificialchromosomes and estimating their objective function values. Each chromosome symbolize apossible solution to a problem and has a fitness, which corresponds to how well the value ofthe generated chromosome fits the solution. In the manner regarding GA, the objectivefunction is also recognized as fitness function.

Fig. 2.19 Scheme of GA and its iterative process and steps

GA performs a scheme of fitness based selection and recombination to generate a heirpopulation, the next generation. During recombination, parent chromosomes are selectedand their genetic material is recombined to produce child chromosomes. These then passinto the successor population. As this scheme is iterated, consecutive successive generationsevolves and the average fitness of the chromosomes tends to increase until some stoppingcriterion is reached. In this way, a GA “evolves” the best solution to a given problem.

35

Chapter 3

Method

3.1 OverviewTo properly analyze soil-structure interaction for traffic induced vibrations in buildings,great consideration has to be taken regarding modeling of the soil. In order to obtain asuitable soil model, the authors concluded it necessary to model using solid elements. Ascan be seen in figure 3.1, an axisymmetric solid model was first developed, where the goalwas to perform the necessary convergence analyses, validate the mathematical model and tocalibrate the material parameters against measurement data from measurements performedin Portugal.

Fig. 3.1 Description of the main parts of the performed work.

36


After the axisymmetric modeling was finalized, a 3D-soil model was developed in order toreproduce the results from the axisymmetric model. As the 3D-model was constructed, ahouse model was analyzed and altered, which served as the structure used when testingthe soil-structure interaction. The last step was a 3D soil-structure model, where thesoil-structure interaction was analyzed. The procedure for developing the axisymmetricmodel and description of the calibration trials are presented in section 3.2. The descriptionof the 3D-model and the reproduction of results from the axisymmetric model is presentedin section 3.3. The work done regarding the used structure in the analysis is found in section3.4. In the last section, section 3.5, the soil-structure interaction analysis is presented.


3.2.1 Overview

The methodology employed for the axisymmetric modeling of the soil follows general FEmodeling approaches found in literature Cook et al. (2001) with some additions, especiallyregarding the calibration and measurement processing. The axisymmetric modeling issummarized in figure 3.2 below.

Fig. 3.2 Employed procedure for the axisymmetric modeling.

37


In this section, a general description of the procedure will be presented. In the subsequentsubsections, each part of the the used method will be described in more detail, includingpartial results from each step.The physical system analyzed, i.e. the measurement site situated in the vicinity of Carregado,Portugal, was analyzed through literature regarding studies carried out in the area by dosSantos et al. (2016), as presented in section 3.2.2. The physical system was then idealizedand simplified in order to facilitate numerical analyses, as presented in section 3.2.3. Afinite element model was then constructed based on the idealization, as presented in section3.2.4. Following the discretization, determination of input parameters to be used in themodel was carried out in accordance with the procedure presented in section 3.2.5. Themodel was then subjected to a robustness analysis, as presented in section 3.2.6. In orderto validate the model, the results were compared to processed measurement results, wherethe processing is described in section 3.2.7, and an attempt at calibrating the parameters ofthe model according to the procedure described in 3.2.8 was performed. The final outputfrom the axisymmetric model is presented in section 3.2.9.

3.2.2 Physical system

The physical system analyzed within this thesis is a private property situated near Carregado,Portugal, closely situated to a railway track. The soil profile at the test site consists of atop layer of embankment clayey sand, followed by clay with organic material, clay withsand intercalations, progressing down to clay with limestone fragments, i.e. weathered rock.This profile is presented in table 3.1, as per the suggested soil profile by the authors of themeasurements report from Portugal, dos Santos et al. (2016).

Table 3.1 Description of soil profile according to SPT tests in Portugal.

Depth [m] Description0 − 2.5 Embankment clayey sand2.5 − 3.5 Clayey sand with organic material3.5 − 6.5 Clay with sand intercalations6.5 − ∞ Clay with limestone fragments

The groundwater level varies throughout the year, with other measurements produced byCosta et al. (2012), indicating a GWL of −3 m in winter and −6 m in summer. The soilprofile is visualized below in figure 3.3.

38


Fig. 3.3 Soil profile from SPT tests in Portugal.

3.2.3 Idealization

In order to simplify the physical system enough to facilitate numerical analyses, someidealizations and assumptions were carried out. The first assumption implied that each soillayer was completely horizontal, meaning no variation of soil thickness in the longitudinaldirections. Furthermore, it was assumed that no heterogeneity was present in the longitudinaldirection of the soil, implying that the soil parameters were held constant for each layer. Thefinal assumption was that the GWL was at −6 m, due to the fact that the measurementspresented in section 3.2.7 were performed during the summer, i.e. the measurements againstwhich the model was calibrated. One last important aspect of the idealization was that itwas assumed that clear distinctions between the soil layers existed.

3.2.4 FE-model

For the finite element model, great consideration has to be taken regarding discretization ofthe mesh and modeling of the boundaries. As presented in chapter 2 regarding numericalconsiderations for elastodynamics problems, the governing differential equations describingwave propagation in elastic media imposes conditions on both the mesh and the boundaryconditions.Due to the layering of the soil, coupled with the risk of spurious wave reflection, the authorsconcluded the appropriate approach regarding mesh was to keep the mesh size constantthroughout the model, meaning no bias was employed. Regarding the mesh size, thecriterion was that the element size should not exceed λ

8 for linear elements and λ2.5 for

quadratic elements (Bažant (1978), Bažant and Celep (1982)), where λ is the wavelengthconsidered. In order to find the design wavelength, first the lowest wave velocity was

39


calculated. As previously shown in the theory section, the lowest wave length belongs tothe Rayleigh wave, and since Rayleigh waves are surface waves, the Rayleigh wave velocityfor the top layer was calculated from

CR = 0.87 + 1.12ν1 + ν

CS (3.1)

With ν = 0.4675, and CS = 101 m/s. The wave length was then calculated from

λ = CR

f(3.2)

Where f is the maximum frequency of interest, i.e. f = 80 Hz. The resulting wavelengthwas λ = 1.20 m, meaning the mesh size should not be larger than h = 0.48 m for quadraticelements and h = 0.15 m for linear elements, as risk of spurious wave reflection otherwisecould pose a problem.Furthermore, regarding mesh size, a criterion was imposed that the mesh size needed be amultiple of 0.5 m, due to the assumed layering of the finite element model, as visible fromthe input data in section 3.2.5.Considering the boundary conditions, the authors concluded the boundaries be modeled asLysmer-Kuhlemeyer elements (i.e. infinite elements), due to their computational efficiencywhen compared to absorbing boundaries. This however requires consideration taken as tothe Sommerfeld radiation condition, i.e. that the incident waves at the boundaries must benormal to the boundary surface to ensure complete absorption of the wave energy.Based on the above, an initial model was constructed, with the geometry being that of alayered, circular half space model with quadratic, axisymmetric elements, and a mesh sizeof 0.125 m, as to completely be assured that the criteria stated above were fulfilled. Theinitial model is presented in figure 3.4 below.

40


Fig. 3.4 Initial proposed model.The model had a radius of 50 m, and was analyzed in the frequency range 1 ≤ f ≤ 80 Hzusing 800 frequency steps initially.

3.2.5 Input

In this section, the input parameters that were used in the FE analysis are presented. Thematerial parameters are based partly on calculations, partly on measurement data fromPortugal, dos Santos et al. (2016). Below, in tables 3.2, 3.3, and 3.4, the measurement dataregarding the shear wave velocities, pressure wave velocities and the damping ratios arepresented.

41


Table 3.2 Variation of CS with depth.

Depth [m] CS [m/s]0 − 1.4 1011.4 − 5.6 1145.6 − 8.10 1278.10 − 11.10 20211.10 − ∞ 318

Table 3.3 Variation of CP with depth.

Depth [m] CP [m/s]0 − 1.9 4091.9 − 5.5 10725.5 − ∞ 1503

Table 3.4 Variation of damping ratio with depth.

Depth [m] ξ [−]0 − 1.10 0.1011.10 − 2.40 0.0522.40 − 4.00 0.0344.00 − ∞ 0.029

From the measurements regarding the shear wave velocities and pressure wave velocities,the value of Poisson’s ratio for each layer was calculated according to equation 3.3 below.

ν = 1 − κ2/21 − κ2 (3.3)

With κ = CP

CS. As no information regarding density was presented, the authors searched the

literature for densities corresponding to the presented soil layers in table 3.1, and the shearmoduli for the layers were calculated according to equation 3.4 below:

G = ρC2S (3.4)

The elastic moduli were then calculated according to equation 3.5:

E = 2(1 + ν)G (3.5)

42


Regarding damping, the stiffness proportional and mass proportional parts of the dampingwere calculated according to equations 3.6 and 3.7 below, as Rayleigh damping was used inthe analyses:

α = 2ωiωjξ(ωj − ωi)ω2

j − ω2i

(3.6)

β = 2ξ(ωj − ωi)ω2

j − ω2i

(3.7)

In the two equations above, the frequencies ωi and ωj were set to 2π rad/s and 160π rad/srespectively, as the analyses were carried out in the frequency range 1 ≤ f ≤ 80Hz.From the descriptions presented in table 3.1, the soil layers were compared to those foundin literature (Geotechdata (2018), Bodare (1997), SGF (2013)),The final initial input parameters, i.e. the parameters used in the convergence analyses, arepresented in table 3.5 below.

Table 3.5 Input parameters used in the FE analysis.

