Numerical Prediction of Ground Vibrations Generated by Road Traffic

Arenberg Doctoral School of Science, Engineering & Technology Faculty of Engineering Department of Civil Engineering

NUMERICAL PREDICTION OF GROUND VIBRATIONS GENERATED BY ROAD TRAFFIC AND PAVEMENT BREAKING

Mohammad Amin LAK

Dissertation presented in partial fulfilment of the requirements for the degree of Doctor of Engineering

February 2013

NUMERICAL PREDICTION OF GROUND VIBRATIONS GENERATED BY ROAD TRAFFIC AND PAVEMENT BREAKING

Mohammad Amin LAK

Jury: Prof. dr. ir. Y. Willems, chair Prof. dr. ir. G. Lombaert, supervisor Prof. dr. ir. G. Degrande, supervisor Prof. dr. ir. A. Beeldens Prof. dr. ir. D. Vandepitte Prof. dr. ir. W. Haegeman (Universiteit Gent, KHBO, and KU Leuven) Prof. dr. ir. A. Holeyman (Université Catholique de Louvain) Prof. dr. ir. J.F. Semblat (Université Paris-Est, LCPC, Paris, France)

February 2013

Dissertation presented in partial fulfilment of the requirements for the degree of Doctor of Engineering

© KU Leuven – Faculty of Engineering Science Kasteelpark Arenberg 40 box 2448, B-3001 Leuven (Belgium) Alle rechten voorbehouden. Niets uit deze uitgave mag worden vermenigvuldigd en/of openbaar gemaakt worden door middel van druk, fotocopie, microfilm, elektronisch of op welke andere wijze ook zonder voorafgaande schriftelijke toestemming van de uitgever. All rights reserved. No part of the publication may be reproduced in any form by print, photoprint, microfilm or any other means without written permission from the publisher. D/2013/7515/5 ISBN 978-94-6018-618-9

Preface . . . , and say, “My Lord! increase me in knowledge.” Quran, sura 20 (Ta-ha), verse 114. The present thesis deals with the numerical prediction and experimental validation of ground vibrations generated by road traffic and pavement breaking. The study has been carried out at the Structural Mechanics Section in the Department of Civil Engineering, KU Leuven, Belgium, under the supervision of Prof. Geert Lombaert and Prof. Geert Degrande. First of all, I wish to express my deep sense of gratitude to Prof. Geert Lombaert for his great support, precise guidance, and valuable advice. Apart from these, his kind words and consistent encouragement throughout these years made the research a more enjoyable experience. I would like to sincerely thank Prof. Geert Degrande who provided me the opportunity to carry out my research at KU Leuven and for his interest in the work and constructive comments. I believe that the high quality research and the international reputation of the section is greatly indebted to his high standards and tireless effort. I gratefully acknowledge the members of the doctoral commission Prof. Anne Beeldens, Prof. Dirk Vandepitte, Prof. Wim Haegeman, Prof. Alain Holeyman, and Prof. Jean-Fran¸cois Semblat for evaluating my work and making valuable remarks and comments. I am also grateful to Prof. Yves Willems for chairing the preliminary and public defence sessions. I would also like to thank Prof. Guido De Roeck, the head of the section, for his helpful suggestions. I am very grateful to my friends and colleagues Stijn, Hamid, Ali, and Mattias for useful discussions and for openly sharing their knowledge. I particularly thank Stijn for translating the summary into Dutch. Throughout these years, I had the opportunity to share my office with many researchers. I express my appreciation to them, namely Arne, Kristof, Pieter, Yuhang, Javier, Zuhal, Lola, Yushu, Bram, i

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Preface

and Soon for providing a warm and friendly atmosphere at the office. I also thank my colleagues at the Structural Mechanics Section Bui, Chaoyi, Daan, David, Edwin, Eliz-Mari, Ellen, Hans, Hernan, Jaime, Jeroen, Jonas, Kai, Katrien, Leqia, ¨ Miche, Ozer, Pieter, Saartje, and Shashank, and many foreign visitors Costas, Ding, Federico, Gang, Ghada, Jiawang, Liang, Pedro, Shiqiang, Suzhen, Ting, and Weiwei for their friendship and help. I must also thank Danielle, the secretary of the section, and other staff members of the department, especially Luc and Frank, for their assistance and support. This research has been performed within the frame of the IWT project VIS-CO 060884 ‘Vibration controlled stabilisation of concrete slabs for durable asphalt overlaying with crack prevention membrane’ in collaboration with the Belgian Road Research Centre and the FWO project G.0397.09 ‘The constitutive behaviour of granular soils under repeated dynamic loading’. The financial support of IWT Vlaanderen (Institute for the Promotion of Innovation by Science and Technology in Flanders) and FWO (Research Foundation Flanders) is gratefully acknowledged. I would like to thank my great friends who made all these years in Belgium a precious memory. They are Ali, Alireza, Amin, Amir, Amir-Abbas, Babak, Farhad, Florent, Hadi, Hakem, Hamid, Hassan, Homayoon, Jalal, Karim, Majid, Michel, Mohammad, Reza, Rozbeh, Saeed, Soheil, Taha, Vahid, and their families. My special thanks go to my mother for her unconditional love, and my brother, sister, in-laws, and other family members for their help, encouragement, and love. I always cherish the memories of my late beloved father who was my best friend, colleague, philosopher, and guide. Finally, and most importantly, I would like to thank my lovely wife for all her understanding, support, and patience during all these years and also our wonderful daughter(s) for bringing joy and happiness into our lives. Without them, it would have been very difficult to reach this stage today. Mohammad Amin Lak, Leuven, February 2013

Summary Poor ride quality and traffic induced noise and vibrations require replacement or rehabilitation of deteriorated pavements. In the case of concrete roads, the pavement is usually broken to prepare it for removal or as the first step in the cracking and seating method of road rehabilitation. The pavement breaking operation generates a high level of ground-borne vibrations. In this study, three aspects of vibrations in the vicinity of concrete pavements are addressed: 1) the prediction of ground vibrations due to traffic on deteriorated jointed concrete pavements, 2) the prediction of vibrations generated by pavement breaking, and 3) the level of vibration reduction gained by road rehabilitation. Several experiments on traffic induced vibrations and vibrations generated by pavement breaking are conducted prior to, during, and after the rehabilitation of a deteriorated jointed concrete pavement. The road is rehabilitated with the cracking and seating technique and application of an asphalt overlay. The prediction of traffic induced vibrations is performed in two stages: first, the dynamic wheel loads due to the passage of a vehicle over road irregularities are estimated; second, these loads are applied to a road-soil system to compute the radiated wave field in the soil. The road unevenness is obtained by means of a high speed profiler and introduced into a three-dimensional vehicle model to compute the dynamic vehicle loads. The loads are subsequently applied to a coupled finite element-boundary element model of the road-soil system to compute ground vibrations. The results are in a good agreement with the experimental data and show a significant reduction of ground vibrations by road rehabilitation. For the estimation of the impact load due to the blow of a falling weight pavement breaker, a numerical model is developed and experimentally validated. The energy dissipated by fracturing of concrete is found to be negligible and, consequently, can be disregarded when predicting ground vibrations. Hence, the linear coupled finite element-boundary element model of the road-soil system is adopted to predict ground vibrations generated by pavement breaking. The strain level in the soil is found to be beyond the linear elastic range. Therefore, the model is further elaborated to an equivalent linear model and subsequently to a non-

iii

iv

Summary

linear model which takes into account inelastic behaviour of the soil and slab uplifting. It is composed of a finite element model for the slab and a part of the soil coupled to viscous boundary conditions. To compute the response outside the finite element domain, the tractions and displacements along a path inside the finite element domain are computed and introduced in the integral representation formulation. Ground vibrations predicted with the non-linear model are compared to the experimental results where a relatively good agreement is observed.

Samenvatting Zowel een slecht rijcomfort als trillingen en geluid afkomstig van verkeer noodzaken het vervangen of het herstel van een beschadigd wegdek. Voor betonwegen wordt het wegdek vaak gebeukt ter vervanging of ter voorbereiding van een overlaging met asfalt. Het beuken van de weg veroorzaakt sterke trillingen die zich voortplanten in de ondergrond. In deze studie worden drie aspecten van trillingen rond betonwegen onderzocht: (1) de voorspelling van trillingen in de grond ten gevolge van verkeer op beschadigde betonwegen, (2) de voorspelling van trillingen als gevolg van het beuken van de weg en (3) het begroten van de trillingsreductie ten gevolge van de herstelling. In het kader van deze studie werden verschillende experimenten uitgevoerd om de trillingen ten gevolge van verkeer en beuken op te meten, voor, tijdens en na de herstelling van een beschadigde betonweg. De weg is hersteld door middel van de “cracking and seating” techniek, waarbij het gebeukte wegdek overlaagd werd met asfalt. De voorspelling van de trillingen ten gevolge van verkeer wordt uitgevoerd in twee fases: eerst worden de dynamische aslasten bij de passage van een voertuig over een oneffen wegdek geschat; ten tweede worden deze lasten aangebracht op het gekoppelde systeem weg-grond om het afgestraalde golfveld in de grond te berekenen. De oneffenheid van het wegdek wordt in situ opgemeten door middel van een lengteprofielanalysator als input voor een driedimensionaal voertuigmodel om de dynamische aslasten te berekenen. De aslasten worden vervolgens aangelegd aan het gekoppelde systeem weg-grond om de trillingen in het vrije veld te berekenen. De resultaten zijn in goede overeenstemming met experimentele resultaten en tonen een dat de herstelling leidt tot een significante reductie van de trillingen. Voor de schatting van de belasting ten gevolge van één enkele impact van de beukmachine wordt een numeriek model ontwikkeld en experimenteel gevalideerd. De energie gedissipeerd bij het breken van het beton blijkt zeer klein te zijn en kan bijgevolg verwaarloosd worden in de voorspelling van trillingen in de ondergrond. Het gekoppelde eindige elementen - randelementenmodel van het systeem weg-grond wordt toegepast om de trillingen ten gevolge van het beuken

v

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Samenvatting

te voorspellen. Hieruit blijkt dat de grootte van de vervormingen in de grond onder de betonplaat buiten het lineair elastisch gebied vallen. Daarom wordt het model uitgebreid met een lineair equivalente benadering en vervolgens een volledig nietlineair model dat rekening houdt met het inelastische gedrag van de grond en het loskomen van de betonplaat van de ondergrond. Het model bestaat uit een eindig elementenmodel voor de plaat en een deel van de grond onder de plaat dat zich nietlineair gedraagt. Om de respons buiten dit inelastische domein te berekenen, wordt het integraalrepresentatietheorema toegepast op de spanningen en verplaatsingen langsheen een pad binnen het eindige elementenmodel. Trillingen berekend met het niet-lineaire model zijn in overeenkomst met de experimentele resultaten.

Contents Preface

i

Summary

iii

Samenvatting

v

Contents

vii

List of Figures

xiii

List of Tables

xxv

List of Symbols

xxvii

1 Introduction

1

1.1

1.2

1.3

Motivation and problem outline . . . . . . . . . . . . . . . . . . . .

1

1.1.1

Environmental impact of vibrations . . . . . . . . . . . . .

1

1.1.2

Road rehabilitation as a vibration reduction measure . . . .

2

State of the art and further needs . . . . . . . . . . . . . . . . . . .

4

1.2.1

Traffic induced vibrations . . . . . . . . . . . . . . . . . . .

4

1.2.2

Vibrations generated by pavement breaking . . . . . . . . .

8

Focus of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.3.1

13

Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

viii

CONTENTS

1.4

1.3.2

Methodology . . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.3.3

Original contributions . . . . . . . . . . . . . . . . . . . . .

15

Organisation of the text . . . . . . . . . . . . . . . . . . . . . . . .

17

2 Numerical estimation of dynamic vehicle loads

21

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.2

Road unevenness . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.2.1

Road unevenness measurement . . . . . . . . . . . . . . . .

22

2.2.2

Road profile processing . . . . . . . . . . . . . . . . . . . .

24

2.2.3

Unevenness of the N9 road prior to and after rehabilitation

27

2.3

Vehicle model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

2.4

Experimental validation of the vehicle model . . . . . . . . . . . .

33

2.4.1

Experimental configuration . . . . . . . . . . . . . . . . . .

33

2.4.2

Initial estimation of the vehicle parameters . . . . . . . . .

34

2.4.3

Calibration of the vehicle parameters . . . . . . . . . . . . .

34

2.4.4

Eigenmodes and eigenfrequencies of the vehicle . . . . . . .

38

2.4.5

Frequency response functions of the vehicle . . . . . . . . .

39

2.4.6

Validation of the vehicle model . . . . . . . . . . . . . . . .

40

2.5

Dynamic vehicle loads . . . . . . . . . . . . . . . . . . . . . . . . .

46

2.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

3 Prediction of traffic induced ground vibrations

51

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.2

Road-soil interaction . . . . . . . . . . . . . . . . . . . . . . . . . .

52

3.2.1

Three-dimensional model of road-soil interaction . . . . . .

52

3.2.2

Two-and-a-half-dimensional model of road-soil interaction .

55

3.2.3

Numerical example . . . . . . . . . . . . . . . . . . . . . . .

58

Ground vibrations generated by moving loads . . . . . . . . . . . .

62

3.3

CONTENTS

ix

3.4

Experimental validation . . . . . . . . . . . . . . . . . . . . . . . .

64

3.4.1

Experimental configuration . . . . . . . . . . . . . . . . . .

64

3.4.2

Soil and road characteristics . . . . . . . . . . . . . . . . . .

65

3.4.3

Validation of the predicted ground vibrations . . . . . . . .

68

3.5

Soil strains due to traffic load . . . . . . . . . . . . . . . . . . . . .

83

3.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

4 Pavement breaking, impact load estimation, and fracturing

87

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

4.2

Pavement breaking . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

4.3

Experimental study . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

4.4

4.5

4.6

4.3.1

Measurement of ground vibrations due to pavement breaking 90

4.3.2

Identification of soil parameters . . . . . . . . . . . . . . . .

93

Estimation of the drop hammer impact load . . . . . . . . . . . . .

95

4.4.1

Impact model . . . . . . . . . . . . . . . . . . . . . . . . . .

95

4.4.2

Numerical verification . . . . . . . . . . . . . . . . . . . . . 100

4.4.3

Experimental validation . . . . . . . . . . . . . . . . . . . . 102

Fracturing of the concrete slabs . . . . . . . . . . . . . . . . . . . . 103 4.5.1

Experimental investigation of fracturing . . . . . . . . . . . 104

4.5.2

Estimation of the fracture energy . . . . . . . . . . . . . . . 105

4.5.3

Estimation of the crack propagation velocity . . . . . . . . 109

4.5.4

Effect of fracturing on ground vibrations . . . . . . . . . . . 110

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5 Linear prediction of ground vibrations due to pavement breaking

115

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.2

Effect of the presence of the pavement breaker . . . . . . . . . . . 116

5.3

Axisymmetric coupled finite element-boundary element model . . . 118

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CONTENTS

5.4

5.5

5.3.1

Model description . . . . . . . . . . . . . . . . . . . . . . . 118

5.3.2

Numerical verification . . . . . . . . . . . . . . . . . . . . . 121

5.3.3

Results obtained with the linear model . . . . . . . . . . . . 122

5.3.4

Comparison with the experimental results . . . . . . . . . . 127

Equivalent linear prediction of ground vibrations . . . . . . . . . . 130 5.4.1

Model description . . . . . . . . . . . . . . . . . . . . . . . 131

5.4.2

Results obtained with the equivalent linear model . . . . . 132

5.4.3

Experimental validation . . . . . . . . . . . . . . . . . . . . 136

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6 Non-linear prediction of ground vibrations due to pavement breaking

143

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.2

Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.3

Numerical verification . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.3.1

Linear elastic halfspace . . . . . . . . . . . . . . . . . . . . 148

6.3.2

Static elasto-plastic SSI with foundation uplift . . . . . . . 150

6.4

Critical state soil parameters . . . . . . . . . . . . . . . . . . . . . 151

6.5

Results obtained with the non-linear model . . . . . . . . . . . . . 154

6.6

Comparison of the models . . . . . . . . . . . . . . . . . . . . . . . 162

6.7

Experimental validation . . . . . . . . . . . . . . . . . . . . . . . . 167

6.8

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7 Conclusions and recommendations for further research 7.1

7.2

171

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.1.1

Traffic induced vibrations . . . . . . . . . . . . . . . . . . . 171

7.1.2

Vibrations generated by pavement breaking . . . . . . . . . 173

Recommendations for further research . . . . . . . . . . . . . . . . 175

Bibliography

177

CONTENTS

A Mass, stiffness, and damping matrices of the 3D vehicle model

xi

199

A.1 Mass matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 A.2 Stiffness matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 A.3 Damping matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 B Critical state soil parameters

203

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 B.2 Unit weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 B.3 Relative density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 B.4 Internal friction angle . . . . . . . . . . . . . . . . . . . . . . . . . 207 B.5 Dilatancy angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 B.6 Shear wave velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 B.7 Cohesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 B.8 Summary of the estimated soil parameters . . . . . . . . . . . . . . 213 Curriculum vitae

217

List of Figures 1.1

Summary of measured vertical PPV generated by pavement breaking as a function of distance. Superimposed are the empirical formula in equation (1.1) with n = 1.1 (dashed line) and n = 1.5 (dashed-dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . .

10

Attenuation of ground vibrations measured on different dynamic soil compaction projects (from [179]). . . . . . . . . . . . . . . . . .

11

Comparison of the recorded vertical PPV due to pavement breaking reported by [235] and [147, 152, 153] to the empirical formula for the vertical PPV due to soil compaction (equation (1.2)) for md = 5900 kg and h = 1.2 m (thick black line) and for md = 600 kg and h = 1.8 m (thick grey line). . . . . . . . . . . . . . . . . . . . . . .

12

2.1

The APL trailers. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.2

Wavenumber response function of the rectangular time window with a baselength lm = 0.30 m (solid line) and the rolling line segment with a baselength lz = 0.16 m (dashed line). . . . . . . . . . . . . .

26

Road profile (black line) of the right wheel track of the (a) deteriorated and (b) rehabilitated N9 recorded by the APL at a speed of 21.6 km/h. Superimposed is the filtered road profile (grey line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

PSD of road displacement (thin black line) and fitted line to the PSD (thick line) of the (a) deteriorated and (b) rehabilitated N9. Superimposed are the road quality classes A-H according to ISO 8608 (grey lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

Ratio of roll displacement spectral density to bounce displacement spectral density of the (a) deteriorated and (b) rehabilitated N9. .

29

1.2 1.3

2.3

2.4

2.5

xiii

xiv

LIST OF FIGURES

2.6

(a) The quarter-car vehicle model (from [216]) and (b) half-car vehicle model (from [159]). . . . . . . . . . . . . . . . . . . . . . . .

30

2.7

Three-dimensional model of a two-axle truck. . . . . . . . . . . . .

31

2.8

(a) Volvo FL180 truck and (b) two accelerometers on the left side of the rear axle and one accelerometer on the left rear side of the body. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

Apparent contact patch of the (a) rear and (b) front tyres of Volvo FL180 in unladen condition. . . . . . . . . . . . . . . . . . . . . . .

36

2.10 PSD of acceleration (a) on the right side of the rear axle for a vehicle speed of 70 km/h and (b) on the right side of the front axle for a vehicle speed of 50 km/h predicted with the initial (dashed black line) and updated (solid black line) parameters. Superimposed are the corresponding experimental results (solid grey line) and the experimental results of the other passages with similar speeds (dashed grey lines). . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

ˆ g uc for the 3D vehicle model of the Volvo 2.11 Modulus of the FRF h k l FL180 truck with the initial (dashed line) and updated (solid line) parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.12 Predicted (black) and measured (grey) (a) time history, (b) narrow band frequency spectrum, (c) running RMS value, and (d) PSD of the acceleration on the right side of the rear axle during a passage of the truck at a speed of 50 km/h on the deteriorated N9. Superimposed are on (a) anti-smoothed profiles of the near (N) and far (F) wheel tracks (black lines) and on (c) and (d) the experimental results for passages with similar speeds (dashed grey lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

2.13 Predicted PA from the road profile recorded at 21.6 km/h (thin line) and 54 km/h (thick line) versus measured PA (plus signs) on the left hand (left) and right hand (right) side of the rear (top) and front (bottom) axle as a function of the vehicle speed prior to the road rehabilitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

2.14 Predicted (black) and measured (grey) (a) time history, (b) narrow band frequency spectrum, (c) running RMS value, and (d) PSD of the acceleration on the right side of the rear axle during a passage of the truck at a speed of 50 km/h on the rehabilitated N9. Superimposed are on (a) anti-smoothed profiles of the near (N) and far (F) wheel tracks (black lines) and on (c) and (d) the experimental results for passages with similar speeds (dashed grey lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

2.9

LIST OF FIGURES

xv

2.15 Predicted PA from the road profile recorded at 21.6 km/h (thin line) and 54 km/h (thick line) versus measured PA (plus signs) on the left hand (left) and right hand (right) side of the rear (top) and front (bottom) axle as a function of the vehicle speed after the road rehabilitation. Superimposed on the graphs are the measured PAs in the passages on the middle lane (crosses). . . . . . . . . . . . . .

45

2.16 Predicted (a) time history and (b) narrow band frequency spectrum of the dynamic load at the right rear wheels (top) and right front wheel (bottom) during a passage of the truck at a speed of 50 km/h on the deteriorated N9. Superimposed on (a) are the anti-smoothed profiles of the near (N) and far (F) wheel tracks (black lines). . . .

47

2.17 Predicted (a) time history and (b) narrow band frequency spectrum of the dynamic load at the right rear wheels (top) and right front wheel (bottom) during a passage of the truck at a speed of 50 km/h on the rehabilitated N9. Superimposed on (a) are the anti-smoothed profiles of the near (N) and far (F) wheel tracks (black lines). . . .

48

3.1

Finite plate model of the road-soil system with a force (a) at the centreline xs = {wt , 0, tr }T and (b) near the edge xs = {wt , L − p, tr }T of the slab. . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.2

Model of a road slab on the soil surface. . . . . . . . . . . . . . . .

53

3.3

(a) Continuous plate model and (b) beam model of the road-soil system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

ˆ iz (xs , x, 0, 0, ω) in the x- (left) and zFree field mobility i ω h direction (right) at x = 4 m (top), x = 16 m (middle), and x = 64 m (bottom) due to a vertical point load on the finite plate (thin black line), continuous plate (thick grey line), and continuous beam (thick black line) model at xs = {1 m, 0, 0.2 m}T . Superimposed is the Green’s function i ω u ˆG iz (1, 0, 0, x, 0, 0, ω) of the soil for a disc load with a diameter of 0.80 m (dashed line). . . . . . . . . . . . . . . .

59

3.4

3.5

ˆ iz (xs , x, y, 0, ω) in the x- (left), y- (middle), Free field mobility i ω h and z-direction (right) at x = 4 m (top), x = 16 m (middle), and x = 64 m (bottom) and y = 0 (solid line) and y = 2.2 m (dasheddotted line) due to a vertical point load on the finite plate at xs = {1 m, 0, 0.2 m}T (black line) and xs = {1 m, 2.2 m, 0.2 m}T (grey line). 61

3.6

ˆ iy (0, x, y, 0, ω) in (a) the x-, (b) y-, and (c) Free field mobility i ω h z-direction at x = 16 m due to a point load in the y-direction at the centre of the finite plate model of the slab. . . . . . . . . . . . . .

62

xvi

LIST OF FIGURES

Free field mobility i ω ˆhiz (0, x, y, 0, ω) in (a) the x-, (b) y-, and (c) z-direction at x = 16 m due to a point load in the z-direction at the centre of the finite plate model of the slab. . . . . . . . . . . . . .

62

Location of the measurement points in the free field during the measurement campaign in Lovendegem. . . . . . . . . . . . . . . .

65

(a) Lithological description of the soil around the measurement site next to the N9 road in Lovendegem and dynamic soil characteristics identified by means of in situ tests at the measurement site: (b) shear wave velocity profile estimated from the SASW test (thick line) and the SCPT (thin line), (c) dilatational wave velocity profile estimated by the seismic refraction test (thick line) and the SCPT (thin line), and (d) material damping ratio estimated from the SASW test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

3.10 Predicted (black) and measured (grey) (a) time history, (b) narrow band frequency spectrum, (c) running RMS value, and (d) PSD of the vertical velocity at x = {4 m, 0, 0}T during a passage of the truck at a speed of 50 km/h along the deteriorated N9. Superimposed on (c) and (d) are the experimental results for passages with similar speeds (dashed grey lines). . . . . . . . . . . . . . . . . . . . . . . .

69

3.11 Predicted (black) and measured (grey) (a) time history, (b) narrow band frequency spectrum, (c) running RMS value, and (d) PSD of the vertical velocity at x = {16 m, 0, 0}T during a passage of the truck at a speed of 50 km/h along the deteriorated N9. Superimposed on (c) and (d) are the experimental results for passages with similar speeds (dashed grey lines). . . . . . . . . . .

70

ˆ zz (wt , 0, tr , x, 0, 0, ω) at (a) x = 4 m, 3.12 Free field vertical mobility i ω h (b) x = 16 m, (c) x = 32 m, and (d) x = 64 m due to a vertical point load on the continuous beam model resting on the soil profile in table 3.1 (thick line), the same soil but an increased damping of 0.044 for the first layer (thin line), and the same soil but a decreased damping of 0.01 for the fifth layer and the halfspace (dashed grey line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

3.13 Predicted (black) and measured (grey) (a) time history, (b) narrow band frequency spectrum, (c) running RMS value, and (d) PSD of the vertical velocity at x = {64 m, 0, 0}T during a passage of the truck at a speed of 50 km/h along the deteriorated N9. Superimposed on (c) and (d) are the experimental results for passages with similar speeds (dashed grey lines). . . . . . . . . . .

72

3.7

3.8 3.9

LIST OF FIGURES

xvii

3.14 Predicted (solid line) and measured (plus signs) vertical PPV at (a) x = {4 m, 0, 0}T , (b) x = {16 m, 0, 0}T , and (c) x = {64 m, 0, 0}T next to the deteriorated N9 as a function of the vehicle speed. . . . . . .

73

3.15 Predicted PPV in the (a) x-, (b) y-, and (c) z-direction in the free field at 11 sections perpendicular to the deteriorated N9 at intervals of 10 m from y = 50 m to y = 150 m as a function of the distance from the road for a passage of the truck at a speed of 50 km/h. Superimposed are measured PPVs at 15 receivers shown in figure 3.8 (plus signs). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3.16 Predicted (black) and measured (grey) vibration velocity in the (a) x-, (b) y-, and (c) z-direction at x = {16 m, 0, 0}T during a passage of the truck at a speed of 50 km/h along the deteriorated N9. . . .

74

3.17 Predicted (solid line) and measured (plus signs) PPV in the (a) x-, (b) y-, and (c) z-direction at x = {16 m, 0, 0}T next to the deteriorated N9 as a function of the vehicle speed. . . . . . . . . .

75

3.18 Predicted (black) and measured (grey) (a) time history, (b) narrow band frequency spectrum, (c) running RMS value, and (d) PSD of the vertical velocity at x = {4 m, 0, 0}T during a passage of the truck at a speed of 50 km/h along the rehabilitated N9. . . . . . . . . . .

76

3.19 (a) Time history, (b) narrow band frequency spectrum, (c) running RMS value, and (d) PSD of the vertical velocity due to ambient excitation measured at x = {4 m, 0, 0}T during the measurement campaign after the road rehabilitation. . . . . . . . . . . . . . . . .

77

3.20 Predicted (black) and measured (grey) (a) time history, (b) narrow band frequency spectrum, (c) running RMS value, and (d) PSD of the vertical velocity at x = {16 m, 0, 0}T during a passage of the truck at a speed of 50 km/h along the rehabilitated N9. . . . . . .

78

3.21 Predicted (black) and measured (grey) (a) time history, (b) narrow band frequency spectrum, (c) running RMS value, and (d) PSD of the vertical velocity at x = {64 m, 0, 0}T during a passage of the truck at a speed of 50 km/h along the rehabilitated N9. . . . . . .

79

3.22 Predicted (solid line) and measured (plus signs) vertical PPV at (a) x = {4 m, 0, 0}T , (b) x = {16 m, 0, 0}T , and (c) x = {64 m, 0, 0}T next to the rehabilitated N9 as a function of the vehicle speed. Superimposed on the graphs are the measured vertical PPVs during the passages on the middle lane (crosses). . . . . . . . . . . . . . .

80

xviii

LIST OF FIGURES

3.23 Predicted PPV in the (a) x-, (b) y-, and (c) z-direction in the free field at 11 sections perpendicular to the rehabilitated N9 at intervals of 10 m from y = 50 m to y = 150 m as a function of the distance from the road for a passage of the truck at a speed of 50 km/h. Superimposed are measured PPVs at 15 receivers shown in figure 3.8 (plus signs and crosses). . . . . . . . . . . . . . . . . . . . . . .

80

3.24 Predicted (black) and measured (grey) vibration velocity in the (a) x-, (b) y-, and (c) z-direction at x = {16 m, 0, 0}T during a passage of the truck at a speed of 50 km/h along the rehabilitated N9. . . .

81

3.25 Predicted (solid line) and measured (plus signs) PPV in the (a) x-, (b) y-, and (c) z-direction at x = {16 m, 0, 0}T next to the rehabilitated N9 as a function of the vehicle speed. Superimposed on the graphs are the measured PPVs during the passages on the middle lane (crosses). . . . . . . . . . . . . . . . . . . . . . . . . . .

82

3.26 Contour plot of the predicted vertical PPV in the free field for a passage of the truck at a speed of 50 km/h along the (a) deteriorated and (b) rehabilitated N9. The PPV is shown in mm/s. . . . . . . .

82

oct 3.27 Maximum octahedral shear strain γmax in the soil under the road at y = 90 m for a passage of the truck at a speed of 50 km/h along the deteriorated N9. . . . . . . . . . . . . . . . . . . . . . . . . . .

84

4.1

(a) Resonant pavement breaker used for rubblisation (from [168]) and (b) impact roller used for pavement breaking. . . . . . . . . .

88

4.2

Guillotine-type pavement breaker (from [121]). . . . . . . . . . . .

89

4.3

Multi-head pavement breaker. . . . . . . . . . . . . . . . . . . . . .

90

4.4

Location of the impact points (P1 to P4) on the road and measurement points in the free field during the measurement campaign in Waarschoot. . . . . . . . . . . . . . . . . . . . . . . .

92

4.5

(a) Lithological description of the soil near the measurement site in Waarschoot and dynamic soil characteristics estimated from the surface wave tests at the measurement site: (b) shear wave velocity, (c) dilatational wave velocity, and (d) material damping ratio profile. 94

4.6

Drop hammer and a slab of the road. . . . . . . . . . . . . . . . . .

96

4.7

(a) Spring-dashpot and (b) mass-spring-dashpot representation of the slab-soil system. . . . . . . . . . . . . . . . . . . . . . . . . . .

96

LIST OF FIGURES

4.8

4.9

(a) Displacement, (b) velocity, and (c) acceleration of the drop hammer for an impact on a massless undamped (light grey line), underdamped (dark grey line), and overdamped (black line) springdashpot system. Superimposed are the responses assuming the drop hammer is attached to the system (dashed lines). . . . . . . . . . .

xix

97

(a) 3D model of the slab-soil system and (b) simplified model by assuming infinite lateral extensions of the slab. . . . . . . . . . . . 101

ˆ 0, 0, ω) 4.10 (a) Real and (b) imaginary part of the dynamic stiffness S(0, of the slab-soil system computed with the layered halfspace model ˆ s , ω) computed with (thin solid line) and the dynamic stiffness S(x the 3D FE-BE model for a load at the centre xs = {0, 0, tr }T (dashed line), near the edge xs = {0.75 m, 0, tr }T (dashed-dotted line), and near the corner xs = {0.75 m, 2.0 m, tr }T (dotted line) of the slab. Superimposed on (a) is the mass inertia md ω 2 of the drop hammer (thick line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.11 Predicted (black line) and measured (grey line) (a) time history and (b) narrow band frequency spectrum of the impact force. . . . . . . 103 4.12 Typical crack pattern due to the operation of the MHB. . . . . . . 104 4.13 Cracks formed around the impact footprints. . . . . . . . . . . . . 104 4.14 The average length of apparent cracks due to one impact is about 1 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.15 Crack tip stress versus crack mouth opening displacement, total fracture energy GF , and initial fracture energy Gf (from [29]). . . . 105 4.16 The three fracture modes. . . . . . . . . . . . . . . . . . . . . . . . 106 4.17 Crack propagation velocity as a function of strain rate (from [155]). 110 4.18 Recorded time history (left) and narrow band frequency spectrum (right) of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road due to 8 consecutive impacts of the MHB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.19 Free field mobility at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road, measured with the sledge hammer impacts on the slab in its original condition (solid line), after the 8 impacts of the MHB (dashed line), and after the MHB fractured the entire length of the slab (dotted line). . . . . . . . . . . . . . . . . . . . . 112

xx

LIST OF FIGURES

5.1

Predicted time history (left) and narrow band frequency spectrum (right) of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road due to an impact of the MHB for the case where the presence of the MHB on the slab is accounted for (dashed line) and the case where it is disregarded (solid line). . 117

5.2

Axisymmetric model of the FE domain Ω1 and the exterior soil domain Ω2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.3

Axisymmetric model of the halfspace: (a) FE mesh and (b) BE mesh (circles) and CHIEF receivers (plus signs). . . . . . . . . . . 122

5.4

Real (solid line) and imaginary (dashed line) part of the vertical displacement due to a harmonic point load computed with the coupled FE-BE model (black line) and the direct stiffness method (grey line) for a receiver at (a) x = {0.1 m, 0, 0}T , (b) x = {2 m, 0, −2 m}T , and (c) x = {10 m, 0, 0}T . . . . . . . . . . . . . . . 123

5.5

Axisymmetric slab-soil model: (a) FE mesh and (b) BE mesh (circles) and CHIEF receivers at a frequency of 200 Hz (plus signs). 124

5.6

Displacement field in the slab and the soil at (a) t = 0.200 s, (b) t = 0.205 s, (c) t = 0.210 s, and (d) t = 0.230 s due to an impact of the MHB. The displacements are scaled by a factor of 500. . . . . 125

5.7

Horizontal vibration velocity at (a) t = 0.200 s, (b) t = 0.205 s, (c) t = 0.210 s, and (d) t = 0.230 s generated by an impact of the MHB. 126

5.8

Vertical vibration velocity at (a) t = 0.200 s, (b) t = 0.205 s, (c) t = 0.210 s, and (d) t = 0.230 s generated by an impact of the MHB. 127

5.9

Trace of the particle displacement at (a) {0.1 m, 0, 0}T , (b) {3 m, 0, 0}T , (c) {0.1 m, 0, −4 m}T , and (d) {3 m, 0, −4 m}T due to an impact of the MHB. The motion is shown from the unloaded state (black) to t = 0.30 s (grey). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

oct 5.10 Peak octahedral shear strain γmax due to an impact of the MHB. . 128

5.11 Time history (left) and narrow band frequency spectrum (right) of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road due to an impact of the MHB predicted with the linear FE-BE model (black line) and compared with the experimental results (grey line). . . . . . . . . . . . . . . . . . . . . 129 5.12 Typical stress-strain relationship for soil under large and small deformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

LIST OF FIGURES

xxi

5.13 (a) Modulus reduction and (b) material damping curves for sandy soils (from [223]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 oct 5.14 (a) Peak octahedral shear strain γmax in the updated model by means of the equivalent linear analysis with the corresponding (b) modulus reduction µ/µ0 and (c) material damping ratio β. . . . . 133

5.15 Displacement field in the slab and the soil at (a) t = 0.200 s, (b) t = 0.205 s, (c) t = 0.210 s, and (d) t = 0.230 s due to an impact of the MHB. The displacements are scaled by a factor of 500. . . . . 134 5.16 Horizontal vibration velocity at (a) t = 0.200 s, (b) t = 0.205 s, (c) t = 0.210 s, and (d) t = 0.230 s generated by an impact of the MHB. 135 5.17 Vertical vibration velocity at (a) t = 0.200 s, (b) t = 0.205 s, (c) t = 0.210 s, and (d) t = 0.230 s generated by an impact of the MHB. 136 5.18 Trace of the particle displacement at (a) {0.1 m, 0, 0}T , (b) {3 m, 0, 0}T , (c) {0.1 m, 0, −4 m}T , and (d) {3 m, 0, −4 m}T due to an impact of the MHB. The motion is shown from the unloaded state (black) to t = 0.50 s (grey). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.19 Time history (left) and narrow band frequency spectrum (right) of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road due to an impact of the MHB, predicted with the linear (solid black line) and equivalent linear (dashed-dotted line) model and compared with the experimental results (grey line). 138 5.20 Narrow band frequency spectrum of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m due to an impulsive unit load on the soil surface with the profile in table 4.1 (solid line) and the same soil profile with a material damping ratio of 0.01 for the third layer and below (dashed line). . . . . . . . . . . . . . . . . . 139 6.1