Layer Depth [m] ρ [kg/m3] E [MPa] ν [−] α [1/s] β [s]1 0 − 1.0 1850 56 0.488 1.254 4 × 10−4

2 1.0 − 1.5 1850 56 0.488 0.645 2 × 10−4

3 1.5 − 2.5 1850 72 0.493 0.645 2 × 10−4

4 2.5 − 3.5 1700 66 0.491 0.422 1.3 × 10−4

5 3.5 − 4.0 2000 78 0.495 0.422 1.3 × 10−4

6 4.0 − 4.5 2000 78 0.495 0.360 1.1 × 10−4

7 4.5 − 5.5 2000 78 0.494 0.360 1.1 × 10−4

8 5.5 − 6.5 2000 96 0.489 0.360 1.1 × 10−4

9 6.5 − 7.5 2100 101 0.487 0.360 1.1 × 10−4

10 7.5 − 8.0 2100 101 0.488 0.360 1.1 × 10−4

11 8.0 − 8.5 2100 255 0.488 0.360 1.1 × 10−4

12 8.5 − 11.0 2100 255 0.489 0.360 1.1 × 10−4

13 11.0 − 12.0 2100 631 0.489 0.360 1.1 × 10−4

14 12.0 − 20.0 2100 633 0.490 0.360 1.1 × 10−4

3.2.6 Convergence analysis

In order to validate the proposed model, and to attempt to reduce the computational timeof the model, a mesh convergence analysis was initiated. The convergence was analyzed for

43


equidistant points of 5 m, starting at 5 m, and ending at 45 m from the load. Below, theperformed analyses are presented:

1. Mesh convergence. Mesh convergence was carried out for the model using bothquadratic and linear elements. For the quadratic elements, the mesh convergence consistedof trying mesh sizes of 0.5, 0.25, and 0.125 m. For the linear elements, mesh sizes of 0.125and 0.0625 m were used.

2. Step reduction. In order to reduce computational time, the frequency step in thesteady state analysis was decreased, until discrepancies of results was observed. Thefrequency step reduction consisted of halving the amount of frequency steps, from theoriginal 800 steps, down to 50 steps.

3. Rectangular approximation. A new, rectangular model was proposed in order toreduce computational time, based on the fact that it can be found in literature (KÄLLATEXAS) that the surface wave propagation mainly depends on the material parametersof the layers closer to the surface, meaning the lower areas of the circular model could beargued unnecessary.

4. Validation of boundaries. In order to validate the boundaries, the depth and theheight of the rectangular model was varied, in order to check for spurious wave reflection atthe boundaries.

5. Time domain comparison. The final step of the convergence analysis consisted ofperforming a time domain analysis of an impulse load, computing the frequency responsefunction of the response, and comparing with the steady state analyses performed earlier.

The final convergence check, for the converged model with reduced computationaltime, compared to the initially proposed model, is presented below in figure 3.5. Thecomputational times for the models were 13 hours for the initial model, and 90 seconds forthe final model. For the full convergence analysis, the reader is referred to Appendix A.

44


Fig. 3.5 Final convergence analysis. The red line is the initial model, the blue is the final.The output model after the convergence analysis can be seen in figures 3.6 and 3.7.

Fig. 3.6 Cross-section of axisymmetric model with infinite elements and mesh size h = 0.5 m

45


Fig. 3.7 Description of the dimensions and layering of the rectangular axisymmetric model.

3.2.7 Measurements

Following the measurements conducted in Portugal, signal processing was performed inorder to obtain good data to be compared with calculations done in the FEA. The mainreason for the measurement processing was to ensure that the effect of ambient noise inthe measurement data remained a minimum. In this section, the procedure that was usedin order to obtain the FRF’s used in the validation of the model is described, as well aspresentation of the frequency response functions that were used in further analysis. In orderto obtain the data to be used, a procedure as outlined in figure 3.8 was employed.

46


Fig. 3.8 Flowchart of the employed procedure used to obtain valid data from the measurements.

As visible from figure 3.8 above, the procedure was initiated by filtering the exciting forceand the response. After the filtering was performed, the frequency response functions andthe coherence functions were calculated for each individual measurement point. After theFRF’s and the coherence functions were obtained, an analysis was performed in order toselect the optimal measurement points based on the coherence of the responses. The finalstep in the measurement processing was to extract the relevant data from the selectedmeasurement points.

Filtering

As an initial step of the measurement processing, filtering of the measured signals wasperformed, the reason being twofold, as described below.

47


1. The first reason was to eliminate aliasing of the signal, before the Fourier transformwas employed for further analysis in the frequency domain.

2. The second reason was to remove signals of frequencies above 80 Hz, as the relevantspectrum of frequencies was between 1 and 80 Hz for the study.

In order to achieve the desired effect, a 2nd order Butterworth filter with a cutofffrequency of 80 Hz was applied on the signal, prior to any further analysis.

Frequency response functions

The filtered signal was subjected to a DFT in order to obtain the linear spectra of theexciting force and the measured response, as depicted in equations 3.8 and 3.9. Index irefers to which measurement the result refers to, as 40 tests for each setup was conducted.Index j refers to which measurement point (accelerometer) the result refers to, and index kindicates which setup the result refers to.

Sxx,i,k(f) = DFT(pi,k(t)), i = 1, 2, ..., 40, k = 1, 2, ..., 5 (3.8)

Syy,i,j,k(f) = DFT(ai,j,k(t)), j = 1, 2, ..., 20 (3.9)

From these linear spectra, the auto power spectra of the exciting force along with the crosspower spectrum for each individual measurement was computed. The equation for the autopower spectrum of the exciting force is presented in equation 3.10, and the equation for thecross power spectra is found in equation 3.11.

Gxx,i,k = Sxx,i,k(f)S∗xx,i,k(f) (3.10)

Gyx,i,j,k(f) = Syy,i,j,k(f)S∗xx,i,k(f) (3.11)

S∗xx,i,k(f) denotes the complex conjugate of Sxx,i,k(f).

As 40 measurements existed for each spectra, the auto power spectra and the cross powerspectra were averaged in order to obtain the average value of the 40 measurements conducted,as presented mathematically in equations 3.12 and 3.13.

Gxx,k(f) = 140

40∑i=1

Gxx,i,k(f) (3.12)

48


Gyx,j,k(f) = 140

40∑i=1

Gyx,i,j,k(f) (3.13)

Succeeding this, the frequency response functions for each measurement point was calculatedby dividing the averaged values of the cross power spectra with the averaged values of theauto power spectra for the exciting force, as shown in equation 3.14.

Hj,k(f) = Gyx,j,k(f)Gxx,k(f)

(3.14)

Coherence functions

Regarding the calculation procedure for the coherence functions, the averaged values ofthe cross power spectrum and autopower spectrum of the exciting force was calculatedidentically to the description in section 3.2.7. The addition was that the auto power spectraof the responses were calculated, as presented in equation 3.15,

Gyy,i,j,k = Syy,i,j,k(f)S∗yy,i,j,k(f) (3.15)

and averaged according to equation 3.16.

Gyy,j,k(f) = 140

40∑i=1

Gyy,i,j,k(f) (3.16)

From these power spectra, the coherence functions were computed according to equation3.17.

γ2j,k(f) =

Gyx,j,k(f)G∗yx,j,k(f)

Gxx,k(f)Gyy,j,k(f)(3.17)

Selection of measurements

The selection of which measurement points j from setups k to be used was based on thepreviously calculated coherence functions. The selection of the measurements was doneaccording to the description below.

1. Coherence functions were calculated in the frequency range 1-80 Hz for each measure-ment point j from each setup k.

49


2. The amount of discrete data points from each coherence function with coherence ofγ2(f) > 0.95 was calculated. This amount was then divided with the total number ofdiscrete data points for each measurement point j from each setup k.

3. This final value revealed the fraction of the response from each measurement pointwhich could be considered of good coherence, i.e. which frequencies from each measurementpoint that contained no ambient noise. A value of 1 indicated that 100 % of the signal wasof good quality, whereas a value of 0 indicated that 0 % of the signal was of good quality.

4. The results were plotted visually on an absolute scale, as seen in figure 3.9. From thesevalues, one measurement was selected for each 5 m interval, with distances from the impactpoint up to and including 45 m. All in all, this resulted in 9 measurement points, presentedin table 3.6.

50


Fig. 3.9 Diagram of the results from the coherence fraction process The different colorsdenote different setups.

From figure 3.9, a total of 9 measurement points were chosen from the measurement data,one point for each 5 meter interval up until and including 41-45 m. This means that onesetup was chosen for each measurement point for the first 9 measurement points. Based onthe diagram presented in figure 3.9, the selected points are presented in table 3.6 below.

51


Table 3.6 Selected measurement points for further analysis.

Name Measurement Point, j Setup, k Distance [m]P1 1 1 1P2 2 3 8P3 3 3 13P4 4 4 19P5 5 4 24P6 6 3 27P7 7 1 31P8 8 1 36P9 9 1 41

Selection of data points

After the measurement selection process was finished, the work of obtaining good datafrom the selected points was initiated. First, the frequency response functions along withthe corresponding coherence functions of the selected points were plotted. The obtainedfrequency response functions and coherence functions for the selected points are presentedin figure 3.10 and figure 3.11 respectively.

52


Fig. 3.10 Frequency response functions for the selected points.

53


Fig. 3.11 Coherence functions for the selected points.

An analysis was performed on the coherence function plots, in order to investigate whichfrequencies from which points contained response of low coherence, i.e. where results notcaused by the exciting force occurred. The results from these investigations are presentedin figure 3.12 and 3.13 below, where the shaded areas indicate frequency ranges of lowcoherence, i.e. γ2(f) < 0.95.