(a) Coupled inelastic and elastic subdomains Ωi ∪ Ωe , (b) the inelastic region and a part of the elastic domain Ω1 coupled to the local absorbing boundary Γv , and (c) the elastic halfspace Ωs . . . . 145

6.2

Finite element domain Ω1 and the absorbing boundary conditions Γv with the position of the path Σ. . . . . . . . . . . . . . . . . . . 149

6.3

Time history of the vertical velocity at (a) x = {1 m, 0, 0}T , (b) x = {2 m, 0, −2 m}T , (c) x = {4 m, 0, −5 m}T , and (d) x = {10 m, 0, 0}T due to an impact of the MHB on the soil surface computed with the FE and the accompanying BE model (black line) and the direct stiffness formulation assuming a frequency dependent damping (grey line). . . . . . . . . . . . . . . . . . . . . . . . . . . 150

xxii

LIST OF FIGURES

6.4

Schematic representation of a rigid structure rocking on a yielding soil (from [90]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.5

Moment-rotation curve, predicted using the FE model (black line) and the result of Gazetas and Apostolou [90] (grey line). . . . . . . 151

6.6

Time history (left) and narrow band frequency spectrum (right) of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road due to an impact of the MHB computed with the non-linear model for a dense sand with φ′cv = 37° and ψ = 11° (solid line) and a loose sand with φ′cv = 30° and ψ = 0 (dashed line). 153

6.7

Time history (left) and narrow band frequency spectrum (right) of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road due to an impact of the MHB computed with the non-linear model for slab radii rr = 1.425 m (solid line) and rr = 0.3 m (dashed line). . . . . . . . . . . . . . . . . . . . . . 155

6.8

Permanent deformation of the soil due to an impact of the MHB for slab radii (a) rr = 1.425 m and (b) rr = 0.3 m. The displacements are scaled by a factor of 20. . . . . . . . . . . . . . . . . . . . . . . 156

6.9

Displacement field in the slab and the soil at (a) t = 1.0 s, (b) t = 1.003 s, (c) t = 1.008 s, (d) t = 1.013 s, (e) t = 1.033 s, (f) t = 1.071 s, (g) t = 1.084 s, and (h) t = 1.500 s due to an impact of the MHB. The displacements are scaled by a factor of 100. . . . . 157

6.10 Predicted normal stress σzz in the slab and the soil (a) before and (b) after the impact. . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.11 Horizontal vibration velocity at (a) t = 1.003 s, (b) t = 1.008 s, (c) t = 1.013 s, (d) t = 1.033 s, (e) t = 1.071 s, and (f) t = 1.084 s generated by an impact of the MHB. . . . . . . . . . . . . . . . . . 159 6.12 Vertical vibration velocity at (a) t = 1.003 s, (b) t = 1.008 s, (c) t = 1.013 s, (d) t = 1.033 s, (e) t = 1.071 s, and (f) t = 1.084 s generated by an impact of the MHB. . . . . . . . . . . . . . . . . . 160 6.13 Trace of the particle displacement at (a) {0.1 m, 0, 0}T , (b) {3 m, 0, 0}T , (c) {0.1 m, 0, −4 m}T , and (d) {3 m, 0, −4 m}T due to an impact of the MHB. The motion is shown from time t = 1.000 s (black) till t = 1.300 s (grey). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.14 Elastic region (grey) and plastic region (black) in the FE domain due to an impact of the MHB. Superimposed is the position of the path Σ (thick line). . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

LIST OF FIGURES

xxiii

oct 6.15 (a) Peak octahedral shear strain γmax due to an impact of the MHB estimated using the non-linear model and its (b) elastic and (c) plastic part. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 √ 6.16 Square root of the second deviatoric stress invariant J2 versus the mean pressure −I1 /3 at (a) x = {0.1 m, 0, 0}T , (b) x = {1 m, 0, 0}T , (c) x = {1.5 m, 0, 0}T , (d) x = {0.1 m, 0, −0.5 m}T , (e) x = {1 m, 0, −0.5 m}T , (f) x = {1.5 m, 0, −0.5 m}T , (g) x = {0.1 m, 0, −1.5 m}T , (h) x = {1 m, 0, −1.5 m}T , (i) x = {1.5 m, 0, −1.5 m}T due to an impact of the MHB. The stress path is shown from time t = 1.000 s (black) till t = 1.100 s (grey). Superimposed is the yield surface (thick grey line). . . . . . . . . . 163

6.17 Trace of the particle displacement at (a) {0.1 m, 0, 0}T , (b) {1.5 m, 0, , 0}T , (c) {3 m, 0, 0}T , (d) {0.1 m, 0, −2 m}T , (e) {1.5 m, 0, −2 m}T , (f) {3 m, 0, −2 m}T , (g) {0.1 m, 0, −4 m}T , (h) {1.5 m, 0, −4 m}T , and (i) {3 m, 0, −4 m}T due to an impact of the MHB computed with the linear (solid line), equivalent linear (dashed-dotted line), and non-linear model (dashed line). . . . . . . . . . . . . . . . . . . . . 164 oct 6.18 Peak octahedral shear strain γmax due to an impact of the MHB computed using (a) the linear, (b) equivalent linear, and (c) nonlinear model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.19 Time history (left) and narrow band frequency spectrum (right) of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road due to an impact of the MHB, predicted with the linear (solid line), equivalent linear (dashed-dotted line), and non-linear model (dashed line). . . . . . . . . . . . . . . . . . . 166 6.20 Time history (left) and narrow band frequency spectrum (right) of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road due to an impact of the MHB predicted with the non-linear model (dashed line) and compared with the experimental results (grey line). . . . . . . . . . . . . . . . . . . . . 168 6.21 Time history (left) and narrow band frequency spectrum (right) of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road due to an impact of the MHB predicted with the non-linear model for the soil profile presented in table 4.1 but damping ratios of 0.01 and 0.005 for the second and deeper layers, respectively (dashed line), compared with the experimental results (grey line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 A.1 Three-dimensional model of a two-axle truck. . . . . . . . . . . . . 199

xxiv

LIST OF FIGURES

B.1 (a) Cone resistance qc , (b) sleeve friction fs , and (c) friction ratio Rf of the soil near the measurement sites as a function of depth, obtained from six CPTs. Superimposed are the average values (thick lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 B.2 (a) Unit weight of the soil as a function of depth, estimated from six CPTs and (b) the computed effective vertical stress. Superimposed on figure (a) is the average value (thick line) and the values recommended by the DOV (dots). . . . . . . . . . . . . . . . . . . 206 B.3 Relative density of the soil as a function of depth estimated using the method suggested by Kulhawy and Mayne [145] (dashed line) and Jamiolkowski et al. [127] (solid line). . . . . . . . . . . . . . . . 207 B.4 Definition of the peak friction angle φ′p and the critical state friction angle φ′cv (from [145]). . . . . . . . . . . . . . . . . . . . . . . . . . 207 B.5 Critical state friction angle φ′cv of the soil as a function of depth estimated using the methods suggested by Meyerhof [180] (solid line), Eurocode 7 [76] (plus signs), and the DOV (dots). . . . . . . 208 B.6 Peak friction angle φ′p of the soil as a function of depth estimated using the methods suggested by Robertson and Campanella [205] (solid line), Durgunoglu and Mitchell [74] (dashed line), and Kulhawy and Mayne [145] (dotted line) (a) without and (b) with the reduction according to equation (B.8). . . . . . . . . . . . . . . 210 B.7 Dilatancy angle of the soil as a function of depth estimated according to Kulhawy and Mayne [145] (thick line), Lee et al. [156] with 0% (thin solid line) and 10% of silt content (dashed line), and Bolton’s relation [36] (dotted line). . . . . . . . . . . . . . . . . . . 212 B.8 Shear wave velocity as a function of depth measured with the SCPT in Lovendegem [15] (thick solid line), surface wave test in Waarschoot [20] (thick dashed line), and SASW test in Lovendegem [151] (thick dotted line) and estimated from the CPT data using the method suggested by Robertson [203] (thin solid line), Hegazy and Mayne [107] (thin dashed line), and Mayne [178] (thin dotted line). 214 B.9 (a) Experimental dispersion curve determined from the active and passive surface wave tests at the site in Waarschoot in terms of the wavelength and (b) initial shear wave velocity profile (from [20]). . 215 B.10 (a) Unit weight, (b) critical state friction angle, and (c) dilatancy angle of the soil near the measurement sites as a function of depth estimated from the CPT data. Superimposed are the averaged values over the thickness of each layer (thick lines). . . . . . . . . . 215

List of Tables 1.1

Experimental data found in the literature on the vertical PPV generated by pavement breaking. . . . . . . . . . . . . . . . . . . .

9

Experimental data on the vertical PPV generated by dynamic soil compaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.3

Organisation of the text. . . . . . . . . . . . . . . . . . . . . . . . .

17

2.1

Updated model parameters of the Volvo FL180. . . . . . . . . . . .

37

2.2

Eigenfrequencies and eigenmodes of the 3D vehicle model of the Volvo FL180 truck. . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

Dynamic soil characteristics at the measurement site next to the N9 road in Lovendegem. . . . . . . . . . . . . . . . . . . . . . . . . . .

67

1.2

3.1

4.1

Dynamic soil characteristics at the measurement site in Waarschoot. 95

4.2

Experimental data on the effect of strain rate on the fracture energy of concrete. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.1

Density ρ, critical state friction angle φ′cv , and dilatancy angle ψ of the soil near the measurement site estimated from the CPT data and Young’s modulus E and Poisson’s ration ν obtained from in situ geophysical tests. . . . . . . . . . . . . . . . . . . . . . . . . . 152

B.1 Soil classification and geotechnical soil properties based on CPT data (proposal of the Database of the Subsoil of Flanders [188] for the national annex of Eurocode 7 - part 2). . . . . . . . . . . . . . 204

xxv

xxvi

LIST OF TABLES

B.2 Intrinsic soil variables QCPT and RCPT proposed by Lee et al. [156]. 211 B.3 Unit weight, critical state friction angle, and dilatancy angle of the soil near the measurement sites estimated from the CPT data. . . 215

List of Symbols The following list provides an overview of symbols used throughout the text. Symbols which are only used locally are not included. The physical meaning of the symbols is explained in the text. Vectors, matrices and tensors are denoted by bold characters. The general symbols, conventions and acronyms are collected in the first two sections. The remaining symbols are categorised in sections referring to the chapters where they are first introduced.

General symbols and conventions ˙ ¨

ˆ ˜

−1

T

0 f i I k t T (x, y, z) δ ( ) ω

first order time derivative of the variable second order time derivative of the variable discretised vector variable in the frequency domain variable in the frequency-wavenumber domain inverse of a matrix transpose of a matrix zero matrix frequency imaginary unit i2 = −1 identity matrix wavenumber time time period Cartesian coordinates Dirac delta function circular frequency

Acronyms 2D

Two-dimensional

xxvii

xxviii

2.5D 3D ABC ACI APL ARAN BE BEMFUN BRRC CHIEF COV CPT CRCP DC DLC DOF DOV EDT FE FE-BE FRF IRI IRRE ISO JPCP JRCP MHB P-wave PA PCC PIARC PML PPV PSD PSR RMS S-wave SASW SCPT

LIST OF SYMBOLS

Two-and-a-half-dimensional Three-dimensional Absorbing Boundary Condition American Concrete Institute in French: Analyseur de Profil en Long, longitudinal profile analyser Automatic Road Analyser Boundary Element Matlab Toolbox for Boundary Elements in Elastodynamics Belgian Road Research Centre Combined Helmontz Integral Equation Formulation Coefficient Of Variation Cone Penetration Test Continuously Reinforced Concrete Pavement Direct Current Dynamic Load Coefficient Degree Of Freedom in Dutch: Databank Ondergrond Vlaanderen, database of the subsoil of Flanders ElastoDynamics Toolbox Finite Element Finite Element-Boundary Element Frequency Response Function International Roughness Index International Road Roughness Experiment International Organization for Standardization Jointed Plain Concrete Pavement Jointed Reinforced Concrete Pavement Multi-Head pavement Breaker Primary wave (dilatational wave) Peak Acceleration Portland Cement Concrete Permanent International Association of Road Congresses Perfectly Matched Layer Peak Particle Velocity Power Spectral Density Present Serviceability Rating Root Mean Square Secondary wave (shear wave) Spectral Analysis of Surface Waves Seismic Cone Penetration Test

LIST OF SYMBOLS

SDOF SQP SSI TRRL

xxix

Single Degree Of Freedom Sequential Quadratic Programming Soil-Structure Interaction Transport and Road Research Laboratory

Numerical estimation of dynamic vehicle loads v c

a ê (ω) a ˆm (ω) a ˆv (ω) C cp i ct i fNyq fobj fs g ˆ av uc h ˆ guc h Ibi K KD kpi kti li lm lz M mai mb 2p r u uai ub uci

subscript referring to the vehicle subscript referring to the contact points measured accelerations of the vehicle predicted accelerations of the vehicle at the measurement points predicted accelerations of the vehicle model damping matrix damping coefficients of the primary suspension on each side of axle i damping coefficients of the tyre(s) on each side of axle i Nyquist frequency objective function sampling frequency dynamic wheel loads frequency response function of the vehicle DOFs acceleration frequency response function of wheel loads mass moments of inertia of the vehicle body with respect to axis i stiffness matrix dynamic stiffness matrix stiffness of the primary suspension on each side of axle i stiffness of the tyre(s) on each side of axle i position of axle i in the y-direction relative to the centre of gravity of the vehicle body baselength of the rectangular moving average window baselength of the rolling line segment mass matrix mass of axle i mass of the vehicle body tyre footprint length residual vector displacement vertical displacement of axle i vertical displacement of the vehicle body vertical displacement of contact point i

xxx

uw/ri v 2waci 2waki 2wp 2wt x x0 θb λ ϕai ϕb

LIST OF SYMBOLS

road profile of wheel track i vehicle speed distance between the dampers on axle i distance between the springs on axle i distance between the chassis beams track width of the vehicle design variables initial estimate of design variables rotation of the vehicle body in the plane of motion wavelength rotation of axle i rotation of the vehicle body in the plane perpendicular to the plane of motion

Prediction of traffic induced ground vibrations B Cp Cs Ec fcm fr h Krr Ksrr L Mrr Nr Ns S T Tq tr Tr ts u U ucz uG ij ur

half of the width of road slab dilatational wave velocity shear wave velocity Young’s modulus of concrete mean compressive strength of concrete for the standard cylindrical sample external load vector of the road road-soil transfer function stiffness matrix of the road dynamic stiffness matrix of the soil half of the length of road slab mass matrix of the road finite element shape function boundary element shape function selection matrix traction boundary element system matrix stress transfer matrix road thickness boundary element traction transfer matrix boundary element tractions displacement vector displacement boundary element system matrix displacement of the centre of the road cross section Green’s displacements displacement field in the road

LIST OF SYMBOLS

Ur us x xs βcy βp βs Γrσ Γsσ Γs∞ ν ρ ρc Σrs ω ˜ Ωr Ωs

xxxi

boundary element displacement transfer matrix displacement field in the soil receiver location source location rotation of the centre of the road cross section hysteretic material damping ratio for P-waves hysteretic material damping ratio for S-waves free boundary of the road domain free boundary of the soil domain outer boundary of the soil domain Poisson’s ratio density density of concrete road-soil interface circular frequency at the source road domain soil domain

Pavement breaking, impact load estimation, and fracturing

i

AF c da df Eimp fd g Gf GF h k m md S t0 t1 , t2 , t3 , . . . td u

superscript referring to mode i fracture area of the fractured surface damping of the road-soil system maximum size of aggregates impact footprint diameter impact energy impact force acceleration of gravity initial fracture energy total fracture energy drop height stiffness of the road-soil system equivalent mass of the road-soil system mass of the drop hammer dynamic stiffness of the halfspace model of road-soil system moment of the first impact between the drop hammer and the road moments of consecutive impacts between the drop hammer and the road impact duration vertical displacement of the road

xxxii

ud uD zz v0 w/c βc ǫ1 , ǫ2 , ǫ3 ǫ˙ ξ

LIST OF SYMBOLS

vertical displacement of the drop hammer vertical Green’s displacement due to a vertical disc load impact velocity of the drop hammer water/cement ratio hysteretic material damping ratio for S- and P-waves in concrete principal strains strain rate viscous damping ratio

Linear prediction of ground vibrations due to pavement breaking 1 2

M

Bb Db KD r Tc α γeff oct γmax µ/µ0 Γ1σ Γ2σ Γ2∞ Σ12 Ω1 Ω2

subscript referring to DOFs inside the finite element domain subscript referring to DOFs on the finite element-boundary element interface subscript referring to multi-head breaker radius of the axisymmetric finite element-boundary element interface depth of the axisymmetric finite element-boundary element interface dynamic stiffness of the road-soil system constraint equation coefficients for corner displacements ratio of maximum effective shear strain to peak octahedral shear strain maximum effective shear strain peak engineering shear strain on an octahedral plane ratio of updated to initial shear modulus free boundary of the interior domain free boundary of the exterior soil domain outer boundary of the exterior soil domain interior-exterior domains interface interior domain exterior soil domain

Non-linear prediction of ground vibrations due to pavement breaking Bb Bs C Db Ds I1

radius of imaginary path Σ radius of interior domain Ω1 Rayleigh damping matrix depth of imaginary path Σ depth of interior domain Ω1 first stress invariant

LIST OF SYMBOLS

J2 rr tp up Γeσ Γe∞ Γiσ Γv η Σ Σie φ ψ ω0 Ωe Ωi

xxxiii

second deviatoric stress invariant radius of axisymmetric model of slab tractions along path Σ displacements along path Σ free boundary of the elastic domain outer boundary of the elastic domain free boundary of the inelastic domain local absorbing boundary normalised material damping ratio imaginary path inside Ω1 inelastic-elastic domains interface internal friction angle of soil dilatancy angle of soil normalisation frequency elastic domain inelastic domain

Chapter 1

Introduction 1.1 1.1.1

Motivation and problem outline Environmental impact of vibrations

People’s concern about ground-borne vibrations generated by human activities goes back more than a century ago [119]. The level of concern has increased over time due to industrialisation and urbanisation. A survey study [244] in 1984 by the Transport and Road Research Laboratory (TRRL) has shown that 55% of occupants who were interviewed were bothered by traffic vibrations because they felt that it could damage their homes while 20% alleged vibration damage had already occurred. 27% of inhabitants avoided using a front room or bedroom because of the problem and 23% were considering moving house. Surprisingly only 11% had complained to the authorities. A recent study in Japan (reported by Xia et al. [259]) shows that the number of complaints about road traffic induced vibrations is four times more than the number of complaints about railway induced vibrations. Generally, people are mostly annoyed by traffic noise rather than by vibration from traffic; however, the numbers of people who are seriously bothered by vibration and noise are similar [24]. Most of the people worry about vibration because they think it can cause damage to their properties [250]. Apart from possible damages to buildings [23, 112, 247, 250], historical monuments [57,65,138,200,246,248], and annoyance [2], environmental vibrations can cause malfunctioning of sensitive equipment such as electron microscopes and microelectronics laboratories [96, 260]. Thus, norms and guidelines recommend vibration limits to avoid damage to buildings [70, 124, 232], discomfort to people [10,39,122,123,233], and malfunctioning of sensitive equipment [234]. These limits are usually imposed on the peak particle velocity (PPV) of the response. The

1

2

INTRODUCTION

threshold of perception is of the order of 0.15 − 0.4 mm/s PPV while it may become unacceptable above a level of 1 − 2.5 mm/s [250]. Depending on the type of structure and excitation frequency, standards consider a wide range of PPVs (3 mm/s to 100 mm/s) as a building damage threshold [138, 187]. Groundborne vibrations might also cause fatigue damage of buildings or damage due to differential settlements. Vibration levels at higher levels in buildings are often larger than those at the foundations [245,250]. In order to reduce the level of traffic induced vibrations, speed limiting measures are imposed, heavy goods vehicles are diverted, pavements are rehabilitated or replaced, and passive vibration reduction measures are implemented [6, 57, 239, 250]. The industrialisation has also resulted in the appearance of machineries and powerful construction equipment. Industrial machineries such as weaving looms, tufting machines, printing presses, and hammers, and construction equipment like pile drivers, pavement breakers, compactors, wrecking balls, excavators, and tunnel boring machines are other sources of environmental vibration nuisance. Some construction activities such as tunnelling, demolition, and pavement breaking are often performed near or in residential areas. These activities generate vibrations that are much above the annoyance limit up to several tens of metres away from the source. A recent study in Japan on the changes of the number of complaints due to different sources of environmental vibrations shows that the number of complaints about construction activities has doubled from 1976 to 2007 and composes about 60% of the total number of complaints [146]. In the same period, the number of complaints about industrial activities has been reduced by a half while the number of complaints about traffic induced vibrations has remained almost unchanged. Since construction activities usually have a short duration - e.g. a few seconds for demolition, several minutes for pavement breaking, and a few days for tunnelling - annoyance is of relatively less concern in comparison to possible damages to the buildings and buried and surface infrastructures in their immediate vicinity. This can result in the employment of an alternative method [186], for example for the case of pile driving, from impact pile driving to bottom driven piles [126] or by preboring the subsurface firm soils, utilizing an open end pile, and utilizing smaller diameter piles [101].

1.1.2

Road rehabilitation as a vibration reduction measure

In 2011, the worldwide network of paved roads was about 26.6 million kilometres [110]. At least 2.1 million kilometres of these roads (8%) are rigid (Portland cement concrete (PCC)) roads. Among other countries, the concrete roads are very common in the USA, China, the Czech Republic, and Belgium. Concrete pavements can be divided into three categories: Jointed Plain Concrete Pavement (JPCP), Jointed Reinforced Concrete Pavement (JRCP), and Continuously Reinforced Concrete Pavement (CRCP) of which JPCP is the most common type.

MOTIVATION AND PROBLEM OUTLINE

3

Belgium has a network of about 154000 kilometres of paved roads where about 26000 kilometres (17%) are in concrete [110]. The history of these concrete roads goes back to 1925 when the Avenue de Lorraine in Brussels was constructed [100]. The early JPCPs were 12 − 15 cm thick with a joint spacing between 10 and 20 metres. Problems over the years with slab cracking and faulting, led to changes in the standard Belgian concrete pavement design. Today, the JPCPs are about 10 cm thicker and the slab lengths are between 4 and 5 metres [100]. Population growth and globalisation have increased the size and number of vehicles and thus accelerated deterioration of pavements. This resulted in stronger groundborne vibrations, hence nowadays, road authorities in Flanders receive more complaints from inhabitants of buildings located near jointed concrete pavements than in the past [197]. In addition to problems due to traffic noise and vibration in the vicinity of these roads, road users complain frequently about poor ride quality. Therefore, the deteriorated concrete pavements are reconstructed or repaired. In the reconstruction procedure - which in the case of severely deteriorated road might be used - the concrete pavement is broken into smaller pieces for easier removal and a new pavement is constructed. The repair of surface damage is applied in varying degrees depending on the level of deterioration. For texture restoration, diamond grinding, jet blasting, or reactive resin coating can be applied. In a pavement with faulted joints1 , dowelling and stitching of joints and cracks and injection of expansion resin are used [25, 56, 87]. Excessive deterioration and severe faulting generally warrant total rehabilitation. In this case, the old pavement is overlaid with bituminous materials. To prevent that the joints are reflected by cracks in the new surface (reflective cracking), the existing pavement is cracked [222] or rubblised [141]. The cracking operation generates a high level of ground vibrations which can be potentially damaging to buried pipelines [190] and infrastructure beside the roads [12]. Therefore, a priori prediction of ground vibrations generated by pavement breaking is very valuable for practical applications. Three aspects of ground-borne vibrations in the vicinity of concrete roads are therefore of interest and addressed in this thesis: 1) the prediction of ground vibrations due to traffic on deteriorated jointed concrete pavements, 2) the level of vibration reduction gained by road rehabilitation, and 3) the prediction of vibrations generated by pavement breaking. 1 Stepping or step faulting is a difference in elevation across a joint or crack of undowelled jointed concrete pavements. The approach slab is usually higher than the leave slab due to pumping which is the most common faulting mechanism. Other causes of faulting are slab settlement, curling, and warping [201].

4

1.2

INTRODUCTION

State of the art and further needs

In recent years, a large amount of research has been performed on environmental vibrations [91, 227, 230, 231, 265]. These studies cover both theoretical and experimental aspects and include a wide range of vibration sources such as earthquake, blasting, road traffic, railway traffic, machineries, and construction activities. In the following, the literature on ground-borne vibrations due to road traffic and pavement breaking is reviewed and further studies required to address the three problems outlined above are set out.

1.2.1

Traffic induced vibrations

Experimental studies Intrinsic complexities in the numerical prediction of traffic induced ground-borne vibrations resulted in mostly experimental studies for about half a century. It has been generally appreciated that traffic induced vibrations are mainly due to local road irregularities. Based on experimental observations, Watts [250] has proposed an empirical formula to compute the maximum vertical PPV due to the passage of a vehicle over a local unevenness as a function of the unevenness height or depth, vehicle speed, distance from the unevenness, and soil properties. Watts also states that the length and detailed shape of surface irregularities are not affecting the ground-borne vibrations generated by vehicles, although a joint in a concrete road gives results markedly different from other road surface features. Hajek et al. [98] state that joint faults (or stepped transverse cracks) exceeding about 4 mm and potholes or bumps more than 25 mm in depth or height and about 150 mm long can overshadow the vibrations generated by global road unevenness. The different criteria for faults and potholes stem from the fact that faults excite all wheels of an axle simultaneously. Al-Hunaidi et al. [5] explain that the dynamic wheel load resulting from a road irregularity is composed of an initial impact load and an oscillating force from the subsequent axle hop of the vehicle. The impact generates peak ground vibrations that are predominant at the natural frequencies of the soil. House [112] discusses different aspects of traffic induced vibrations and states that the road-soil interaction in the case of rigid pavements can affect traffic induced vibrations and therefore should be studied in more detail. Watts [245] suggests that a concrete slab might move as a whole as a result of the passage of a heavy vehicle and this movement with consequent secondary impacts on the road base can generate a large level of vibration. Watts [250] has observed a linear increase of the vertical PPV with vehicle speed, gross vehicle weight, and profile height. Decreasing the load carried on a particular vehicle, however, does not necessarily reduce the PPV, which has also been

STATE OF THE ART AND FURTHER NEEDS

5

reported by Al-Hunaidi and Tremblay [7]. In some cases an empty lorry can produce higher levels than when fully laden. In addition, the response of a soft soil (peat) can be up to two orders of magnitude larger than a very hard ground (chalk rock). Watts and Krylov [252] have performed experimental studies on ground-borne vibrations generated by different types of vehicles driving over road humps and speed cushions. Based on these studies, guidance is provided on the siting of these surface profiles to avoid disturbance for different types of soil. Regarding the effect of road rehabilitation on the reduction of traffic induced vibrations, Al-Hunaidi and Tremblay [7] have found that road resurfacing can result in the reduction of vibration levels by a factor of at least 6. In an earlier study [210] into the effect of road resurfacing on traffic induced vibrations, it was observed that the level of vibrations reduces by about 60% with each unit improvement of the road smoothness on a 0 to 5 scale defined in the Present Serviceability Rating (PSR) index. In the last few years, a relatively small number of experimental studies on road traffic induced vibrations has been reported. As an example, the work of Sutherland and Cenek [236] and Cenek [47] can be mentioned where experimental and theoretical studies have been performed to assess vibration levels arising from road traffic and construction activities on two proposed new roadways in New Zealand. The objective of these studies was to identify potential adverse vibration effects at an early stage so that vibration reduction measures could be advanced. They found that vibrations induced by traffic on new roads are unlikely to be perceived by occupants of buildings as long as the maximum roughness of the road surface is maintained to the limits recommended by the national standards. Numerical prediction models The prediction of ground-borne vibrations due to road traffic is usually performed in two stages: first, estimation of the dynamic wheel loads and second, application of the estimated loads on the road and prediction of ground-borne vibrations. The dynamic wheel loads are mainly due to road irregularities while the contribution of running engine vibration, wheel imbalance, and impact forces of the individual parts of the tyre tread is usually small [92,98]. The prerequisites for the estimation of the dynamic wheel loads are thus the road unevenness, the vehicle model, and the vehicle speed. In some of the existing numerical prediction models, a statistical description of global road unevenness by a power spectral density (PSD) function has been employed [54, 103, 116]. In practice, however, peak wheel loads are mainly caused by local road unevenness [5,249]. Lombaert et al. [162] and Pyl et al. [199] consider the deterministic geometry of a traffic plateau and a limited stretch of a concrete pavement with a few number of faulted joints. The only practical way to obtain

6

INTRODUCTION

the profile of a rather long stretch of a road (e.g. a few hundreds of metres or more) is by means of a road profiler. The road profilers are usually employed to evaluate the ride quality and condition of a pavement. They obtain a representation of the true profile within a limited range of wavelengths where some filtering effects can also be present [216]. The wavelengths of interest in the monitoring of a pavement - e.g. 1.2 − 30 m for the International Roughness Index (IRI) - do not cover the whole range of relevant wavelengths for traffic induced vibrations - e.g. 0.2 − 30 m. Two very common vehicle models used for the estimation of the dynamic wheel loads are the ‘quarter-car’ and ‘half-car’ or pitch-plane vehicle models. The quarter-car model has two degrees of freedom (DOFs) corresponding to the vertical motion of the sprung and unsprung masses. It can represent the axle hop mode and the bounce mode. The half-car model contains four DOFs representing the vertical motion of the unsprung masses and the vertical motion and in-plane rotation of the sprung mass. The quarter-car and half-car models disregard the coupling between the left and right side of the vehicle and, consequently, the sprung and unsprung roll modes. Gillespie and Karamihas [92] have shown that this can result in a nominal error of 20% in the root-mean-square (RMS) value of the estimated dynamic wheel loads in comparison with a three-dimensional (3D) vehicle model. The suspension system and tyres are usually modelled as linear spring-dashpots. This arises from the fact that the non-linear behaviour of the tyres and suspension systems is complex, usually not provided by the manufacturer, difficult to identify experimentally, and there is a large variation between different vehicles. When the dynamic wheel loads have been estimated, they are applied to the road in order to compute the radiated wave field in the soil. Many researchers disregard the presence of the road so the wheel loads are immediately applied to the soil surface. Hao and Ang [103] have developed an analytical model to predict the PSD of traffic induced ground vibrations. They use the quarter-car vehicle model and consider the road-soil system as a homogeneous halfspace where only Rayleigh waves are accounted for. To account for the road filtering effects, rather large damping ratios (0.1-0.3) which are varying with distance from the source have been considered. Hunt [116], Krylov [139,140], and Hung and Yang [115] have also used the halfspace model for the prediction of ground vibrations generated by road traffic. Hao et al. [104] compare the free field mobilities obtained experimentally by hammer impacts on the road and on the soil and conclude that the road filtering effect is significant. They disregard, however, the fact that the soil deformations for an impact on the soil are inelastic so the differences in the mobilities are not solely due to the road filtering effect. Aubry et al. [17] and Lombaert et al. [162] account for the road-soil interaction by representing the road as an infinitely long beam on a layered elastic halfspace. Lombaert et al. [159, 162] use the half-car vehicle model to calculate the dynamic wheel loads due to a local unevenness. Next, the road-soil interaction problem is solved using an efficient two-and-a-half dimensional (2.5D) procedure [102, 118] in


7

the frequency-wavenumber domain. It is based on a substructure method using a boundary element (BE) method for the soil and an analytical beam model for the road. The dynamic reciprocity theorem is subsequently employed to compute ground vibrations due to moving wheel loads. The model has been successfully validated by several experiments on continuous asphalt roads [160, 161]. In these experiments, ground vibrations as well as the response of the axles of two test trucks during the passage of the trucks on an artificial unevenness have been measured simultaneously. Using this model, Lombaert et al. [163] have observed a significant influence of soil stratification on the road-soil transfer functions and, consequently, traffic induced ground-borne vibrations. Schevenels et al. [220] have also used this model and studied the effect of ground water table on traffic induced ground-borne vibrations. It is found that a moderate seasonal variation of the depth of the ground water table has an insignificant effect on peak ground vibrations generated by road traffic which is in agreement with the experimental observations of Al-Hunaidi and Tremblay [7]. Fran¸cois et al. [85] have developed a general 2.5D coupled finite element-boundary element (FE-BE) methodology for the computation of the dynamic interaction between a longitudinally invariant structure and a layered halfspace. They model the road-soil system as an infinitely long flexible plate on the surface of a halfspace. Mhanna et al. [181] use the half-car vehicle model to estimate dynamic vehicle loads due to passage over a traffic plateau. The loads are then introduced into a FLAC3D model - which is based on the finite difference method - to predict ground vibrations. Pyl et al. [197–199] have used the numerical model developed by Lombaert et al. [162] and coupled it to a receiver model to predict traffic induced vibrations in a building. The model has been validated by measurement of the response of a test truck and ground and building vibrations beside a jointed concrete pavement and a traffic plateau. For the case of concrete pavement, the height of faulted joints are obtained by surveying and modelled by step functions. To account for tyre envelopment, the wavenumbers larger than 1.7 m−1 are filtered out. This corresponds to a large tyre footprint of 0.6 m that is used to avoid high frequency components in the predictions. They observed that the truck response is independent of the vehicle speed for the passages on the joints, whereas, for the passages on the traffic plateau, the vibration increases with the vehicle speed. A joint with a height of 7 mm causes a larger PPV in the free field than the traffic plateau with a height of 54 mm. Ju [128] uses a finite element model to study traffic induced vibrations due to global road unevenness. In order to reduce building vibrations generated by moving trucks, it is suggested to reduce road roughness, to slow down trucks, and to avoid the coincidence of the first natural frequency of buildings and the vertical natural frequency of trucks.

8

INTRODUCTION

Further needs The above review of the literature reveals that the road unevenness, vehicle properties, road characteristics, and soil profile are important parameters in the problem of traffic induced ground-borne vibrations. The large variability of these parameters results in the shortcoming of empirical prediction models. Since local road irregularities are responsible for peak ground vibrations, the deterministic geometry of road unevenness is required especially in the case of roads with local distresses such as faulted concrete pavements. Practically, this can only by obtained by road profiling. In order to simultaneously account for the unevenness of the left and right wheel tracks, a 3D vehicle model should be employed. This model is more accurate than the common quarter-car and half-car vehicle models where an error of 20% in the RMS of estimated dynamic wheel loads and, consequently, a larger error in the peak loads has been observed [92]. Most of the existing prediction models disregard or only approximately account for the presence of the road and thus road-soil interaction. The model developed by Lombaert et al. [162] considers the road-soil interaction by assuming the road as an infinitely long beam on a layered elastic halfspace. For jointed concrete pavements, the discontinuity and flexibility of the slabs violate the underlying assumptions of the beam model. Therefore, the road-soil interaction must be studied in more detail for jointed concrete pavements. The reduction of ground vibrations due to road rehabilitation must be further investigated. This should preferably be carried out in such a way that the influence of other affecting parameters such as vehicle and soil characteristics on the generated ground vibrations is diminished or minimised.

1.2.2

Vibrations generated by pavement breaking

Most of the literature on construction induced vibrations has been dedicated to vibrations due to impact and vibratory pile driving [73, 136, 171, 174]. Theoretical studies on ground vibrations due to pavement breaking are not found in the literature and experimental data are rather limited. The vibration level for new cracking and seating projects can therefore only be estimated from limited empirical data. This lack of information results in fear for architectural/structural damage and, consequently, a limited use of the cracking and seating technique near residential areas, buried pipelines, and surface installations. In densely populated regions like Belgium, this limitation can be a major drawback. Ferahian and Hurst [78] are among the first researchers who have reported measurements on ground vibrations due to the operation of a pavement breaker.