54


Fig. 3.12 Coherence functions for the selected points.

55


Fig. 3.13 Frequency response functions for the selected points.

As the measurement data was finalized as shown in figure 3.13, the measurement processingwas considered done. For the further analysis, the shaded areas from the frequency responsefunctions were discarded from the comparisons between the calculated and the measuredresults. This choice was motivated by the fact that only signals in absence of noise canaccurately be subjected to comparison between calculations and measurements.

56


3.2.8 Calibration

Overview

As the original model presented discrepancies between the measured results and thecalculated results, seen in figure 3.14 the authors deemed it necessary to calibrate thematerial parameters of the numerical model in an attempt to match the response functionsbetween the measurements and the calculations. The optimization scheme consisted of firstsimplifying the problem, followed by determination of which parameters to be calibrated,and determination of objective function to be minimized.

Fig. 3.14 Results from the converged model compared with measurement results. Thecalculated results are in blue and the measured results are in red.

Simplifications

In order to increase the chances of acquiring better soil parameters, two major simplificationswere incorporated in the optimization, namely:

1. Reduction of number of measurement points. This reduction consisted of selecting the4 best cohering measurements, with the exception of the first point at 1 m, as the first

57


point requires careful consideration being taken to the near-field effects. This resulted inthe points at 8 m, 19 m, 27 m, and 31 m being considered in the optimization.

2. Reduction of layers. In order to reduce the amount of parameters being considered,the amount of layers were reduced using weighted averages, from 14 layers down to 5. Itis shown in Appendix B that this reduction did not introduce discrepancies between the14 layered model and the 5 layered model. Following geometry and cross-section of soilis obtained with corresponding material parameters as described in table 3.7. The crosssection of the soil with dimensions is presented in figure 3.15.

Fig. 3.15 Cross-section of soil after reduction of parameters and geometry simplifications.Table 3.7 Material properties for 5-layer model

Soil-layer Depth [m] Density [kg/m3] Cs [m] E [MPa] ν ξ α β

1 1.5 1850 101 55.39 0.4675 0.0847 1.0512 0.000332 4 1887.5 114 73.31 0.4943 0.0366 0.4546 0.000143 2.5 2060 127 99.44 0.4964 0.029 0.3599 0.000114 3 2100 202 255.49 0.4908 0.029 0.3599 0.000115 9 2100 318 627.13 0.4766 0.029 0.3599 0.00011

The main reason for these simplifications is the time gain from reducing the amount ofvariables considered, as the time taken for performing one optimization trial is in themagnitude of days, even for the simplified models.

Objective Function

The optimization scheme consisted of using the built-in Genetic Algorithm (GA) provided inMATLAB. In order to progress with the algorithm, the program needs an objective function

58


to be minimized. For every iteration in GA, the Frequency Response Assurance Criterion(FRAC) and the Frequency Amplitude Assurance Criterion (FAAC) was calculated, whichare both functions defined between 0 and 1, where 0 means that no correlation betweentwo response functions can be observed, and 1 means full correlation between responsefunctions. The FRAC compares the shape of the functions subjected to comparison, andFAAC compares the amplitudes of the functions. The functions are defined in equations3.18 and 3.19 below:

FRACi = |H(ω)ex,iH(ω)num,i|2

(H(ω)ex,iH(ω)ex,i)(H(ω)num,iH(ω)num,i)(3.18)

FAACi = 2|H(ω)ex,iH(ω)num,i|(H(ω)ex,iH(ω)ex,i) + (H(ω)num,iH(ω)num,i)

(3.19)

Where H(ω)ex is the frequency response of the measurements, H(ω)num is the frequencyresponse of the calculations and index i denotes which point is being considered. After thecalculation of the response criteria, the objective function to be minimized was calculatedaccording to equation 3.20 below.

Obj =n∑

i=1[(FRACi − 1)2 + (FAACi − 1)2] (3.20)

This means that the assurance criteria were calculated for every measurement point,thereafter the objective function was calculated. The algorithm was set to run untilconvergence was achieved (i.e. when further runs didn’t yield better results).

Parameters of interest

In total, 4 optimization trials were executed, where each trial focused on varying differentparameters, with the goal to achieve the best possible correspondence between the calculatedand the measured results. These trials are outlined in the paragraphs below.

1. First trialThe basis of the first optimization trial was that the shear wave velocity and pressurewave velocity is not 100 % certain, as it is shown in literature (Marosi and Hiltunen(2004)) that SASW tests does not produce unique solutions for a given set of measurementdata. Based on this, CP and CS were varied according to table 3.8 below, where the soilprofile from section 3.2.3 was assumed and the wave velocities were varied according to thecorresponding soil in the layers (SGF (2013), Bodare (1997)). In the result table of table

59


3.8, the original values, minimum/maximum values used in the calibration and the valuesfrom the converged solution of the Genetic algorithm are presented.

Table 3.8 Variation of wave velocity used in the first optimization trial.

LayerCurrent Minimum Maximum FinalCS CP CS CP CS CP CS CP

1 101 409 100 185 250 450 174 2802 114 1072 100 150 500 1000 330 6583 127 1503 80 1450 180 1550 83 14614 260 1503 180 1500 300 1700 299 1522

The resulting response function from the first calibration trial is presented below in figure3.16.

60


Fig. 3.16 Resulting response functions (blue) compared to measured results (red) from thefirst optimization trial.As is clearly visible above in figure 3.16, the first optimization trial was unsuccessful, why anew trial was set up using other parameters.

2. Second trialFor the second trial, the stiffness E, and damping ratio ξ, was adjusted, based on theresults from the sensitivity analysis. For each layer, a factor 2 was used as the base for theminimum and maximum values, i.e. the minimum value was 50 % of the original value,and the maximum value was 200 % of the original value. This was motivated by the factthat soil carries an intrinsic difficulty regarding the accuracy of parameter determination(Phoon and Kulhawy (1999)). Furthermore, the sensitivity analysis (presented in AppendixB) showed that the results are independent of the damping in the bottom layer, meaningonly 9 parameters were varied in the optimization: The E-moduli for the 5 layers and thedamping ratios for the top 4 layers. The parameters are presented below in table 3.9, andthe results are seen in figure 3.17. An important note for this trial is that rate-independencywas assumed, meaning a constant value of the damping was used in the calculations, i.e.structural damping was used in ABAQUS.

Table 3.9 Variation of stiffness and damping for the second optimization trial.

LayerCurrent Minimum Maximum Final

E ξ E ξ E ξ E ξ

1 55.39 0.0847 27.70 0.0424 110.78 0.1694 36.71 0.06162 73.31 0.0366 36.66 0.0183 146.62 0.0732 131.94 0.03953 99.44 0.029 49.72 0.0145 198.88 0.058 165.17 0.01944 255.49 0.029 127.75 0.0145 510.98 0.058 397.62 0.02315 627.13 0.029 313.57 0.029 1254.26 0.029 920.69 0.029

61


Fig. 3.17 Results from the second optimization trial using structural damping.

3. Third trialAs the second trial proved unsuccessful, a new trial was set up, where rate dependency ofthe damping was assumed, through the use of mass-, and stiffness-proportional damping. Acombination of these two were allowed, meaning a layer could exhibit both mass proportionaland stiffness proportional damping, i.e. Rayleigh damping. The same variation of parameterswas assumed as in the second optimization trial, meaning the stiffness was varied by afactor 2, and the damping ratio ξ was varied by a factor two, only this time the values ofα and β were calculated from the damping ratio instead of applying the damping ratiodirectly into ABAQUS. The parameters from the third optimization trial are presented intable 3.10, and the results are visualized in figure 3.17.

Table 3.10 Variation of stiffness and damping for the second optimization trial.

62


LayerCurrent Minimum Maximum Final

E α β E α β E α β E α β

1 55.4 1.05 3.3 × 10−4 27.7 2.10 6.6 × 10−4 110.8 0 0 31.2 0.92 3.0 × 10−4

2 73.3 0.45 1.4 × 10−4 36.7 0.91 2.8 × 10−4 146.6 0 0 129.0 0.05 7.0 × 10−6

3 99.4 0.36 1.1 × 10−4 49.7 0.72 2.2 × 10−4 198.9 0 0 109.9 0.03 2.8 × 10−6

4 255.5 0.36 1.1 × 10−4 127.8 0.72 2.2 × 10−4 511.0 0 0 260.6 0.04 4.1 × 10−6

5 627.1 0.36 1.1 × 10−4 313.6 0.72 2.2 × 10−4 1254.3 0 0 919.0 0.02 2.4 × 10−5

Fig. 3.18 Results from the optimization trial using Rayleigh damping.

4. Fourth trialThe fourth trial consisted of dividing up the calibration in frequency intervals, where eachinterval covered frequencies of 4 Hz. This means that for the frequencies between 10−40 Hz,

63


a total of 8 frequency intervals were used. The parameter approach was to use structuraldamping, and varying the stiffness and damping according to the variation described forthe second optimization trial. The resulting response functions from the calibration arepresented in figure 3.18 below.

Fig. 3.19 Results from the optimization trial using frequency intervals.As is evident from the 4 trials, the calibration of the soil parameters proved unsuccessful,why the authors concluded that the initial soil model was to be used in the further analysispresented in the following sections.

3.2.9 Output

The output model from the axisymmetric modeling is the rectangular model with numericalproperties described in subsection 3.2.6, with material parameters according to the 5-layeredmodel presented in subsection 3.2.8, as presented in figure 3.20 and table 3.11 respectively.

64

3.3 3D Soil Model

Fig. 3.20 Cross section of the final axisymmetric model.