Author

md [kg]

Ferahian and Hurst [78] 500

h [m]

9

r PPV Remarks and (label in figure 1.1) [m] [mm/s]

3

1 12

7.6 1.4

Measurement on the foundation wall of a building. (F)

Ames et al. [12]

NA

NA

3 12

73 7

EMSCO pavement breaker. (A)

Wiss [256]

NA

1.5

10

8

(W)

Dowding [73]

500

3.5

10

2.5

(D)

Hendriks [108]

NA

NA

3

73

Data might be taken from [12]. (H)

Jones & Stokes [235]

5900

1.2

12 27 34 63

32 11 7.4 2.1

Walker Megabreaker 8-13000. (J)

Lak et al. [147, 153]

600

1.8

4 8 16 32 64

45 9 3 1.8 0.6

Multi-head breaker. (L1)

Lak et al. [152]

600

1.8

5 25 57 105

22-31 1.2 0.5 0.17

Multi-head breaker. (L2)

Table 1.1: Experimental data found in the literature on the vertical PPV generated by pavement breaking. They have measured the PPV on the foundation walls of two buildings. Table 1.1 lists the specifications of the pavement breaker employed as well as the vertical PPVs. Table 1.1 also shows some other experimental results reported in the literature where the mass md of the drop hammer, the drop height h, and the measured PPVs at different distances r from the source are given. Figure 1.1 shows the vertical PPV as a function of distance corresponding to the data listed in table 1.1. Ames et al. [12] state that “it is highly improbable that construction equipment, other than pavement breakers, would create sufficient vibrations to approach the

10

INTRODUCTION

2

Vertical PPV [mm/s]

10

H

1

10

W

A J1

D J F

0

10

J2 L1 L2

−1

10

0

10

1

10

2

10

Distance [m]

Figure 1.1: Summary of measured vertical PPV generated by pavement breaking as a function of distance. Superimposed are the empirical formula in equation (1.1) with n = 1.1 (dashed line) and n = 1.5 (dashed-dotted line).

architectural damage level.” A PPV of 5 mm/s has been considered as the lower limit of architectural damage. Hendriks [108] remarks that pile driving, pavement breaking, blasting, and demolition of structures are among the human activities that generate the highest level of vibration. He also suggests that these operations are potentially damaging to buildings at distances less than 7.5 m from the source. In a report prepared by Jones & Stokes Company for the California Department of Transportation [235], an empirical formula for the estimation of the PPV as a function of distance has been suggested. This formula is based on a simple exponential attenuation rule which, in SI units, reads as: PPV = PPVref (7.6/r)n

(1.1)

where PPVref is the reference PPV at 7.6 m from the source and is suggested to be equal to 61 mm/s for pavement breaking, r is the distance from the source, and n is the attenuation rate with a suggested range of 1.1 to 1.5. Figure 1.1 shows two lines corresponding to equation (1.1) with n = 1.1 (J1) and 1.5 (J2). Vibrations generated by dynamic soil compaction Dynamic soil compaction is to some extent similar to the pavement breaking operation. In the dynamic compaction, a heavy weight of several tonnes is dropped from a height of 1 − 20 m on the soil surface which generates a high level of ground vibrations. Figure 1.2 presents the PPV data as a function of scaled distance


11

collected from several dynamic compaction projects by Mayne et al. [179]. The scaled distance is defined as the source to receiver distance divided by the square root of the impact energy.

Figure 1.2: Attenuation of ground vibrations measured on different dynamic soil compaction projects (from [179]).

Table 1.2 summarises some other experimental data reported in the literature on ground vibrations due to dynamic soil compaction. It also lists the data reported by Adam et al. [3] on ground vibrations due to the operation of a rapid impact compactor. This device drops a weight between 5000 kg and 12000 kg from a height of 1.2 m on a circular impact foot with a diameter of 1.5 m. Mayne et al. [179] suggest the following formula as a conservative upper limit for the preliminary estimation of the PPV generated by dynamic soil compaction: !1.4 p h md /1000 PPV ≤ 70 (1.2) r where the PPV is in mm/s, md is the mass of the drop hammer in kg, h is the drop height in m, and r is the distance from the impact point in m. In dynamic soil compaction, a large amount of energy is dissipated by inelastic deformation in craters while in pavement breaking this is not the case. Figure 1.3 compares the PPV predicted by equation (1.2) to the recorded PPVs due to pavement breaking labelled J, L1, and L2 in figure 1.1. It can be observed that the empirical formula underestimates the recorded PPVs although it is assumed to provide a conservative value. Therefore for a similar impact energy, it is expected that pavement breaking generates a higher level of ground vibrations than dynamic soil compaction.

12

INTRODUCTION

Author

md [kg]

h [m]

r [m]

PPV [mm/s]

Hwang and Tu [117]

25000

5-20

10 50

20-90 3-7

Massarch [176]

21000

15-20

10 17.5

65 25

Adam et al. [3]

9000

1.2

5 10 20

45-90 12-70 4-20

7000

1.2

5 10 20

100 20 4

Table 1.2: Experimental data on the vertical PPV generated by dynamic soil compaction. 2

Vertical PPV [mm/s]

10

1

10

J 0

10

L1 L2 −1

10

0

10

1

10

2

10

Distance [m]

Figure 1.3: Comparison of the recorded vertical PPV due to pavement breaking reported by [235] and [147, 152, 153] to the empirical formula for the vertical PPV due to soil compaction (equation (1.2)) for md = 5900 kg and h = 1.2 m (thick black line) and for md = 600 kg and h = 1.8 m (thick grey line).

Further needs The experimental data on ground vibrations generated by pavement breaking are rather limited. A comprehensive experimental study on this matter is therefore

FOCUS OF THE THESIS

13

needed. This study should identify the exciting force at source, the characteristics of the source-receiver transmission path, and the response at the receiver. There is no numerical model for the prediction of ground-borne vibrations due to pavement breaking. Since the flexibility of the pavement plays an important role in the response, the existing numerical models for the prediction of ground vibrations due to pile driving [173, 175] and dynamic soil compaction [3] cannot be used. Therefore, a numerical prediction model must be developed and experimentally validated. Such a model is very valuable for practical applications and can be used in conducting parametric studies to support guidelines with respect to the effect of vibrations on building damage and annoyance to people.

1.3 1.3.1

Focus of the thesis Objectives

There are three objectives for the present research. The first objective is the development and experimental validation of a numerical model for the prediction of traffic induced ground-borne vibrations. This work builds further on the consideration of deterministic road unevenness in the estimation of the dynamic wheel loads as well as the case of jointed concrete pavements where joint faulting and independent behaviour of the slabs distinguish them from other pavements. The second objective is to evaluate numerically and experimentally the success of road rehabilitation in the reduction of traffic induced ground-borne vibrations. The third objective is the development and experimental validation of a numerical model for the prediction of ground vibrations generated by pavement breaking.

1.3.2

Methodology

The prediction of ground-borne vibrations generated by road traffic is performed in two stages employing the model developed by Lombaert et al. [162]. In the first stage, the dynamic wheel loads due to the passage of a vehicle over road irregularities are estimated. For this purpose, local and global road unevenness is obtained by means of an appropriate profiler. To account for the enveloping effect of the tyres of road vehicles, the recorded road profiles of the left and right wheel tracks are filtered. The resulting effective road profiles are simultaneously introduced into a 3D vehicle model and the dynamic wheel loads are computed. In the second stage, the road-soil interaction problem is considered. For the case of jointed concrete pavements, a 3D coupled FE-BE model of the road-soil system is proposed. In this model, a slab of the road is modelled as a flexible plate with

14

INTRODUCTION

the FE method. It is coupled to a BE model for the soil. The soil is represented by a horizontally layered elastic halfspace. The 3D model is compared to the 2.5D plate [85] and beam [162] models. The dynamic wheel loads estimated in the first stage are introduced into the road-soil model to compute traffic induced ground-borne vibrations. A comprehensive set of experiments [150] has been performed to validate the predicted results and to gain more insight into the problem. These experiments have been performed during the rehabilitation of the N9 road in Lovendegem (Belgium). In these experiments, an instrumented test truck is driven over the jointed concrete pavement of the deteriorated N9 and the new asphalt pavement of the rehabilitated N9 where the free field vibrations as well as the accelerations of the body and the axles of the truck are measured simultaneously. The experimental results are used to validate the vehicle model and the predicted ground vibrations as well as to experimentally evaluate the effectiveness of the road rehabilitation work on the reduction of traffic induced vibrations. The prediction of ground-borne vibrations generated by a falling weight pavement breaker is also divided into two subproblems: estimation of the impact load due to the blow of the drop hammer on the road surface and prediction of ground vibrations due to the impact load. The flexibility of the pavement results in the shortcoming of the single degree of freedom (SDOF) models that are commonly used for the prediction of the force due to impact pile driving or the force applied to the foundation of presses and hammers. In addition, the short duration of the impact results in the contribution of very high frequency components in the force and thus difficulties to solve a complex model. Therefore, an efficient numerical model is developed that represents the road-soil system by a layered elastic halfspace. The model is validated experimentally. In the pavement breaking operation, a part of the impact energy is transferred to the soil and causes ground-borne vibrations. Prior to the prediction of ground vibrations, the amount of energy dissipated due to fracturing of the concrete slabs is estimated. This is done by experimental investigation of the crack pattern and estimation of the fracture energy from similar fracture tests in the literature. For the prediction of ground vibrations, the previously developed 3D FE-BE model of the road-soil system is coupled to a SDOF model of the pavement breaker to investigate the effect of the pavement breaker-road-soil interaction on the dynamics of the road and, consequently, on the generated ground vibrations. The 3D FE-BE model of the road-soil system is simplified to an axisymmetric model to reduce the computational costs and allow extension to high frequencies. The linear axisymmetric model is elaborated to an equivalent linear model to approximately account for non-linear behaviour of the soil at large strains. At a final stage, an efficient non-linear road-soil interaction model is developed. The model is composed of two uncoupled subdomains. The first subdomain is

FOCUS OF THE THESIS

15

composed of the road and a part of the soil coupled to viscous boundary conditions. It is analysed in the time domain using the non-linear finite element method. The second subdomain is a linear elastic halfspace representing the rest of the soil. The displacements and tractions along a path in the linear region of the first subdomain are computed and used in the second subdomain where the integral representation theorem is employed to compute the radiated wave field at large distances. This is done using a boundary element discretisation in the frequency domain. A comprehensive set of experiments [152] on pavement breaking induced vibrations has been performed during the second phase of the rehabilitation of the N9 road in Waarschoot (1200 m north of the site in Lovendegem). In these experiments, a multi-head pavement breaker (MHB) is employed where the applied force of the drop hammers as well as the generated ground vibrations is measured. The experimental results are used for model validation while they also enrich the current state of the art on ground vibrations generated by pavement breaking.

1.3.3

Original contributions

The research performed within the frame of this thesis is situated in a broader body of research on dynamic soil-structure interaction and environmental vibrations that is carried out at the Structural Mechanics Section of the Department of Civil Engineering, KU Leuven. In-house Matlab toolboxes developed in these studies [84, 219, 221] have been employed for some of the computations made in this thesis. In the following, the original contributions of this work are listed in order of their appearance in the text. • For the prediction of traffic induced ground-borne vibrations, the road profile recorded by a profiler is used. Two methods to filter road profiles accounting for tyre envelopment are evaluated. The dynamic wheel loads are estimated more accurately by employing a 3D vehicle model that considers the unevenness of the left and right wheel tracks simultaneously. For the case of jointed concrete pavements, the effect of the independent behaviour of the slabs as well as the flexibility of the slabs on the road-soil transfer functions is studied. • The effect of road rehabilitation on the reduction of traffic induced vibrations is studied numerically and experimentally. This study gives a clear insight into the effect of local and global road unevenness on the dynamic vehicle response and ground-borne vibrations. • In pavement breaking, a part of the impact energy is dissipated by fracturing of the concrete slabs and the remaining part is transferred to the ground. By means of experimental studies and a survey of the literature on fracture

16

INTRODUCTION

mechanics, the amount of energy dissipated by fracturing of a concrete pavement is estimated. • A numerical model for the estimation of the impact load of a falling weight on a plate resting on a halfspace is developed. The model is used to estimate the impact load due to the blow of a drop hammer on the road. The model is validated experimentally. • An efficient two step non-linear model for the prediction of ground vibrations due to pavement breaking is developed, numerically verified, and experimentally validated. The model can also be used in other cases where a semi infinite non-linear medium is under a short duration loading and the response at large distances is required, e.g. the prediction of ground vibrations due to blasting, pile driving, and dynamic soil compaction. • Two comprehensive vibration measurement campaigns on traffic induced ground-borne vibrations have been performed. In these measurements, the accelerations of the axles and the body of a test truck and the free field response are measured simultaneously for several passages with different speeds. In the first campaign, the test truck passes over a deteriorated jointed concrete pavement. In the second campaign, the same truck passes on the same road after it has been rehabilitated. These measurements give a clear insight into the effect of local and global road unevenness on the dynamic vehicle response and traffic induced ground-borne vibrations. It also provides information about the level of vibration reduction gained by road rehabilitation. • A comprehensive experimental study on the ground vibrations generated by pavement breaking has been carried out. In this experiment the ground vibrations and the acceleration of the drop hammer of the pavement breaker - from which the impact force can be estimated - have been recorded. The results are used to validate the numerical model for the estimation of the impact load and the non-linear model for the prediction of ground vibrations. Apart from the aforementioned contributions, the effect of pavement type on traffic induced vibrations has been studied. In this study, the unevenness of six roads with different types of pavements as well as the response of a test truck due to passages over these pavements are recorded. The ground vibrations beside these roads are predicted and correlated to some common road unevenness indices. The findings of this study have been published in reference [148] and are not repeated in this text.

ORGANISATION OF THE TEXT

1.4

17

Organisation of the text



The text can be divided into two main parts. The first part is composed of chapters 2 and 3 and deals with the problem of traffic induced vibrations. Chapters 4, 5, and 6 are dealing with the problem of ground vibrations generated by pavement breaking. As tabulated in table 1.3, in each part the exciting force at source is identified first; second, the transmission of the excitation to the free field is discussed. Identification of the exciting force at source

Transmission of the excitation from the source to the free field

Identification of the dynamic wheel loads by the analysis of the vehicle-road interaction problem. Chapter 2: Numerical estimation of dynamic vehicle loads

Analysis of the road-soil interaction problem to compute the road-soil transfer functions. The motion of the source is taken into account. Chapter 3: Prediction of traffic induced ground vibrations

Estimation of the impact load Analysis of the pavement breakerof the pavement breaker by the road-soil interaction problem to analysis of the drop hammer-road- compute the road-soil transfer soil interaction problem. functions. Analysis of the effect of fracturing Approximate consideration of the on the dissipation of the impact non-linear behaviour of the soil energy and its influence on the by means of the equivalent linear change of ground vibrations. analysis. Chapter 4: Pavement breaking, Chapter 5: Linear prediction of impact load estimation, and frac- ground vibrations due to pavement turing breaking Non-linear analysis of the roadsoil interaction problem. Chapter 6: Non-linear prediction of ground vibrations due to pavement breaking Table 1.3: Organisation of the text.

18

INTRODUCTION

Chapter 1 introduces the thesis by motivating the subject and outlining the problem. The state of the art is reviewed and further needs are identified. Next, the objectives of the present study are stated and the methodology is explained. Finally, the own contributions are highlighted and the organisation of the text is clarified. Chapter 2 deals with the estimation of dynamic vehicle loads. It starts by discussing the road unevenness including the relevant range of wavelengths, a practical method of profiling, and two techniques to process the recorded road profiles. The profiles of the N9 road prior to and after rehabilitation are presented. The vehicle-road interaction problem is considered and a 3D vehicle model is presented. Next, the results of the measurement campaign along the deteriorated N9 are used to calibrate and validate the 3D vehicle model of the test truck. Finally, the dynamic wheel loads due to the passage of the truck over the deteriorated and rehabilitated N9 are estimated. Chapter 3 studies the road-soil interaction problem to predict traffic induced ground vibrations. A 3D coupled FE-BE model of the road-soil system is presented first. The continuous plate model and the beam model are also briefly presented and the effect of the road model on the transfer functions is investigated. Then, the generation of ground vibrations due to moving dynamic loads is considered. Next, the transfer functions are combined with the dynamic vehicle loads estimated in the previous chapter to compute ground vibrations. The predicted ground vibrations are validated experimentally. The effectiveness of road rehabilitation work on the reduction of traffic induced ground-borne vibrations is evaluated. Additionally, the soil strains due to traffic load are estimated. Chapter 4 starts with introducing the pavement breaking operation and different types of breakers. Then, an overview of some experiments on ground vibrations due to pavement breaking is presented. Next, a numerical model for the estimation of the impact load due to the blow of a falling weight pavement breaker is presented, verified, and experimentally validated. Finally, the effect of fracturing of the concrete slabs on the generated ground vibrations is investigated. Chapter 5 presents a linear model for the prediction of ground vibrations generated by pavement breaking. The effect of the mass of the pavement breaker on the road response and induced vibrations is investigated by coupling the 3D FE-BE model of the slab-soil system presented in chapter 3 to a simple model of the pavement breaker. The 3D FE-BE model is then simplified to an axisymmetric model which is verified and used to predict ground vibrations generated by pavement breaking. The model is further elaborated to an equivalent linear model and validated experimentally. Chapter 6 uses the model presented in chapter 5 and advances it to a non-linear road-soil interaction model. The model is verified numerically and used to predict pavement breaking induced vibrations, taking into account inelastic behaviour

ORGANISATION OF THE TEXT

19

of the soil. The results predicted with the non-linear model are compared to the results of the linear and equivalent linear models. The non-linear model is validated experimentally. Chapter 7 summarises the main conclusions of this work and gives recommendations for further research.

Chapter 2

Numerical estimation of dynamic vehicle loads 2.1

Introduction

The study of environmental vibrations can usually be performed in two stages: the identification of the exciting force at the source and the estimation of the source-receiver transfer function where a linear system is assumed. In the case of traffic induced ground-borne vibrations, the exciting forces are the dynamic wheel loads on the road surface. This chapter presents a study on the estimation of these loads due to the passage of a vehicle over road irregularities. In the next chapter, the estimated wheel loads are applied on the road-soil system to compute induced ground-borne vibrations. Throughout this chapter and the next chapter, the experimental data of the measurement campaign along the N9 road in Lovendegem [150] are used to validate the predicted results. In this chapter, the road unevenness that is the main cause of dynamic vehicle loads is considered first. The relevant range of road unevenness in the problem of traffic induced vibrations is estimated. Some practical methods to obtain road profiles are reviewed and their limitations and characteristics are discussed. Two techniques to process the recorded road profiles are evaluated. The profiles of the N9 road prior to and after rehabilitation are presented. It is shown how the differences between the left and right wheel tracks of a pavement can affect the dynamics of a vehicle and, consequently, the dynamic wheel loads. Second, the vehicle-road interaction problem is addressed. Two simple vehicle models which are frequently used in the prediction of traffic induced vibrations

21

22

NUMERICAL ESTIMATION OF DYNAMIC VEHICLE LOADS

are introduced. A more elaborate 3D vehicle model is presented to simultaneously account for the unevenness of the left and right wheel tracks. Third, the experiment carried out along the N9 in Lovendegem is explained. The results of the prerehabilitation experiment are used to calibrate the 3D vehicle model of the test truck. The eigenfrequencies, eigenmodes, and frequency response functions of the truck are presented. The vehicle model is validated by means of the experiments performed on the deteriorated and rehabilitated N9. Finally, the dynamic wheel loads due to the passage of the truck over the deteriorated and rehabilitated N9 are estimated. These loads are used in chapter 3 to predict induced ground vibrations.

2.2

Road unevenness

The input to each wheel of a road vehicle is described by a longitudinal profile. A longitudinal road profile may include different sorts of information such as the geometry of very long vertical curves, roughness information, or even texture data. Roughness or unevenness - which are used interchangeably in the literature - is defined as the deviations of a pavement surface from a true planar surface with characteristic dimensions that affect vehicle dynamics, ride quality, dynamic loads, and drainage [11]. These deviations include distinct localised pavement failures, such as faults and potholes, and also random deviations that reflect the practical limit of precision to which a pavement can be constructed and maintained [212]. The range of wavelengths λ that affect vehicle dynamics depends on the vehicle speed v. The frequency range of interest in the study of vehicle dynamics is between f = 1 Hz and 25 Hz [46,59]. For a vehicle speed range v = 30 − 120 km/h, the wavelengths of interest are thus between 0.33 m and 33 m. In the case of traffic induced ground-borne vibrations, the frequency content of vibrations extends from a few Hz up to about 80 Hz, which for a similar range of vehicle speeds corresponds to wavelengths from 0.1 m up to 33 m. Wavelengths shorter than the size of tyre footprints (0.15 − 0.25 m) are strongly filtered out by the tyre envelopment. These short wavelengths are mainly responsible for tyre/road noise inside and outside vehicles except local irregularities such as faults and potholes which cause peak ground vibrations [5, 148, 196, 249]. The largest wavelengths are only relevant for motorways with high vehicle speeds.

2.2.1

Road unevenness measurement

The road irregularities are recorded using different types of instruments including the usual rod and level, Dipstick, profilometers, ARAN, APL,. . . Using the rod and level and Dipstick - which is called ‘static’ profiling - is slow, labour intensive,

ROAD UNEVENNESS

23

interrupts traffic, and has many potential sources of human error. They are therefore only applicable for profiling a short stretch of a road. High-speed inertial profilers or so called ‘profilometers’ as well as Automatic Road Analyser (ARAN) [88] use a vehicle instrumented with an accelerometer and a height sensor. The accelerometer senses the vertical motions of the vehicle body while the height sensor measures the distance between the vehicle and the road surface. The signals from the accelerometer and the height sensor are used together to compute the profile of the road [212]. The APL (in French: Analyseur de Profil en Long, longitudinal profile analyser) is a trailer towed by a vehicle (figure 2.1) and a data acquisition system. Two trailers are used in the case where the profiles of both the left and right wheel tracks are required.

Figure 2.1: The APL trailers. The APL trailer consists of three mechanical elements: a frame that acts as a sprung mass, a follower wheel, and a horizontal pendulum. The vertical movements of the follower wheel result in angular movements of the frame. The low frequency pendulum provides the inertial reference basis for the translation of the angular movement of the frame into road profile height values [165, 214]. The trailer frame and the suspension are designed to keep the follower wheel on the ground even on very rough roads. Usually, the wheels of the towing vehicle leave the pavement before the trailer will. Thus, the system can be used for very rough pavements and unpaved roads [212]. Several studies such as the International Road Roughness Experiment (IRRE) [214] which led to the development of the IRI and studies performed by the Permanent International Association of Road Congresses (PIARC) [1] have shown that the APL is one of the most accurate profilers. The APL has also advantage over non-contact inertial profilers because it senses the

24


road with a follower wheel which has a footprint of a few centimetres1 [130]. In this way the tyre envelops macrotexture and megatexture and bridges over concave features that are much smaller than the tyre footprint. The APL records the road profile with a sample interval of 0.05 m. It obtains a signal proportional to the profile over a frequency range 0.4 − 25 Hz [62, 165]. The recorded waveband is determined by the measurement speed which can be from a few km/h up to 144 km/h. The usual towing speeds are 21.6 km/h, 54 km/h, and 72 km/h where the APL can measure without distortion and attenuation wavelengths in the range of 0.25 − 15 m, 0.6 − 37 m, and 0.8 − 50 m, respectively. The frequency range of interest in the study of vehicle dynamics is 1−25 Hz [46,59] and in the study of traffic induced vibrations is about 1−80 Hz. For a vehicle speed of 50 km/h, corresponding wavelengths are 0.6 − 14 m and 0.2 − 14 m, respectively. Therefore, depending on the vehicle speed, the appropriate APL speed must be selected. At some vehicle speeds, profiling with more than one speed can be required. From this brief review of profiling instruments, it is deduced that the APL is an appropriate profiler to be used in the study of traffic induced vibrations as well as vehicle dynamics.

2.2.2

Road profile processing

The tyre enveloping acts as a low-pass filter for the road roughness input to a vehicle. It depends on the size and construction of the tyre as well as on the wheel load. In order to study the enveloping effect, several tyre models have been developed. These models differ in terms of complexity, accuracy, bandwidth, and efficiency requirements. Some of them can be applied as an equivalent profile processing filter. The simplest tyre model is the point contact model which follows the road profile exactly, so it does not filter small sharp obstacles or bridge over small concave features in the road. This lack of proper filtering and bridging disqualifies the point contact model for use on natural road profiles. The model can be used in situations where only low frequency results are of interest. At higher frequencies, it can overestimate transmitted tyre forces by up to an order of magnitude [45]. 1 The deflection ∆ of a tyre with a stiffness k under a load F is ∆ = F/k . Approximating t t √ 2 the p contact patch in 2D as a chord of a circle, the contact length becomes 2p = 2 2R∆ − ∆ ≈ 2 2RF/kt where R is the radius of the tyre. The APL trailer weighs 1200 N of which 450 N is carried by the towing vehicle and the remaining F = 750 N is transferred to the pavement through the follower wheel [62]. The motorcycle-type tyre of the follower wheel has a radius R = 0.27 m and a stiffness of about kt = 150 kN/m [241]. This gives a deflection ∆ =p F/kt = 0.005 m under the self weight. The length of the contact patch is thus equal to 2p = 2 2RF/kt = 0.10 m.

ROAD UNEVENNESS

25

Another very common tyre model is the fixed footprint tyre model. This model interacts with the terrain through a footprint of constant size, independent of the tyre deflection. The finite footprint provides this model with the ability to envelop small irregularities through local deformations within the footprint. In fact, this model is equivalent to a point contact model in which the local terrain elevation is replaced with a smoothed profile obtained from a moving average filter. The appropriate moving average baselength lm must be approximately 50% longer than the length 2p of the contact patch between the tyre and the road [214]. Other advanced tyre models such as the radial spring model, ring model, and finite element model have been developed for the prediction of vehicle ride and durability [130]. More information about different tyre models can be found in the studies of Captain et al. [45], Zegelaar [264], and Pacejka [191]. Considering the trade-off between the accuracy and complexity of different tyre models, the fixed footprint model is usually used in the study of vehicle dynamics. This is the model that has also been implemented in the IRI [214] where a baselength of 0.25 m has been considered for a passenger car. A baselength of 0.30 m is typically used for the wheels of trucks [212]. Gillespie et al. [95] suggest to filter the road profile with a U-shaped moving average window instead of the rectangular window. They found that the response function of the U-shaped window is very close to that of a representative tyre obtained experimentally. Zegelaar [264] suggests a methodology to obtain the effective road profile input to the vehicle from the true profile. He proposes that the road profile is first filtered with a rigid wheel model to get the ‘basic curve’. This results in bridging concave features and rounding and broadening sharp obstacles. The effective road profile is obtained by rolling a two-point contact model (a line segment) over the basic curve. The distance between the two points should be about 80% of the contact length. In essence, this method is very similar to the application of the U-shaped moving average filter as both methods give a larger weight to the two ends of the contact patch. Figure 2.2 compares the wavenumber response functions of the rolling line segment and the rectangular moving average. The wavenumber response function of the rolling line segment with a baselength lz is equal to | cos (ky lz /2)| and the one of the rectangular moving average with a baselength lm is equal to | sin(ky lm /2)/(ky lm /2)| where ky = 2π/λy is the wavenumber. For a tyre footprint length 2p = 0.20 m, the baselength of the rolling line segment is thus equal to lz = 0.80 × 2p = 0.16 m and the baselength of the rectangular moving average is lm = 1.5 × 2p = 0.30 m. The response function of the rolling line segment is found to be closer to that of the U-shaped window and the representative tyre than the response function of the rectangular moving average. The first zero values of both filters occur approximately at the same cyclic wavenumber ky0 ≈ 1/(3p). Below this wavenumber, the response functions are very similar while above it, the difference is large. Assuming a typical tyre footprint length 2p ≈ 0.20 m results in the first


Wavenumber response function [−]

26

1 0.8 0.6 0.4 0.2 0

0

2 4 6 Cyclic wavenumber [1/m]

8

Figure 2.2: Wavenumber response function of the rectangular time window with a baselength lm = 0.30 m (solid line) and the rolling line segment with a baselength lz = 0.16 m (dashed line). zero value wavenumber ky0 ≈ 3.3 m−1 . For a vehicle speed v, the two filters give similar responses at frequencies below f = ky0 v ≈ 3.3v. In the study of vehicle dynamics, the response is mainly situated below a frequency of 25 Hz. Therefore for vehicle speeds above v = 7.5 m/s, the two filters essentially give similar results. Assuming an upper frequency limit of 80 Hz in the analysis of traffic induced vibrations results in the similarity between both filters at vehicle speeds above v = 24 m/s. For vehicle speeds below 24 m/s, the filtered road profiles will be different for frequencies above f = ky0 v ≈ 3.3v while the profile filtered with the rolling line segment represents better the response function of the representative tyre. Zegelaar’s method can be easily applied to the road profile recorded by the APL. The wheel of the APL approximately acts as a rigid wheel so the recorded road profile represents the ‘basic curve’. The road profile is then filtered by averaging the height of two points at a distance equal to 80% of the contact length of the vehicle tyre. Due to the tyre width there is also an enveloping effect in the lateral direction. The narrower footprint of the APL senses a wheel path which is narrower than the wheel path sensed by the trucks. The wider footprint of the trucks can bridge concave features in the road surface, such as potholes, that are recorded by the APL, or on the other hand, it might experience convex features, like small bumps, that have not been sensed by the APL.

ROAD UNEVENNESS

27

Statistical representation of road unevenness Statistical representation of the road roughness in the form of spectral density has been recognised as one of the best available diagnostic tools for interpreting pavement properties and measurement capabilities of profilers [215]. The PSD function of a road profile is the mean-square value of the profile per unit wavenumber bandwidth. ISO 8608 [125] recommends that the road profile should be described by the PSD of its vertical displacement. Some researchers have used the PSD function of a road profile to compute the PSD function of ground vibrations [103, 116, 260]. However, local irregularities cause peak ground vibrations [5, 148, 249]. Dodds and Robson [72] state that road surfaces can be considered as random with a normal probability distribution if large local irregularities, such as potholes, are isolated and treated separately. Rouillard et al. [209], however, have found that the distribution of road roughness deviates from the normal distribution. Therefore, for roads with mainly local irregularities like jointed concrete pavements and pavements composed of small concrete or stone pavers, the actual road profile should be used.

2.2.3

Unevenness of the N9 road prior to and after rehabilitation

In the following, the road profiles of the N9 road between kilometre points 61.052 and 60.852 prior to and after rehabilitation are presented. The N9 was originally a deteriorated jointed concrete pavement while it has been rehabilitated by applying a new asphalt overlay. The deteriorated road was profiled on 19 March 2008 and the rehabilitated road on 9 May 2008 [40]. They were profiled by the APL at two speeds of 21.6 km/h and 54 km/h. Figure 2.3 shows the longitudinal profiles measured at 21.6 km/h on the nearside wheel track of the roads for a stretch of 200 m. The deteriorated N9 (figure 2.3a) is a rather rough road while the rehabilitated N9 (figure 2.3b) is quite smooth. Figure 2.3 also shows the high-pass filtered (antismoothed [216]) road profiles which have been obtained by subtracting a moving average (smoothed profile) computed with a base length of 0.30 m from the original profiles. Local irregularities in the road surface such as joints, faults, or potholes can be identified as sharp peaks in the filtered road profile [215]. The peaks in the filtered profile of the deteriorated N9 in figure 2.3a correspond to joint faulting. The filtered profile of the rehabilitated N9 (figure 2.3b) is quite smooth, showing that irregularities and slab misalignments have been eliminated. Figure 2.4 situates the power spectral density of the road displacement with respect to the road quality classes A-H of ISO 8608 [125]. The PSDs have been computed using Welch’s method [253] because of its minimum bias and variance [68]. The deteriorated N9 is situated between class B and C while the rehabilitated N9 is of


0.02

0.02

0.01

0.01 Unevenness [m]

Unevenness [m]

28

0 −0.01 −0.02

0

50

(a)

100 150 Distance [m]

0 −0.01 −0.02

200

0

50

(b)

100 150 Distance [m]

200

Figure 2.3: Road profile (black line) of the right wheel track of the (a) deteriorated and (b) rehabilitated N9 recorded by the APL at a speed of 21.6 km/h. Superimposed is the filtered road profile (grey line).

−3

10

H G F E D C B A

−5

10

−7

10

−9

10

−1

10

(a)

PSD of road unevenness [m3]

PSD of road unevenness [m3]

class A. Figure 2.4b shows that the rehabilitation eliminates local irregularities so reduces large wavenumber unevenness. A large reduction at wavenumbers about 0.13 − 0.4 m−1 is due to the elimination of irregularities related to the independent behaviour of each slab such as slab misalignments and curling. The roads have also been profiled at a speed of 54 km/h where unevenness at smaller wavenumbers down to 0.027 m−1 is measured. As expected, the PSDs of the unevenness showed that the rehabilitation does not significantly reduce very long wavelength undulations [149]. In figure 2.4b, sharp peaks are observed at multiples of wavenumber 0.6 m−1 , corresponding to a wavelength of 1.7 m, which is close to the circumference of the APL wheel, and probably due to a nonuniformity in the wheel [213, 214].

0

10 Cyclic wavenumber [1/m]

1

10

−3

10

H G F E D C B A

−5

10

−7

10

−9

10

−1

10

(b)

0


1

10

Figure 2.4: PSD of road displacement (thin black line) and fitted line to the PSD (thick line) of the (a) deteriorated and (b) rehabilitated N9. Superimposed are the road quality classes A-H according to ISO 8608 (grey lines).

ROAD UNEVENNESS

29

Differences between unevenness of two wheel tracks of a road excite the rolling mode of passing vehicles. To study these differences, the nearside and offside profiles (uw/r1 and uw/r2 ) are transformed into a vertical bounce displacement (uw/r1 + uw/r2 )/2 and a roll displacement (uw/r1 − uw/r2 )/2. The ratio of roll spectral density to bounce spectral density for these pavements is shown in figure 2.5. 2 Roll PSD/Bounce PSD [−]

Roll PSD/Bounce PSD [−]

2 1.5 1 0.5 0

−1

10

(a)

0


1.5 1 0.5 0

1

10

−1

10

(b)

0


1

10

Figure 2.5: Ratio of roll displacement spectral density to bounce displacement spectral density of the (a) deteriorated and (b) rehabilitated N9. The ratio is close to zero at small wavenumbers and increases at higher wavenumbers which is typical of many road surfaces [46, 131]. In most of the vehicles, the typical range of roll mode natural frequencies are 1 − 2 Hz for the sprung mass and 8−20 Hz for the unsprung masses. Considering a vehicle travelling at a speed of 50 km/h, a range of wavenumbers 0.07 − 0.14 m−1 corresponds to the sprung mass roll mode frequency and a range of 0.58 − 1.44 m−1 corresponds to the unsprung mass roll mode frequency. The value of the roll/bounce ratio at 0.07 − 0.14 m−1 is below 0.5, and so there is comparatively little roll excitation of the vehicle. On the other hand, the roll/bounce ratio at 0.58 − 1.44 m−1 is above 0.5 and, therefore, the unsprung mass roll modes will be significantly excited at 50 km/h. The unsprung mass modes will normally always be excited at wavenumbers where the ratio of roll to bounce excitation is close to one which is the case for most road surfaces [131]. To accurately account for the excitation of the roll mode of the unsprung masses, the nearside and offside profiles must be introduced in a 3D vehicle model. The development, calibration, and experimentally validation of such a 3D model is the subject of the following section.

30

2.3


Vehicle model

Irregularities in a road profile subject the sprung and unsprung masses of vehicles to vertical oscillations that cause dynamic loads exerted on the pavement. One of the simplest and commonest vehicle models used for the simulation of the dynamic wheel loads as well as vehicle ride comfort is the so called ‘quarter-car’ model (figure 2.6a). This model has two DOFs corresponding to the vertical motion of the sprung and unsprung masses. The suspension system and tyres are modelled as spring-dashpots. It can represent the axle hop mode and the bounce mode. The quarter-car model has also been used in the definition of the IRI [214]. A more advanced model for a two axle vehicle is the ‘half-car’ or pitch-plane vehicle model as shown in figure 2.6b. It contains four DOFs representing the vertical motion of the unsprung masses and the vertical motion and in-plane rotation of the sprung mass. This model considers the coupling between the rear and front side of a vehicle while the coupling between the left and right side, and consequently the sprung and unsprung roll modes, are disregarded. The RMS values of dynamic wheel loads estimated with the quarter-car and half-car vehicle models can have a nominal error of 20% in comparison with a 3D model [92].

(a)

(b)

Figure 2.6: (a) The quarter-car vehicle model (from [216]) and (b) half-car vehicle model (from [159]). In order to estimate the dynamic vehicle loads more accurately and to take into account the differences of the nearside and offside profiles, a 3D vehicle model is required. Figure 2.7 shows a 3D model of a two-axle vehicle. The model represents a single unit truck or a bus where the axles and body can usually be considered as rigid elements [93]. Single unit trucks are also called rigid or straight trucks in contrast to articulated vehicles. The vehicle itself has 7 DOFs and 4 DOFs describe the vertical displacements of the contact points. The sprung mass (vehicle body) is considered as a rigid body with three DOFs: vertical displacement ub (bounce), rotation θb in the plane of motion (yz-plane, pitch), and rotation ϕb in the plane

VEHICLE MODEL

31

perpendicular to the plane of motion (xz-plane, roll). The mass of the vehicle body is denoted by mb , and the mass moments of inertia with respect to the x- and y-axis are denoted as Ibx and Iby , respectively. The unsprung masses consist of the wheels, axle, brakes, steering knuckle, and portions of the suspension linkage directly connected to them. They are considered as slender rigid bars with two DOFs: vertical displacement uai and rotation ϕai in the xz-plane (axle roll), where i is 1 or 2 for the rear or the front axle, respectively. The axle mass is denoted by mai and the mass moment of inertia with respect to the y-axis is denoted as Iai . The suspension system and tyres are modelled as linear spring-dashpot systems. For each contact point of the vehicle and the road, one vertical DOF is considered, and referred to as uci , where i ranges from 1 to 4 for the left rear wheel(s), right rear wheel(s), left front wheel, and right front wheel (figure 2.7). The vehicle model is a simple model that is used for the prediction of traffic induced vibrations, it therefore does not contain the detailed suspension non-linearities and complexities of the body motion [92].