Table 3.11 Material properties for the final axisymmetric model.

Soil-layer Depth [m] Density [kg/m3] Cs [m] E [MPa] ν α β

1 1.5 1850 101 55.39 0.4675 1.0512 0.000332 4 1887.5 114 73.31 0.4943 0.4546 0.000143 2.5 2060 127 99.44 0.4964 0.3599 0.000114 3 2100 202 255.49 0.4908 0.3599 0.000115 9 2100 318 627.13 0.4766 0.3599 0.00011

3.3 3D Soil Model

3.3.1 Overview

After that the axisymmetric modeling was finished, a 3D-model was developed in orderto create a 3D representation using 3D-solid elements, comparable to the results from theaxisymmetric model. The goal of the development of the 3D model was to obtain a modelwith sufficient dimensions that minimizes the magnitude of spurious wave reflection. Themethod used was to develop a model with equal depth to the axisymmetric model, wherethe width of the model was varied in order to compute the response. It was here assumedthat the convergence analysis of the axisymmetric model served as a viable convergenceanalysis for the 3D solid model.

65

3.3 3D Soil Model

3.3.2 Modeling approach

A main goal of the 3D-model was to obtain a model with sufficient dimensions to eliminatethe impact of spurious wave reflections. Therefore, the proposed model was a model with asquare surface of dimensions 70x70 m, with depth 20 m. The proposed model is visualizedin figures 3.21-3.23.

Fig. 3.21 Cross section of 3D solid model.It can be noted that layer properties fromthe axisymmetric model are assigned to thismodel as well.

Fig. 3.22 Top view of the soil

Fig. 3.23 3D model of the soil

Choice of geometry

Material parameters and layer-section for three-dimensional model has been assignedaccording to properties corresponding to the axisymmetric model, described in section3.2.9. The idea was to obtain a solid model that would generate the highest correlation

66

3.3 3D Soil Model

between axisymmetric and 3D regarding its response.

In order to fulfill the radiation condition at the new boundary close to the impact point,a quarter circle was modeled with a radius of 20 m, corresponding to the depth of themodel. The motivation for this can be visualized graphically in figures 3.24 and 3.25. Ascan be observed, the implemented model should theoretically exhibit better wave absorbingproperties at the boundaries than the alternative model. The spurious waves seen in cyancolor in the figures have already been shown in the previous convergence analyses to be anon-issue, as they dissipate entirely within the model.

Fig. 3.24 Implemented model with radius 20 m and radius 9 m at short ends.

Fig. 3.25 Alternative model with radius 9 m at both short ends

67

3.3 3D Soil Model

An alternative procedure regarding the new boundary would be to impose a symmetryboundary condition, however this would in the later analyses including the structure proveto be erratic, as the structure only exists on one side of the boundary.

Convergence

Initially, the 3D-model was run using the supercomputer situated at KTH, however themodel proved too computationally heavy with the mesh size h = 0.5 m, why the mesh sizewas increased to h = 1.0 m. The two models with different mesh sizes can be observed infigure 3.3.2. The resulting comparison between the 3D-results and the axisymmetric resultscan be observed in figure 3.27.

=⇒

Fig. 3.26 Model after mesh enlargement from h = 0.5 m to h = 1.0 m

68

3.3 3D Soil Model

0 10 20 30 40

Frequency [Hz]

0

0.1

0.2

0.3

Accele

rance [m

/s2/k

N]

P1, 5m

0 10 20 30 40

Frequency [Hz]

0

0.05

0.1

0.15

0.2

Accele

rance [m

/s2/k

N]

P2, 10m

0 10 20 30 40

Frequency [Hz]

0

0.05

0.1

Accele

rance [m

/s2/k

N]

P3, 15m

0 10 20 30 40

Frequency [Hz]

0

0.02

0.04

0.06

Accele

rance [m

/s2/k

N]

P4, 20m

0 10 20 30 40

Frequency [Hz]

0

0.02

0.04

0.06A

ccele

rance [m

/s2/k

N]

P5, 25m

0 10 20 30 40

Frequency [Hz]

0

0.01

0.02

0.03

Accele

rance [m

/s2/k

N]

P6, 30m

0 10 20 30 40

Frequency [Hz]

0

0.01

0.02

0.03

Accele

rance [m

/s2/k

N]

P7, 35m

0 10 20 30 40

Frequency [Hz]

0

0.01

0.02

0.03

Accele

rance [m

/s2/k

N]

P8, 40m

Comparison between 3D Model (70m x 70m) and Axisymmetric d=0.5

0 10 20 30 40

Frequency [Hz]

0

0.005

0.01

0.015

0.02

Accele

rance [m

/s2/k

N]

P9, 45m

3D Model

Axisymmetric

Fig. 3.27 Comparison between 3D model with mesh size h = 1.0 m and axisymmetric modelwith mesh size h = 0.5 m

As can be predicted - and also seen - in the results, the models lack agreement at frequenciesabove 30 Hz, due to the conditions discovered by Bašant and presented in the theory section.Therefore, the frequencies of interest were limited with an upper bound of 30 Hz, and anew axisymmetric model with mesh size h = 1.0 m was calculated and compared with the3D-results. The results are presented in figure 3.28.

69

3.4 3D Structure Model

0 10 20 30 40

Frequency [Hz]

0

0.1

0.2

0.3

Accele

rance [m

/s2/k

N]

P1, 5m

0 10 20 30 40

Frequency [Hz]

0

0.05

0.1

0.15

0.2

Accele

rance [m

/s2/k

N]

P2, 10m

0 10 20 30 40

Frequency [Hz]

0

0.05

0.1

Accele

rance [m

/s2/k

N]

P3, 15m

0 10 20 30 40

Frequency [Hz]

0

0.02

0.04

0.06

Accele

rance [m

/s2/k

N]

P4, 20m

0 10 20 30 40

Frequency [Hz]

0

0.02

0.04

0.06A

ccele

rance [m

/s2/k

N]

P5, 25m

0 10 20 30 40

Frequency [Hz]

0

0.01

0.02

0.03

Accele

rance [m

/s2/k

N]

P6, 30m

0 10 20 30 40

Frequency [Hz]

0

0.01

0.02

0.03

Accele

rance [m

/s2/k

N]

P7, 35m

0 10 20 30 40

Frequency [Hz]

0

0.01

0.02

0.03

Accele

rance [m

/s2/k

N]

P8, 40m

Comparison between 3D Model (70m x 70m) and Axisymmetric d=1.0

0 10 20 30 40

Frequency [Hz]

0

0.005

0.01

0.015

0.02

Accele

rance [m

/s2/k

N]

P9, 45m

3D Model

Axisymmetric

Fig. 3.28 Comparison between 3D model with mesh size h = 1.0 m and axisymmetric modelwith mesh size h = 1.0 m

It is there seen that correspondence between the axisymmetric and the 3D-model is evident,however due to the lack of convergence for frequencies exceeding 30 Hz, the frequency rangeof interest was decreased to frequencies between 1 − 30 Hz for the further analyses.


3.4.1 Overview

In order to obtain structures for testing in the soil-structure interaction analyses, threesimple structures were modeled. All structures had a mesh size of h = 0.5 m and consistedsolely of quadratic elements.

70


3.4.2 Geometry of the structures

The developed structures consisted of 3-, 5-, and 8 story buildings, which can be seen infigure 3.29.

CLAYSTEELCONCRETE

8-FLOORSTRUCTURE

5-FLOORSTRUCTURE

3-FLOORSTRUCTURE

Fig. 3.29 Original geometry setup

The structure models were implemented in two various geometry setups. One which issituated on ground and an another lowered in soil. What also can be noted, is that thereare different thicknesses implemented on ground bearing slabs. The height on each floor areset to 2.75m. Besides this, there are no deviations regarding the geometrical consistency ofthe structures. The two setups are schematically presented in figures 3.29-3.30.

71


CLAYSTEELCONCRETE

8-FLOORSTRUCTURE

5-FLOORSTRUCTURE

3-FLOORSTRUCTURE

Fig. 3.30 Modified geometry setup

The buildings were designed with a slab foundation, with a slab of dimensions 9 × 9 × 0.5 m,and floor plates of size 8×8×0.5 m. The supporting pillars were of height 3 m and consistedof box sections with dimensions 0.25 × 0.25 m and thickness 0.04 m. For simplicity, onlythe structural components were included, meaning non-bearing components were omittedin the analyses. The slabs profiles are illustrated in figure 3.4.2 and the steel pillar crosssection is seen in figure 3.32

72


GROUND BEARINGSLAB FLOOR SLAB

Fig. 3.31 Dimensions of ground-bearing slab (left) and floor slabs (right).

Fig. 3.32 Cross section of steel-column

3.4.3 Material properties

The building slabs are made of concrete with the following material parameters:

• Young’s modulus: Ec = 40 GPa

• Poisson ratio: νc = 0.2

• Damping ratio: ξc = 0.005

• Density: ρc = 2650 kg/m3

Vertical pillars are made of steel with the following properties:

• Young’s modulus: Es = 210 GPa

• Poisson ratio: νs = 0.3

• Damping ratio: ξs = 0.0015

• Density: ρs = 7800 kg/m3

73

3.5 3D Soil-Structure Model

3.4.4 Modeling of structure

Element and mesh

The whole structure had a mesh size ofh = 0.5 and consisted solely of quadraticelements. For columns, Timoshenko beamelements were implemented, due to their ap-propriate application on slender beams andallowance of transverse shear deformations,Hibbitt et al. (2006). Slab elements werecomposed by Mindlin thick shell elements.Mindlin solution was entirely consistent withthe 3D solution since it explicitly accountsfor the shear stresses τzx.