Figure 2.7: Three-dimensional model of a two-axle truck. The equation of motion for the 3D vehicle model is written as: #( ) ( ) #( ) " " #( ) " 0 Kvv Kvc uv (t) Mvv 0 u ¨ v (t) Cvv Cvc u˙ v (t) = + + Kcv Kcc uc (t) Ccv Ccc u˙ c (t) g(t) 0 0 u ¨ c (t) (2.1)

32


T

The vector uv = {ub , θb , ϕb , ua1 , ϕa1 , ua2 , ϕa2 } collects the DOFs of the vehicle, T while the vector uc = {uc1 , uc2 , uc3 , uc4 } contains the displacements of the contact points. A dot above a variable denotes differentiation with respect to the time t. Mvv represents the mass matrix of the vehicle, and Kkl and Ckl stand for the stiffness and damping matrices, where k and l are equal to ‘v’ or ‘c’ for the DOFs of the vehicle and the contact points, respectively. These matrices are given in T appendix A. The vector g = {g1 , g2 , g3 , g4 } collects the dynamic vehicle loads. In the following, it is assumed that the road deformations are small compared to the vehicle displacements, so that the displacements of the contact points uc are equal to the road unevenness at the position of the contact points. This decouples the vehicle-road interaction from the road-soil interaction. The validity of this assumption has been confirmed by studies of Cebon [46], Gillespie [93], Lombaert [159], and Pyl [197]. On the other hand, Watts [245] has speculated that a concrete slab moves as a whole as a result of the passage of a heavy vehicle. To investigate this supposition, the vertical displacements (rotations) of more than 200 slabs of the deteriorated N9 have been measured during the passage of a heavy truck at a very low speed [42,150]. At most of the joints, the relative displacement of the slab ends was very small and much smaller than the height of the joint faulting. In the frequency domain, equation (2.1) can be rewritten as: ) ) ( #( " D 0 u ˆ v (ω) KD vv (ω) Kvc (ω) = D g ˆ (ω) u ˆ c (ω) KD cv (ω) Kcc (ω)

(2.2)

2 where KD kl (ω) = −ω Mkl + i ω Ckl + Kkl is the dynamic stiffness matrix, ω is the angular frequency, and i is the imaginary unit defined as i2 = −1. A hat above a variable denotes its representation in the frequency domain. The first row of ˆ av uc (ω) equation (2.2) allows computing the frequency response function (FRF) h between the accelerations a ˆv (ω) of the vehicle and the displacements u ˆ c (ω) of the contact points as:

ˆ av uc (ω) = ω 2 KD −1 (ω) KD (ω) h vv vc

(2.3)

In order to estimate the dynamic wheel loads that cause ground vibrations, the ˆ guc (ω) between the dynamic wheel loads g ˆ (ω) and the displacements u ˆ c (ω) FRF h of the contact points is found from the second row of equation (2.2) as: ˆ guc (ω) = −KD (ω) KD −1 (ω) KD (ω) + KD (ω) h cv vv vc cc

(2.4)

The 3D vehicle model is experimentally validated in the following for the passage of the test truck over the irregularities of the N9 road (figure 2.3).

EXPERIMENTAL VALIDATION OF THE VEHICLE MODEL

2.4 2.4.1

33

Experimental validation of the vehicle model Experimental configuration

In the measurement campaign along the N9 road in Lovendegem [150], a Volvo FL180 truck (FL612L-4x2 rigid rear air suspension, figure 2.8a) has been instrumented and driven over the road prior to and after the road rehabilitation. At the same time, vibrations are measured in the free field. The truck has a wheel base of 4.00 m, an average track width 2wt = 1.92 m, and a tare mass of 6420 kg. The masses carried by the front and rear tyres are 3465 kg and 2955 kg, respectively. The front suspension system consists of tapered multi-leaf springs and additional shock absorbers. The rear suspension system is composed of trailing arms - to which the axle tube is clamped - connected to air springs (air bags) and additional shock absorbers.

(a)

(b)

Figure 2.8: (a) Volvo FL180 truck and (b) two accelerometers on the left side of the rear axle and one accelerometer on the left rear side of the body.

In the prerehabilitation experiments, vibration measurements have been performed for 34 passages of the truck with speeds between 30 km/h and 70 km/h. After rehabilitation, 21 passages with speeds between 30 km/h and 80 km/h have been measured. The vehicle response has been recorded by a total of 12 accelerometers mounted on both the axles and body of the truck. On each axle, two accelerometers are installed on the right side of the axle and two accelerometers are installed on the left side. The body response is measured by means of four accelerometers mounted on the two chassis beams, one on top of each axle. Figure 2.8b shows two accelerometers on the left side of the rear axle and one accelerometer on the left rear side of the body. The truck speed is measured by means of a GPS-based VBOX mini speed measurement, which records the truck speed every 0.1 s with a precision of ±0.1 km/h. On each side of the truck a set of a photoelectric sensor and a reflector is installed to synchronise vibration measurements on the truck

34


and in the free field. The photoelectric sensors give an impulse when the truck passes in front of a reflector installed next to the measurement setup in the free field. All the sensors and VBOX are connected to a NI PXI-1050 data acquisition system coupled to a portable computer. The analogue to digital conversion is performed at a sampling rate fs = 1000 Hz, the corresponding Nyquist frequency thus equals fNyq = 500 Hz. The data recording is started and stopped manually so a different number of data points is recorded for each passage. The signals are trimmed by a rectangular time window centred at the pulse recorded by the photoelectric sensors. The signals from the accelerometers are filtered with a third order band-pass Chebyshev type I with a high-pass frequency of 0.5 Hz, a low-pass frequency of 100 Hz, and a ripple of 0.1 dB to remove the low frequency Direct Current (DC)-component and high frequency components. More details about this experiment can be found in [150].

2.4.2

Initial estimation of the vehicle parameters

Within the frame of a study on the influence of the road surface on the vehicle response [64, 164], the parameters of the half-car model (figure 2.6b) of the Volvo FL180 truck have been approximately estimated based on data obtained by weighing or data provided by the truck manufacturer as: mb = 5270 kg, Ib = 23835 kg m2 , ma1 = 750 kg, ma2 = 400 kg, l1 = −2.326 m, l2 = 1.674 m, kp1 = 672 kN/m, cp1 = 7500 Ns/m, kp2 = 460 kN/m, cp2 = 1000 Ns/m, kt1 = 4000 kN/m, ct1 = 12600 Ns/m, kt2 = 2000 kN/m, and ct2 = 6300 Ns/m. The mass moment of inertia of the vehicle body about the y-axis Iby is estimated by assuming the mass of the body mb = 5270 kg to be uniformly distributed over the width of the vehicle wv = 2.45 m, leading to a value Iby = mb wv2 /12 = 2636 kg m2 . This gives a radius of gyration of 0.71 m for the body which is close to the typical value of 0.74 m mentioned in the literature [77]. In a similar way, the mass moments of inertia of the axles are estimated as Ia1 = ma1 wv2 /12 = 375 kg m2 and Ia2 = ma2 wv2 /12 = 200 kg m2 . The distances between the chassis beams 2wp , between the springs 2waki , and between the dampers 2waci on axle i are obtained from the drawing of the chassis assembly of the truck as 2wp = 0.82 m, 2wak1 = 2wac1 = 1.04 m, 2wak2 = 0.82 m, and 2wac2 = 1.13 m.

2.4.3

Calibration of the vehicle parameters

The damping constants for the suspension system cp estimated in reference [64] are about 2 to 5 times lower than the values reported by Cebon [46], Hardy and Cebon [106], and Lombaert et al. [161] for similar vehicles. In addition, the considered damping constant of the tyres is too high when compared to the value of 1050 Ns/m suggested by Gillespie et al. [93] and also used for one tyre of a Volvo FE7 [159] and a zero value assumed for the damping of the tyres of a Volvo FL6 truck


35

[161]. Furthermore, the values for the stiffness of the suspension system and tyres have been roughly estimated. In order to adjust the aforementioned vehicle parameters, the stiffness and damping parameters are calibrated by minimizing the difference between the vehicle response predicted from the measured road profile of the deteriorated N9 and the measured vehicle response. The FRFs of the vehicle body acceleration are strongly sensitive to the stiffness of the suspension system and moderately to the damping of the suspension system. The FRFs of the axles acceleration are strongly sensitive to the stiffness of the tyres and moderately to the damping of the suspension system and tyres. The force FRFs are, in descending order, sensitive to the stiffness of the tyres, damping of the tyres, and damping of the suspension system. The parameters considered in the calibration of the vehicle model are the stiffness and damping of the suspension system and tyres which are collected in the vector x = {kp1 , kp2 , cp1 , cp2 , kt1 , kt2 , ct1 , ct2 }T . Since the rear axle has dual wheels and the front axle has single wheels with similar tyres, it is assumed that 1.6 ≤ kt1 /kt2 ≤ 2.4 and 1.6 ≤ ct1 /ct2 ≤ 2.4. In the calibration procedure, the difference between the PSD function of the measured accelerations a ê (ω) and predicted accelerations a ˆm (ω) is minimised. The PSD has a uniform resolution in the entire frequency range and is therefore more appropriate than the previously used [149] one-third octave band RMS spectrum. Other representations, such as the time history of the response, are not well suited for the optimisation problem since they contain too many data points or are very sensitive to the synchronisation of the measured and predicted vehicle response. Herein, the PSD function is computed for a frequency range of 1 Hz to 30 Hz at a frequency step of 0.5 Hz. For the characterisation of the measured vehicle response, the accelerations of all measurement points on the axles and body are used. The optimisation is performed for 13 passages prior to the road rehabilitation with different speeds between 30 km/h to 70 km/h. The other passages are not used since they took place in the opposite direction or on different wheel paths. For each passage, the PSD function of the measured acceleration is computed on a time window of 20 seconds. The predicted acceleration of the vehicle is computed from the frequency content of the displacements of the contact points u ˆ c (ω) by means of the FRFs ˆ a u (ω) presented in equation (2.3): h v c ˆ av uc (ω) u ˆ c (ω) a ˆv (ω) = h

(2.5)

where u ˆ c (ω) is obtained from the frequency content of the measured road profile u ˆ w/r (ω). In order to cover the entire frequency range of the vehicle response (1 − 25 Hz), the road profile recorded by the APL at 21.6 km/h is used for the passages with speeds of 30 km/h, 40 km/h, and 50 km/h, and the road profile recorded at 54 km/h is used for the other passages with speeds of 60 km/h and 70 km/h. The road profiles are filtered by rolling a two-point contact model as explained in section 2.2.2. The distance lz between the two points is 60% of the

36


contact length 2p. This is slightly smaller than the 80% suggested by Zegelaar [264] to approximately compensate for the filtering effect of the APL wheel. The tyres of the Volvo FL180 are of the type 245/70R19.5 which has a radius R = 0.42 m. The static load on the rear wheels on each side of the vehicle is Fr = 14.8 kN and p their stiffness is about kt1 = 2000 kN/m, so the contact length becomes 2pr = 2 2RFr /kt1 = 0.16 m and for a front wheel Ff = 17.3 kN and kt2 = 1000 kN/m, so 2pf = 0.24 m. These contact lengths are in agreement with the apparent contact patches of the rear and front tyres shown in figures 2.9a and 2.9b, respectively. The lengths of the rolling line segments are consequently found as lzr = 0.10 m and lzf = 0.15 m.

(b)

(a)

Figure 2.9: Apparent contact patch of the (a) rear and (b) front tyres of Volvo FL180 in unladen condition.

Equation (2.5) gives the accelerations a ˆv (ω) of the 7DOFs of the vehicle model shown in figure 2.7. The accelerations a ˆm (ω) at the 12 measurement points are obtained from the predicted accelerations a ˆv (ω) by applying a transformation matrix Tu ∈ RNm ×NDOF where Nm = 12 is the number of measurement points and NDOF = 7 is the number of degrees of freedom: a ˆm (ω) = Tu a ˆv (ω)

(2.6)

The difference between the PSD of the predicted and measured vehicle response is collected in the residual vector r: r(ω) = a ê (ω) − a ˆm (ω)

(2.7)

The objective function fobj is defined as the p-norm of r with p = 0.5. The 0.5norm is less sensitive to outliers than other common norms such as the Euclidean (L2) norm. fobj = krk0.5 =

Nm X ×NPSD i=1

|ri |0.5

(2.8)


37

In the equation above, NPSD = 59 is the number of frequencies between 1 Hz and 30 Hz at a frequency step of 0.5 Hz. For the vehicle body, only the response below 4 Hz is considered, since the response at higher frequencies is affected by the flexible body modes [92, 150]. The calibration of the vehicle model is a constrained non-linear optimisation aimed at finding the vehicle parameters x that minimise the objective function fobj subject to constraints on the allowable values of x. For this purpose, a Sequential Quadratic Programming (SQP) is used which is a quasi-Newton updating procedure. The optimisation is performed using the Matlab function FMINCON [177], that attempts to find a constrained minimum of a scalar function of a vector x based on an initial estimate x0 . The values for the lower and upper bounds of the constraints are based on values reported in the literature for similar vehicles [30,46,77,159,161,184]. Since the values reported for the damping constant of tyres are very different, the maximum value for the damping constant of one tyre is set at a value of 2500 Ns/m, which is more than two times of what is considered by Lombaert for a Volvo FE7 truck [159]. The initial values and the lower and upper bounds of the parameters are shown in table 2.1. Design variables x kp 1 kp 2 cp1 cp2 kt 1 kt 2 ct1 ct2 [kN/m] [kN/m] [Ns/m] [Ns/m] [kN/m] [kN/m] [Ns/m] [Ns/m] Initial value Lower bound Upper bound Updated value COV

336 200 600 236 0.16

230 160 300 179 0.15

3750 3000 20000 9946 0.20

500 1000 10000 2470 0.17

2000 1400 2200 1889 0.07

1000 700 1100 922 0.06

6300 1000 5000 4529 0.12

3750 500 2500 2302 0.11

Table 2.1: Updated model parameters of the Volvo FL180.

Table 2.1 also shows the average as well as the coefficient of variation (COV) of the updated model parameters obtained for the 13 passages. The COV is the ratio of the standard deviation to the mean. The updated parameters are in the range of values that have been reported by Fancher et al. [77] from many experimental tests. Fancher et al. give a range kp1 = 100 − 600 kN/m for the stiffness of air suspension systems, kp2 = 175 − 220 kN/m for the stiffness of multi-leaf tapered springs, and kt2 = 770 − 1015 kN/m for the stiffness of a single tyre. Compared to the initial values of the 2D vehicle model, the stiffness of the suspension system kp and tyres kt has decreased. The damping constant of the tyres ct has also reduced while a large increase is observed for the damping constants of the suspension (cp1 and cp2 ). The COVs are rather small showing a relatively small variability of the

38


updated parameters in different passages. In the following, the average values of the updated parameters are used to predict the vehicle response.

2

10

Acceleration PSD [(m/s2)2/Hz]


Figure 2.10 compares the PSD of the measured vehicle accelerations a ê to the predicted accelerations a ˆm with the initial and updated parameters. It shows how the calibration of the vehicle model parameters leads to a better agreement between the measured and predicted accelerations. On these figures, the measured accelerations for different passages with a similar speed are shown as well. Variations in the measured responses can be due to the fact that the wheel tracks are slightly different or due to small differences in the vehicle speed.

1

10

0

10

−1

10

(a)

0

10 20 Frequency [Hz]

30

(b)

2

10

1

10

0

10

−1

10

0


30

Figure 2.10: PSD of acceleration (a) on the right side of the rear axle for a vehicle speed of 70 km/h and (b) on the right side of the front axle for a vehicle speed of 50 km/h predicted with the initial (dashed black line) and updated (solid black line) parameters. Superimposed are the corresponding experimental results (solid grey line) and the experimental results of the other passages with similar speeds (dashed grey lines).

2.4.4

Eigenmodes and eigenfrequencies of the vehicle

Table 2.2 shows the eigenmodes and eigenfrequencies of the calibrated vehicle model. The first mode with a natural frequency of 1.1 Hz is the body roll mode. The second mode has a natural frequency of 1.5 Hz and is a combined pitch and bounce mode that results in a bouncing motion of the front side of the vehicle. The third mode is similar to the second mode and is a bouncing mode of the rear side. The fourth and fifth modes are the axle hop modes of the front and rear axle, respectively. The last two modes are the roll modes of the front and rear axle. The computed eigenfrequencies are in good agreement with the frequencies at which peaks are observed in the narrow band spectrum of the measured vehicle response [147].


Eigenfrequency Mode [Hz] ub 1 Body roll 1.1 0 2 Pitch and bounce (front) 1.5 1.00 3 Pitch and bounce (rear) 2.1 1.00 4 Front axle hop 11.8 −0.01 5 Rear axle hop 12.0 −0.02 6 Front axle roll 14.9 0 7 Rear axle roll 15.6 0

39

Eigenmode θb ϕb ua1 ϕa1 0 1.00 0 0.03 0.54 0 −0.03 0 −0.39 0 0.22 0 0 0 −0.01 0 0.01 0 1.00 0 0 0 0 0 0 0 0 1.00

ua2 ϕa2 0 0.03 0.31 0 0.06 0 1.00 0 0.02 0 0 1.00 0 0

Table 2.2: Eigenfrequencies and eigenmodes of the 3D vehicle model of the Volvo FL180 truck.

2.4.5

Frequency response functions of the vehicle

ˆ g u that represent the frequency content of the Figure 2.11 shows the FRFs h k cl force at contact point k due to an impulse excitation at contact point l for both the initial and updated vehicle model. Only 6 FRFs out of a total of 16 for all input-output combinations are shown because the remaining FRFs can be derived from these FRFs by considering symmetry and reciprocity. Figure 2.11a shows the FRF of the force at the left rear wheels for excitation at the same contact point. The FRF shows a large peak at about 15 Hz that corresponds to the eigenfrequencies of the hop mode and roll mode of the rear axle (table 2.2) and a small peak at low frequencies corresponding to the pitch and bounce modes and the roll mode of the vehicle body. Figure 2.11b shows the FRF of the force at the left wheel of the rear axle due to excitation at the right wheel of the same axle. Low frequency excitation of the right wheel (figure 2.11b) leads to similar forces as excitation at the left wheel (figure 2.11a). Above 30 Hz, the coupling between the two contact points is very small. Figures 2.11c and 2.11d show that only a weak coupling exists between the rear and front wheels. This implies that two 2D roll-plane vehicle models - one for the front and one for the rear side - could also be used instead of the 3D model. One can further deduce that the half-car pitch plane vehicle model does not give a much more accurate estimation of the dynamic wheel loads in comparison to the quarter-car model. The FRFs of the force at the left wheel of the front axle (figures 2.11e and 2.11f) follow a similar trend as the previously discussed FRFs of the wheel of the rear axle (figures 2.11a and 2.11b). A slightly larger amplitude at the pitch and bounce frequencies arises from the fact that the centre of gravity of the vehicle body is closer to the front side, whereas the smaller amplitude at the eigenfrequencies of the axle is due to the lower mass

40


5 Modulus of FRF [N/m]

Modulus of FRF [N/m]

6

x 10

4 3 2 1 0

0

20


3 2 1 0

20

5 Modulus of FRF [N/m]


3 2 1

4 3 2 1 20


ˆ g1 uc (d) h 4

0

20

80

5

3 2 1 0

80

6

x 10

4

0


ˆ g1 uc (c) h 3

6

x 10

0

4

0

80


6

0


x 10

ˆ g1 uc (b) h 2

ˆ g1 uc (a) h 1 5

5

4

0

80

6

x 10


6

5

20


ˆ g3 uc (e) h 3

80

x 10

4 3 2 1 0

0

20


80

ˆ g3 uc (f) h 4

ˆ g uc for the 3D vehicle model of the Volvo Figure 2.11: Modulus of the FRF h k l FL180 truck with the initial (dashed line) and updated (solid line) parameters. of the front axle. Comparing the FRFs of the initial model with the FRFs of the updated model shows a smaller amplitude at the natural frequencies due to the increased damping of the suspension. The FRFs also show that the natural frequencies have shifted to a slightly lower value due to the decreased stiffnesses of the suspension system and the tyres.

2.4.6

Validation of the vehicle model

In order to validate the vehicle model, the acceleration of the vehicle a ˆm (ω) is computed from the profiles of the deteriorated and rehabilitated N9 using equations (2.5) and (2.6). The time history am (t) is obtained subsequently by an inverse Fourier transform of a ˆm (ω). Since the dynamic wheel loads that induce ground-borne vibrations are mainly determined by the axle motion, as well as for the sake of brevity, only the acceleration of the rear axle for one speed is compared to the experimental results in the following; the results for the other speeds as well as the results of the front axle can be found in reference [149].


41

Passages on the deteriorated N9 Figure 2.12 compares the predicted and measured acceleration on the right side of the rear axle for a passage at 50 km/h on the deteriorated jointed concrete pavement of the N9 road. Figure 2.12a shows the time history of the acceleration as well as the anti-smoothed road profile with a base length of 0.30 m. Sharp peaks in the filtered road profile correspond to local irregularities on the road surface such as faults and potholes. Comparing the filtered road profile and the predicted acceleration of the axle shows that whenever the truck passes over a local unevenness, a peak occurs in the acceleration of the axle. A good agreement is observed between the measured and predicted accelerations. Small time shifts are due to small variations of the vehicle speed. A large peak in the predicted response at t ≈ 2.5 s is not observed in the measured response. This is due to a local unevenness which has been measured by the APL but has not been traversed by the truck in the experiments or has been filtered out by the wider footprint of the tyres of the truck. Figure 2.12c shows the running RMS value of the acceleration computed with a time window of 1 s. Except for the peak at t ≈ 2.5 s, the predicted response matches very well with the measured response. On this figure, the accelerations measured during the other passages with a similar speed are shown as well. Variations in the measured responses can be due to the fact that the wheel tracks are slightly different or due to small variations of the vehicle speed. Low values at the first and last 0.5 s of the signals are not valid and arising from the 1 s duration of the running time window. Figures 2.12b and 2.12d compare the narrow band frequency spectrum and PSD of the predicted and measured acceleration. The predicted and measured response follow a similar trend with the same dominant frequency and similar amplitudes. The underestimation of the response at high frequencies is attributed to two facts. First, as indicated in section 2.2.1, the road profile recorded at 21.6 km/h contains wavelengths in the range 0.25 − 15 m. Thus, for the vehicle speed v = 50 km/h, the range of excitation frequencies is limited to 0.9 − 56 Hz. Second, the flexibility of the axles and the body of the vehicle as well as other sources of vibration are disregarded in the model. Figure 2.13 compares the predicted and measured peak acceleration (PA) on the axles. To avoid the aforementioned irregularity in the measured unevenness of the nearside wheel track, the PAs have been computed for a travel distance of 120 m from 20 m before the measurement line (y = 80 m in figure 2.3a) till 100 m after it (y = 200 m). The predicted PAs have been computed from both road profiles recorded by the APL at 21.6 km/h and 54 km/h. The road profile recorded at 21.6 km/h includes wavelengths of 0.25 − 15 m so it covers the dominant frequency range (1 − 25 Hz) of the vehicle response for the vehicle speeds of 21.6 − 54 km/h. Similarly, the road profile recorded at 54 km/h completely covers the vehicle speeds of 54 − 135 km/h. Hence at low vehicle speeds, the PA predicted from the road

42


15

0 −20 −40 N

Acceleration [m/s2/Hz]

20

Filtered road profile

Acceleration [m/s2]

40

10

5

F 0

2

4

6

8

Time [s]

(a)

(c)


Acceleration [m/s2]

15 10 5

0

2

4

6 Time [s]

0

20 40 60 Frequency [Hz]

80

0


80

(b)

20

0

0

10

8

10

2

10

1

10

0

10

−1

10

(d)

Figure 2.12: Predicted (black) and measured (grey) (a) time history, (b) narrow band frequency spectrum, (c) running RMS value, and (d) PSD of the acceleration on the right side of the rear axle during a passage of the truck at a speed of 50 km/h on the deteriorated N9. Superimposed are on (a) anti-smoothed profiles of the near (N) and far (F) wheel tracks (black lines) and on (c) and (d) the experimental results for passages with similar speeds (dashed grey lines). profile recorded at 21.6 km/h is usually larger than the PA predicted from the road profile recorded at 54 km/h while at high vehicle speeds the opposite is true. There is a good agreement between the predicted and measured PAs. Higher vehicle speeds generally result in larger values of the predicted and measured PAs. This is due to the fact that the time between subsequent excitations by local irregularities in the road surface decreases with increasing vehicle speed. As a result, the axle has a higher initial vibration level when it is excited again. The excitation can be in-phase or out-of-phase, leading to an amplification or reduction of the axle vibrations. Since only small changes in the vehicle speed result in a change of an in-phase excitation to an out-of-phase excitation, the PAs do not monotonously increase with the vehicle speed, as can be clearly observed in the predicted PAs. The influence of the vehicle speed on the PA of the axle is in line with the observations of Cebon [46] and Gillespie et al. [93]. The PAs on the left hand side of the axles are slightly higher than the PAs on the right hand side


43

100 Peak acceleration [m/s2]

Peak acceleration [m/s2]

100 80 60 40 20 0

0

(a)

10 20 30 40 50 60 70 80 90 Vehicle speed [km/h]

40 20 0


0


(b) 100 Peak acceleration [m/s2]


(c)

60

0

100 80 60 40 20 0

80

0


80 60 40 20 0

(d)

Figure 2.13: Predicted PA from the road profile recorded at 21.6 km/h (thin line) and 54 km/h (thick line) versus measured PA (plus signs) on the left hand (left) and right hand (right) side of the rear (top) and front (bottom) axle as a function of the vehicle speed prior to the road rehabilitation. because the left wheel track is slightly rougher than the right wheel track [148]. Passages on the rehabilitated N9 Figure 2.14 compares the predicted and measured acceleration on the right side of the rear axle for a passage at 50 km/h on the new asphalt pavement of the rehabilitated N9 road. Figure 2.14a shows the time history of the acceleration as well as the anti-smoothed road profile with a base length of 0.30 m. The filtered road profile is quite smooth without any distinct irregularity. The time history of the acceleration of the axle does not show any large peak and is much lower than the one corresponding to the prerehabilitation passage. This indicates that slab misalignments and local unevenness have been eliminated and the vehicle vibration is due to the global road roughness and, according to Rouillard et al. [209], engine vibrations, wheel imbalance, and other sources of excitation. Figure 2.14c shows that the predicted and measured running RMS values of the accelerations are very similar. Comparing the frequency content of the predicted

44


1.5

0 −2 −4 N

Acceleration [m/s2/Hz]

2


Acceleration [m/s2]

4

1

0.5

F 0

2

4

6

8

Time [s]

(a)

0


80

0


80

(b) 0

10

2 2

Acceleration PSD [(m/s ) /Hz]

2 Acceleration [m/s2]

0

10

1.5 1 0.5 0

(c)

0

2

4

6 Time [s]

8

10

−1

10

−2

10

−3

10

(d)

Figure 2.14: Predicted (black) and measured (grey) (a) time history, (b) narrow band frequency spectrum, (c) running RMS value, and (d) PSD of the acceleration on the right side of the rear axle during a passage of the truck at a speed of 50 km/h on the rehabilitated N9. Superimposed are on (a) anti-smoothed profiles of the near (N) and far (F) wheel tracks (black lines) and on (c) and (d) the experimental results for passages with similar speeds (dashed grey lines). and measured response (figure 2.14b) reveals that the agreement is not perfect, however. This arises from the fact that the vehicle model has been calibrated by the measured response prior to the road rehabilitation, while, due to the nonlinearity of the suspension system and tyres, the parameters of the simplified linear vehicle model depend on the response level [94]. In the passages on the rehabilitated road the force applied to the suspension system is small and the nonlinear suspension system shows a higher stiffness, so the eigenfrequencies of the axles increase [46, 150]. Moreover, the damping of the suspension system strongly depends on the deformation velocity. Figure 2.14d compares the PSD of the predicted and measured acceleration. The agreement between the predicted and measured response is not as good as prerehabilitation. The dominant frequency of the predicted acceleration is smaller than the dominant frequency of the measured acceleration. Furthermore, in the predicted response, narrow peaks occur at multiples of 8.3 Hz. This is due to the peaks at multiples of ky = 0.6 m−1 in the


45

wavenumber content of the road unevenness measured after rehabilitation (figure 2.4b). This corresponds to the first harmonic runout of the follower wheel of the APL. Figure 2.15 compares the predicted and measured peak acceleration on the axles for the passages on the rehabilitated N9. In the experiments, some of the passages took place on the middle lane, so a distinction is made between the passages on the nearside lane and on the middle lane. A relatively good agreement is observed between the predicted and measured PAs of the axles. Both of them generally increase with increasing vehicle speed. The PAs are about one tenth of the PAs in the passages on the deteriorated N9 (figure 2.13) thanks to the new asphalt overlay. In general, the agreement between the predicted and measured PAs is considered to be reasonable. 10 Peak acceleration [m/s2]


10 8 6 4 2 0

0

(a)


4 2 0


0


(b) 10 Peak acceleration [m/s2]


(c)

6

0

10 8 6 4 2 0

8

0


8 6 4 2 0

(d)

Figure 2.15: Predicted PA from the road profile recorded at 21.6 km/h (thin line) and 54 km/h (thick line) versus measured PA (plus signs) on the left hand (left) and right hand (right) side of the rear (top) and front (bottom) axle as a function of the vehicle speed after the road rehabilitation. Superimposed on the graphs are the measured PAs in the passages on the middle lane (crosses).

46

2.5


Dynamic vehicle loads

The dynamic vehicle loads g ˆ (ω) are estimated from the measured road unevenness ˆ guc (ω) computed in equation (2.4). In order to cover the by means of the FRFs h relevant frequency range in the study of traffic induced vibrations (1 − 80 Hz), for all vehicle speeds between 30 km/h and 70 km/h the road profile recorded at the lower speed of 21.6 km/h is used. This profile, however, still does not include very short wavelengths that correspond to high frequency excitations at vehicle speeds smaller than 70 km/h. Passages on the deteriorated N9 Figure 2.16 shows the time history and narrow band frequency spectrum of the predicted load at the right wheels of the rear and front axles during a passage at 50 km/h on the deteriorated N9. Figure 2.16a shows that the peak loads are approximately twice the static loads. This is in agreement with the observations of Mitchell and Gyenes [183] and Cebon [46]. The large peak at t ≈ 2.5 s is due to the aforementioned local unevenness in the measured road profile. During some other passages with a higher vehicle speed, the predicted upward dynamic load becomes larger than the static load at a few locations along the road. In these cases, the assumption of a perfect contact between the vehicle and the road is no longer valid as a loss of contact would occur in reality. Figure 2.16a also shows that the rear wheels apply larger dynamic loads than the front wheel, although the static load on the front wheel is larger. From the narrow band spectrum of the vehicle load it is observed that the dominant frequencies correspond to the eigenfrequencies of the vehicle that are situated around 1 − 3 Hz and 10 − 18 Hz. The lower eigenfrequencies contribute more to the dynamic loads of the front wheel as expected from the FRFs (figure 2.11). The Dynamic Load Coefficient (DLC) is frequently used to characterise the magnitude of dynamic wheel loads. It is defined as the ratio of the RMS value of the dynamic wheel load to the static wheel load. The DLC of the right rear wheels is 0.293 which is close to the top limit of the typical range 0.1 − 0.3 [46, 71, 183]. This arises from the facts that the deteriorated N9 is quite rough and that the vehicle is unladen, so the static wheel load is small. The DLC of the right front wheel is 0.162. This is smaller than the DLC of the rear wheels mainly because the front axle is lighter than the rear axle (so its dynamic load is smaller) and the static load of the front wheels is larger than the rear wheels. The DLCs of the left rear and front wheels are equal to 0.305 and 0.168, respectively.

DYNAMIC VEHICLE LOADS

47

10

−10 −20 −30 −40 N F 0

2

4

6

8

8 Force [kN/Hz]

Force [kN]

0


10

6 4 2 0

10

0


80

0


80

Time [s]

10

−20 −30 −40 N F 0

(a)

2

4

6 Time [s]

8

10

8 Force [kN/Hz]

Force [kN]

0 −10


10

6 4 2 0

(b)

Figure 2.16: Predicted (a) time history and (b) narrow band frequency spectrum of the dynamic load at the right rear wheels (top) and right front wheel (bottom) during a passage of the truck at a speed of 50 km/h on the deteriorated N9. Superimposed on (a) are the anti-smoothed profiles of the near (N) and far (F) wheel tracks (black lines). Passages on the rehabilitated N9 Figure 2.17 shows the time history and narrow band frequency spectrum of the predicted load at the right wheels of the rear and front axles during a passage at 50 km/h on the rehabilitated N9. These figures show that the level of the dynamic loads after the road rehabilitation is about 10% of the level prior to rehabilitation. The dominant frequencies of the estimated forces are about 1−3 Hz and 10 − 18 Hz. The peaks in the measured road profile at multiples ky = 0.6 m−1 are also present in the frequency content of the predicted forces. The DLCs of the rear and front wheels are between 0.012 and 0.034. These values are very small in comparison with the typical range of 0.1 − 0.3 thanks to the new asphalt overlay. Non-uniformities in the tyres of the trucks can usually generate a similar or even larger dynamic loads [92].


N F 0

2

4

6

8

Force [kN/Hz]

1.5

−12 −13 −14 −15 −16 −17 −18


Force [kN]

48

1

0.5

0

10

0


80

0


80

N F 0

(a)

2

4

6 Time [s]

8

Force [kN/Hz]

1.5

−12 −13 −14 −15 −16 −17 −18


Force [kN]

Time [s]

1

0.5

0

10

(b)

Figure 2.17: Predicted (a) time history and (b) narrow band frequency spectrum of the dynamic load at the right rear wheels (top) and right front wheel (bottom) during a passage of the truck at a speed of 50 km/h on the rehabilitated N9. Superimposed on (a) are the anti-smoothed profiles of the near (N) and far (F) wheel tracks (black lines). Comparison of the 3D and half-car vehicle models Herein, the half-car vehicle model shown in figure 2.6b is used to compute the dynamic vehicle loads. The results are compared with the results of the 3D vehicle model to evaluate the importance of the unsprung masses roll modes in the estimation of the dynamic vehicle loads. The DLCs of the half-car model are computed in two cases: operated separately on profiles from each wheel track of the road and on the point-by-point average of the profiles from both wheel tracks. The relative difference between the DLCs computed with the half-car model and the 3D model for the vehicle speeds between 30 km/h and 70 km/h driven over the deteriorated N9 is less than 9%, 6%, and 17% for the right wheels, left wheels, and the point-by-point average of the left and right profiles, respectively. These values are, respectively, 12%, 16%, and 36% for the passages over the rehabilitated N9. The relative difference between the peak dynamic wheel loads is less than 16% for the passages over the deteriorated N9 computed from two wheel tracks and 38%

CONCLUSION

49

for the averaged wheel track. These values are, respectively, 24% and 44% for the passages over the rehabilitated N9. The larger differences between the half-car and the 3D vehicle models for the passages along the rehabilitated N9 is in agreement with the larger roll/bounce ratio of the rehabilitated road observed in figure 2.5. The differences observed above are in agreement with the findings of Gillespie and Karamihas [92] where a nominal error of 20% in the RMS values of the dynamic wheel loads and, consequently, in the DLCs were found.

2.6

Conclusion

In this chapter, the problem of estimating dynamic wheel loads due to the passage of a vehicle over road irregularities has been studied. First, the range of road irregularities that are relevant in vehicle dynamics and traffic induced vibrations is discussed. Some of the available techniques for road profiling are reviewed. The APL is found to be able to measure the required range of wavelengths provided an appropriate profiling speed. The effect of tyre enveloping that filters small road irregularities is discussed. Two methods for processing a recorded road profile are evaluated. The first method is the common rectangular moving average filter. The second filtering method consists of two steps. First, the true profile is smoothed by a rigid wheel model to get the basic curve. Then the effective road profile is obtained by rolling a line segment over the basic curve. The wavenumber response function of the second method is found to be closer to the experimental response function of a representative tyre. The unevenness of the deteriorated jointed concrete pavement of the N9 prior to rehabilitation and the new asphalt pavement of the rehabilitated N9 profiled by the APL are presented. It is shown how the differences in the unevenness of the left and right wheel tracks can result in the excitation of the axle roll modes of passing vehicles. This requires employment of a 3D vehicle model for the estimation of the dynamic vehicle loads. Next, the vehicle-road interaction problem is discussed. First, the common quartercar and half-car vehicle models are reviewed. Then, a more elaborate 3D vehicle model is presented which takes into account the profiles of the left and right wheel tracks simultaneously. The governing equations of the model are given. Afterward, the configuration of the measurement setup on the instrumented test truck in the experiments in Lovendegem is presented. The parameters of the 3D model of the test truck are adjusted by means of the experiment on the deteriorated N9. The eigenfrequencies, eigenmodes, and frequency response functions of the truck are presented. The accelerations of the axles are computed and compared to the experimental results for the passages over the deteriorated and rehabilitated N9. The predicted accelerations agree relatively well with the experimental results.

50


The non-linear behaviour of the vehicle and shortcomings in the recorded road profile - such as lateral tracking errors and the narrow footprint of the APL - are thought to be the source of discrepancies between the predicted and measured response. Finally, the dynamic vehicle loads during the passage of the truck over the deteriorated and rehabilitated N9 are estimated. The peak dynamic loads on the deteriorated N9 are found to be due to local road irregularities. These peak loads can be as large as the static load. The dynamic loads on the rehabilitated N9 are much smaller and caused by the global road roughness. The dynamic load coefficient for the unladen truck driven on the deteriorated N9 road can be as large as 0.3, while it is less than 0.04 for the same truck driven over the new asphalt road. The difference between the DLCs computed with the half-car and 3D vehicle models can be up to 16%.

Chapter 3

Prediction of traffic induced ground vibrations 3.1

Introduction

In this chapter, the road-soil interaction problem is addressed and the transfer functions between a point load on the road and the vibration velocities in the free field are computed. The road-soil interaction is a 3D dynamic soil-structure interaction (SSI) problem. When only the dynamic response of the road is of interest - for example in the study of pavement damage - frequently used models of the road-soil system are a layered halfspace (with the top layer representing the pavement) [137, 169] and an elastic beam [106] or plate [144] on a Winkler foundation or on a halfspace [18, 44, 48]. These models as well as simpler models such as a homogeneous halfspace have also been used for the prediction of ground-borne vibrations. Krylov [139, 140] has used the halfspace model for the prediction of ground vibrations generated by vehicles travelling over road humps and accelerating and braking vehicles. Hao and Ang [103] and Hunt [116] have also used the halfspace model for the prediction of ground vibrations generated by road traffic. Hung and Yang [115] have studied vibrations in a visco-elastic halfspace due to various vehicle loads. Aubry et al. [17] and Lombaert et al. [162] have used a more elaborate model for the prediction of ground-borne vibrations where the road is represented by an infinitely long elastic beam on a layered elastic halfspace. In this model, the road-soil interaction problem is solved using an efficient 2.5D procedure [102, 118] in the frequency-wavenumber domain. The model has been successfully validated by several experiments on continuous asphalt roads [160,161]. The model of an infinitely long beam appears to be reasonable for continuous pavements such as a continuously reinforced concrete pavement. It has also been 51

52

PREDICTION OF TRAFFIC INDUCED GROUND VIBRATIONS

used for jointed concrete pavements [197,199] where the discontinuity between the slabs is disregarded. Yang and Hung [262] have used a 2.5D finite/infinite element approach for modelling visco-elastic bodies subjected to moving loads. They divide the domain into a near field which consists of loads and structures (pavement) and a semi-infinite far field. The near field is simulated by finite elements and the far field covering the soil extending to infinity by infinite elements. A comprehensive review of the literature on the road-soil interaction problem is given by Beskou and Theodorakopoulos [32]. In the following, a more refined model for jointed concrete pavements is presented first. The model consists of a 3D flexible plate representing the pavement coupled to a layered elastic halfspace for the soil. Then, a continuous plate model and the beam model are presented. The effect of the road model on the transfer functions is investigated and demonstrated by a numerical example. Next, the generation of ground vibrations due to moving dynamic loads is considered. The transfer functions are combined with the dynamic vehicle loads estimated in the previous chapter to compute ground vibrations. The predicted ground vibrations are experimentally validated. The effect of road rehabilitation on vibration reduction is evaluated. Finally, the soil strains due to traffic loading are estimated to assess the validity of linear behaviour assumed for the soil.

3.2 3.2.1

Road-soil interaction Three-dimensional model of road-soil interaction

In the past, rigid pavements were analysed using the Westergaard theory [254]. Westergaard developed analytical equations for stresses in rigid pavements by modelling the road slab as a thin, infinite or semi-infinite plate resting on a Winkler foundation. In recent years, the finite element method is widely used for the analysis of rigid pavements where the slab is considered as a finite plate on a Winkler foundation [93, 185]. Herein, the jointed concrete slabs are modelled by a sequence of rectangular flexible plates resting on top of a horizontally layered elastic halfspace representing the soil (figure 3.1). In the following, the transfer function between a point load on the slab and the free field vibrations is calculated. For the definition of positions, a right-handed Cartesian frame of reference is defined with the origin at the soil surface and at the lane centre as shown in figure 3.1. The y-axis is parallel with the road in the driving direction, the x-axis is perpendicular to the road, and the z-direction is pointing upwards. Since the width of the concrete slabs is usually equal to the lane width and the vehicle is moving on the road, two extreme load positions can be assumed. Figure 3.1a

ROAD-SOIL INTERACTION

(a)

53

(b)

Figure 3.1: Finite plate model of the road-soil system with a force (a) at the centreline xs = {wt , 0, tr }T and (b) near the edge xs = {wt , L − p, tr }T of the slab. shows the first load position where a single slab subjected to a point load on the transverse centreline at the point xs = {wt , 0, tr }T where wt is half of the wheel track of a vehicle (figure 2.7) and tr is the slab thickness. Figure 3.1b shows the second load position where the same slab is subjected to a point load near its edge at the point xs = {wt , L − p, tr }T , with L half the length of the slab and p half the tyre footprint. Figure 3.2 shows a finite rectangular plate Ωr representing the concrete slab bounded by the free surface Γrσ and the slab-soil interface Σrs . The soil domain Ωs is a horizontally layered halfspace bounded by the free surface Γsσ , the slab-soil interface Σrs , and the outer boundary Γs∞ . The slab is subjected to a point load at an arbitrary position xs = {xs , ys , zs }T .

Figure 3.2: Model of a road slab on the soil surface. The slab is modelled using the FE method and coupled to a BE model for the soil. Welded boundary conditions are assumed at the interface of the FE and BE models. Continuous boundary elements are used. The weak form of the

54


equilibrium equation of the coupled slab-soil system gives the following system of equations in the frequency domain [58]: ˆ srr (ω)] u [−ω 2 Mrr + Krr + K ˆ r (ω) = ˆf r (ω)

(3.1)

where Mrr and Krr are the mass and stiffness matrices of the slab. The vector u ˆ r (ω) collects the degrees of freedom of the slab. The force vector ˆf r (ω) contains a unit force at the point xs = {xs , ys , zs }T in the negative z-direction. The dynamic ˆ srr (ω) of the soil is computed as [82]: stiffness matrix K Z ′ ′ ′ ′ ′ ′ s ˆ ˆ NT (3.2) Krr (ω) = r (x , y , z = 0)ts (Ns )(x , y , z = 0, ω) dS Σrs

The matrix Ns (x′ , y ′ , z ′ = 0) collects the globally defined shape functions for the BE mesh and is defined as Ns = S Nr , where S is a matrix that selects the translation DOFs of the FE shape functions Nr . In equation (3.2), the prime on the coordinates refers to a point on the interface. The matrix ˆts (Ns )(x′ , y ′ , z ′ = 0, ω) contains the soil tractions at the interface Σrs imposed by the shape functions Ns (x′ , y ′ , z ′ = 0) of the BE mesh. Following a regularisation procedure proposed by Fran¸cois [82], the BE system of equations is found as: h i ˆ ˆ ˆts (ω) (3.3) T(ω) +I u ˆ s (ω) = U(ω)

ˆ ˆ where T(ω) and U(ω) are the BE system matrices and I is the identity matrix. The BE system matrices are computed based on the Green’s functions of a horizontally layered halfspace by means of the Matlab toolbox BEMFUN developed by Fran¸cois et al. [84]. The Green’s functions are computed with the direct stiffness method using the Matlab toolbox EDT 2.2 developed by Schevenels et al. [219, 221]. The tractions ˆts (ω) are computed from the solution of equation (3.3) as: h i ˆ −1 (ω) T(ω) ˆ ˆts (ω) = U (3.4) +I u ˆ s (ω) ˆ s (ω) is found by introducing equation (3.4) in The dynamic soil stiffness matrix K rr equation (3.2): Z h i T ′ ′ ′ ′ ′ ′ ˆ s (ω) = ˆ −1 (ω) T(ω) ˆ K N (x , y , z = 0)N (x , y , z = 0) dS U +I s rr r Σrs

(3.5)

The integral in the right hand side of equation (3.5) is independent of the frequency and is denoted as a stress transfer matrix Tq : h i ˆ −1 (ω) T(ω) ˆ s (ω) = Tq U ˆ K +I (3.6) rr


55

ˆ s (ω) from equation (3.6) in equation (3.1), Introducing the dynamic soil stiffness K rr allows computing the displacement u ˆ r (ω) of the slab and the displacement u ˆ s (ω) of the soil. The tractions on the slab-soil interface are computed from equation (3.4). The integral representation theorem is applied to compute the radiated wave field from the tractions ˆts (ω) and displacements u ˆ s (ω) at the FE-BE interface: ˆ r (ω) ˆt (ω) − T ˆ r (ω) u ˆ s (ω) u ˆ (ω) = U s

(3.7)

where the vector u ˆ (ω) collects the displacement components at all receiver ˆ r (ω) and T ˆ r (ω) are boundary element transfer matrices. The locations and U transfer matrices are computed based on the Green’s functions of a horizontally ˆ s , x, ω) = u layered halfspace. The road-soil transfer function is equal to h(x ˆ (ω) T T where xs = {wt , 0, tr } or xs = {wt , L − p, tr } .

3.2.2

Two-and-a-half-dimensional model of road-soil interaction

Continuous plate model If the discontinuity of the slabs is disregarded, the 3D model of the road-soil system becomes invariant in the y-direction along the road. This allows for a transformation of the coordinate y to the wavenumber ky by means of a forward Fourier transform and an efficient solution of the equations of the coupled roadsoil system in the frequency-wavenumber domain (x, ky , z, ω) [85, 162]. Another advantage of this assumption is that the motion of the force in the forward direction can be replaced by a reverse motion of the receivers. This allows to compute the response due to a moving load from the transfer functions for a fixed source point.

(a)

(b)

Figure 3.3: (a) Continuous plate model and (b) beam model of the road-soil system.

56


Figure 3.3a shows the case of an infinitely long flexible plate subjected to a vertical point load. As the model is longitudinally invariant, the load is shifted in the y-direction to obtain a source position ys = 0. The 2.5D FE model of the road consists of shell elements with 4 DOFs at each node corresponding to 3 translations and one rotation around the y-axis. The boundary element mesh is chosen to match the FE mesh at the interface. Welded boundary conditions are assumed at the interface. The road displacement vector ur (x′ , y ′ , z ′ = 0, t) can be discretised as [85]: ur (x′ , y ′ , z ′ = 0, t) = Nr (x′ , z ′ = 0)ur (y ′ , t)

(3.8)

where Nr (x′ , z ′ = 0) are the two-dimensional boundary element interpolation functions defined on the road-soil interface and ur (y ′ , t) collects the DOFs of the boundary element mesh. As the model is longitudinally invariant, the y ′ coordinate can be transformed to the wavenumber ky by means of a forward Fourier transform. This allows for the formulation of equation (3.8) in the frequencywavenumber domain: u ˜ r (x′ , ky , z ′ = 0, ω) = Nr (x′ , z ′ = 0)˜ ur (ky , ω)

(3.9)

The equilibrium equation in the frequency-wavenumber domain can be elaborated from the virtual work equation as [85]: ˜ rr (ky , ω) + K ˜ s (ky , ω) u (3.10) S ˜ r (ky , ω) = ˜f r (ky , ω) rr

where ˜f r (ky , ω) is a unit force vector in the frequency-wavenumber domain and ˜ rr (ky , ω) is the dynamic stiffness matrix of the road in the frequency-wavenumber S domain which is equal to [38, 89]: ˜ rr (ky , ω) = −ω 2 Mrr + K(0) − iky K(1) − k 2 K(2) + ik 3 K(3) + k 4 K(4) (3.11) S rr rr y rr y rr y rr (0)

(4)

The finite element mass matrix Mrr and the stiffness matrices Krr to Krr are independent of the wavenumber ky and the frequency ω. The dynamic stiffness of ˜ s (ky , ω) is equal to: the soil K rr Z ′ ′ ′ ′ ˜ s (ky , ω) = ˜ NT (3.12) K r (x , z = 0)ts (Ns )(x , ky , z = 0, ω) dS rr Σrs

Since the BE mesh matches the FE mesh on the road-soil interface, the BE interpolation functions Ns correspond to the FE shape functions Nr on the roadsoil interface. The 2.5D BE method is based on the integral equation that relates the displacements in the soil domain to the displacements and tractions on the road-soil interface [85]. This leads to the following BE system of equations: ˜ y , ω) + I u ˜ y , ω)˜t (ky , ω) T(k ˜ s (ky , ω) = U(k (3.13) s


57

˜ y , ω) and T(k ˜ y , ω) are fully populated unsymmetric BE system matrices. where U(k Their evaluation requires the Green’s displacements u ˜G ij (x, ky , z, ω) and tractions G t˜ij (x, ky , z, ω) of the layered halfspace which are computed with the direct stiffness method [134,135]. The tractions ˜ts (ky , ω) are found as the solution of the boundary element system of equations (3.13): ˜ y , ω) + I u ˜ −1 (ky , ω) T(k ˜ts (ky , ω) = U ˜ s (ky , ω) (3.14) ˜ s (ky , ω) is found by introducing equation (3.14) The dynamic soil stiffness matrix K rr in equation (3.12): Z ′ ′ ′ ′ ˜ srr (ky , ω) = ˜ −1 (ky , ω) T(k ˜ y , ω)+I K NT (x , z = 0) N (x , z = 0) dS U r r Σrs

˜ y , ω) + I ˜ −1 (ky , ω) T(k = Tq U

(3.15)

R ′ ′ ′ ′ The matrix Tq = Σrs NT r (x , z = 0) Nr (x , z = 0) dS is independent of the ˜ s (ky , ω) wavenumber and frequency. Introducing the dynamic soil stiffness K rr from equation (3.15) in equation (3.10) the displacement u ˜ r (ky , ω) of the slab and therefore the boundary element model u ˜ s (ky , ω) is computed. Once the equilibrium equation (3.10) for the dynamic road-soil interaction problem has been solved, tractions on the slab-soil interface are computed from equation (3.14). The integral representation theorem is applied to compute the radiated wave field from the tractions ˜ts (ky , ω) and displacements u ˜ s (ky , ω) at the road-soil interface: ˜ r (ky , ω) ˜t (ky , ω) − T ˜ r (ky , ω) u u ˜ (ky , ω) = U ˜ s (ky , ω) s

(3.16)

where the vector u ˜ (ky , ω) collects the displacement components at all receiver locations. The free field displacements in the space-frequency domain are obtained by an inverse Fourier transform from the wavenumber ky to the horizontal coordinate y: Z +∞ 1 (3.17) u ˆ (y, ω) = u ˜ (ky , ω) exp (−iky y) dky 2π −∞ The road-soil transfer function for the source position xs = {wt , 0, tr }T is equal to ˆ s , x, ω) = u h(x ˆ (y, ω). Continuous beam model If the flexibility of the cross section of the plate is disregarded, the beam model is obtained [17, 159, 162]. This model only accounts for the bending and torsional

58


deformations of the road. Figure 3.3b shows the case of an infinitely long beam subjected to a point load. In this case, a 2.5D methodology can be applied as well for the calculation of the transfer function. The vertical road displacements urz (x, y, t) are now decomposed into bending and torsional modes: urz (x, y, t) = ucz (y, t) + xβcy (y, t) = Nr (x)ur (y, t)

(3.18)

where Nr (x) = {1, x}T collects the displacement modes and ur (y, t) contains the displacement ucz (y, t) and rotation βcy (y, t) of the centre of the road cross section. ˆ s , x, ω) is obtained by solving equations (3.10) to (3.17) The transfer function h(x for the case of the continuous beam.

3.2.3

Numerical example

In order to compare the three models of the coupled road-soil system, the transfer functions between an impulse load on the road and the vibration velocity in the free field are compared. In the analysis, the presence of the adjacent lanes is disregarded and only a single lane is considered. The road properties of the N9 road and the dynamic soil characteristics of the measurement site next to the N9 in Lovendegem are used. These parameters are presented in section 3.4.2. The width of the road is 2B = 2.85 m and the length of the finite plate model is 2L = 4.6 m. The finite plate model of the slab is discretised by a mesh of 22 × 36 4-node rectangular shell elements with 6 DOFs per node. This FE mesh allows for an accurate representation of the first few modes of the plate. A conforming BE mesh is chosen that matches the FE mesh on the road-soil interface where welded boundary conditions are assumed. Considering that the lowest shear wave velocity in the soil is 81 m/s, the resulting BE mesh is fine enough to accurately represent propagating waves in the soil in the frequency range up to 100 Hz. For the continuous plate model, the cross section of the road is discretised by 32 line elements. The boundary element model for both the continuous plate and the beam model is also composed of 32 line elements. A convergence study with finer FE and BE meshes showed that the considered meshes are sufficient for frequencies up to more than 100 Hz. The computation time for the analysis of the finite plate model is about 40% longer than the time required for the analysis of the continuous plate or the beam model. The transfer functions are also compared with the Green’s functions of the soil for a disc load with a diameter of 0.80 m. The size of the disc is roughly estimated from the load distribution pattern in the pavement [148]. The positions of the source and receivers are defined with respect to the right-handed Cartesian frame of reference shown in figure 3.1. Figure 3.4 compares the free field mobility in the x- and z-direction, at 4 m, 16 m, and 64 m from the road due to a vertical impulse load on the road. The mobility in the y-direction is zero because the source and receiver are at the same y-coordinate and the models are symmetric with respect to the xz-plane. Up to a frequency


59

x

z

−7

8

6

Mobility [m/s/N]

4m

Mobility [m/s/N]

8

−7

x 10

4 2 0

0

20


6 4 2 0

80

−7

1.5

1

0.5

0

0

20


0

80

80

20


80

20


80

x 10

0 −9

x 10

5

4 3 2 1 0


0.5

Mobility [m/s/N]

Mobility [m/s/N]

64m

20

1

−9

5

0 −7

x 10

Mobility [m/s/N]

16m

Mobility [m/s/N]

1.5

x 10

x 10

4 3 2 1

0

20


80

0

0

ˆ iz (xs , x, 0, 0, ω) in the x- (left) and z-direction Figure 3.4: Free field mobility i ω h (right) at x = 4 m (top), x = 16 m (middle), and x = 64 m (bottom) due to a vertical point load on the finite plate (thin black line), continuous plate (thick grey line), and continuous beam (thick black line) model at xs = {1 m, 0, 0.2 m}T . Superimposed is the Green’s function i ω u ˆG iz (1, 0, 0, x, 0, 0, ω) of the soil for a disc load with a diameter of 0.80 m (dashed line). of 50 Hz the transfer functions are similar for all receiver positions. At higher frequencies and small distances from the road, the differences between the models are more pronounced. This is due to the fact that, at these frequencies, the wavelength of the dominant waves in the soil is of the same order of magnitude as the dimensions of the slab. The free field mobilities in the x- and z-direction at 4 m from the road show a similar trend while the mobility in the x-direction has a slightly smaller amplitude at low frequencies up to about 20 Hz. At 16 m, the mobility in the x-direction at about 20 Hz (where the eigenfrequencies of the axles are situated) is considerably larger than the mobility in the z-direction. At 64 m, the mobilities in the x- and z-direction have a similar amplitude. Since only a small difference is found between the transfer functions of the continuous plate and the continuous beam model, the influence of the flexibility of the road cross

60


section is small. This is in agreement with the findings of Fran¸cois et al. [85]. In the frequency range up to 40 Hz, a good agreement is also observed between the Green’s function of the soil and the transfer functions. At higher frequencies, the distribution of the load by the road affects the free field response and a difference is found between the transfer functions and the Green’s function. Since the free field mobilities for all three road models are similar in the frequency range below 50 Hz that dominates the frequency content of the dynamic vehicle loads (figures 2.16 and 2.17), the beam model is used for the calculation of the free field response in the following. Effect of load position Since vehicles are moving along a road, the dynamic vehicle loads are applied at different positions along the wheel tracks. This is especially important on the jointed concrete pavements since the difference between two cases where the loads are applied on the edge or on the centre of a slab can be large due to the rigid body rotation of the slab. Figure 3.5 compares the free field mobility in the x-, y-, and z-direction, at 4 m, 16 m, and 64 m from the road due to a vertical impulse load on the centreline or near the edge of the finite plate model (figure 3.1). Two receiver lines are also considered: one along the centreline of the plate and one along the edge of the plate. The free field mobilities are only different at the smallest distance of 4 m from the road. The load on the edge of the plate generates a higher level of ground vibrations than the load on the centre. This is mostly apparent at frequencies above 20 Hz and arises from the fact that the load on the edge is transferred to a smaller area of the soil so there is less filtering effect of the road. The mobilities in the x- and z-direction have a similar amplitude while the one in the y-direction is smaller. The load on the edge generates vibrations in the y-direction at the receivers located along the edge of the plate in the free field due to the asymmetry of loading. Figures 3.4 and 3.5 reveal that there is a difference between the beam model and the finite plate model for the case where the load is moving along the road. The beam model will still be used in the rest of this chapter because the difference between the beam model and the finite plate model is limited to the high frequency content at small distances from the road. Effect of longitudinal loads The dynamic vehicle loads in the longitudinal direction generated by the passage of a vehicle over road irregularities can be as large as the dynamic loads in the vertical direction [79, 80, 166, 191, 264]. In order to investigate the effect of these longitudinal loads on the free field response, the free field mobilities due to an


61

y

x −7

8

4 2

20


6 4 2 0

80

−7

1.5

1

0.5

0

0

20


0

80

2

0

5

4 3 2

1.5

20


0

0

20


80

80

20


80

20


80

x 10

0 −9

5

2

0


0.5

x 10

3

1

20

1

80

4

1

0 −7

x 10

−9

x 10

0

4

0

80

0.5

Mobility [m/s/N]

Mobility [m/s/N]

64m


1

−9

5

20

x 10

6

−7

x 10

Mobility [m/s/N]

16m

Mobility [m/s/N]

1.5

0

Mobility [m/s/N]

0

8

Mobility [m/s/N]

0

z −7

x 10

Mobility [m/s/N]

6

Mobility [m/s/N]

4m

Mobility [m/s/N]

8

−7

x 10

x 10

4 3 2 1

0

20


80

0

0

ˆ iz (xs , x, y, 0, ω) in the x- (left), y- (middle), Figure 3.5: Free field mobility i ω h and z-direction (right) at x = 4 m (top), x = 16 m (middle), and x = 64 m (bottom) and y = 0 (solid line) and y = 2.2 m (dashed-dotted line) due to a vertical point load on the finite plate at xs = {1 m, 0, 0.2 m}T (black line) and xs = {1 m, 2.2 m, 0.2 m}T (grey line). impulsive vertical and longitudinal load are compared. Figures 3.6 and 3.7 show the free field mobility at x = 16 m as a function of the y-coordinate for an impulse load in, respectively, the y- and z-direction on the centre of the finite plate model. Figure 3.6 shows that longitudinal forces on the road surface generate a much larger response in the horizontal directions than in the vertical direction. When the source and receiver are at the same y-coordinate, the response in the y-direction shows a very large peak while the response in the x- and z-direction vanishes. The maximum vibration velocities in the x-direction due to a longitudinal impulsive force occur at about 10 m from the source. For a vertical load, the largest response is found in the x- and z-direction, while the response in the y-direction is relatively small (figure 3.7). When the source and receiver are at the same y-coordinate, the responses in the x- and z-direction


30 20 10 0

0

20


80

(a)

40

y−coordinate of receiver [m]

40



62

30 20 10 0

0

20


80

(b)

m/s/N −8 x 10 5

40

4

30

3

20 2

10 0

1

0

20


0

80

(c)

ˆ iy (0, x, y, 0, ω) in (a) the x-, (b) y-, and (c) Figure 3.6: Free field mobility i ω h z-direction at x = 16 m due to a point load in the y-direction at the centre of the finite plate model of the slab.

30 20 10 0

0

(a)

20


80

40


40



show peaks while the response in the y-direction becomes zero. These observations are used later in section 3.4.3 to interpret the predicted and measured free field response in the horizontal and vertical directions.

30 20 10 0

0

(b)

20


80

m/s/N −8 x 10 5

40

4

30

3

20 2

10 0

1

0

20


80

0

(c)

ˆ iz (0, x, y, 0, ω) in (a) the x-, (b) y-, and (c) Figure 3.7: Free field mobility i ω h z-direction at x = 16 m due to a point load in the z-direction at the centre of the finite plate model of the slab.

3.3

Ground vibrations generated by moving loads

Since the beam model has been chosen for the road, the following procedure which is only valid for a longitudinally invariant road model - can be employed. In the fixed frame of reference, the distribution of nc wheel loads is written as the summation of the product of Dirac functions that determine the time-dependent

GROUND VIBRATIONS GENERATED BY MOVING LOADS

63

position {xs , yk+vt, zs }T and the time history gk (t) of the k-th wheel load [159,162]: ρb(x, y, z, t) =

nc X

k=1

δ(x − xs )δ(y − yk − vt)δ(z − zs )gk (t)ez

(3.19)

where ρb(x, y, z, t) is the body load, nc is the number of contact points, yk is the initial position of the contact point k that moves with the vehicle speed v along the y-axis and ez denotes the vertical unit vector. Accounting for the invariance of the road-soil system in the longitudinal y-direction, the Betti-Rayleigh reciprocal theorem allows to derive the following expression for the response ui (x, y, z, t) due to the moving wheel loads: nc Z t X ui (x, y, z, t) = gk (τ ) hiz (x, y − yk − vτ, z, t − τ ) dτ (3.20) k=1

−∞

where hiz (x, y, z, t) is the transfer function between a vertical point load on the road at {xs , 0, zs }T and the free field response in the i-direction. The response due to a moving load can therefore be calculated from the response for a concentrated impulse load at a fixed position {xs , 0, zs }T . A double forward Fourier transformation allows to derive the following expression in the frequencywavenumber domain: u ˜i (x, ky , z, ω) =

nc X

k=1

˜ iz (x, ky , z, ω) exp (iky yk ) gˆk (ω − ky v)h

(3.21)

The frequency content of the displacement uî (x, y, z, ω) is found as the inverse Fourier transform of its representation in the frequency-wavenumber domain: nc Z +∞ 1 X ˜ iz (x, ky , z, ω) exp [−iky (y − yk )] dky gˆk (ω − ky v)h u î (x, y, z, ω) = 2π −∞ k=1

(3.22)

A change of variables according to ky = (ω − ω ˜ )/v moves the frequency shift from the frequency content of the moving load to the wavenumber content of the transfer function: nc Z +∞ 1 X ˜ ˜ iz (x, ω − ω u î (x, y, z, ω) = , z, ω) gˆk (˜ ω )h 2πv v −∞ k=1

ω−ω ˜ (y − yk ) d˜ ω exp −i v

(3.23)

and illustrates that traffic induced vibrations are caused by dynamic vehicle loads that cause wave propagation in the soil. The solution in the time domain is finally found as the inverse Fourier transformation with respect to ω.

64

3.4


Experimental validation

In this section, the free field vibrations generated by the passage of the truck on the N9 road are computed. The computation is based on the results obtained for the dynamic vehicle loads g ˆ (ω) (figures 2.16 and 2.17) and the road-soil ˆ s , x, ω) (figure 3.4). The results obtained for the passages transfer functions h(x along the deteriorated and rehabilitated N9 are compared to the in situ vibration measurements to validate the predictions. They are also compared against each other to evaluate the amount of vibration reduction gained by road rehabilitation.

3.4.1

Experimental configuration

During the passages of the Volvo FL180 on the N9 road prior to and after the rehabilitation, ground vibrations have been measured simultaneously [150]. Figure 3.8 shows the road and the setup of the accelerometers in the free field. The vertical ground accelerations are measured by means of 11 accelerometers located at the ground surface (z = 0 m) and at distances from 4 m up to 64 m from the road. At 16 m and 32 m, the horizontal accelerations perpendicular to the road (xdirection) and parallel with the road (y-direction) are measured as well. In order to detect the passage of the truck and also to synchronise vibration measurements in the free field and on the truck, a set of photoelectric sensor and reflector is installed in the free field (figure 3.8). The photoelectric sensor detects the passage of the reflectors on the truck while the reflector in the free field is detected by the photoelectric sensors on the truck (section 2.4.1). The experimental configuration in the free field consists of 15 accelerometers, one photoelectric sensor, and an LMS SCADAS III data acquisition system, coupled to a portable computer. The analogue to digital conversion is performed at a sampling rate fs = 2048 Hz, the corresponding Nyquist frequency thus equals fNyq = 1024 Hz. The measurement is started and stopped manually so a different number of data points is recorded for each passage. The signals are trimmed by a time window of length T = 10 s centred at the pulse recorded by the photoelectric sensor when the reflector on the truck passes the photoelectric sensor in the free field. To ensure that all signals are initially zero and vanish at the end, the time window corresponds to the modulus of the frequency content of a second order Butterworth filter with a pass band between 0.01T and 0.99T . The DC-component is removed with a fifth order Chebyshev type I high-pass filter with a cut-off frequency of 2 Hz and a ripple of 0.1 dB. The accelerations are integrated by means of the trapezium rule to obtain the velocity time histories.

EXPERIMENTAL VALIDATION

65

Figure 3.8: Location of the measurement points in the free field during the measurement campaign in Lovendegem.

3.4.2

Soil and road characteristics

Dynamic soil characteristics The database of the subsoil of Flanders (in Dutch: Databank Ondergrond Vlaanderen (DOV)) [188] contains data for a number of boreholes on or close to this site. Figure 3.9a shows the lithological description of the soil identified from three borehole samples (kb13d39e-B277, B/154/21/3, and kb13d39e-B2) at different locations up to 1200 m from the measurement site. These samples reveal the presence of a layer of grey sandy loam with a thickness of 0.8 m, a layer of yellow loose sand with a thickness of 2.2 m, and a layer of fine grey sand with a thickness of 3 m. Below 6 m, mainly grey sand is found. The dynamic soil characteristics have been identified by means of in situ geophysical tests. A Spectral Analysis of Surface Waves (SASW) test was performed to identify the dynamic soil characteristics of the shallow layers [151]. In addition, a Seismic Cone Penetration Test (SCPT) was performed to obtain information about the dynamic soil characteristics at a larger depth [15]. Figure 3.9b compares the shear wave velocity Cs in the soil as a function of depth obtained by means of the SASW test and the SCPT. Up to a depth of about 7 m, a good agreement between both results is observed. The results obtained from the SASW test at larger depths are not reliable due to the limited length of the measurement line and the limited energy of the excitation source [218]. At depths larger than 7 m, the shear wave velocity profile identified from the SCPT is therefore considered in the following. The dilatational wave velocity Cp of the soil layers is estimated from

66


0 0.8

Grey sandy loam Yellow loose sand

0

0

0

2

2

2

4

4

4

6

6

6

8

8

Depth [m]

Depth [m]

10 12

10 12

Depth [m]

Fine grey sand 6

Depth [m]

3

8 10 12

Grey sand

20

(a)

14

14

14

16

16

16

18

18

18

20

20

20

(b)

0 200 400 600 800 Shear wave velocity [m/s]

(c)

0 500 1000 1500 2000 Dilatational wave velocity [m/s]

(d)

0

0.01 0.02 0.03 0.04 0.05 Damping [−]

Figure 3.9: (a) Lithological description of the soil around the measurement site next to the N9 road in Lovendegem and dynamic soil characteristics identified by means of in situ tests at the measurement site: (b) shear wave velocity profile estimated from the SASW test (thick line) and the SCPT (thin line), (c) dilatational wave velocity profile estimated by the seismic refraction test (thick line) and the SCPT (thin line), and (d) material damping ratio estimated from the SASW test. a seismic refraction test and is shown in figure 3.9c. The relatively large difference between the estimated dilatational wave velocities from the seismic refraction test and the SCPT could be due to anisotropic behaviour of the soil that results in different phase velocities for horizontally and vertically propagating dilatational waves. The large increase of the dilatational wave velocity at a depth of 2.1 m is attributed to the presence of water table. The soil densities ρ are obtained from the data of six Cone Penetration Tests (CPT) near the measurement site as explained in appendix B. The material damping of the soil is modelled using the frequency independent hysteretic damping ratio. The application of the correspondence principle results in the use of complex Lamé coefficients (λ + 2µ)(1 + 2βp i) and µ(1 + 2βs i) and, consequently, the complex shear and dilatational wave velocities are equal to Cs∗ ≈ (1 + iβs )Cs and Cp∗ ≈ (1 + iβp )Cp , respectively. The material damping ratio βs for shear waves is estimated from the experimental attenuation curve using the half-power bandwidth method [19]. In this method, the wavefield recorded in the experiments is transformed to the frequency-wavenumber domain. The peak of the spectrum corresponds to the fundamental Rayleigh wave. The attenuation curve is obtained from the width of the peak using the half-power bandwidth method. The material damping ratio profile of the soil is obtained by solving an optimisation problem that tries to fit the attenuation curve of a synthetic profile to the experimental attenuation curve. The material damping ratio βp for the dilatational waves is assumed to be equal to βs . Poisson’s ratio ν has been computed from the values obtained for the shear and dilatational wave


67

velocities. Table 3.1 summarises the identified soil parameters. Layer 1 2 3 4 5 6

Thickness [m] 0.5 1.6 0.2 1.0 3.7 ∞

Cs [m/s] 81 155 155 206 206 260

Cp [m/s] 405 405 1029 1029 1664 1664

ν [-] 0.47 0.38 0.48 0.47 0.49 0.48

ρ [kg/m3 ] 1850 1900 2010 1980 1900 2000

βs and βp [-] 0.014 0.044 0.029 0.029 0.029 0.029

Table 3.1: Dynamic soil characteristics at the measurement site next to the N9 road in Lovendegem.

Mechanical properties of the road The N9 road prior to rehabilitation is composed of concrete slabs without any base or subbase layer. It has three lanes for cars and one cycling lane on each side. Each lane consists of concrete slabs with a width 2B = 2.85 m, a thickness tr = 0.20 m, and different lengths from 4 m up to 12 m, where a length 2L = 4.60 m has been considered in this study. Between the concrete slabs, expansion joints are present. The mechanical properties of the slabs have been identified from three cylindrical concrete core samples with a diameter of 113 mm [41, 149] according to Eurocode 2 [75]. The average compressive strength of these samples is equal to 105 MPa. The mean compressive strength for the standard cylindrical sample is estimated from the core samples as fcm = 0.74 × 105 = 78 MPa [75]. The dynamic modulus of elasticity is estimated from the compressive strength as Ec = 1.05 × 22 (fcm/10)0.3 = 43000 MPa [75]. The average density of the samples is equal to ρc = 2400 kg/m3 [41]. Poisson’s ratio is assumed to be equal to 0.20 as recommended for uncracked concrete [75]. These parameters give a shear wave velocity Cs = 2732 m/s and a dilatational wave velocity Cp = 4462 m/s. During the rehabilitation, the concrete slabs of the deteriorated N9 have been transformed into a suitable foundation for the new asphalt overlay by the cracking and seating method. The rehabilitated N9 is composed of a layer of cracked concrete with a thickness of 0.20 m, a crack prevention membrane, and an asphalt layer with a thickness of about 0.10 m. The width of the road is about 8.6 m, but since the longitudinal expansion joints between the slabs are still present, a width 2B = 2.85 m is considered as for the original road. Watts [251] and Lombaert [159] have shown that the thickness and mechanical properties of the road layers have

68


a small influence on the free field vibrations. Therefore, the road-soil transfer functions of the original N9 are also used for the rehabilitated road.