Fig. 3.33 Meshed 5-storey building with struc-tural parts


3.5.1 Overview

In this section, the work done regarding the soil-structure interaction is presented. The scopeof the analysis consisted of modeling different structures, while varying material parametersand modeling submerged/non-submerged structures. The force used to depict the trainload is described in section 3.5.3, and the parameter variations are described in section3.5.4. The extracted results from the analyses consisted of frequency response functionsfor non-normalized acceleration in the foundation slab of the structures, compared withfree-field vibrations and theoretical soil-structure interaction curves (Administration (2006)),which are presented in section 3.5.5. Furthermore, the vibrations in points surrounding thestructure were extracted, in order to obtain data regarding how the structure affects thesurrounding ground vibrations.

3.5.2 Geometry

The modeled site is presented below in figure 3.34. The building was placed with the closestedge of the foundation slab located 15 m from the loaded area.

74


Fig. 3.34 Layout of the numerical site as depicted from above.The measurement points in figure 3.34 are as follows:• P1 and P2 were located 4 and 2 m in front of the building, respectively.• P3 and P4 were located 2 and 4 m behind the building, respectively.• P5 and P6 were located 2 and 4 m beside the center of the building, respectively.• P7, P8, and P9 were located on the free field for the simulation done without the building,and on equivalent points at the top of the foundation slab for the cases with buildings. P8was located at the center of the foundation slab, whereas points P7 and P9 were located 3m away from the center in the horizontal direction in the figure.

75


Fig. 3.35 3D-Layout of the numerical site as depicted from above.

3.5.3 Modeling of the force

In the soil-structure model, the force has been modeled as a uniformly distributed loadwith the magnitude 100 kPa, aiming to reproduce the induced pressure from a train. Dueto the linear nature of the problem, and the fact that very accurate representations of traininduced vibrations is an unsolved problems, the authors have concluded that a uniformlydistributed load proved sufficient for the performed analyses. The force was applied in anarea covering 0.5 × 70 m.

3.5.4 Variation of parameters

The analyses consisted of performing a steady-state analysis in ABAQUS using the previouslydescribed 3D-model from section 3.3, adding the buildings from section 3.4 and a frequencyrange of 1 − 30 Hz with a frequency step of 0.8 Hz. Below are the performed simulationspresented:

Original buildings

The original trial consisted of applying the three structures (3, 5, and 8 stories) directly onthe soil, with a tie constraint between the soil and the bottom slab.

Submerged buildings

The second trial consisted of submerging the structure in the soil, so that the bottomface of the foundation slab was situated 2 m below the ground surface. This was done in

76


order to investigate the effect of submerging a structure in order to decrease the vibrationstransmitted into the foundation.

Variation of mass

The following trials consisted of applying the structure on top of the soil, and varying themass of the structures. First, the mass was doubled for each structure, followed by trialswhere the mass was halved for all the structural components.

Variation of foundation slab thickness

The final simulations consisted of increasing the thickness of the foundation slab, to 1 mas opposed to the original buildings’ slab thickness of 0.5 m. This was done for both thestructures situated at ground level, and the structures situated 2 m below the groundsurface.

3.5.5 Curves from literature

Curves from literature were obtained from present theory regarding soil structure interaction(Administration (2006)), where the relationship between free field vibrations and foundationvibrations are presented as depicted in figure 3.36 below.

77


Fig. 3.36 Soil-structure interaction curves from literature for different types of buildings,presented with decibels plotted against 1/3 octave band center frequencies.For the conducted study, the curve corresponding to the Single Family Residential buildingshave been used in the comparisons, due to the lightweight nature of the modeled structures.Curves from literature used in the study have been calculated as a piecewise function, withthe division occurring at 8 Hz and 16 Hz. For values below 8 Hz, the transfer functionfrom literature has been assumed to be of unity. Between 8 and 16 Hz, it has been linearlyinterpolated between unity and −5 dB, i.e. 10−0.25. For frequencies above 16 Hz, it hasbeen assumed to be constant with a value of 10−0.25.

78

Chapter 4

Results

In this section results regarding soil-structure interaction and free field relationship ispresented, as well as acceleration plots between original structures and modified structureswith varying parameters. Major findings will be appointed and described more in-depth, andfurther relevant discoveries will briefly be introduced. All results are illustrated as FRF:swith acceleration in vertical direction on y-axis and frequency range on x-axis. Moreover,the frequency ranges are limited to 30 Hz as acknowledged in section 3.3. The purposewith a free field relationship for this study, reflects a 3D model of the soil without anyinteracting structure, which serve as references curves. Remaining plots describes eachdetailed differences when models are modified.

4.1 Comparison between free field and SSIHere, the relationship and differences between free field and 3-, 5- and 8 story structureson soil are presented. As declared in section 3.5, acceleration in vertical direction inmeasurements points P1-P6 are evaluated relative to frequency. First, FRF:s for originalmodels are demonstrated, followed by doubled and halved mass in structure and finallyoriginal structure lowered in soil are presented.

79

4.1 Comparison between free field and SSI

0 5 10 15 20 25 30

Frequency [HZ]

0

2

4

6

8

Accele

ration [m

/s2]

2 m before slab

0 5 10 15 20 25 30

Frequency [HZ]

0

2

4

6

8

Accele

ration [m

/s2]

4 m before slab

Free Field

3 Story

5 Story

8 Story

0 5 10 15 20 25 30

Frequency [HZ]

0

1

2

3

4

5

6

Accele

ration [m

/s2]

2 m after slab

0 5 10 15 20 25 30

Frequency [HZ]

0

1

2

3

4

5

6

Accele

ration [m

/s2]

4 m after slab

0 5 10 15 20 25 30

Frequency [HZ]

0

2

4

6

8

10

Accele

ration [m

/s2]

2 m beside slab

Original Buildings

0 5 10 15 20 25 30

Frequency [HZ]

0

2

4

6

8

Accele

ration [m

/s2]

4 m beside slab

Fig. 4.1 Original buildings in comparison with free fieldIn figure 4.1, original soil-structure interaction are depicted. Generally, a similar level ofacceleration in the soil is measured with and without the structure. It is worth noting thatthe accelerations next to slab, measurement points P1-P6, are similar to each other whencomparing with free field and SSI. An another observation is that the acceleration afterground slab is vastly changed when the simulations are compared.

80


0 5 10 15 20 25 30

Frequency [HZ]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

2 m before slab

0 5 10 15 20 25 30

Frequency [HZ]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

4 m before slab

Free Field

3 Story

5 Story

8 Story

0 5 10 15 20 25 30

Frequency [HZ]

0

1

2

3

4

5

Acce

lera

tio

n [

m/s

2]

2 m after slab

0 5 10 15 20 25 30

Frequency [HZ]

0

1

2

3

4

5

6

Acce

lera

tio

n [

m/s

2]

4 m after slab

0 5 10 15 20 25 30

Frequency [HZ]

0

2

4

6

8

10

Acce

lera

tio

n [

m/s

2]

2 m beside slab

Double Mass Buildings

0 5 10 15 20 25 30

Frequency [HZ]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

4 m beside slab

Fig. 4.2 Double mass on structures in relation to free fieldWhen doubling the mass of the structures, no substantial differences emerged in comparisonto the original models. Possibly some adjustments regarding amplitude levels, but they areslightly low.

81


0 5 10 15 20 25 30

Frequency [HZ]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

2 m before slab

0 5 10 15 20 25 30

Frequency [HZ]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

4 m before slab

Free Field

3 Story

5 Story

8 Story

0 5 10 15 20 25 30

Frequency [HZ]

0

1

2

3

4

5

6

Acce

lera

tio

n [

m/s

2]

2 m after slab

0 5 10 15 20 25 30

Frequency [HZ]

0

1

2

3

4

5

6

Acce

lera

tio

n [

m/s

2]

4 m after slab

0 5 10 15 20 25 30

Frequency [HZ]

0

2

4

6

8

10

Acce

lera

tio

n [

m/s

2]

2 m beside slab

Halved Mass Buildings

0 5 10 15 20 25 30

Frequency [HZ]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

4 m beside slab

Fig. 4.3 Halved mass on structures in comparison with free fieldHalved mass on structures yields almost the same response as when masses are doubled.As figure 4.2 and 4.3 illustrates a minor changes on the amplitude levels. However, theform and shape of the response are similar to the originals.

82


Lowered structures have a greater impact on the response around the surrounding soil thenthe other interventions presented above. Both amplitudes levels and shape of response areaffected by the lowered structure, which is presented in figure 4.4. One can remark thelower accelerations behind the structure, 2 and 4 m after slab, in contrast to originals. Theresponse 2 m before ground slab, in the frequency range 1 − 20 Hz are damped and insteadbetween 20 − 30 Hz a higher acceleration occur.

0 5 10 15 20 25 30

Frequency [HZ]

0

2

4

6

8

10

12

Acce

lera

tio

n [

m/s

2]

2 m before slab

0 5 10 15 20 25 30

Frequency [HZ]

0

2

4

6

8

10

Acce

lera

tio

n [

m/s

2]

4 m before slab

Free Field

3 Story

5 Story

8 Story

0 5 10 15 20 25 30

Frequency [HZ]

0

1

2

3

4

5

Acce

lera

tio

n [

m/s

2]

2 m after slab

0 5 10 15 20 25 30

Frequency [HZ]

0

1

2

3

4

Acce

lera

tio

n [

m/s

2]

4 m after slab

0 5 10 15 20 25 30

Frequency [HZ]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

2 m beside slab

Lowered Buildings

0 5 10 15 20 25 30

Frequency [HZ]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

4 m beside slab

Fig. 4.4 Lowered structures and free field

83

4.2 SSI - Ground Slab Acceleration

4.2 SSI - Ground Slab AccelerationBelow, acceleration curves are illustrated for each case, and as in previous section resultsare plotted as response in acceleration with relation to frequency. As described in section3.5.5, curves from literature are added in order to embody the differences between thenumerically calculated response and the response calculated according to current literature,(Administration (2006)), and are represented by green and cyan lines.