3.4.3

Validation of the predicted ground vibrations

In the following, the ground vibrations generated by the passage of the test truck at a speed of 50 km/h along the deteriorated and rehabilitated N9 are predicted and compared with the experimental results. The comparison is made for the vertical velocity at x = 4 m, 16 m, and 64 m, and for the horizontal velocity in the x- and y-direction at x = 16 m. In addition, the predicted and measured PPVs are compared. A time window of 10 s of the free field response centred at the time the truck passes in front of the measurement line is considered. This time window corresponds to a distance of 139 m travelled by the vehicle. Passages on the deteriorated N9 Figure 3.10 shows the time history, narrow band frequency spectrum, running RMS value, and PSD of the predicted and measured vertical velocity at 4 m from the road. Since the ground response is not stationary, the PSD of the vibration depends on the length of the time window considered. The PSDs of the predicted and measured response are therefore computed and compared on a same time window of 10 s. Figure 3.10a shows that the response is composed of several peaks that correspond to excitations by faulted joints. As the vehicle approaches, the vibration level increases and decreases after the vehicle passes in front of the measurement line. Comparing the time histories of the predicted and measured vibrations shows that the general trend is well predicted and an acceptable agreement is observed between the predicted and measured amplitudes. The vertical PPV reaches a value of about 0.8 mm/s. For the calculation of the running RMS value, a time window of 1 s is used (figure 3.10c). The predicted and measured running RMS values of the vertical velocity follow a similar trend and show a peak when the truck passes in front of the measurement line. The prediction underestimates the experimental results, however. This is attributed to the fact that the peak response at small distances is mostly sensitive to the unevenness in a stretch of a few metres of the road in front of the measurement line. So, a small error in the profiling of this section of the road - such as a lateral tracking error can result in a discrepancy between the predicted and measured responses. In addition, errors in the identified soil characteristics of the shallow layers and the soil near and under the road should be pointed out. Figures 3.10b and 3.10d show that the dominant frequency of the vertical velocity is situated at about 15 Hz corresponding to the eigenfrequencies of the axles. At frequencies below 40 Hz


69

0.12 Velocity [mm/s/Hz]

Velocity [mm/s]

1 0.5 0 −0.5 −1

0

2

4

6

8

0.04 0.02 0

0


80

0


80

−2

Velocity PSD [(mm/s)2/Hz]

RMS velocity [mm/s]

(c)

0.06

(b)

0.3

0.2

0.1

0

0.08

10

Time [s]

(a)

0.1

0

2

4

6 Time [s]

8

10

10

−4

10

−6

10

−8

10

−10

10

(d)

Figure 3.10: Predicted (black) and measured (grey) (a) time history, (b) narrow band frequency spectrum, (c) running RMS value, and (d) PSD of the vertical velocity at x = {4 m, 0, 0}T during a passage of the truck at a speed of 50 km/h along the deteriorated N9. Superimposed on (c) and (d) are the experimental results for passages with similar speeds (dashed grey lines).

the response is underestimated while a relatively good agreement is observed at higher frequencies. The vibration velocities recorded during different passages with similar speeds are rather similar showing a reasonable repeatability of the experiment. Figure 3.11 shows the vertical velocity at 16 m from the road. When compared to the response at 4 m, it is observed that the duration of the vibration has increased while the PPVs are smaller. As opposed to the free field velocity at 4 m (figure 3.10), the PPVs are slightly overestimated. At time t ≈ 3 s, the predicted response is smaller than the experimental results which is more apparent in the running RMS value in figure 3.11c. This cannot be attributed to the underestimation of the dynamic vehicle loads, because at this time, the predicted acceleration of the rear axle and consequently the dynamic wheel loads have been overestimated (figure 2.12c). At t = 3 s, the truck is located at a distance of 32 m from the receiver at 16 m. The underestimation of the ground vibration is thought to be caused by

70


the large damping ratios that have been identified for the deep soil layers (table 3.1). Figure 3.12 compares the transfer functions of the soil profile in table 3.1 with the transfer functions of the same soil profile but a lower damping of 0.01 for the fifth layer and the halfspace. A lower damping of the deep layers does not affect the free field mobility at small distances of 4 m and 16 m while it increases the mobility at 32 m and 64 m in the frequency range of 10 − 40 Hz. This confirms the argument presented above. 0.12 Velocity [mm/s/Hz]

Velocity [mm/s]

0.4 0.2 0 −0.2 −0.4

0

2

4

6

8

0.02 0


RMS velocity [mm/s]

0.04

0


80

0


80

−2

0.1

0.05

(c)

0.06

(b)

0.15

0

0.08

10

Time [s]

(a)

0.1

0

2

4

6 Time [s]

8

10

10

−4

10

−6

10

−8

10

−10

10

(d)


Figures 3.11a and 3.11c show that the vertical velocity at t ≈ 5 s - when the truck is at the smallest distance from the receiver - is overestimated. Figures 3.11b and 3.11d show that the response at frequencies above 30 Hz has mainly been overestimated. This can be attributed to the road-soil transfer function. At high frequencies, the road-soil transfer function is very sensitive to the material damping ratio βs in the soil. Since the value of βs for the shallow top layer is relatively small (table 3.1), the overestimation of the free field response in this frequency range


71

might be due to an underestimation of βs . Figure 3.12 also compares the transfer functions of the soil profile in table 3.1 for two different damping ratios of the first layer βs1 = 0.014 (the identified damping ratio) and βs1 = 0.044 (similar to the second layer). A larger damping of the first layer results in a significant decrease of the mobility at frequencies above 40 Hz (figure 3.12b). This argument is in agreement with the experimental observations of Yang et al. [261]. They found that the soil damping is large at low stresses (smaller than 25 kPa) corresponding to the conditions in the top 1 − 2 m of soil. They also found that the soil damping at high confining pressures and under low amplitude vibration is very small (about 1%). The PSD of the predicted vibrations (figure 3.11d) shows a dip at 40 Hz as observed for the road-soil transfer function at 16 m (figure 3.4). −7

−7

x 10

1.5

6

Mobility [m/s/N]

Mobility [m/s/N]

8

4 2 0

0

20

(a)


1

0.5

0

80

Mobility [m/s/N]

Mobility [m/s/N]

(c)

8

2 1

0

20


80

20


80

−9

x 10

3

0

0

(b)

−8

4

x 10

20


6 4 2 0

80

(d)

x 10

0

ˆ zz (wt , 0, tr , x, 0, 0, ω) at (a) x = 4 m, Figure 3.12: Free field vertical mobility i ω h (b) x = 16 m, (c) x = 32 m, and (d) x = 64 m due to a vertical point load on the continuous beam model resting on the soil profile in table 3.1 (thick line), the same soil but an increased damping of 0.044 for the first layer (thin line), and the same soil but a decreased damping of 0.01 for the fifth layer and the halfspace (dashed grey line).

At 64 m from the road (figure 3.13), the vibration level in the time window of 10 s is nearly stationary. This stems from the fact that the distance between the

72


source and receiver does not significantly change during this period (a variation between 64 m and 94 m). When the narrow band spectrum of the free field velocity at 64 m from the road is compared to the results at 4 m (figure 3.10b) and 16 m (figure 3.11b), it is observed that the high frequency components are more strongly attenuated, so that the low frequency vibrations dominate the free field response. Figure 3.13c shows that the numerical model underestimates the experimental results. This is mainly due to the underestimation of the frequency components below 20 Hz (figure 3.13b) and is attributed to the large damping ratios that have been identified for the deep soil layers as shown and discussed in figure 3.12d. 0.03 Velocity [mm/s/Hz]

Velocity [mm/s]

0.06 0.03 0 −0.03 −0.06

0

2

4

6

8

0.005 0


RMS velocity [mm/s]

0.01

0


80

0


80

−2

0.02

0.01

(c)

0.015

(b)

0.03

0

0.02

10

Time [s]

(a)

0.025

0

2

4

6 Time [s]

8

10

10

−4

10

−6

10

−8

10

−10

10

(d)


Figure 3.14 compares the predicted and measured vertical PPV in the free field as a function of the vehicle speed. Both the predicted and measured PPVs generally increase with the vehicle speed although not monotonically. It can be observed that a small change of the vehicle speed can result in the increase or decrease of the PPVs. The PPV is slightly overestimated at 16 m and underestimated at 4 m


73

and 64 m. In general, the agreement between the predicted and measured PPVs is acceptable.

1 0.5

0.6

0.1

0.5

0.08

PPV [mm/s]

1.5

PPV [mm/s]

PPV [mm/s]

2

0.4 0.3 0.2

0

0

10 20 30 40 50 60 70 80 90

(a)

0

0

10 20 30 40 50 60 70 80 90

(b)

Vehicle speed [km/h]

0.04 0.02

0.1 0

0.06

0

10 20 30 40 50 60 70 80 90

(c)



5

5

1 0.5

1 0.5

1 0.5

0.1 0.05 0.01 0.005

(a)

4

8

16 24 32 Distance [m]

48 64

PPV [mm/s]

5

PPV [mm/s]

PPV [mm/s]

Figure 3.14: Predicted (solid line) and measured (plus signs) vertical PPV at (a) x = {4 m, 0, 0}T , (b) x = {16 m, 0, 0}T , and (c) x = {64 m, 0, 0}T next to the deteriorated N9 as a function of the vehicle speed.

0.1 0.05 0.01 0.005

(b)

4

8


48 64

0.1 0.05 0.01 0.005

(c)

4

8


48 64

Figure 3.15: Predicted PPV in the (a) x-, (b) y-, and (c) z-direction in the free field at 11 sections perpendicular to the deteriorated N9 at intervals of 10 m from y = 50 m to y = 150 m as a function of the distance from the road for a passage of the truck at a speed of 50 km/h. Superimposed are measured PPVs at 15 receivers shown in figure 3.8 (plus signs). The peak ground vibrations are usually caused by local road irregularities. To assess the variation of ground vibrations along the deteriorated N9, the PPVs in the x-, y-, and z-direction are computed at 11 sections perpendicular to the road at intervals of 10 m from y = 50 m to y = 150 m. Figure 3.15 shows these PPVs as a function of the distance from the road for a vehicle speed of 50 km/h. The measured PPVs have been recorded during several passages of the vehicle. At x = 8 and 16 m the vertical PPVs recorded by the additional sensors (FF02zb, FF02zc, FF03zb, and FF03zc in figure 3.8) are also shown. The large variation of the PPV along the road is due to the presence of faulted joints at different locations. The ratio of the largest PPV to the smallest PPV is about 5 at small distances and reduces to 2 at large distances. The PPV generally decreases with the distance as expected, although not necessarily monotonically. This is attributed to two phenomena. First, the dominant velocity of the soil particles may change from the vertical direction to the horizontal direction or vice versa as

74


0.4

0.4

0.2

0.2

0.2

0 −0.2

0

2

4 6 Time [s]

8

4 6 Time [s]

8

4 6 Time [s]

8

0.05

0

10

0.12

0.1

0.1

0.08 0.06 0.04 0.02 0

20


2

4 6 Time [s]

8


−4

10

−6

10

−8

10

−10

0

20


80

0.02 0

20


8

10

0

2

4 6 Time [s]

8

10

0.1 0.08 0.06 0.04 0.02 0

80

0

20


80

0

20


80

−2

10

−4

10

−6

10

−8

10

(b)

4 6 Time [s]

0.12

0.04

−10

10

2

0.05

0

−2

10

0

0.1

10

0.06

80

−2

0

0.08

0

−0.2

0.15 RMS velocity [mm/s]

RMS velocity [mm/s] 2

0

−0.4

10

0.1

0.12

(a)

2

Velocity [mm/s/Hz]

Velocity [mm/s/Hz]

0

Velocity [mm/s/Hz]

RMS velocity [mm/s]

0.05

10

0

0.15

0.1

0


−0.2 −0.4

10

0.15

0

0


−0.4

Velocity [mm/s]

0.4 Velocity [mm/s]

Velocity [mm/s]

a function of the distance to the source. For example, at x ≈ 5 m the horizontal velocity is larger than the vertical velocity while at x ≈ 10 m the vertical velocity is dominant. This effect is mainly determined by the soil stratification. Second, a constructive or destructive interference may occur between the contributions of the wheels to the free field velocity. A good correspondence between the predicted and measured PPVs in the x-direction is observed while the PPV in the y-direction is underestimated. Comparing the measured and predicted vertical PPVs shows that, even when the response is slightly overestimated at small distances and underestimates at large distances, the agreement is reasonable.

0

20


80

10

−4

10

−6

10

−8

10

−10

10

(c)

Figure 3.16: Predicted (black) and measured (grey) vibration velocity in the (a) x-, (b) y-, and (c) z-direction at x = {16 m, 0, 0}T during a passage of the truck at a speed of 50 km/h along the deteriorated N9. Figure 3.16 shows the time history, narrow band frequency spectrum, running RMS value and PSD of the predicted and measured horizontal and vertical velocities at


75

16 m from the road. At the time when the truck passes in front of the measurement line, the predicted and measured vibrations in the x- and z-direction show a peak while the vibrations in the y-direction show a dip. This is similar to what has been observed in figure 3.7 with the horizontal and vertical free field mobilities due to a vertical load: when the vertical load approaches the receiver, the vibration velocities in the x- and z-direction increase, while the one in the y-direction decreases. The measured velocity in the x-direction has the largest amplitude while the smallest amplitude is found in the z-direction. In the predictions, however, the velocity in the z-direction has the largest amplitude and the smallest amplitude is observed in the y-direction. A number of researchers have shown that the longitudinal wheel forces generated by the passage of a vehicle over faults might have a similar order of magnitude as the vertical forces [79,80,166,191,264]. The underestimation of the horizontal vibrations, however, cannot be attributed to the fact that these longitudinal forces have been disregarded in the model because of the following reasons: first, figure 3.6 shows that these longitudinal forces would generate large vibrations in the y-direction but can not make up for the underestimation of the response in the x-direction; second, measurements of the free field vibrations during the operation of a pavement breaker at this measurement site [147,153] show that the vertical impacts of the pavement breaker generate higher vibration levels in the horizontal direction than in the vertical direction. The underestimation of the horizontal response is therefore probably due to errors in the identified soil profile or shortcomings in the model of the coupled road-soil system. Figure 3.16 also shows that, at frequencies above 40 Hz, the measured horizontal vibrations are considerably larger than the measured vertical vibrations.

0.6

0.6

0.5

0.5

0.5

0.4 0.3 0.2

0.4 0.3 0.2

0.1

0.1

0

0

(a)

0

10 20 30 40 50 60 70 80 90


PPV [mm/s]

0.6

PPV [mm/s]

PPV [mm/s]

Figure 3.17 compares the predicted and measured horizontal and vertical PPVs at x = 16 m as a function of the vehicle speed. The measured PPV in the x-direction is larger than in the two other directions. The vertical PPV is slightly overestimated whereas the horizontal PPVs are underestimated. Both the horizontal and vertical PPVs generally increase with the vehicle speed.

(b)

0.4 0.3 0.2 0.1

0

0

10 20 30 40 50 60 70 80 90


(c)

0

10 20 30 40 50 60 70 80 90


Figure 3.17: Predicted (solid line) and measured (plus signs) PPV in the (a) x-, (b) y-, and (c) z-direction at x = {16 m, 0, 0}T next to the deteriorated N9 as a function of the vehicle speed.

76


Passages on the rehabilitated N9 Figure 3.18 shows the time history, narrow band frequency spectrum, running RMS value, and PSD of the predicted and measured vertical velocity at 4 m from the rehabilitated N9. Figure 3.18a shows that, when the vehicle approaches, the vibration level increases where the PPV reaches to a maximum of about 0.15 mm/s - which is barely perceptible [123] - and decreases after the vehicle passes in front of the measurement line. The duration of the vibrations as well as the PPV is relatively well predicted. Figures 3.18a and 3.18c show that the vibration velocity recorded at t < 3 s and t > 7 s is larger than the prediction due to the background noise. 0.02 Velocity [mm/s/Hz]

Velocity [mm/s]

0.2 0.1 0 −0.1 −0.2

0

2

4

6

8

0.01 0.005 0

10

Time [s]

(a)

0.015

0


80

0


80

(b) −4


RMS velocity [mm/s]

0.06

0.04

0.02

0

(c)

0

2

4

6 Time [s]

8

10

10

−6

10

−8

10

−10

10

−12

10

(d)

Figure 3.18: Predicted (black) and measured (grey) (a) time history, (b) narrow band frequency spectrum, (c) running RMS value, and (d) PSD of the vertical velocity at x = {4 m, 0, 0}T during a passage of the truck at a speed of 50 km/h along the rehabilitated N9.

In order to obtain the level of ambient vibrations, the vertical velocity recorded at 4 m from the road from 15 s till 5 s before the truck passes in front of the measurement line are shown in figure 3.19. During this period of time, the truck is located between 208 m and 70 m from the measurement line so the level of induced


77

ground vibrations in the receivers is negligible. Comparing figures 3.18b and 3.19b reveals that the underestimation of the response at frequencies below 10 Hz arises from the ambient vibrations. The underestimation of the vibration velocities in the frequency range between 20 Hz and 30 Hz is attributed to the underestimation of the acceleration of the axle (figure 2.14b) and, consequently, the dynamic wheel loads in this range. The overestimation of the response above 40 Hz is, firstly, due to the spurious peaks in the dynamic vehicle loads (figure 2.17) which are arising from the peaks observed in the recorded road profile at multiples of wavenumber 0.6 m−1 (figure 2.4) and, secondly, due to the small material damping ratio that has been identified for the top soil layer which results in the overestimation of the road-soil transfer function at high frequencies. Comparing figure 3.18 to figure 3.10 shows that the vibration level is about one-sixth of the level prior to the road rehabilitation. 0.02 Velocity [mm/s/Hz]

Velocity [mm/s]

0.2 0.1 0 −0.1 −0.2

0

2

4

6

8


RMS velocity [mm/s]

0

0


80

0


80

−4

0.04

0.02

(c)

0.005

(b)

0.06

0

0.01

10

Time [s]

(a)

0.015

0

2

4

6 Time [s]

8

10

10

−6

10

−8

10

−10

10

−12

10

(d)

Figure 3.19: (a) Time history, (b) narrow band frequency spectrum, (c) running RMS value, and (d) PSD of the vertical velocity due to ambient excitation measured at x = {4 m, 0, 0}T during the measurement campaign after the road rehabilitation.

Figure 3.20 shows the vertical velocity at 16 m from the road. The duration of the vibration is well predicted while the PPV is overestimated. Figures 3.20b and

78


3.20d show that the predicted vibrations agree well with the experimental results in the frequency range between 10 Hz and 30 Hz. Similar to the response of the receiver at 4 m, below this frequency range, the response is underestimated due to ambient excitations while at frequencies higher than 30 Hz it is overestimated due to the spurious peaks in the estimated dynamic vehicle loads and the small material damping ratio of the top soil layer. 0.015 Velocity [mm/s/Hz]

Velocity [mm/s]

0.06 0.03 0 −0.03 −0.06

0

2

4

6

8

0.005

0

10

Time [s]

(a)

0.01

0


80

0


80

(b) −4


RMS velocity [mm/s]

0.02 0.015 0.01 0.005 0

(c)

0

2

4

6 Time [s]

8

10

10

−6

10

−8

10

−10

10

−12

10

(d)


Figure 3.21 shows that the vibration level at 64 m from the road is barely above the level of background noise. The PPV is as small as 0.01 mm/s while the peak running RMS value is below 0.005 mm/s. These are about one-fifth of what has been observed prior to the road rehabilitation (figure 3.13). Comparing figure 3.21d with figure 3.19d shows that the level of background noise is higher at the small distance from the road suggesting that the sources of ambient vibrations are located near the road. In the other passages over the rehabilitated N9, a higher level of background noise was observed which was found to be caused by ongoing construction works on the road [149, 150].


79

−3

6 Velocity [mm/s/Hz]

Velocity [mm/s]

0.02 0.01 0 −0.01 −0.02

0

2

4

6

8

Time [s]

(a)

2

0

3 2 1 0


80

0


80

−4


RMS velocity [mm/s]

(c)

x 10

4

0

4

(b)

−3

6

5

0

10

x 10

2

4

6 Time [s]

8

10

10

−6

10

−8

10

−10

10

−12

10

(d)


Figure 3.22 compares the predicted and measured PPV in the free field as a function of the vehicle speed. In the experiments, some of the passages took place on the middle lane, so a distinction is made between the passages on the nearside and the middle lane. A relatively good agreement is observed between the predicted and measured PPVs at 4 m and 16 m from the road during the passages on the nearside lane. In the passages on the middle lane, the distance between the vehicle and the receivers is about 3 m larger so smaller PPVs have been measured. At these receivers, the PPV increases with increasing vehicle speed. Figure 3.22c shows that the PPV at 64 m from the road is underestimated due to the high level of background noise and thus it is almost independent of the vehicle speed. The PPVs are about 5 to 10 times smaller than the PPVs during the passages on the deteriorated N9 (figure 3.14) thanks to the new asphalt overlay. In general, the agreement between the predicted and measured PPVs is considered to be reasonable. To assess the variation of ground vibrations along the rehabilitated N9, the PPVs

80


0.1

0.2 0.1

PPV [mm/s]

0.3

0

0.02

0.08

PPV [mm/s]

PPV [mm/s]

0.4

0.06 0.04 0.02

0

0

10 20 30 40 50 60 70 80 90

(a)

0

0.01 0.005 0

10 20 30 40 50 60 70 80 90

(b)


0.015

0

10 20 30 40 50 60 70 80 90

(c)



Figure 3.22: Predicted (solid line) and measured (plus signs) vertical PPV at (a) x = {4 m, 0, 0}T , (b) x = {16 m, 0, 0}T , and (c) x = {64 m, 0, 0}T next to the rehabilitated N9 as a function of the vehicle speed. Superimposed on the graphs are the measured vertical PPVs during the passages on the middle lane (crosses).

5

5

1 0.5

1 0.5

1 0.5

0.1 0.05 0.01 0.005

(a)

4

8


48 64

PPV [mm/s]

5

PPV [mm/s]

PPV [mm/s]

at 11 sections perpendicular to the road at intervals of 10 m from y = 50 m to y = 150 m are computed similar to what has been shown in figure 3.15 for the deteriorated N9. Figure 3.23 shows these PPVs as a function of the distance from the road for a vehicle speed of 50 km/h. In contradiction to what has been observed for the deteriorated N9, the variation of the PPV along the rehabilitated road is small because local irregularities have been eliminated by the new asphalt layer and the PPVs are mainly due to the global road unevenness.

0.1 0.05 0.01 0.005

(b)

4

8


48 64

0.1 0.05 0.01 0.005

(c)

4

8


48 64

Figure 3.23: Predicted PPV in the (a) x-, (b) y-, and (c) z-direction in the free field at 11 sections perpendicular to the rehabilitated N9 at intervals of 10 m from y = 50 m to y = 150 m as a function of the distance from the road for a passage of the truck at a speed of 50 km/h. Superimposed are measured PPVs at 15 receivers shown in figure 3.8 (plus signs and crosses). Figure 3.24 shows the time history, narrow band frequency spectrum, running RMS value, and PSD of the predicted and measured horizontal and vertical velocities at 16 m from the road. Similar to the results of the passages along the deteriorated N9 (figure 3.16), the measured velocity in the x-direction has the largest amplitude, while in the predictions, the velocity in the z-direction has the largest amplitude and the smallest amplitude is observed in the y-direction. The vertical PPV is overestimated while the horizontal PPVs are underestimated. This is found to be


81

0.06

0.06

0.03

0.03

0.03

0 −0.03

0

2

4 6 Time [s]

8

0.005

0

2

4 6 Time [s]

8

0.005

0

20


0.005

0

2

4 6 Time [s]

8


−6

10

−8

10

−10

10

−12

0

20


80

0

20


10

0

2

4 6 Time [s]

8

10

0.01

0.005

0

0

20


80

0

20


80

−4

−6

10

−8

10

−10

10

−12

10

8

0.005

80

10

(b)

4 6 Time [s]

0.01

0

−4

10

2

0.015

0.005

0

0

0.015

10

0.01

80

−0.03

0.02

Velocity [mm/s/Hz]

0.01

0

−0.06

10


Velocity [mm/s/Hz]

8

0.01

0

−4


4 6 Time [s]

0.015

10

0.015

10

2

RMS velocity [mm/s]

RMS velocity [mm/s]

RMS velocity [mm/s]

0.01

(a)

0

0.02

0.015

0

−0.03 −0.06

10

0.02

0

0


−0.06

Velocity [mm/s]

0.06 Velocity [mm/s]

Velocity [mm/s]

partly due to the larger level of background noise in the horizontal directions than in the vertical direction at frequencies below 15 Hz.

0

20


80

10

−6

10

−8

10

−10

10

−12

10

(c)

Figure 3.24: Predicted (black) and measured (grey) vibration velocity in the (a) x-, (b) y-, and (c) z-direction at x = {16 m, 0, 0}T during a passage of the truck at a speed of 50 km/h along the rehabilitated N9. Figure 3.25 compares the predicted and measured horizontal and vertical PPVs at x = 16 m as a function of the vehicle speed. The measured PPV in the xdirection is larger than in the two other directions. The PPVs measured during the passages of the truck on the middle lane are smaller than the PPVs measured during the passages on the nearside lane because of the larger source to receiver distance. Similar to figure 3.17, the vertical PPV is slightly overestimated whereas the horizontal PPVs are underestimated.


0.1

0.1

0.08

0.08

0.06 0.04

0.06 0.04

0.02

0.02

0

0

0

10 20 30 40 50 60 70 80 90

(a)


PPV [mm/s]

0.1 0.08

PPV [mm/s]

PPV [mm/s]

82

0.06 0.04 0.02

0

0

10 20 30 40 50 60 70 80 90

(b)

0

10 20 30 40 50 60 70 80 90

(c)



Figure 3.25: Predicted (solid line) and measured (plus signs) PPV in the (a) x-, (b) y-, and (c) z-direction at x = {16 m, 0, 0}T next to the rehabilitated N9 as a function of the vehicle speed. Superimposed on the graphs are the measured PPVs during the passages on the middle lane (crosses). Effect of road rehabilitation on vibration reduction 60

0.05

0.05

50

0.15

30

0.15

0.15

20 0.5

0.3

0.3

0.5

1

1

1 1

20

40

60

.5

0.5

22 80

1 100 y [m]

(a)

0.3

0.30

120

140

1

0.5

10 1 1 0 0

0.15

1

x [m]

40

1

1 160

180

200

60 50

x [m]

40 30 20

0.05

10

0.05

0.0

5

0.0

5

0.15

0

(b)

0

20

40

0.15 60

80

100 y [m]

120

140

160

180

200

Figure 3.26: Contour plot of the predicted vertical PPV in the free field for a passage of the truck at a speed of 50 km/h along the (a) deteriorated and (b) rehabilitated N9. The PPV is shown in mm/s.

SOIL STRAINS DUE TO TRAFFIC LOAD

83

Figure 3.26 shows the contour plots of the vertical PPV beside the deteriorated and rehabilitated N9. The large variation of the PPV along the deteriorated road that was presented in figure 3.15 can be better appreciated in this figure. At some locations along the deteriorated N9, the PPV is above 1 mm/s up to more than 10 m from the road. This level of vibration is very annoying to people [250]. The PPV is above the perception level (0.15 mm/s) up to about 30 m from the road. The large PPVs around y = 70 m are due to the large road irregularity at that point as shown in figure 2.3a. For the rehabilitated road, the PPV does not significantly vary along the road and is usually below the perception level even next to the road. Comparing figures 3.26a and 3.26b reveals a considerable reduction in traffic induced ground-borne vibrations achieved by road rehabilitation.

3.5

Soil strains due to traffic load

In this section, the level of strains in the soil due to traffic loading is evaluated to verify the validity of assuming a linear behaviour for the soil. Since the dynamic vehicle loads for the passages on the deteriorated road are larger than the dynamic loads after rehabilitation, only the former case is considered. In order to evaluate the strain level, the peak octahedral shear strain is computed as: q 2 2 2 2 oct (3.24) γmax = max (ǫ1 (t) − ǫ2 (t)) + (ǫ2 (t) − ǫ3 (t)) + (ǫ3 (t) − ǫ1 (t)) 3 where ǫ1 , ǫ2 and ǫ3 are the principal strains. Figure 3.27 shows the contour plot of the maximum octahedral shear strain in the soil under the road at y = 90 m for a passage of the truck at a speed of 50 km/h. At y = 90 m there is a large faulted crack with a step height of 0.011 m in the road surface (figure 2.3a). The strain distribution under the road is asymmetrical because the dynamic vehicle loads on the left and right wheel tracks are different. The large shear strains under the edges of the road are due to stress concentration at these points. In reality, the abrupt change and high value of tractions under the edge results in a very localised inelastic deformation of the soil in that region and, consequently, smaller strains in its vicinity. There are also large shear strains at the interface between the soft top layer and much stiffer second layer at a depth of 0.5 m (table 3.1). It is appreciated that the soft top layer has been compacted by traffic loading during many years operation of the road. Thus, the real shear strains in the top layer at the interface will be smaller than what has been predicted. Shear strains in the second layer also show peaks under the edges of the road and reach a maximum of 7 × 10−5 . This peak strain is still within the limits where the linear elastic behaviour assumption for the soils is valid.

84


0.2 0

3e−005

3e−005 5e−005 5e−005

5e−0

05

3e−005

5e−00

5

−1

3e−0

3e−005 3e−

−2 −2

01 5e−005

3e−005

005

3e−005 3e−005 −1.5 −1

00

0

5 3e−00 3e−005

05

−1.5

0.

.0001

05

z [m]

−0.5

.00 00.00010.0

5e−005

3e−0

0.0 00 002 1

01

−0.5

0 x [m]

0.5

1

1.5

2

oct Figure 3.27: Maximum octahedral shear strain γmax in the soil under the road at y = 90 m for a passage of the truck at a speed of 50 km/h along the deteriorated N9.

Shear strains due to the static wheel loads can increase the estimated octahedral shear strains due to the dynamic wheel loads. The peak load of the rear wheels due to this faulted crack is 35 kN where 15 kN is the static part and 20 kN is the dynamic part; therefore, the maximum octahedral shear strain due to the static load is about 5 × 10−5 and is similar to the shear strain due to the dynamic load. The total shear strain is therefore estimated to be about 1.2 × 10−4 which is still within the linear elastic limits as will be further investigated in section 5.4.1. Since high values of shear strain are only found in a small region, this small region does not affect traffic induced vibrations.

3.6

Conclusion

In this chapter, the numerical prediction of ground vibrations generated by dynamic vehicle loads is studied. First, the literature on dynamic road-soil interaction is briefly reviewed. Then, a 3D model of the road-soil system is presented. The model is composed of a finite element model for the slab coupled to a boundary element model for the layered elastic halfspace representing the soil. By disregarding the presence of the joints between the concrete slabs, the model becomes longitudinally invariant. Therefore, a 2.5D methodology is used to model the interaction between the infinitely long flexible plate and the soil. By assuming a rigid cross section for the plate, the model is further simplified to the available

CONCLUSION

85

beam model with bending and torsional modes. The transfer functions of the three road-soil interaction models are compared by means of a numerical example. In this example, the experimentally identified properties of the N9 road and the soil in the measurement site in Lovendegem are used to compute the free field mobilities at three distances from the road. It is shown that the free field mobilities computed with all models are very similar in the frequency range of interest so the road structure does not significantly affect the road-soil transfer functions. In addition, the free field mobilities are compared for a vertical load at the centreline and at the edge of the finite plate model. The results are only different at high frequency components of the response of near receivers. Furthermore, the free field mobilities are compared for a vertical and a horizontal load at the centre of the finite plate model. Next, the generation of ground vibrations due to moving dynamic loads is considered. Considering the dynamic Betti-Rayleigh reciprocal theorem, the motion of the source is replaced by an equivalent motion of the receiver position. Finally, the dynamic wheel loads estimated in the previous chapter and the road-soil transfer functions computed in this chapter are used to predict traffic induced ground-borne vibrations. The configuration of the experiment performed in Lovendegem is presented. In this experiment, the position and response of the vehicle and the free field response are measured simultaneously. The dynamic characteristics of the concrete road and the soil are identified by means of additional experiments. The ground vibrations at three distances of 4 m, 16 m, and 64 m from the deteriorated and rehabilitated N9 road are predicted. They are compared to the experimental results where a relatively good agreement is observed. The vertical vibration velocities are slightly underestimated at 4 m and 64 m while overestimated at 16 m. The observed discrepancies between the prediction and the measurement are attributed to inaccuracies in the estimated dynamic vehicle loads as well as errors in the identified soil parameters especially the material damping ratios of the soil layers. Numerical investigations showed that an accurate estimation of the material damping ratio in the soil is necessary for an accurate prediction of ground vibrations. The horizontal ground vibrations have a similar or even larger amplitude than the vertical vibrations. The PPV generally increases with the vehicle speed. The experimental and numerical results show a large variation of the PPV along the jointed concrete pavement of the deteriorated N9. The level of ground vibrations due to the passages along the rehabilitated N9 is about 5 to 10 times smaller than the level prior to the road rehabilitation. The induced vibrations are barely perceptible at a small distance of 4 m from the rehabilitated road while at 64 m, they hardly exceed the level of background noise. The PPV next to the rehabilitated road is mainly due to global road unevenness rather than local irregularities, thus its variation along the road is small. Apart from a small zone of the soil under the edges of the road where an abrupt

86


change of tractions occurs, shear strains due to traffic loading are of the order of 10−4 which is within the limits where the soil behaviour is usually assumed to be linear elastic. This assumption will be further investigated in section 5.4.1.

Chapter 4

Pavement breaking, impact load estimation, and fracturing 4.1

Introduction

Pavement breaking is a common construction activity at the end of the service life of rigid pavements and runways. It is performed to prepare the pavement for removal or as the first step in road rehabilitation. In this chapter, the pavement breaking operation is first introduced. Then, an overview of experiments on ground vibrations due to pavement breaking that have been performed in the frame of this study is presented. Next, a simple model is developed for the prediction of the impact load of a falling-weight pavement breaker. The model is numerically verified and experimentally validated. Finally, the effect of fracturing of the concrete slabs on the change of ground vibrations is investigated.

4.2

Pavement breaking

Concrete pavements are broken for several purposes such as the removal of the pavement, preparation for a new overlay, and burial of pipelines, sewer systems, and cables. For removal, the pavement is broken to small pieces for easier transportation and disposal. The size of the broken pieces depends on the project specifications and is between 15 cm by 15 cm to 60 cm by 60 cm. In preparation for a new overlay, an existing deteriorated concrete pavement is broken to smaller parts to eliminate reflective cracks in asphalt overlay [86]. Two techniques are usually employed: crack and seat method and rubblisation. In the crack and seat

87

88

PAVEMENT BREAKING, IMPACT LOAD ESTIMATION, AND FRACTURING

method, the concrete pavement is cracked so as to produce hairline cracks at a nominal spacing between 45 cm to 90 cm [170]. Then, a heavy roller - e.g. a 50 tonnes rubber tired roller - passes two or three times over the cracked pavement to seat the broken pieces on the base. Finally, an asphalt layer is overlaid. The procedure is called ‘break and seat’ in the case of reinforced concrete pavements such as JRCPs. The rubblisation method is usually employed when the pavement is severely deteriorated and has reached the end if its service life. In this method, the concrete pavement is intensely broken and reduced into rubble [141]. This results in a crushed, high-quality aggregate base for the new pavement. The rubblisation technique is also used to debond and remove reinforcing steel in the breaking for removal of reinforced concrete pavements. Depending on the required intensity of breaking, pavement thickness, and the amount of work, different types of pavement breakers may be employed. One of the smallest types of pavement breakers is the hand-held pneumatic drill or jackhammer. It is often used in breaking for removal in small scale projects. The operation of the jackhammer generates a series of transient excitations - e.g. 19 blows per second - that is perceived as pseudo-steady state ground vibrations. The level of ground vibrations generated by the jackhammer is smaller than for other types of pavement breakers (for example a vertical PPV of 0.9 mm/s at 7.6 m) [105, 256], but they are of more concern to the operators’ health. A pavement breaker frequently used for rubblisation is the resonant breaker shown in figure 4.1a. It blows the pavement at a low amplitude - e.g. 9 kN - and at a high frequency - e.g. 44 Hz - which results in severely broken pavement [51,168]. There is no information in the literature about the level of ground vibrations generated by this type of breaker. It is thought, however, that the vibration level is small as the impact force is rather small and also very concentrated. The main part of the impact energy is dissipated by severe crushing of the concrete to rubblised particles in the range of 2 cm to 8 cm.

(a)

(b)

Figure 4.1: (a) Resonant pavement breaker used for rubblisation (from [168]) and (b) impact roller used for pavement breaking.

PAVEMENT BREAKING

89

A less common method of pavement breaking is by means of an impact roller (figure 4.1b). In this method, a shaped drum (e.g. triangular ellipsoid) is towed over the road where the impacts of the drum crack the pavement. The most commonly used breakers impact the pavement with one or more fallingweight drop hammers. These breakers are generally in two types: guillotine type (figure 4.2) and multi-head breaker (figure 4.3). The guillotine breakers drop a single hammer from a height up to 3 m on the road surface. The hammer has a width between 1.8 m to 2.5 m and a narrow footprint of about 4 cm to concentrate the impact energy. Figure 4.2 shows two types of guillotine breakers with a drop hammer of about 6 tonnes [121].

Figure 4.2: Guillotine-type pavement breaker (from [121]).

The multi-head breaker has several cylindrical hammers (4 to 16) that blow the pavement one after another (figure 4.3). The mass of the drop hammers is usually between 400 kg and 700 kg each and the maximum drop height is about 2 m. By changing the equipment speed and drop height, the falling weight pavement breakers are able to create a broad range of crack pattern densities. For instance, a multi-head breaker with a large number of drop hammers can also be used for rubblisation. The operation of falling-weight pavement breakers generates a high level of vibrations that can be potentially damaging to the adjacent built environment (see section 1.2.2). In the following, an overview of an experiment on ground vibrations generated by the operation of a multi-head pavement breaker is given.

90


Figure 4.3: Multi-head pavement breaker.

4.3

Experimental study

As presented in section 1.2.2, a limited amount of literature considers ground vibrations generated by pavement breaking. None of these studies provides detailed information about the pavement breaker, pavement specifications, and soil characteristics. Therefore, in the frame of the present research a comprehensive measurement of ground vibrations generated by pavement breaking has been performed. This experiment has been designated for model validation where the applied force of the breaker as well as the generated ground vibrations has been measured. In addition, some other experiments have been conducted to identify the properties of the pavement and the soil. In the following, the experimental configuration in the vibration measurement campaign is described first. Next, the soil parameters identified from in situ tests are presented.