0 10 20 30

Frequency [Hz]

0

2

4

6

8

10

Accele

ration [m

/s2]

3 Stories, Point 7

0 10 20 30

Frequency [Hz]

0

2

4

6

8

10

Accele

ration [m

/s2]

5 Stories, Point 7

0 10 20 30

Frequency [Hz]

0

2

4

6

8

10

Accele

ration [m

/s2]

8 Stories, Point 7

Original

Double Mass

Free Field

Small House

2-4 Story House

0 10 20 30

Frequency [Hz]

0

2

4

6

8

Accele

ration [m

/s2]

3 Stories, Point 8

0 10 20 30

Frequency [Hz]

0

2

4

6

8

Accele

ration [m

/s2]

5 Stories, Point 8

0 10 20 30

Frequency [Hz]

0

2

4

6

8

Accele

ration [m

/s2]

8 Stories, Point 8

0 10 20 30

Frequency [Hz]

0

1

2

3

4

5

Accele

ration [m

/s2]

3 Stories, Point 9

0 10 20 30

Frequency [Hz]

0

1

2

3

4

5

Accele

ration [m

/s2]

5 Stories, Point 9

Acceleration comparison between original and double mass

0 10 20 30

Frequency [Hz]

0

1

2

3

4

5

Accele

ration [m

/s2]

8 Stories, Point 9

Fig. 4.5 Acceleration between original structure and double mass at point P7, P8 and P9

84


On previous page, differences in acceleration, between original structures and when structuresmass are doubled, are presented. It can be mentioned that there is significant level ofdamping in acceleration between free field and with a structure. However, if one study themidpoint (P8 ) on ground slab, there are no considerable changes found in accelerationbetween initial and modified model.

0 10 20 30

Frequency [Hz]

0

2

4

6

8

10

Acce

lera

tio

n [

m/s

2]

3 Stories, Point 7

0 10 20 30

Frequency [Hz]

0

2

4

6

8

10

Acce

lera

tio

n [

m/s

2]

5 Stories, Point 7

0 10 20 30

Frequency [Hz]

0

2

4

6

8

10

Acce

lera

tio

n [

m/s

2]

8 Stories, Point 7

Original

Half Mass

Free Field

Small House

2-4 Story House

0 10 20 30

Frequency [Hz]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

3 Stories, Point 8

0 10 20 30

Frequency [Hz]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

5 Stories, Point 8

0 10 20 30

Frequency [Hz]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

8 Stories, Point 8

0 10 20 30

Frequency [Hz]

0

1

2

3

4

5

Acce

lera

tio

n [

m/s

2]

3 Stories, Point 9

0 10 20 30

Frequency [Hz]

0

1

2

3

4

5

Acce

lera

tio

n [

m/s

2]

5 Stories, Point 9

Acceleration comparison between original and half mass

0 10 20 30

Frequency [Hz]

0

1

2

3

4

5

Acce

lera

tio

n [

m/s

2]

8 Stories, Point 9

Fig. 4.6 Acceleration between original structure and halved mass

85


In figure 4.6 it can be seen that when structures mass were halved, there were no remarkableeffects noted in comparison with original model, if the mid point (P8) is studied.

0 10 20 30

Frequency [Hz]

0

2

4

6

8

10

Acce

lera

tio

n [

m/s

2]

3 Stories, Point 7

0 10 20 30

Frequency [Hz]

0

2

4

6

8

10

Acce

lera

tio

n [

m/s

2]

5 Stories, Point 7

0 10 20 30

Frequency [Hz]

0

2

4

6

8

10

Acce

lera

tio

n [

m/s

2]

8 Stories, Point 7

Original

Lowered

Free Field

Small House

2-4 Story House

0 10 20 30

Frequency [Hz]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

3 Stories, Point 8

0 10 20 30

Frequency [Hz]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

5 Stories, Point 8

0 10 20 30

Frequency [Hz]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

8 Stories, Point 8

Original

Lowered

Free Field

Small House

2-4 Story House

0 10 20 30

Frequency [Hz]

0

1

2

3

4

5

Acce

lera

tio

n [

m/s

2]

3 Stories, Point 9

0 10 20 30

Frequency [Hz]

0

1

2

3

4

5

Acce

lera

tio

n [

m/s

2]

5 Stories, Point 9

Acceleration comparison betweeen original and lowered building

0 10 20 30

Frequency [Hz]

0

1

2

3

4

5

Acce

lera

tio

n [

m/s

2]

8 Stories, Point 9

Original

Lowered

Free Field

Small House

2-4 Story House

Fig. 4.7 Acceleration between original structure and lowered structure at point P7, P8 andP9

Results from the lowered buildings are presented in figure 4.7. It can be seen that forfrequencies between 15 − 25 Hz, the response is generally greater for the lowered case.

86


0 10 20 30

Frequency [Hz]

0

2

4

6

8

10

Acce

lera

tio

n [

m/s

2]

3 Stories, Point 7

0 10 20 30

Frequency [Hz]

0

2

4

6

8

10

Acce

lera

tio

n [

m/s

2]

5 Stories, Point 7

0 10 20 30

Frequency [Hz]

0

2

4

6

8

10

Acce

lera

tio

n [

m/s

2]

8 Stories, Point 7

Original

Thick Slab

Free Field

Small House

2-4 Story House

0 10 20 30

Frequency [Hz]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

3 Stories, Point 8

0 10 20 30

Frequency [Hz]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

5 Stories, Point 8

0 10 20 30

Frequency [Hz]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

8 Stories, Point 8

0 10 20 30

Frequency [Hz]

0

1

2

3

4

5

Acce

lera

tio

n [

m/s

2]

3 Stories, Point 9

0 10 20 30

Frequency [Hz]

0

1

2

3

4

5

Acce

lera

tio

n [

m/s

2]

5 Stories, Point 9

Acceleration comparison between original and thick slab

0 10 20 30

Frequency [Hz]

0

1

2

3

4

5

Acce

lera

tio

n [

m/s

2]

8 Stories, Point 9

Fig. 4.8 Acceleration between original structure and structure with thick slab at point P7,P8 and P9

Above in figure 4.8 the results from the simulation with the thick ground slab are presented.It can here be seen that the vibrations at the center of the slab vary for different frequencies,sometimes exhibiting a dampening effect, sometimes increasing the accelerations.

87


0 10 20 30

Frequency [Hz]

0

2

4

6

8

10

Acce

lera

tio

n [

m/s

2]

3 Stories, Point 7

0 10 20 30

Frequency [Hz]

0

2

4

6

8

10

Acce

lera

tio

n [

m/s

2]

5 Stories, Point 7

0 10 20 30

Frequency [Hz]

0

2

4

6

8

10

Acce

lera

tio

n [

m/s

2]

8 Stories, Point 7

Original

Thick Slab, Lowered

Free Field

Small House

2-4 Story House

0 10 20 30

Frequency [Hz]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

3 Stories, Point 8

0 10 20 30

Frequency [Hz]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

5 Stories, Point 8

0 10 20 30

Frequency [Hz]

0

2

4

6

8

Acce

lera

tio

n [

m/s

2]

8 Stories, Point 8

0 10 20 30

Frequency [Hz]

0

1

2

3

4

5

Acce

lera

tio

n [

m/s

2]

3 Stories, Point 9

0 10 20 30

Frequency [Hz]

0

1

2

3

4

5

Acce

lera

tio

n [

m/s

2]

5 Stories, Point 9

Acceleration comparison between comparison original and thick slab, lowered building

0 10 20 30

Frequency [Hz]

0

1

2

3

4

5

Acce

lera

tio

n [

m/s

2]

8 Stories, Point 9

Fig. 4.9 Acceleration between lowered structure and structure with thick slab at point P7,P8 and P9

Lastly, in figure 4.9 above, the results from the lowered buildings with thick ground slabsare presented. It can here be visualized that the thick slab presents a damping effect forthe lowered building compared to the original building for all observable frequencies in themidpoint of the slab.

88

Chapter 5

Discussion and Conclusions

5.1 Meshing in steady-state dynamics problemsAs has been shown within this thesis, it can be concluded that using quadratic solid elementsproves more computationally efficient than using linear solid elements when performingdynamic analyses, due to the fast rate of convergence observed for the quadratic elementswhen compared to linear elements. The difference can be observed in figure 5.1 below. Inthe figure, it can be seen that the quadratic element (left) gives 8 nodes, and the linearelements results in 81 nodes, if the same result is sought after.

Fig. 5.1 Difference in number of nodes for quadratic elements (left) and linear elements(right) for an axisymmetric element. The two elements in the figure yield the same resultsas per the convergence analysis performed in this thesis.

89

5.2 Considerations regarding soil calibration

5.2 Considerations regarding soil calibrationAs discovered by the authors, great difficulties are inherent when calibrating a layered soilmedium to measurement data, based on several parameters:

1. The effect of layer depth for the soil plays an important role in the frequency responsefunctions (Nazarian and Kenneth H. Stokoe (1986)). This however imposes difficultiesregarding meshing of the model, as the criteria of Bazant regarding using constant meshsize becomes a challenge when layer depth is manipulated, especially when automationis necessary as is the case for the calibration. In addition, extraction of data becomes achallenge, as changing mesh size results in a different numbering of the nodes in the model,resulting in a method needed to find the correct nodes to extract data from.

2. It is difficult to determine where one layer ends and another layer begins when assuminga numerical model to represent a gradually changing soil with depth.

3. Soils are inherently heterogenous, meaning a change of material properties and layerthickness can occur throughout the horizontal direction of the layer.

4. Soils are intrinsic non-linear, meaning viscoelasticity should be accounted for. Howeverthis results in computationally heavy simulations when compared to the linear model.

5. Regarding the objective function and fitness functions; several assurance criteria existin addition to the FAAC and FRAC. Furthermore, evaluating the assurance criteriaindependently in the objective function has not been tested within this thesis.

5.3 Effect of a structure on surrounding soilThe effect of placing a structure in a soil has minimal impact on the response in pointslocated next to the buildings, as can be seen in section 4.1 in the results chapter. The effectof placing a structure does however result vastly altered vibrations behind the structure.The authors conclude this is due to the fact that the structure becomes excited by thevibrations, in turn exciting the ground, acting as a secondary harmonic force producingvibrations further away from the original source.

90

5.4 Important aspects of soil-structure interaction

5.4 Important aspects of soil-structure interactionAs can be seen from the response functions in the foundation slab, where the numericallyevaluated response is compared to free-field response and response curves found in literature,it is evident that difficulties are present as to which curve should be chosen from literaturewhen approximating foundation vibrations, as discussed in the introduction chapter of thisthesis.Regarding the most important factor, it can be observed from the results that increasing thethickness of the foundation slab while submerging the building proves the most favorableeffect regarding reduction of the transmitted vibrations into the foundation of the buildings.

91

Chapter 6

Recommendations

Substantial effort of this project has struggled with a severe complexity regarding modelinga representation of the soil in Portugal. Some of the main and central difficulties havebeen addressed in previous sections in this thesis. To pursue some of this issues, generalrecommendations will be introduced to pass forward to other researchers a good basis foranalyzing and exploring further about soil-structure interaction in buildings. Followingparts below will treat remarks regarding both consultation and future research.

6.1 Propositions for consultationFrom the results presented in this report, there are foremost two recommendations worthmentioning regarding consultation. Following two measure can be taking into account whenexecuting both field and numerical analysis for real projects:

• Location of accelerometers on soil shall be placed at least 2 m above or beside thestructure, in horizontal direction against in loaded surface, in order to obtain aadequate response.

• Regarding soil modeling in commercial finite element softwares, such as ABAQUS,one is strongly advised of implementing quadratic solid elements to enhance bothcomputation time and also to minimize the level of spurious waves within the model.

92

6.2 Recommendations for future research

6.2 Recommendations for future researchAs the optimizations trialled within this thesis proved unsuccessful, the authors recommendfuture researchers who aim to calibrate a soil the following:

• Include layer thickness as a parameter when performing the calibration.

• Uncouple the FRAC and FAAC, meaning calibrating the shape of the response functionaccording to the response assurance criteria by varying stiffness and layer thickness,thereafter calibrating the damping against the amplitude assurance criterion.

This however does not imply a guarantee for success regarding calibration.

93

References

Administration, F. T. (2006). Transit noise and vibration impact assessment. StandardForm 298, page 193.

Bažant, Z. P. (1978). Spurious reflection of elastic waves in nonuniform finite element grids.Computer Methods in Applied Engineering and Mechanics, 16(1):91–100.

Bažant, Z. P. and Celep, Z. (1982). Spurious reflection of elastic waves in nonuniform meshesof constant and linear strain unite elements. Computers and Structures, 15(4):451–459.

Bodare, A. (1997). Jord och Bergdynamik. KTH, Stockholm.

Bodén, H., A. K. and Carlsson, U. (2014). Applied Signal Analysis. KTH Farkost & Flyg.

Chopra, A. K. (2014). Dynamics of Structures: Theory and Applications to EarthquakeEngineering, 4th edition. Pearson Prentice Hall, Upper Saddle River, New Jersey.

Cook, R. D., Malkus, D. S., Plesha, M. E., and Witt, R. J. (2001). Concepts and Applicationsof Finite Element Analysis. John Wiley and Sons, Inc., New Jersey, USA.

Costa, P. A., Calcada, R., and Cardoso, A. S. (2012). Track-ground vibrations inducedby railway traffic: In-situ measurements and validation of a 2.5d fem-bem model. SoilDynamics and Earthquake Engineering, (32):111–128.

dos Santos, N. C., Colaço, A., Costa, P. A., and Calçada, R. (2016). Experimental analysisof track-ground vibrations on a stretch of the portuguese railway network. Soil Dynamicsand Earthquake Engineering, 90(10):358–380.

Geotechdata (2018). Dry unit weight.

Hewlett-Packard (1994). The fundamentals of signal analysis. Application Note 243, pages1–67.

Hibbitt, Karlsson, and Sorensen (2006). ABAQUS/Standard User’s Manual, volume 6thEdition. Pawtucket.

Holland, J. (1992). Genetic algorithms - computer programs that evolve in ways thatresemble natural selection can solve complex problems even their creators do not fullyunderstand. Scientific American.

Landsting, S. L. Nya tunnelbanan.

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References

Landsting, S. L. Regional utvecklingsplan för Stockholmsregionen - RUFS 2050 - Europasmest attraktiva storstadsregion.

Marosi, K. T. and Hiltunen, D. R. (2004). Characterization of spectral analysis of sur-face waves shear wave velocity measurement uncertainty. Journal of Geotechnical andGeoenvironmental Engineering, 130(10):1034–1041.

Mathworks (2018). Global optimization toolbox user’s guide. The Mathworks Inc.

Nazarian, S. and Kenneth H. Stokoe, I. (1986). In situ determination of elastic moduli ofpavement systems by spectral-analysis-of-surface-waves method (theoretical aspects).

Negrut, D., Ottarsson, G., Rampalli, R., and Sajdak, A. (2006). On an implementation ofthe hilber-hughes-taylor method in the context of index 3 differential-algebraic equationsof multibody dynamics.

Phoon, K.-K. and Kulhawy, F. H. (1999). Characterization of geotechnical variability.Canadian Geotechnical Journal, 36(4):612–624.

Rao, S. S. (2009). Engineering Optimization: Theory and Practice, Fourth Edition. JohnWiley and Sons, Inc., New Jersey.

SGF (2013). Markvibrationer. Svenska Geotekniska Föreningen, Göteborg and Stockholm.

Svanström, S. (2015). Urbanisering – från land till stad. Välfärd, A05(1):26–27.

Tschebotarioff, G. P. (1962). Dynamics of Bases. McGraw-Hill Book Company Inc., NewYork.

Vretblad, A. (2003). Fourier Analysis and its Applications. Springer-Verlag, 175 FifthAvenue, New York, NY 10010, USA.

95

96

Appendix A

Convergence analysis forAxisymmetric model

Half space in comparison with rectangular model

0 20 40 60 80

Frequency [Hz]

0

0.1

0.2

0.3

Accele

rance [m

/s2/k

N]

P1, 5m

0 20 40 60 80

Frequency [Hz]

0

0.05

0.1

0.15

Accele

rance [m

/s2/k

N]

P2, 10m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

0.08

Accele

rance [m

/s2/k

N]

P3, 15m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

Accele

rance [m

/s2/k

N]

P4, 20m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

0.04

Accele

rance [m

/s2/k

N]

P5, 25m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Accele

rance [m

/s2/k

N]

P6, 30m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Accele

rance [m

/s2/k

N]

P7, 35m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Accele

rance [m

/s2/k

N]

P8, 40m

Comparison between element types for Half Space model with mesh size 0.125 m

0 20 40 60 80

Frequency [Hz]

0

0.005

0.01

0.015

0.02

Accele

rance [m

/s2/k

N]

P9, 45m

Quadratic

Linear

Fig. A.1 Convergence between quadratic and linear elements

97

0 20 40 60 80

Frequency [Hz]

0

0.1

0.2

0.3

Acce

lera

nce

[m

/s2/k

N]

P1, 5m

0 20 40 60 80

Frequency [Hz]

0

0.05

0.1

0.15

Acce

lera

nce

[m

/s2/k

N]

P2, 10m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

0.08

Acce

lera

nce

[m

/s2/k

N]

P3, 15m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

Acce

lera

nce

[m

/s2/k

N]

P4, 20m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

0.04

Acce

lera

nce

[m

/s2/k

N]

P5, 25m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P6, 30m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P7, 35m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P8, 40m

Comparison between Half Space model and Rectangular model, mesh size 0.125 m

0 20 40 60 80

Frequency [Hz]

0

0.005

0.01

0.015

0.02A

cce

lera

nce

[m

/s2/k

N]

P9, 45m

Half Space

Rectangular

Fig. A.2 Comparison among half space and rectangular model with mesh size h = 0.125

98

Mesh convergence for rectangular model

0 20 40 60 80

Frequency [Hz]

0

0.1

0.2

0.3

Acce

lera

nce

[m

/s2/k

N]

P1, 5m

0 20 40 60 80

Frequency [Hz]

0

0.05

0.1

0.15

Acce

lera

nce

[m

/s2/k

N]

P2, 10m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

0.08

Acce

lera

nce

[m

/s2/k

N]

P3, 15m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

Acce

lera

nce

[m

/s2/k

N]