4.3.1

Measurement of ground vibrations due to pavement breaking

The experiments for model validation have been performed at two sites along the N9 road in Belgium [147,152,153]. In both measurements, a multi-head pavement breaker has been used as shown in figure 4.3. In the first measurement campaign in Lovendegem, ground vibrations at 4 m up to 64 m from the road have been measured. More information about this measurement can be found in reference [153]. Herein, the second measurement campaign [152] is presented which is more comprehensive and has been performed in Waarschoot.

EXPERIMENTAL STUDY

91

In the first stage of this experiment, the multi-head breaker, which is equipped with 4 drop hammers, stands still and applies 8 individual impacts (each hammer impacts two times) at 4 points of one particular slab. During these impacts, the response of the drop hammers, the slab, and the free field is measured. The purpose of this experiment is to obtain the free field response for one individual impact as well as the measurement of the drop hammer acceleration and, consequently, the impact force. In the second stage, the free field response during continuous operation of the MHB over a length of 29 m of the road is measured. This measurement provides more information about the vibration levels in the free field due to the operation of the MHB, and complements the previous measurement in Lovendegem [153]. In addition to these two stages, an instrumented sledge hammer has been used to determine the transfer functions between a point force on the road and the vibration velocity in the free field. The transfer functions have been measured in three conditions of the concrete slab: 1) the slab in its original condition, 2) after 8 impacts of the MHB, and 3) after the entire length of the slab has been fractured by the MHB. At each stage, a series of hammer impacts is applied on the slab. For each hammer impact, the force and the response of the slab and the free field are measured. The aim of these measurements is to investigate how the transfer of vibrations is influenced by the condition of the concrete slab. Figure 4.4 shows the measurement site as well as the location of the measurement points in the free field. A right-handed Cartesian frame of reference is defined with the origin at the centre of the slab on the soil surface, the x-axis perpendicular to the road, the y-axis parallel with the road, and the z-axis pointing upwards. The r-axis is in the direction of the measurement line and makes an angle of 13° with the x-axis. The free field vertical acceleration is measured by means of 16 accelerometers which are located on the ground surface at distances from 5 m up to 105 m from the road. Two shock accelerometers are installed on the drop hammers of the MHB. The shock sensors are ceramic shear accelerometers that can measure up to 50000 m/s2. They are screwed into nuts that are glued on the drop hammers. Figure 4.4 also shows the four points (P1 to P4) at the centreline of a slab next to the free field where the eight consecutive impacts of the MHB have been applied. The experimental configuration consists of 16 accelerometers in the free field, 8 accelerometers on the road, 1 accelerometer on the rightmost and 1 accelerometer on the leftmost drop hammer, and a NI PXI-1050 data acquisition system, coupled to a portable computer. The analogue to digital conversion is performed at a sampling rate fs = 5000 Hz, the corresponding Nyquist frequency thus equals fNyq = 2500 Hz. The measurement is started before the first impact and stopped after the last impact so the responses due to the 8 impacts have been recorded continuously. During post-processing, 8 time windows with a length T = 3 s have been applied to separate the impacts. The time windows are applied such that the peak of the signal of the first sensor on the slab is situated at 0.2 s. After the

92


Figure 4.4: Location of the impact points (P1 to P4) on the road and measurement points in the free field during the measurement campaign in Waarschoot.

remaining 2.8 s, the vibration level in all measurement points has reduced to the level of background noise. To ensure that all signals are initially zero and vanish at the end, a second time window is applied. It has a time history that corresponds to the modulus of the frequency content of a second order Butterworth filter with a pass band between 0.01T and 0.99T . To omit the DC-component from the recorded signals, the mean value or any possible linear trend is removed. The low frequency components of the accelerations are removed by applying a third order Chebyshev type I high-pass filter with a ripple of 0.1 dB. Since different types of accelerometers have different low frequency limits, the filter has different cut-off (high-pass) frequencies. For the signals of the sensors on the slab and the first sensor in the free field at 5 m, the high-pass frequency is fh = 2 Hz while for the remaining sensors in the free field fh = 0.5 Hz. The accelerations are integrated by means of the trapezoidal rule to obtain velocity time histories. More details about this experiment can be found in [152]. Properties of the drop hammers The drop hammers of the MHB are four steel cylinders with a diameter of 0.30 m, a height of 1.0 m, and a mass md = 600 kg (figure 4.3). The MHB can release the drop hammers from two different heights of 1.3 m and 1.8 m; a height h = 1.8 m is considered herein. During its fall, the drop hammer has to return the hydraulic lift cylinder of the MHB to its original position. This prevents the free fall of the drop

EXPERIMENTAL STUDY

93

hammer and reduces the impact velocity. It was found that the impact velocities cannot be estimated accurately by the integration of the recorded accelerations of the drop hammers [152]. This arises from the fact that the acceleration of the drop hammer during the falling phase is almost constant. The constant acceleration generates a DC signal which is out of the operational frequency range of the accelerometers. The impact velocity is therefore estimated from a movie recorded during the operation of the MHB. The camera takes 24 images per second. Two consecutive images recorded just before the impact show a displacement of 0.23 ± 0.01 m of the drop hammer. Divided by the time interval between the images, this gives the impact velocity v0 = 5.5 ± 0.24 √ m/s. The estimated speed v0 = 5.5 m/s is about 92% of the free fall velocity ( 2gh = 5.94 m/s). The bottom of the drop hammer is not completely flat, so the impact footprint is a circle with a diameter df = 0.13 m. After the impact, the drop hammer rebounds off the road and a mechanism prevents reimpacting. The height up to which the hammer rebounds is very small.

4.3.2

Identification of soil parameters

In addition to the measurements explained above, some other experiments have been performed to obtain the properties of the road and the soil. The mechanical properties of the concrete pavement of the N9 road have been obtained from three core samples as explained in section 3.4.2. The slabs of the N9 have different lengths between 4 m up to 12 m. Herein, a length 2L = 4.60 m is considered which corresponds to the length of the slab shown in figure 4.4. Since the measurement site of Waarschoot is located at about 1200 m from the measurement site in Lovendegem, the dynamic soil characteristics have been obtained from additional experiments. Figure 4.5a shows the lithological description of borehole sample kb1314-B60 [188], 1100 m northeast of the measurement site. It reveals the presence of a layer of organic soil with a thickness of 0.5 m, a layer of brown sand with a thickness of 9.5 m, and a layer of grey sand with a thickness of 5 m. Below 15 m, several layers of fine and coarse grey sand are observed. Borehole sample kb13d39e-B278 with a depth of 4.0 m is the nearest borehole to the measurement site at about 200 m west. It reveals a layer of grey sandy loam with a thickness of 2.5 m overlaying grey loose sand with a thickness of at least 1.5 m. The maximum difference between the elevations of the ground surface at these boreholes is 2 m. The dynamic soil characteristics have been identified by means of in situ geophysical tests. Three methods have been combined to determine the dynamic soil properties: the seismic refraction method, the active SASW method, and the passive SASW method [20, 151]. Figures 4.5b and 4.5c show the shear wave velocity Cs and the dilatational wave velocity Cp as a function of depth as obtained

94


0.5 3.6

Organic soil

0

0

5

5

5

10

10

10

15

15

15

20

20

20

Brown sand

10

0

Grey sand 15

25 30

25 30

Depth [m]

Depth [m]

Fine grey sand Shell

Depth [m]

Depth [m]

Coarse grey sand 21 24 25

25 30

35

35

35

40

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40

45

45

45

50

50

Grey sand

50

(a)

(b)

0 200 400 600 Shear wave velocity [m/s]

(c)

0 500 1000 1500 2000 Dilatational wave velocity [m/s]

50

(d)

0 0.01 0.02 0.03 0.04 0.05 Material damping ratio [−]

Figure 4.5: (a) Lithological description of the soil near the measurement site in Waarschoot and dynamic soil characteristics estimated from the surface wave tests at the measurement site: (b) shear wave velocity, (c) dilatational wave velocity, and (d) material damping ratio profile. from these tests. The identified shear wave velocity profile (figure 4.5b) is rather similar to what has been identified in Lovendegem as shown in figure 3.9b. The large increase of the dilatational wave velocity at a depth of 3.6 m is attributed to the presence of the ground water table. The soil densities ρ of the first four layers are obtained from the data of six CPTs near the measurement site as explained in appendix B. The density of the deeper layers is assumed to be equal to ρ = 2000 kg/m3. The material damping ratio βs for shear waves is estimated from the experimental attenuation curve using the half-power bandwidth method [19]. The same values are used for the material damping ratio βp for the dilatational waves (figure 4.5d). Poisson’s ratio ν has been computed from the values obtained for the shear and dilatational wave velocities. Table 4.1 summarises the identified soil parameters. From fast and slow cyclic tests on sand, Bolton and Wilson [37] have found that the stiffness and damping of the soil are independent of the strain rate. Therefore, the identified soil parameters in table 4.1 are not modified and used in the following to validate a numerical model for the estimation of the impact load and for the prediction of ground vibrations generated by pavement breaking.

ESTIMATION OF THE DROP HAMMER IMPACT LOAD

Layer 1 2 3 4 5 6 7 8 Halfspace

Thickness [m] 3.6 2.3 2.2 4.1 4.0 2.7 7.4 12.5 ∞

Cs [m/s] 159 159 240 240 250 255 328 400 500

Cp [m/s] 562 1465 1465 1579 1829 2000 2000 2000 2000

95

ν [−] 0.46 0.49 0.49 0.49 0.49 0.49 0.49 0.48 0.47

ρ 3 [kg/m ] 1920 1860 2000 1960 2000 2000 2000 2000 2000

βs and βp [−] 0.031 0.038 0.039 0.040 0.040 0.040 0.040 0.040 0.040

Table 4.1: Dynamic soil characteristics at the measurement site in Waarschoot.

4.4 4.4.1

Estimation of the drop hammer impact load Impact model

In the literature, the modelling of the impact load due to impulsive sources such as impact pile driving [67], falling weight dynamic soil compaction [206, 237, 240], and hammers and presses [50] has usually been studied using a simple model consisting of a rigid mass, a dashpot, and/or a spring. Bycroft [43] has studied theoretically and experimentally the impact of a rigid body on an elastic halfspace. Chow et al. [55] have developed a 1D FE model for the simulation of the pounder deceleration in the dynamic soil compaction analysis. In the study of a pavement breaker impact, however, the flexibility of the road must be taken into account. A numerical model for this purpose is presented in this section. First, a simplified numerical model is developed that predicts the force applied by the drop hammers on the road surface. Second, the coupled FE-BE model presented in section 3.2.1 is used to verify the model. Finally, the results obtained with the model are validated experimentally. Figure 4.6 shows a drop hammer and a slab of the road. The hammer is released at time t = 0 from a height h and impacts the slab at t0 . A previous study [152] has shown that the drop hammer is much stiffer than the road, so it can be assumed as a rigid mass in comparison with the road-soil system. The vertical displacement of the drop hammer is denoted as ud (t) where t represents the time. The impact force fd (t) is computed from the mass md of

96


Figure 4.6: Drop hammer and a slab of the road. the drop hammer and its acceleration u ¨d (t) as: fd (t) = −md u ¨d (t)

(4.1)

If the road-soil system is approximated as a massless spring-dashpot system with stiffness k and damping c (figure 4.7a), the equation of motion of the drop hammer becomes: u ¨d (t) = −g

for 0 ≤ t < t0

md u¨d (t) + c u˙ d (t) + k ud (t) = −md g

for t0 ≤ t ≤ t0 + td

(4.2a)

(4.2b) √ with initial condition ud (0) = h which gives the impact velocity u˙ d (t0 ) = − 2 g h. td is the impact duration and obtained from the inequality c u˙ d (t) + k ud (t) ≤ 0 which implies that the road-soil system cannot apply a downward force on the drop hammer.

(a)

(b)

Figure 4.7: (a) Spring-dashpot and (b) mass-spring-dashpot representation of the slab-soil system. The problem is first solved for the case of an undamped system (c = 0). The mass of the drop hammer is assumed aspmd = 600 kg and the drop height h = 1.8√m, which gives the impact time t0 = 2h/g = 0.606 s and the impact velocity − 2 g h = −5.94 m/s. The stiffness k = 8 × 108 N/m is an approximation of the


97

dynamic stiffness of a layered halfspace model for the road-soil system, obtained as the average value over the frequency range from 0 to 400 Hz, as will be shown later in figure 4.10a. Figure 4.8 shows the displacement, velocity, and acceleration of the drop hammer. Since the duration of the free fall is much longer than the impact duration, the lower limit of the time axis is shifted to t0 for better readability. It can be observed that the drop hammer follows half a cycle of the free vibration of an undamped SDOF systemp (figure 4.8a). The duration of the impact is td = T /2 = 0.0027 s where T = 2π md /k is the natural period of the system. Figure 4.8b shows that the impact velocity reduces from the negative initial velocity to zero and then increases. At the time when the drop hammer leaves the system, its velocity is as large as the initial velocity but in the opposite direction. The drop hammer continues to bounce on the spring while the bounce height is the same as the drop height because the system is undamped. The impact force is proportional to the acceleration of the drop hammer as seen inpequation (4.1). The acceleration increases from −g to a maximum value of v0 k/md − g at t = t0 + T /4 and reduces to −g when the drop hammer leaves the system (figure 4.8c). −3

x 10

6

2 0 −2 −4

(a)

−6 0.606

15000

4

Acceleration [m/s2]

4 Velocity [m/s]

Displacement [m]

6

2 0 −2 −4

0.608

0.61 Time [s]

0.612

0.614

(b)

−6 0.606

10000 5000 0 −5000

0.608

0.61 Time [s]

0.612

0.614

(c)

0.606

0.608

0.61 Time [s]

0.612

0.614

Figure 4.8: (a) Displacement, (b) velocity, and (c) acceleration of the drop hammer for an impact on a massless undamped (light grey line), underdamped (dark grey line), and overdamped (black line) spring-dashpot system. Superimposed are the responses assuming the drop hammer is attached to the system (dashed lines). Now, the damping of the system is taken √ into account. First, an underdamped system with a damping ratio ξ = c/ 4 k md = 0.4 is considered. Figure 4.8 also shows the displacement, velocity, and acceleration of the drop hammer for the underdamped case. The displacement of the drop hammer is smaller and the impact duration is shorter than for the undamped system. After the impact, the drop hammer springs back the spring-dashpot system and moves upward and continues bouncing on the system. Figure 4.8b shows that the velocity of the drop hammer reduces from the negative initial velocity to zero and then increases to a positive value. The acceleration of the drop hammer increases from an initial p value equal to −g − 2 ξ v0 k/md , reaches a peak and decreases to −g where the drop hammer leaves the system (figure 4.8c).

98


Second, an overdamped system with a damping ratio ξ = 1.1 is considered. In this case, the drop hammer moves downward and then returns to its original position and stops p (figure 4.8a). The drop hammer has a positive initial acceleration (−g − 2 ξ v0 k/md ) which exponentially reduces to zero (figure 4.8c).

Figure 4.7b shows the case where the road-soil system is approximated by a massspring-dashpot system. In this case, the contact between the drop hammer and the road occurs in instants (the contact duration is zero). Apart from the impact moments, the only force that is exerted on the drop hammer is its weight so the equation of motion of the drop hammer is: u ¨d (t) = −g

for t 6= t0 , t1 , t2 , . . .

(4.3)

with initial condition ud (0) = h. The times at which the instantaneous contacts between the drop hammer and the system occur are denoted by t0 , t1 , t2 , . . . . Assuming that the mass-spring-dashpot system is initially at rest, the displacement of the system before the impact becomes: u(t) = −

mg k

for 0 ≤ t < t0

(4.4)

where u(t) is the vertical displacement of the system and m is its mass. The equation of motion of the mass after the first impact except for the moments of impact is: mu ¨(t) + c u(t) ˙ + k u(t) = −m g

for t0 < t and t 6= t1 , t2 , . . .

(4.5)

At the moments of impact, the equations of motion of the drop hammer and the system are coupled, so the law of conservation of momentum reads as: md u˙ d (t− ) + m u(t ˙ − ) = md u˙ d (t+ ) + m u(t ˙ +)

for t = t0 , t1 , t2 , . . .

(4.6)

In addition, the law of conservation of energy can be used to compute the response of the drop hammer and the system. Since the impacts are instantaneous and the positions of the drop hammer and the system in the infinitesimally small time of the impact do not change, the potential energy at the moments of impact does not change as well. Therefore, the law of conservation of energy results in the conservation of the kinetic energy at each impact as: 1 1 1 1 md u˙ 2d (t− ) + m u˙ 2 (t− ) = md u˙ 2d (t+ ) + m u˙ 2 (t+ ) for t = t0 , t1 , t2 , . . . 2 2 2 2 (4.7) Equations (4.6) and (4.7) give the velocities of the masses after each impact. If the mass of the drop hammer is smaller than the mass of the system, the drop hammer


99

rebounds and the road moves downward with a velocity smaller than the impact velocity. Otherwise, the road moves downward with an initial velocity higher than the impact velocity of the drop hammer and the drop hammer continues its downward motion with a velocity lower than the impact velocity. Since the soil restrains the downward motion of the road, an infinite number of instantaneous contacts between the road and the drop hammer occurs. The dynamic stiffness of the road-soil system depends on the frequency, however, and the aforementioned simple models cannot give an accurate estimation of the impact force. By taking into account the dynamic stiffness of the road-soil system, equation (4.2) can be written more precisely as: u ¨d (t) = −g md u¨d (t) +

Z

for 0 ≤ t < t0

(4.8a)

for t0 ≤ t ≤ t0 + td

(4.8b)

t

t0

S(t − τ ) ud (τ ) dτ = −md g

with initial condition ud (0) = h. S(t) is the dynamic stiffness of the roadsoil system. The solution of equation (4.8) gives the acceleration u ¨d (t) of the drop hammer and, consequently, the impact force fd (t). The computation of the dynamic stiffness S(t) of the slab-soil system in the time domain is not straightforward. Equation (4.8b) can easily be transformed to the frequency domain if the slab-soil system is linear with zero initial conditions. In this case, the convolution in the time domain becomes a multiplication in the frequency domain and the dynamic stiffness is easier to obtain. Since the acceleration of the drop hammer after the impact (t0 < t) is much larger than the acceleration of gravity −g before the impact (t < t0 ), the initial acceleration can be approximated as u ¨d (t− 0 ) ≈ 0. At a time t1 during the course of impact (t0 < t1 ≤ t0 + td ), the mobilised mass of the slab-soil system becomes larger than the mass of the drop hammer. At this time, the velocity of the drop hammer will become zero. Using a procedure proposed by Humar [114], Martins et al. [172], and Clouteau and Aubry [58], the initial conditions can be replaced by a pseudo-force. By approximating the impact duration ∆t = (t1 − t0 ) → 0 and considering Newton’s second law, the pseudo-force is equal to: p(t) =

d (md u˙ d (t)) = md v0 δ(t − t0 ) dt

(4.9)

where δ(t) is the Dirac delta function. The pseudo-force p(t) is an impulsive force that produces a velocity change from 0 to v0 . Equation (4.8b) with initial − conditions u˙ d (t− ¨d (t− 0 ) = −g t0 and u 0 ) ≈ 0 is replaced by the following equation with zero initial conditions: Z t S(t − τ ) ud (τ )dτ = md v0 δ(t − t0 ) for t0 ≤ t ≤ t0 + td md u¨d (t) + t0

(4.10)

100


In the frequency domain, equation (4.10) reads as: ˆ − md ω 2 uˆd (ω) + S(ω) uˆd (ω) = md v0

(4.11)

Consequently, equation (4.11) and the frequency domain representation of equation (4.1) (fˆd (ω) = md ω 2 u ˆd ) give the impact force: fˆd (ω) =

m2d ω 2 v0 ˆ −md ω 2 + S(ω)

(4.12)

ˆ The dynamic stiffness S(ω) of the slab-soil system at the impact point can be computed as the reciprocal of the vertical displacement uˆz (xs , ω) at the impact point xs = {xs , ys , zs }T due to a unit impulsive force δ(x − xs )δ(y − ys )δ(z − zs )δ(t). ˆ s , ys , zs , ω) = S(x

1 u ˆz (xs , ys , zs , ω)

(4.13)

The vertical displacement u ˆz (xs , ys , zs , ω) is computed by means of a 3D model of the slab-soil system where the soil is considered as a horizontally layered elastic halfspace and modelled with the BE method. The BE model of the soil is coupled to a FE model of the slab. Since the duration of the impact is very short, the frequency content of the impact force will extend to very high frequencies. This requires a very fine BE mesh which is computationally very demanding. In the short duration of the impact, the stress waves in the slab and the soil propagate up to a limited distance. It can therefore be anticipated that the finite size of the slab might not play an important role during the course of impact. The slab may therefore be approximated by a layer with infinite lateral extensions on the soil surface (figure 4.9). In this case, the 3D slab-soil system is simplified to a horizontally layered halfspace. This approximation allows computing the dynamic ˆ s , ys , zs , ω) of the slab-soil system more efficiently by the direct stiffness stiffness S(x method [134, 135] as: ˆ 0, tr , ω) = S(0,

1 u ˆD zz (0, 0, tr , ω)

(4.14)

where u ˆD zz (0, 0, tr , ω) is the vertical Green’s function of the simplified slab-soil system for a vertical disc load with a size equal to the impact footprint. The impact force is now approximately estimated from equation (4.12).

4.4.2

Numerical verification

ˆ 0, tr , ω) In order to verify the simplified model, the dynamic stiffness S(0, computed with the horizontally layered model is compared to the dynamic stiffness


(a)

101

(b)

Figure 4.9: (a) 3D model of the slab-soil system and (b) simplified model by assuming infinite lateral extensions of the slab. ˆ s , ys , zs , ω) as computed with the 3D coupled FE-BE road-soil interaction model S(x (section 3.2.1). The verification is limited to the frequency range below 400 Hz because at high frequencies the 3D model is computationally very demanding and the contribution of high frequency components in ground vibration is small. The model parameters are taken from the measurement campaign along the N9 road in Waarschoot [152]. These parameters include the properties of the multihead breaker and the soil which were presented in section 4.3 and the N9 road as given in section 3.4.2. In the layered halfspace model of the road-soil system, a damping ratio βc = 0.03 is assumed for both the shear and dilatational waves in the concrete. The FE model of the slab is composed of 48 × 70 4-node shell elements with 6 DOFs at each node. A conforming BE mesh is chosen that matches the FE mesh on the slab-soil interface. It is composed of 4-node quadrilateral elements with linear shape functions. According to Kuhlemeyer and Lysmer [143] at least six 4-node elements per wavelength are required. The present mesh is therefore sufficiently fine to accurately compute the response in the frequency range up to 400 Hz. Since the MHB moves along the road and applies impacts at different positions, the dynamic stiffness at the centre, at a point near the edge, and at a point near the corner of the 3D model of the slab is computed. ˆ 0, tr , ω) Figure 4.10 shows the real and imaginary part of the dynamic stiffness S(0, of the slab-soil system at the impact point computed with the simplified model. ˆ s , ω) computed with These results are compared with the dynamic stiffness S(x the 3D FE-BE model for a load at the centre xs = {0, 0, tr }T , at a point xs = {0.75 m, 0, tr }T near the edge, and at a point xs = {0.75 m, 2.0 m, tr }T near the corner of the slab. Below 100 Hz, the difference between the dynamic stiffnesses computed with the layered halfspace model and the 3D model for an impact at the centre of the

102


9

9

x 10

3 Dynamic stiffness [N/m]

Dynamic stiffness [N/m]

3 2.5 2 1.5 1 0.5 0

2.5 2 1.5 1 0.5 0

0

(a)

x 10

100


400

0

(b)

100


400

ˆ 0, 0, ω) Figure 4.10: (a) Real and (b) imaginary part of the dynamic stiffness S(0, of the slab-soil system computed with the layered halfspace model (thin solid line) ˆ s , ω) computed with the 3D FE-BE model for a load and the dynamic stiffness S(x at the centre xs = {0, 0, tr}T (dashed line), near the edge xs = {0.75 m, 0, tr }T (dashed-dotted line), and near the corner xs = {0.75 m, 2.0 m, tr }T (dotted line) of the slab. Superimposed on (a) is the mass inertia md ω 2 of the drop hammer (thick line). slab is small. Above 100 Hz, the difference increases and decreases again around 200 Hz. The difference for an eccentric impact is larger as the effect of the shape of the slab becomes more significant. Figure 4.10a also shows the inertia md ω 2 of the drop hammer. It can be observed that at high frequencies the inertia of the drop hammer dominates the dynamic stiffness of the drop hammer-road-soil system as defined in the denominator of equation (4.12). Therefore, the difference between different road-soil models becomes less important at high frequencies. In the following section, the simplified layered halfspace model is evaluated by comparing the predicted force to the experimental results.

4.4.3


In order to estimate the impact force, the recorded acceleration of the drop hammer is multiplied by its mass md . Figure 4.11 compares the experimental impact force with the force predicted by the simplified slab-soil model. Figure 4.11a shows that the duration of the impact is well predicted while the peak value of the force is slightly overestimated. Figure 4.11b shows that the frequency content of the experimental force is mainly situated below 900 Hz. The dip at 900 Hz is related to the impact duration td as 2/td ≈ 900 Hz. The predicted force follows a similar trend as the experimental force. The sharp peaks in the narrow band frequency spectrum of the measured force at frequencies of about 1100 Hz and 2400 Hz correspond to the natural vibration modes of the drop hammer [152] and are not observed in the prediction based on the rigid mass model. In general, the

FRACTURING OF THE CONCRETE SLABS

103

predicted force agrees very well with the experimental result and it is concluded that the simplified model can be used to estimate the impact load. 6

1

x 10

5000 4000 Force [N/Hz]

Force [N]

0 −1 −2 −3

(a)

2000 1000

−4 −5 0.195

3000

0.2

0.205 Time [s]

0

0.21

0

500

(b)

1000 1500 2000 Frequency [Hz]

2500

Figure 4.11: Predicted (black line) and measured (grey line) (a) time history and (b) narrow band frequency spectrum of the impact force. The peak value of the load is about 3300 kN. Considering that the impact footprint is a circle with a diameter df = 0.13 m, the contact stress reaches a value σ = 250 MPa. The strain of the concrete slab at the impact point is of the order of 0.006. This strain level is reached in about 0.001 s (figure 4.11a) so the strain rate is about 6 s−1 . In the following section, the amount of energy dissipated due to the fracturing of the concrete slabs is estimated, taking into account the computed strain rate.

4.5

Fracturing of the concrete slabs

During the operation of the MHB, the impacts of the drop hammers generate hairline cracks in the concrete slabs. The generation of these cracks dissipates a part of the impact energy. The other part of the impact energy is transferred into the soil and generates ground vibrations. In order to evaluate whether the fracturing of the concrete slabs must be accounted for in a prediction model, the amount of energy dissipated due to fracturing is estimated. For this purpose, the characteristics of the cracks generated by the drop hammers are investigated. The dissipated energy is estimated using values and relations recommended in the literature. The formation of cracks might affect the distribution of stresses on the soil surface during the impact. Therefore, the effect of crack formation on the distribution of stresses on the soil surface and, consequently, on the change of ground vibrations is also investigated.

104


4.5.1

Experimental investigation of fracturing

The MHB produces hairline cracks that are not readily visible on the pavement surface. To enhance visual inspection of the cracking pattern following pavement breaking, water is applied to dampen the road surface [222]. Figure 4.12 shows a typical crack pattern after the operation of the MHB. Cracks are propagating in all directions indicating that the fracturing is a 3D problem. Cracks mostly pass through the impact footprints revealing that the fracturing is mainly due to bending. Figure 4.13, however, shows that cracks occasionally form around the impact points resulting from shear forces. Local effects are generally small and shear plug formation is rarely observed. The average length of the apparent cracks generated by a single impact is about 1 m (figure 4.14). With a slab thickness tr = 0.20 m, the average fracture area for one impact is equal to AF = 0.20 m2. In the following, the fracture energy per unit area is estimated from literature for a concrete with a compressive strength of about 78 MPa under a strain rate of about 6 s−1 .

Figure 4.12: Typical crack pattern due to the operation of the MHB.

Figure 4.13: Cracks formed around the impact footprints.


105

Figure 4.14: The average length of apparent cracks due to one impact is about 1 m.

4.5.2

Estimation of the fracture energy

Basic fracture mechanics of concrete The fracture energy of concrete depends on many parameters including the concrete tensile and compressive strength, strain rate, size of aggregates, fracture type, sample size and shape, and moisture content [35]. A distinction is made between the total fracture energy GF and the initial fracture energy Gf . Figure 4.15 shows the crack tip stress versus the crack mouth opening displacement; GF represents the area under the complete curve while Gf represents the area under the initial tangent of the softening curve. Baˇzant et al. [29] suggest a very approximate relation GF /Gf ≈ 2.5. The total fracture energy is considered as the amount of dissipated energy.

Figure 4.15: Crack tip stress versus crack mouth opening displacement, total fracture energy GF , and initial fracture energy Gf (from [29]). There are three modes of fracture as shown in figure 4.16: mode I or opening mode, when the plane of the crack is normal to the applied tensile stress, mode II or sliding mode, when a shear stress acts parallel to the plane of the crack and

106


perpendicular to the crack front, and mode III or tearing mode, when a shear stress acts parallel to the plane of the crack and parallel to the crack front.

Figure 4.16: The three fracture modes. Since the cracks mostly pass through the impact points and occasionally form around them, the type of fracture is probably mostly mode I and partially mode II around the impact footprint. For the impacts on the edge of the slab, mode III is also possible. Fracture energy Hillerborg [109] reports a wide range of values between 90 J/m2 and 370 J/m2 for the fracture energy GF , based on the three point bending test of many samples performed by 9 laboratories. The CEB-FIP Model Code 1990 [61] recommends the following formula for GF : GF = 0.0469d2a − 0.5da + 26

fcm 10

0.70

(4.15)

where GF is in J/m2 , da is the maximum size of aggregates in mm, and fcm is the mean compressive strength in MPa. For a mean compressive strength fcm = 78 MPa and a range of maximum aggregate sizes da = 20 − 30 mm, equation (4.15) gives a total fracture energy GF = 146−224 J/m2. Baˇzant and Becq-Giraudon [26] suggest the following empirical formula for the initial fracture energy under static loading: Gf = 1.12

fcm 0.058

0.40 0.43 da w −0.18 1+ 1.94 c

(4.16)

where w/c is the water/cement ratio and the units are similar to equation (4.15). Assuming fcm = 78 MPa, a range of maximum aggregate sizes da = 20 − 30 mm, and a range of the water/cement ratios w/c = 0.30 − 0.70, the initial fracture energy becomes Gf = 60 − 83 J/m2 and the total fracture will be GF ≈ 2.5Gf =


107

151 − 207 J/m2. The following equation was also extracted from another set of data [26]: 0.46 0.22 fcm da w −0.30 Gf = 1.44 1+ (4.17) 0.051 11.27 c with similar units as in equation (4.15). For da = 20−30 mm and w/c = 0.30−0.70, equation (4.17) gives a range of fracture energy GF ≈ 2.5Gf = 146 − 200 J/m2. Baˇzant and Pfeiffer [27] and Baˇzant and Prat [28] suggest that the fracture energy of mode II and III is, respectively, about 25 and 3 times larger than the fracture energy of mode I; e.g. for a static test (ε˙ < 10−4 s−1 ) of a concrete with fcm = 2 III 2 36 MPa, GIf = 33 J/m2 , GII f = 1040 J/m and Gf = 106 J/m . According to ACI 446.4R-04 [9] and ACI 446.1R-91 [8] there is still a lack of information, however, about the fracture energy of mode II and III. Baˇzant and Pfeiffer [27] also state that the ratio of the size of the specimen to the aggregate size affects the fracture energy. Effect of strain rate The available information regarding the effect of strain rate on fracture energy is very diverse and incomplete [61]. Watstein [243] tested cylindrical samples of a concrete with fcm = 46.4 MPa with a diameter of 76 mm and a height of 152 mm in a drop hammer test with a strain rate of 6.7 s−1 . He observed an increase of 85% in the compressive strength compared to the static strength and a strain energy absorption (fracture toughness) of 146 kJ/m3. The ratio of the dynamic to static toughness is 2.17. Table 4.2 summarises the experimental data found in the literature on the effect of strain rate on the fracture energy of concrete for the relevant strain rates around 6 s−1 . The result of Lambert and Ross [154] (table 4.2) can be introduced in the CEB0.70 FIP formula [61] GF ∝ (fcm /10) to obtain a value of GF = 267 J/m2 for fcm = 78 MPa. It can be observed in table 4.2 that the fracture energies reported by Banthia et al. [21] are very different from the observations of other researchers. Ross et al. [207] observed the strain-rate dependency of concrete mainly above a strain rate of 5 s−1 for tension and 60 s−1 for compression. Rossi [208] reported that the strain-rate dependency of concrete is mainly due to the presence of free water. Local damage Apart from the apparent cracks on the slab surface, the impact can cause local effects such as shear plug formation and scabbing. Scabbing is the peeling off of the material from the bottom side of the slab as the compressive wave propagates to the

108


Author

fcm ε˙ [MPa] [s−1 ]

Watstein [243]

46

Takeda and Tachikawa [238] -

GF Remarks [J/m2 ]

6.7

-

Dynamic to static toughness: 2.17. Energy absorption: 146 kJ/m3 . Dynamic to static compressive strength: 1.85. Drop hammer test of φ76×152 mm specimen.

1

-

Dynamic to static fracture energy: 1.2 − 1.8.

Atchley and Howard [16]

34.5

4.5

-

Dynamic to static toughness: 1.42. Energy absorption: 76 kJ/m3 . Dynamic to static compressive strength: 1.63. Drop hammer test of φ152×305 mm specimen.

Lambert and Ross [154]

43

6

176

Split Hopkinson pressure bar test.

-

0.01 1

170 280

Taken from Oh and Chung (1989).

Ruiz et al. [211]

74 74

< 10−4 0.04

200 249

Banthia et al. [21]

42 42 82 82

Static 0.6 Static 0.6

ACI 446.4R-04 [9]

920 Drop hammer test. 15000 470 12500

Table 4.2: Experimental data on the effect of strain rate on the fracture energy of concrete. bottom side and is reflected. According to empirical formulas of the UMIST and the NDRC [157], scabbing is not very probable. Also according to CEB No187 [60], scabbing and shear plug formation is very unlikely. Hughes and Al-Dafiry [113] state that local effects are small and the energy dissipated in the contact zone is small compared to the total impact energy. Conclusions of the estimation of the fracture energy The following conclusions are made from the experimental investigation and the literature review.


109

• The fracturing is mainly due to bending, so the type of fracture is mostly mode I. The average fracture area for one impact is equal to AIF = 0.20 m2 . • The average fracture energy for mode I is about GStat = 200 J/m2 for a FI concrete under static loading with fcm = 78 MPa. • Since the fracture energy of mode II is about 25 times larger than for mode 2 I, it is estimated as GStat FII = 5000 J/m . It is also assumed that the type of fracture around the impact footprint is mode II. The area of this cylindrical 2 crack is AII F = πdf tr = 0.082 m . • The fracture energy at a strain rate of 6 s−1 is about two times the static fracture energy. Therefore, at a strain rate of 6 s−1 the fracture energy of mode I is GDyn = 400 J/m2 and the fracture energy of mode II is GDyn FI FII = 10000 J/m2. • The total fracture energy due to one impact is estimated as: II Dyn GF = AIF GDyn FI + AF GFII = 900 J

Considering that the velocity of the drop hammer at the time of impact is v0 = 5.5 m/s, the impact energy becomes Eimp = 1/2md v02 = 9075 J. Therefore, only approximately 10% (GF /Eimp = 900/9075) of the impact energy is used to fracture the concrete slabs and the remaining 90% is transferred into the soil. Although fracturing dissipates only a small part of the impact energy, it divides the slabs into smaller parts and thus can modify the distribution of the impact load on the soil surface. This change of loading might affect the ground vibrations. In the following, the velocity of crack propagation is estimated to investigate whether the cracks develop fast enough to divide the slabs into smaller parts during the course of impact or fracturing occurs after the impact load transferred into the soil.

4.5.3

Estimation of the crack propagation velocity

Mindess and Bentur [182] and Bentur et al. [31] (from [97]) used a large-scale dropweight test and measured crack velocities of 75 − 115 m/s. Shah and John [228] (from [97]) observed a crack velocity of 200 m/s at a strain rate ε˙ = 1 s−1 for a concrete with a shear wave velocity of 2600 m/s. Guo et al. [97] reported a maximum crack speed of 178 m/s at a strain rate of about ε˙ = 0.013 s−1 for a concrete with fcm = 18 MPa. Based on the results of several researchers, ACI 446.4R-04 [9] concludes that the crack velocity is limited to approximately 10% of the Rayleigh wave speed in concrete, which gives a value of about 240 m/s for the considered concrete with fcm = 78 MPa. Zhang et al. [266] report that in a concrete with fcm = 103 MPa and E = 31 GPa at a strain rate ε˙ = 1 s−1 , the

110


crack propagation velocity is about 20% of the Rayleigh wave speed. Experimental results of crack propagation velocity by different authors have been collected by Larcher [155] and are presented in figure 4.17.