P4, 20m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

0.04

Acce

lera

nce

[m

/s2/k

N]

P5, 25m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P6, 30m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P7, 35m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P8, 40m

Mesh convergence analysis, quadratic elements

0 20 40 60 80

Frequency [Hz]

0

0.005

0.01

0.015

0.02

Acce

lera

nce

[m

/s2/k

N]

P9, 45m

Mesh 0.125 m

Mesh 0.25 m

Mesh 0.5 m

Fig. A.3 Convergence study regarding mesh size for quadratic elements

99

0 20 40 60 80

Frequency [Hz]

0

0.1

0.2

0.3

Acce

lera

nce

[m

/s2]

P1, 5m

0 20 40 60 80

Frequency [Hz]

0

0.05

0.1

0.15

Acce

lera

nce

[m

/s2]

P2, 10m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

0.08

Acce

lera

nce

[m

/s2]

P3, 15m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

Acce

lera

nce

[m

/s2]

P4, 20m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

0.04

Acce

lera

nce

[m

/s2]

P5, 25m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2]

P6, 30m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2]

P7, 35m

0 20 40 60 80

Frequency [Hz]

0

0.005

0.01

0.015

0.02

Acce

lera

nce

[m

/s2]

P8, 40m

Mesh convergence analysis, linear elements

0 20 40 60 80

Frequency [Hz]

0

0.005

0.01

0.015

0.02A

cce

lera

nce

[m

/s2]P9, 45m

Mesh 0.125 m

Mesh 0.0625 m

Fig. A.4 Convergence study regarding mesh size for linear elements

100

0 20 40 60 80

Frequency [Hz]

0

0.1

0.2

0.3

Acce

lera

nce

[m

/s2]

P1, 5m

0 20 40 60 80

Frequency [Hz]

0

0.05

0.1

0.15

Acce

lera

nce

[m

/s2]

P2, 10m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

0.08

Acce

lera

nce

[m

/s2]

P3, 15m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

Acce

lera

nce

[m

/s2]

P4, 20m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

0.04

Acce

lera

nce

[m

/s2]

P5, 25m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2]

P6, 30m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2]

P7, 35m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2]

P8, 40m

Comparison between linear mesh and quadratic mesh of equivalent convergence

0 20 40 60 80

Frequency [Hz]

0

0.005

0.01

0.015

0.02

Acce

lera

nce

[m

/s2]P9, 45m

Mesh 0.125 m quad

Mesh 0.0625 m lin

Fig. A.5 Convergence between linear and quadratic elements

101

Convergence regarding geometry for rectangular model

0 20 40 60 80

Frequency [Hz]

0

0.1

0.2

0.3

Acce

lera

nce

[m

/s2/k

N]

P1, 5m

0 20 40 60 80

Frequency [Hz]

0

0.05

0.1

0.15

Acce

lera

nce

[m

/s2/k

N]

P2, 10m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

0.08

Acce

lera

nce

[m

/s2/k

N]

P3, 15m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

Acce

lera

nce

[m

/s2/k

N]

P4, 20m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

0.04

Acce

lera

nce

[m

/s2/k

N]

P5, 25m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P6, 30m

Width 70 m

Witdh 50 m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P8, 40m

Comparison between 50 m width and 70 m width, rectangular model, mesh 0.5 m

0 20 40 60 80

Frequency [Hz]

0

0.005

0.01

0.015

0.02

Acce

lera

nce

[m

/s2/k

N]

P9, 45m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P7, 35m

Fig. A.6 Convergence check between a model with a width 70 m and 50 m

102

0 20 40 60 80

Frequency [Hz]

0

0.1

0.2

0.3

Acce

lera

nce

[m

/s2/k

N]

P1, 5m

0 20 40 60 80

Frequency [Hz]

0

0.05

0.1

0.15

Acce

lera

nce

[m

/s2/k

N]

P2, 10m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

0.08

Acce

lera

nce

[m

/s2/k

N]

P3, 15m

Depth 30 m

Depth 20 m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

Acce

lera

nce

[m

/s2/k

N]

P4, 20m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

0.04

Acce

lera

nce

[m

/s2/k

N]

P5, 25m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P6, 30m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P7, 35m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P8, 40m

Comparison between 20 m depth and 30 m depth, rectangular model, mesh 0.5 m

0 20 40 60 80

Frequency [Hz]

0

0.005

0.01

0.015

0.02

Acce

lera

nce

[m

/s2/k

N]P9, 45m

Fig. A.7 Convergence check between a model with a depth of 20 m and 30 m

103

Convergence check for step sizes for rectangular model

0 20 40 60 80

Frequency [Hz]

0

0.1

0.2

0.3

Acce

lera

nce

[m

/s2/k

N]

P1, 5m

0 20 40 60 80

Frequency [Hz]

0

0.05

0.1

0.15

Acce

lera

nce

[m

/s2/k

N]

P2, 10m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

0.08

Acce

lera

nce

[m

/s2/k

N]

P3, 15m

800 steps

400 steps

200 steps

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

Acce

lera

nce

[m

/s2/k

N]

P4, 20m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

Acce

lera

nce

[m

/s2/k

N]

P5, 25m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P6, 30m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P7, 35m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P8, 40m

Comparison different step sizes. Rectangular model, mesh 0.5 m

0 20 40 60 80

Frequency [Hz]

0

0.005

0.01

0.015

0.02

Acce

lera

nce

[m

/s2/k

N]

P9, 45m

Fig. A.8 Different increment step sizes, 800-200

104

0 20 40 60 80

Frequency [Hz]

0

0.1

0.2

0.3

Acce

lera

nce

[m

/s2/k

N]

P1, 5m

0 20 40 60 80

Frequency [Hz]

0

0.05

0.1

0.15

Acce

lera

nce

[m

/s2/k

N]

P2, 10m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

0.08

Acce

lera

nce

[m

/s2/k

N]

P3, 15m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

Acce

lera

nce

[m

/s2/k

N]

P4, 20m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

Acce

lera

nce

[m

/s2/k

N]

P5, 25m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P6, 30m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P7, 35m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P8, 40m

Comparison different step sizes. Rectangular model, mesh 0.5 m

0 20 40 60 80

Frequency [Hz]

0

0.005

0.01

0.015

0.02

Acce

lera

nce

[m

/s2/k

N]P9, 45m

200 steps

100 steps

50 steps

Fig. A.9 Different increment step sizes, 200-50

105

20 40 60 80

Frequency [Hz]

0

0.05

0.1

0.15

0.2

0.25

Acce

lera

nce

[m

/s2/k

N]

P1, 5m

20 40 60 80

Frequency [Hz]

0

0.05

0.1

0.15

Acce

lera

nce

[m

/s2/k

N]

P2, 10m

20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

0.08

Acce

lera

nce

[m

/s2/k

N]

P3, 15m

20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

Acce

lera

nce

[m

/s2/k

N]

P4, 20m

20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

0.04

Acce

lera

nce

[m

/s2/k

N]

P5, 25m

20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P6, 30m

20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P7, 35m

20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P8, 40m

Comparison between time domain analysis and frequency domain analysis

20 40 60 80

Frequency [Hz]

0

0.005

0.01

0.015

0.02

Acce

lera

nce

[m

/s2/k

N]P9, 45m

Time domain

Frequency domain

Fig. A.10 Comparison between time domain and frequency domain analysis

106

0 20 40 60 80

Frequency [Hz]

0

0.1

0.2

0.3

Acce

lera

nce

[m

/s2/k

N]

P1, 5m

0 20 40 60 80

Frequency [Hz]

0

0.05

0.1

0.15

Acce

lera

nce

[m

/s2/k

N]

P2, 10m

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

0.08

Acce

lera

nce

[m

/s2/k

N]

P3, 15m

Rectangular 0.5 quad

Half Space 0.125 quad

0 20 40 60 80

Frequency [Hz]

0

0.02

0.04

0.06

Acce

lera

nce

[m

/s2/k

N]

P4, 20m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

0.04

Acce

lera

nce

[m

/s2/k

N]

P5, 25m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P6, 30m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P7, 35m

0 20 40 60 80

Frequency [Hz]

0

0.01

0.02

0.03

Acce

lera

nce

[m

/s2/k

N]

P8, 40m

Comparison between the best model and the chosen model

0 20 40 60 80

Frequency [Hz]

0

0.005

0.01

0.015

0.02

Acce

lera

nce

[m

/s2/k

N]

P9, 45m

Fig. A.11 Convergence check between Half Space and Rectangular model with mesh sizeh = 0.125 respective h = 0.5, quadratic elements

107

Appendix B

Sensitivity check for Axisymmetricmodel

Material properties

108

Fig. B.1 Comparison between 14- and 5 layer soil model109

110

Fig. B.2 Effect of increased damping by factor 2 for all layers111

112

Fig. B.3 Effect of decreased damping by factor 2 for all layers113

114

Fig. B.4 Effect of no damping for all layers115

116

Fig. B.5 Effect of structural damping on all layers117

118

Fig. B.6 Effect of increased density of 200kg/m3 for all layers119

120

Fig. B.7 Effect of decreased density of 200kg/m3 for all layers121

122

Fig. B.8 Effect of GWL when levels is on -3 m123

124

Fig. B.9 Effect of GWL when levels is on -6 m125

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Soil-structure interaction for traffic induced vibrations ...1237322/FULLTEXT01.pdf · EXAMENSARBETE INOM SAMHÄLLSBYGGNAD, AVANCERAD NIVÅ, 30 HP STOCKHOLM, SVERIGE 2018 Soil-structure