Figure 4.17: Crack propagation velocity as a function of strain rate (from [155]). From the literature review, the crack propagation velocity in the considered concrete at the strain rate of 6 s−1 is estimated to be about 240 m/s. This speed is about 9% of the shear wave velocity. During the course of impact (0.002 s as shown in figure 4.11a), cracks can therefore propagate over the entire thickness of the slab tr = 0.20 m and divide the slab into smaller parts. Thus, they can affect the distribution of stresses on the soil surface.

4.5.4

Effect of fracturing on ground vibrations

In order to investigate the effect of fracturing on ground vibrations, the vibrations measured during 8 consecutive impacts of the MHB at four points P1 to P4 of a single slab (figure 4.4) are compared in figure 4.18. The measured ground vibrations for these eight impacts are very similar which means that further fracturing does not affect the resulting ground vibration. The slightly larger difference at 5 m from the road is partly due to the larger difference in the relative distances from the source points. Figure 4.19 shows measured free field mobility due to the sledge hammer impacts on the slab in three conditions: the slab in its original condition, after the 8 impacts of the MHB, and after the MHB fractured the entire length of the slab. The free field mobility follows a similar trend in the three states of the slab, but with slightly larger values for the fractured slab. After the operation of the MHB, the free field mobility at 5 m from the road increases by a factor of about 1.5. At


111


Velocity [mm/s]

40 20 0 −20 −40

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

0.3

20

0.25

0 −10 −20 −30

0

0.2

0.4

0.6

0.8

(b)

1 1.2 Time [s]

1.4

1.6

1.8

Velocity [mm/s/Hz]

Velocity [mm/s]

2 0 −2 −4 0

0.2

0.4

0.6

0.8

(c)

1 1.2 Time [s]

1.4

1.6

1.8

Velocity [mm/s/Hz]

Velocity [mm/s]

250

300

0

50


250

300

0

50


250

300

0

50


250

300

0

50


250

300

0.1 0.05

0.06 0.04 0.02

0.04

0.5 0 −0.5

0

0.2

0.4

0.6

0.8

(d)

1 1.2 Time [s]

1.4

1.6

1.8

0.03 0.02 0.01 0

2


0.2 0.1 0 −0.1 −0.2


0.2

0

2

1

−1

50

0.08

4

−6

0

0.15

0

2

6

Velocity [mm/s]

0.2

30

10

(e)

0.4

0

2

Velocity [mm/s/Hz]

Velocity [mm/s]

(a)

0.6

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

2

0.015 0.01 0.005 0

Figure 4.18: Recorded time history (left) and narrow band frequency spectrum (right) of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road due to 8 consecutive impacts of the MHB.

112


farther receivers the increase is smaller. The increase of mobility can partly arise from the compaction of the soil under the slab after the impacts of the MHB. −7

−8

x 10

8

3

Mobility [m/s/N]

Mobility [m/s/N]

4

2 1 0

0

50

(a)


6 4 2 0

200

Mobility [m/s/N]

Mobility [m/s/N]

2

2 1

0

50


200

50


200

−8

x 10

3

0

0

(b)

−8

4

x 10

50

(c)


200


200

x 10

1.5 1 0.5 0

(d)

0

−8

Mobility [m/s/N]

1

x 10

0.8 0.6 0.4 0.2 0

(e)

0

50

Figure 4.19: Free field mobility at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road, measured with the sledge hammer impacts on the slab in its original condition (solid line), after the 8 impacts of the MHB (dashed line), and after the MHB fractured the entire length of the slab (dotted line).

CONCLUSION

113

It is therefore concluded that the fracturing of the slabs slightly affects the free field mobility at low amplitude excitations. For the large force of the pavement breaker, the effect of fracturing on the variation of ground vibrations is negligible especially at large distances.

4.6

Conclusion

This chapter starts by introducing the pavement breaking operation. Then, the configuration of an experiment on ground vibrations generated by pavement breaking is described. Next, a numerical model for the prediction of the impact load due to the impact of a falling-weight pavement breaker is developed. Finally, the effect of fracturing of the concrete slabs on the change of ground vibrations is investigated. Falling-weight pavement breakers generate a high level of vibrations. Available experimental studies on this matter are, however, limited and incomplete. Therefore in the frame of this study, two sets of measurement campaigns on the ground vibrations generated by a multi-head pavement breaker have been carried out. These experiments have been designated for model validation while they also contribute to the scarce literature on ground vibrations generated by pavement breaking. The experimental configuration of one of these experiments is described where the impact force of the drop hammer as well as the generated ground vibrations has been measured. To identify the properties of the road and soil, additional experiments have been conducted. For the estimation of the impact load applied by the blow of the drop hammer on the road surface, a simple model that consists of a SDOF system for the road is presented first. This model is frequently used for the estimation of the impact load due to impulsive sources such as impact pile driving and foundation of hammers and presses. The model is then elaborated by employing the 3D roadsoil interaction model presented in section 3.2.1. Since the impact duration is very short, the frequency content of the acceleration of the drop hammer extends to high frequencies. Analysis of the 3D model at high frequencies is very demanding, so a simplified model is proposed that takes advantage of the short duration of impact. In this model, the road is approximated by a layer with infinite lateral extension on the soil surface thus the road-soil interaction problem is simplified to the analysis of a horizontally layered elastic halfspace. The model is verified against the 3D model where a small difference between the dynamic stiffnesses computed with the simplified model and the 3D model is found. The predicted impact force agrees very well with the experimental result. The impacts of the drop hammers generate hairline cracks in the concrete slabs. The fracture energy is estimated from similar experimental studies in the literature

114


to be only 10% of the impact energy and can therefore be disregarded in the prediction of ground vibrations. The crack propagation velocity is found to be about 9% of the shear wave velocity in the concrete. This is, however, fast enough to divide the slab into smaller parts during the course of impact. Experimental investigations reveal that the size of the slab has a negligible effect on the free field response at large distances from the road, while at small distances it might be important to a limited extent. The effect of the slab size on the ground vibration will be further investigated in section 6.5 using a non-linear road-soil interaction model.

Chapter 5

Linear prediction of ground vibrations due to pavement breaking 5.1

Introduction

The review of the literature in section 1.2.2 revealed that no numerical model is available for the prediction of ground vibrations generated by pavement breaking while the experimental data are rather limited. This lack of information has resulted in a limited use of heavy breakers near residential areas and surface and underground installations. The study on the fracturing of the concrete slabs in section 4.5 showed that the effect of fracturing on the ground vibrations is negligible and, consequently, can be disregarded in a prediction model. Therefore in this chapter, a linear model is developed to predict ground vibrations due to the impacts of the multi-head breaker. First, the 3D linear road-soil interaction model presented in section 3.2.1 is coupled to a simple model of the MHB to investigate the effect of the MHB-road-soil interaction on the dynamics of the road and the generated ground vibrations. The 3D model is then modified to an axisymmetric model and employed to predict ground vibration velocities and strains. The results are used to obtain the level of soil deformations and to assess the possibility of non-linear behaviour. Finally, an equivalent linear model is presented that approximately takes into account non-linear behaviour of the soil at large strains generated by the impacts.

115

116

5.2

LINEAR PREDICTION OF GROUND VIBRATIONS DUE TO PAVEMENT BREAKING

Effect of the presence of the pavement breaker

The MHB weighs about eight tonnes. A part of this weight is supported by the towing tractor which is several metres away from the impact point and therefore usually located on the adjacent slab. The other part, which is estimated to be about six tonnes, is in contact with the slab through two wheels (figure 4.3). In the following, the effect of this large mass on the dynamic response of the slab and the predicted ground vibration is investigated. For this purpose, the 3D coupled FE-BE model of the slab-soil system (section 3.2.1) is coupled to a simple model of the MHB. The left and right side of the MHB are modelled as mass-spring systems. The masses mM correspond to half of the mass of the MHB and the springs correspond to the stiffness kM of the wheels. Equation (3.1) is rewritten as: ˆD ˆ r (ω) = ˆf r (ω) K rr (ω) u

(5.1)

2 ˆD ˆs where K rr (ω) = −ω Mrr + Krr + Krr (ω) represents the dynamic stiffness of the slab-soil system. Adding the two mass-spring models of the MHB results in the equation of motion of the MHB-slab-soil system as: !( " ) ( # ) ˆ D (ω) 0 ˆf (ω) u ˆ r (ω) K rr r D ˆ = + kM (ω)SM (5.2) 0 0 u ˆ M (ω) 0 D where kˆM (ω) = −ω 2 mM + kM is the dynamic stiffness of the mass-spring model of the MHB and SM is a selection matrix consisting of ones and zeros that selects the DOFs of the MHB and the coupling DOFs between the MHB and the slab. The vector u ˆ M (ω) collects the two DOFs of the MHB model. Following the same procedure as for equation (3.1), equation (5.2) is solved and ground vibrations are computed.

The model parameters for the slab and the soil have been presented in section 4.3.2. For each of the mass-spring models of the MHB, a mass mM = 3000 kg is assumed. The spring coefficient is mainly due to the stiffness of the tyres which is estimated to be kM = 1000 kN/m [77]. This results in a natural frequency of 2.9 Hz for the mass-spring model of the MHB. The track width of the MHB is equal to 2wMHB = 2.30 m. Figure 5.1 compares the predicted vertical velocities with and without taking into account the presence of the MHB on the slab. It can be observed that the presence of the MHB only slightly affects the predicted vibrations at small distances from the road. For the receiver at 5 m, this effect is mainly situated between 40 Hz and 120 Hz. Since the difference is not very large, the presence of the MHB on the road is disregarded in the remainder of this text. The negligible effect of the presence of the MHB on the ground vibrations is also confirmed by the experimental results [152,153]. When the MHB moves along the road and impacts at different positions

EFFECT OF THE PRESENCE OF THE PAVEMENT BREAKER

117

Velocity [mm/s/Hz]

20 0 −20 −40

(a) Velocity [mm/s]

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(c)

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10

(e)

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0

2

Velocity [mm/s/Hz]

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40

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

2

0.015 0.01 0.005 0

Figure 5.1: Predicted time history (left) and narrow band frequency spectrum (right) of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road due to an impact of the MHB for the case where the presence of the MHB on the slab is accounted for (dashed line) and the case where it is disregarded (solid line).

118


of the slabs, the recorded level of ground vibration does not significantly change with respect to the presence of the MHB on the same slab that is cracked or on an adjacent slab. This also shows that the level of induced vibrations is similar for impacting on the centre or edge of a slab.

5.3

Axisymmetric coupled finite element-boundary element model

The results presented in section 5.2 showed that the presence of the MHB on the slab does not have a significant effect on the predicted ground vibration and can therefore be disregarded. In the following, the rectangular shape of the slab is approximated by a disc, so that the slab-soil interaction problem becomes axisymmetric. The axisymmetric linear road-soil interaction model is numerically verified for the case of harmonic loading and compared with the experimental results for the impacts of the MHB.

5.3.1

Model description

As shown in figure 4.11b, the frequency content of the impact force extends to very high frequencies. This makes the analysis of the 3D FE-BE model of the slab-soil interaction computationally very demanding. Two changes are therefore made to the previously presented 3D model of the slab-soil system. First, the 3D model is simplified to a 2D axisymmetric model by approximating the rectangular slab by a disc with a diameter equal to the width 2B of the slab and a thickness tr equal to the thickness of the slab. This reduces the dimension of the problem by one and hence the computational cost. Second, the soil domain in the near vicinity of the slab is incorporated into the FE model by shifting the FE-BE interface from the slab-soil interface into the soil domain. This allows relaxing the requirements for a fine BE mesh as, far away from the source, the high frequency components of ground vibrations are significantly attenuated. Nevertheless, a larger boundary must be meshed. Including a part of the soil close to the impact point in the FE model has the advantage of allowing an easy extension to a non-linear model for the soil at a later stage. Figure 5.2a shows the interior domain Ω1 and the exterior soil domain Ω2 of the axisymmetric coupled FE-BE model. The interior domain Ω1 consists of a circular plate representing the slab and a part of the soil domain bounded by the free surface Γ1σ and the FE-BE interface Σ12 . The exterior soil domain Ω2 is bounded by the free surface Γ2σ , the FE-BE interface Σ12 , and the outer boundary Γ2∞ .

AXISYMMETRIC COUPLED FINITE ELEMENT-BOUNDARY ELEMENT MODEL

119

Figure 5.2: Axisymmetric model of the FE domain Ω1 and the exterior soil domain Ω2 . The formulation of the model is similar to the one of the 3D model presented in section 3.2.1 except that the DOFs of the BE model coincide with the DOFs of the FE model on the interface Σ12 (figure 5.2). The equation of motion of the slab-soil system is adapted from equation (3.1) as: #! # " " # " 0 0 M r1 r1 M r1 r2 K r1 r1 K r1 r2 2 + −ω ˆ sr r (ω) 0 K M r2 r1 M r2 r2 K r2 r1 K r2 r2 2 2 ×

(

u ˆ r1 (ω) u ˆ r2 (ω)

)

=

(

ˆf r (ω) 1 0

)

(5.3)

where the mass matrix Mrr and the stiffness matrix Krr of the FE model are subdivided into block matrices according to the degrees of freedom u ˆ r1 (ω) inside the FE domain Ω1 and the degrees of freedom u ˆ r2 (ω) on the FE-BE interface Σ12 . The material damping of the slab and the soil in the FE model is accounted for by considering a complex stiffness matrix (section 3.4.2). The force vector ˆf r1 contains the impact force at the corresponding DOFs of the FE model. The matrix ˆ s (ω) is the dynamic stiffness of the BE model. K r2 r2 ˆ s (ω) is the most computationally The calculation of the dynamic stiffness K r2 r2 demanding part in the solution of equation (5.3). If a rectangular FE domain is used, the dynamic stiffness matrix can be computed more efficiently. This arises from the fact that the collocation points of the boundary elements at the bottom side of the interface are situated at the same depth and, therefore, the Green’s functions can be computed with the source at one depth. In this case ˆ s (ω) is slightly different from the computation of the dynamic stiffness matrix K r2 r2 what was presented in equations (3.2) to (3.6) because the normal vector on the corner of the BE mesh is discontinuous. As a result, the traction vector exhibits

120


discontinuities as well. In the case of a nodal collocation, the use of a single node at the corner is insufficient to represent these discontinuities in the traction vector. Solution of the corner problem To solve the corner problem, an additional slave node on the corner of the BE mesh is introduced as proposed by Fran¸cois et al. [84]. This node allows to account for the traction discontinuity. However, the addition of the slave node on the corner results in an underdetermined BE system as the boundary integral equation is now multiply collocated at the same point. The traction vector ˆts (ω) is subdivided into the traction vector ˆts1 (ω) of the BE nodes (master nodes) and the traction vector ˆts (ω) of the slave node on the corner. Using the constitutive equation, the traction 2 ˆts (ω) of the slave node is related to the displacement as: 2 ˆts (ω) = Tc u ˆ s (ω) 2

(5.4)

where Tc is a matrix that collects the partial derivatives of the BE shape functions Ns with respect to the local coordinates of the corner node [84]. Equation (3.3) can be rewritten as: " #( ) h i ˆ 11 (ω) U ˆ 12 (ω) ˆts (ω) U 1 ˆ (5.5) = T(ω) +I u ˆ s (ω) ˆ 21 (ω) U ˆ 22 (ω) ˆts (ω) U 2 Introducing ˆts2 (ω) from equation (5.4) into equation (5.5) allows to compute the traction ˆts1 (ω) of the BE nodes as: h i ˆ −1 (ω) U ˆ 12 (ω) Tc + T(ω) ˆ ˆts (ω) = U (5.6) + I u ˆ s (ω) 11 1 Fictitious eigenfrequencies When the excitation frequency coincides with the eigenfrequencies of the interior domain, the boundary integral equation has a nonunique solution and spurious modes occur. To overcome this problem, the Combined Helmholtz Integral Equation Formulation (CHIEF) method [84,217] is used. In this method, a number of randomly distributed receiver points is introduced in the interior domain where zero displacements are enforced. Applying the integral representation theorem for these points results in the following boundary element system of equations: ˆ r0 (ω) ˆt (ω) = T ˆ r0 (ω) u U ˆ s (ω) s

(5.7)

ˆ r0 (ω) and T ˆ r0 (ω) are the BE system matrices of the CHIEF receivers. where U Adding equation (5.7) to the system of equations (3.3) results in the following


overdetermined system of equations: " # " # ˆ ˆ U(ω) T(ω) + I ˆts (ω) = u ˆ s (ω) ˆ r0 (ω) ˆ r0 (ω) U T

121

(5.8)

which is solved by means of a least squares procedure. The distances between the CHIEF receivers should be a fraction - for example one-sixth - of the shortest wavelength in the domain to avoid the occurrence of spurious modes [82]. Combining equation (5.8) with equation (5.5) allows to compute tractions accounting for both the corner problem and fictitious eigenfrequencies. The tractions ˆts1 (ω) are subsequently used to compute the dynamic stiffness of the BE ˆ s (ω) is substituted in equation (5.3) to model. The dynamic stiffness matrix K r2 r2 find the displacements u ˆ r (ω) of the FE model and thus the displacements u ˆ s (ω) of the BE model. Once the equilibrium equation (5.3) has been solved, the tractions on the BE mesh are computed from equation (5.6). Having the tractions ˆts (ω) and displacements u ˆ s (ω) at the FE-BE interface, equation (3.7) is used to compute the radiated wave field u ˆ (ω).

5.3.2

Numerical verification

In the following, the axisymmetric coupled FE-BE model is verified. For this purpose, a homogeneous halfspace is modelled and the response due to a harmonic point load δ(x)δ(y)δ(z) exp(−iωt) at an excitation frequency ω is compared to the Green’s functions computed with the direct stiffness method. The comparison is made for several receivers located in Ω1 and Ω2 . The halfspace has properties similar to the ones of the first layer of the soil profile in table 4.1 except for the material damping which is disregarded in the FE domain. The dimensions of the FE domain are arbitrarily chosen as a width and depth Bb = Db = 2 m. Figure 5.3a shows the FE mesh consisting of 400 8-node quadrilateral elements with at least 6 elements per wavelength at 100 Hz. 8-node quadratic elements are chosen because of their smaller numerical dispersion error in comparison with lower order 4-node linear elements for the same number of DOFs in the model [69, 120, 225]. The FE mesh is finer near the source and gets coarser at larger distances. The BE mesh is composed of 3-node line elements with quadratic shape functions that matches the FE mesh on the interface (figure 5.3b). The size of all boundary elements is equal to 0.10 m. This provides more than 6 elements per wavelength at 100 Hz. Six CHIEF receivers per wavelength at 100 Hz are considered. This gives a total number of 57 CHIEF receivers in the interior domain as shown in figure 5.3b. The CHIEF receivers are located at the same depths as the collocation points of the BE mesh. This reduces the calculation time as the same Green’s functions that have been computed for the BE model can be used.


0

0

−0.5

−0.5 z [m]

z [m]

122

−1

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−2

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0

−1

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0.5

1 x [m]

1.5

−2 0

2

0.5

(b)

1 x [m]

1.5

2

Figure 5.3: Axisymmetric model of the halfspace: (a) FE mesh and (b) BE mesh (circles) and CHIEF receivers (plus signs). Figure 5.4 compares the vertical Green’s functions at three receiver points obtained by means of the FE-BE model with the results obtained with the direct stiffness method. The first point x = {0.1 m, 0, 0}T is located at the free surface of the FE domain, the second point x = {2 m, 0, −2 m}T at the interface of the FE-BE models on the corner, and the third point x = {10 m, 0, 0}T on the ground surface outside the FE domain. The results of the FE-BE model are in good agreement with the Green’s functions. At the closest receiver (x = {0.1 m, 0, 0}T , figure 5.4a), the displacements computed with the coupled FE-BE model are slightly smaller than the results obtained with the direct stiffness method. This arises from the fact that the FE discretisation always overestimates the stiffness of the system. At other receivers this difference is less noticeable. At high frequencies, the agreement is less good because the number of elements per wavelength reduces and also the material damping, which is disregarded in the FE domain, plays a more important role. In general, the agreement between the results of the model and the Green’s functions is very good.

5.3.3

Results obtained with the linear model

In the following, the axisymmetric coupled FE-BE model is used to predict the ground response to an impact of the MHB on the road surface. The model parameters presented in section 4.3.2 are used. The FE model has a width Bb = 4 m and a depth Db = 5 m. A preliminary analysis showed that the size of the FE domain is large enough so that the high frequency components of the response at the interface with the BE mesh have sufficiently attenuated. Therefore, a relatively coarse BE mesh is sufficient. Figure 5.5a shows the FE mesh which is composed of 8-node quadrilateral elements and has been generated in ANSYS [13].


−8

−9

x 10

1 Displacement [m/Hz]

Displacement [m/Hz]

3 2 1 0 −1

0

123

20

(a)


80


80

0.5 0 −0.5 −1

100

(b)

x 10

0

20


80

100

−10

Displacement [m/Hz]

3 2 1 0 −1 −2 −3

(c)

x 10

0

20

100

Figure 5.4: Real (solid line) and imaginary (dashed line) part of the vertical displacement due to a harmonic point load computed with the coupled FE-BE model (black line) and the direct stiffness method (grey line) for a receiver at (a) x = {0.1 m, 0, 0}T , (b) x = {2 m, 0, −2 m}T , and (c) x = {10 m, 0, 0}T . The FE mesh is finer near the impact point and gets coarser at larger distances from the source. It provides at least 4 elements (8 DOFs) per wavelength up to a frequency of 500 Hz. Kuhlemeyer and Lysmer [143] suggest using at least six 4-node elements per wavelength. In the present case, four 8-node elements provide more DOFs than the six 4-node elements. In addition, convergence has been verified at four frequencies of 0, 50, 100, and 500 Hz using finer FE and BE meshes with three times more DOFs. The refined model gave similar results as the considered model. The impact force is applied to the FE nodes considering an impact footprint with a radius df /2 = 0.065 m. The BE mesh is composed of 3-node line elements with quadratic shape functions and matches the FE mesh on the FE-BE interface (figure 5.5b). At each frequency, the number of CHIEF receivers is determined by requiring 6 receivers per shortest wavelength in the FE domain. At frequencies below 100 Hz, an additional number of 100 receivers is added to ensure that the distances between the randomly

124


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−2 z [m]

0

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0

−3

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−5

(a)

0

1

2 x [m]

3

−5

4

(b)

0

1

2 x [m]

3

4

Figure 5.5: Axisymmetric slab-soil model: (a) FE mesh and (b) BE mesh (circles) and CHIEF receivers at a frequency of 200 Hz (plus signs). distributed receivers are a fraction of the shortest wavelength. Figure 5.5b shows 1140 CHIEF receivers that are used at a frequency of 200 Hz. These receivers are randomly distributed in the soil domain at the same depths as the collocation points. Figure 5.6 shows the displacement field due to an impact of the MHB (figure 4.11). The snap shots are taken at the time of impact t = 0.200 s and at times t = 0.205 s, 0.210 s, 0.230 s. Figure 5.6a shows that, after the impact, the centre of the slab (the impact point) moves about 2 mm downward while the edge of the slab tends to lift up from the soil surface. Figures 5.6a to 5.6d clearly show the propagation of the body and surface waves in the domain. Since the medium is elastic, the slab and soil return to their initial positions. Figures 5.7 and 5.8 show snap shots of the horizontal and vertical vibration velocity, respectively. These figures clearly show wave propagation and attenuation in the soil domain. The horizontal and vertical vibration velocities have the same order of magnitude. Up to about 2 m from the impact point, the vibration velocity is larger than 500 mm/s. In figures 5.7b and 5.8b, two wave fronts can be observed. The wave front closest to the source corresponds to the S-waves and the farthest wave front corresponds to the faster propagating P-waves.


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−2 z [m]

0.2 0

z [m]

0.2 0

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1

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(d)

Figure 5.6: Displacement field in the slab and the soil at (a) t = 0.200 s, (b) t = 0.205 s, (c) t = 0.210 s, and (d) t = 0.230 s due to an impact of the MHB. The displacements are scaled by a factor of 500.

126


(a)

(b) [mm/s] 500 400 300 200 100 0 −100 −200 −300 −400 −500

(c)

(d)

Figure 5.7: Horizontal vibration velocity at (a) t = 0.200 s, (b) t = 0.205 s, (c) t = 0.210 s, and (d) t = 0.230 s generated by an impact of the MHB. Figure 5.9 shows the traces of the particle displacement at four locations on the surface and a depth of 4 m and at radial distances of 0.1 m and 3 m. The traces are shown from the unloaded state to 0.10 s after the impact (t = 0.30 s). The displacement at {0.1 m, 0, 0}T is larger than 2 mm (figure 5.9a). It reduces to about 0.4 mm at a distance of 3 m at the soil surface (figure 5.9b) and to about 0.2 mm at a depth of 4 m (figures 5.9c and 5.9d). The particle motion near the axis of symmetry is mostly vertical (figures 5.9a and 5.9c) while at a radial distance of 3 m, the horizontal displacements have a similar amplitude as the vertical displacements (figures 5.9b and 5.9d). The computed particle motions are not very smooth because the time step is rather large. This indicates that the predicted peak values of the displacements might be slightly smaller than the real peak values. At very small distances from the source, the response extends slightly beyond the maximum frequency of 500 Hz. Computation at higher frequencies requires finer FE and BE meshes and is computationally very demanding. Furthermore, a high accuracy at very small distances is not necessary, therefore, the maximum frequency of 500 Hz is considered to be sufficient.


(a)

127

(b) [mm/s] 500 400 300 200 100 0 −100 −200 −300 −400 −500

(c)

(d)

Figure 5.8: Vertical vibration velocity at (a) t = 0.200 s, (b) t = 0.205 s, (c) t = 0.210 s, and (d) t = 0.230 s generated by an impact of the MHB. oct Figure 5.10 shows the peak octahedral shear strain γmax in the FE domain due to oct an impact of the MHB. It can be observed that γmax reaches to a few millistrain near the slab. The maximum octahedral shear strain in the slab is about 0.003 and the maximum strain rate is 6 s−1 which is in agreement with the strain rate estimated in section 4.4.3. The maximum octahedral shear strain in the soil reaches a value of 0.005 and the maximum strain rate is 12 s−1 .

5.3.4

Comparison with the experimental results

In the following, the results of the linear model are compared to the experimental results measured in the first stage of the experiments in Waarschoot (section 4.3.1). Figure 5.11 compares the predicted and measured vertical vibration velocity due to an impact of the MHB. Up to 25 m from the impact point, the model overestimates

128


0.2 z−displacement [mm]

z−displacement [mm]

1 0 −1 −2 −3 −4

−2

(a)

0 2 x−displacement [mm]

0 −0.2 −0.4 −0.6 −0.4

(b)

0 −0.2 −0.4 −0.6 −0.4

(c)



0.2

−0.2 0 0.2 0.4 x−displacement [mm]


0 −0.2 −0.4 −0.6 −0.4

(d)


Figure 5.9: Trace of the particle displacement at (a) {0.1 m, 0, 0}T , (b) {3 m, 0, 0}T , (c) {0.1 m, 0, −4 m}T , and (d) {3 m, 0, −4 m}T due to an impact of the MHB. The motion is shown from the unloaded state (black) to t = 0.30 s (grey).

0.01

0.008

0.006

0.004

0.002

0

oct Figure 5.10: Peak octahedral shear strain γmax due to an impact of the MHB.

the peak value while it underestimates the level of vibration after the peak. At 57 m and 105 m, the peak value is underestimated and the predicted vibrations



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0

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1 1.2 Time [s]

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0.015 0.01 0.005 0

Figure 5.11: Time history (left) and narrow band frequency spectrum (right) of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road due to an impact of the MHB predicted with the linear FE-BE model (black line) and compared with the experimental results (grey line).

130


decay more rapidly than the experimental results. The arrival time of the P-waves as well as the time delay between the P-waves and the slower waves (Rayleigh and S-waves) is rather well predicted. The peak value of the P-waves and their duration are overestimated while the duration of the slower waves is underestimated. The agreement in the frequency domain is not good especially at high frequencies for small distances and at low frequencies for large distances. In general, the agreement between the numerical and experimental results is not satisfactory. This is thought to be caused by the non-linear behaviour of the soil at large strains as well as possible errors in the identified soil parameters. As shown in figure 5.10, the soil strains are of the order of a few millistrain. The soil behaviour at such large strain levels can no longer be assumed to be linear and the results of the linear model are therefore far from the experimental results. An equivalent linear model is presented in the following to approximately account for the non-linear behaviour of the soil at large strains.

5.4

Equivalent linear prediction of ground vibrations

Shear stress [MPa]

In seismic engineering, the equivalent linear method has been proposed as a technique to approximately account for the non-linear behaviour of soils at large strains. The rationale behind this method is the fact that the shear modulus µ and the material damping ratios βs and βp of the soil are functions of the strain level [224]. Figure 5.12 demonstrates the typical constitutive behaviour of soils which exhibit a softening non-linearity, or a decrease in the shear modulus as strain increases, and progressively larger hysteresis in the stress-strain relation, or an increase of the material damping ratio. 0.02 0.01 0 −0.01 −0.02 −2

−1 0 1 2 Shear strain [ − ] x 10−3

Figure 5.12: Typical stress-strain relationship for soil under large and small deformations. Figure 5.13 shows the variation of the shear modulus and the material damping ratio of sandy soils with the shear strain. These curves are average values of several experimental studies reported by Seed and Idriss [223]. A wide variety of procedures exists to determine the shear moduli and damping characteristics

EQUIVALENT LINEAR PREDICTION OF GROUND VIBRATIONS

131

such as: direct determination of stress-strain relationship by means of triaxial compression or simple shear tests, free and forced vibration tests, and field measurement of ground vibrations [133, 223].

(a)

(b) Figure 5.13: (a) Modulus reduction and (b) material damping curves for sandy soils (from [223]).

5.4.1

Model description

The equivalent linear method is based on an iterative linear analysis and therefore has the advantage that it can be performed in the frequency domain using the present FE-BE approach. In this method, the shear modulus and material damping ratio are modified after each iteration to account for their dependency on the strain level. The first step in the equivalent linear method is the analysis

132


of the model using the small strain soil properties. Then, the maximum effective shear strain in the soil is computed as [63, 99]: oct γeff = α γmax

(5.9)

oct where α is a coefficient and γmax is the maximum octahedral shear strain (equation (3.24)). A wide range of values between 0.2 to 1.0 has been reported for α in the literature [189, 263]. A value of α = 0.65 is frequently used in engineering practice [63,258,263] and adopted in this study as well. The responses are, however, not very sensitive to the particular value assumed for α. The computed effective shear strain γeff is used to estimate the equivalent shear modulus µ and damping ratios βs and βp of the soil. These values should be obtained experimentally. In the present study, the modulus reduction and material damping curves suggested by Seed et al. [223, 224] for sandy soils (figure 5.13) are used. The average curve in figure 5.13b is shifted vertically to have the material damping ratio at a small strain of 10−6 equal to the damping ratio identified for each soil layer as listed in table 4.1. Once the soil parameters are updated, they are used in the second step of the analysis where new effective shear strains are obtained. The iteration continues until the difference between the updated parameters of two successive steps becomes smaller than an assumed tolerance. A tolerance of a few percent (e.g. 3% [63]) usually gives satisfactory results.

In section 3.5, the peak octahedral shear strain due to traffic loading has been oct = 12 × 10−5 , the effective strain is therefore equal to γeff = estimated as γmax −5 8 × 10 . As can be observed in figure 5.13, the shear modulus of the soil may reduce to 83% of small strain modulus. This results in 9% reduction of the shear wave velocity which is in the range of accuracy of estimated soil parameters by the SASW test. This confirms the efficiency of linear prediction of traffic induced ground-borne vibrations and validates the linear behaviour assumption that is generally considered for the prediction of traffic induced vibrations.

5.4.2

Results obtained with the equivalent linear model

The equivalent linear analysis outlined above has been carried out element-wise instead of layer-wise as it is most commonly performed. The reason is that large shear strains are only found in a limited region under the slab and not all over a soil layer. The effective shear strain is computed as the average of the strains at the eight nodes of each element. A convergence tolerance of 4% between two successively estimated shear moduli and damping ratios is considered. The analysis converged after nine iterations. Figure 5.14 shows the peak octahedral shear strain oct γmax in the updated model as well as the corresponding modulus reduction µ/µ0 and material damping ratio β. Comparing figure 5.14a with figure 5.10 shows that, in the updated model, the strains are more concentrated under the slab due to the softening behaviour. The maximum octahedral shear strain exceeds a value of


133

0.01 in a relatively large area under the slab. This value is larger than the limits of the modulus reduction and material damping curves in figure 5.13. Figures 5.14b and 5.14c show that the shear modulus has reduced to less than 10% of the initial value while the material damping ratio has increased up to 0.25 in the region near the slab. 0.01

1

0.008

0.8

0.006

0.6

0.004

0.4

0.002

0.2

0

0

0.3 0.25 0.2 0.15 0.1

(a)

(b)

0.05 0

(c)

oct Figure 5.14: (a) Peak octahedral shear strain γmax in the updated model by means of the equivalent linear analysis with the corresponding (b) modulus reduction µ/µ0 and (c) material damping ratio β.

Figure 5.15 shows the displacement field in the slab and the soil computed with the updated soil parameters. The snap shots are taken at the time of impact t = 0.200 s and at times t = 0.205 s, 0.210 s, 0.230 s. The displacement of the slab after the impact (figure 5.15a) is larger than the displacement computed with the initial soil parameters (figure 5.6a). This arises from the smaller shear modulus of the updated model. Since the shear modulus has been reduced in part of the domain, the medium is inhomogeneous and the wave propagation in the domain is not as clear as in the case of the initial soil parameters (figure 5.6). Compared to the linear model, the vibrations are attenuated more slowly so that 0.030 s after the impact (t = 0.230 s), the soil still shows noticeable deformations (figure 5.15d). Since the model is basically elastic, there is no permanent deformation due to the impact. Figures 5.16 and 5.17 show snap shots of the horizontal and vertical vibration velocity, respectively, due to an impact of the MHB. A strong decay of vibration velocity with increasing distance from the source is observed which is due to the increased material damping ratio (figure 5.14c). Compared to the model with the initial parameters (figures 5.7 and 5.8), the level of vibration is much smaller. The vibration velocity is smaller than 200 mm/s in the most of the domain. It can also be observed that the body and surface waves are propagating at a lower speed due to the reduction of the shear modulus.

134


−1

−1

−2

−2 z [m]

0.2 0

z [m]

0.2 0

−3

−3

−4

−4

−5 0

1

2 x [m]

3

(a)

1

2 x [m]

3

4

1

2 x [m]

3

4

(b)

−1

−1

−2

−2 z [m]

0.2 0

z [m]

0.2 0

−3

−3

−4

−4

−5 0

(c)

−5 0

4

1

2 x [m]

3

−5 0

4

(d)

Figure 5.15: Displacement field in the slab and the soil at (a) t = 0.200 s, (b) t = 0.205 s, (c) t = 0.210 s, and (d) t = 0.230 s due to an impact of the MHB. The displacements are scaled by a factor of 500.


(a)

135

(b) [mm/s] 200 150 100 50 0 −50 −100 −150 −200

(c)

(d)

Figure 5.16: Horizontal vibration velocity at (a) t = 0.200 s, (b) t = 0.205 s, (c) t = 0.210 s, and (d) t = 0.230 s generated by an impact of the MHB. Figure 5.18 shows the traces of the particle displacement at four locations computed with the updated soil parameters. The traces are shown from the unloaded state to 0.30 s after the impact (t = 0.50 s). Comparing figures 5.18a and 5.9a shows that the displacement at {0.1 m, 0, 0}T is larger than the displacement computed with the initial soil parameters. This is due to the decreased stiffness (shear modulus) of the soil near the slab and consequently concentration of the deformations under the slab. At larger distances, however, the displacements computed with the updated model are smaller. It can also be observed that the traces are much smoother since the high frequency components of the displacements have been attenuated more strongly by the increased damping ratios.

136


(a)

(b) [mm/s] 200 150 100 50 0 −50 −100 −150 −200

(c)

(d)

Figure 5.17: Vertical vibration velocity at (a) t = 0.200 s, (b) t = 0.205 s, (c) t = 0.210 s, and (d) t = 0.230 s generated by an impact of the MHB.

5.4.3


Figure 5.19 compares the vertical vibration velocity predicted with the equivalent linear method to the experimental results as well as the results of the linear model. The results of the linear and equivalent linear method are very different at small distances up to 25 m while at larger distances of 57 m and 105 m, the results are similar. The equivalent linear model reduces the very sharp peaks in the time history predicted by the linear model. This is due to the increase of the material damping ratio. Since the shear modulus has been reduced in the updated model, the equivalent linear model predicts a later arrival of shear and surface waves. At small distances, the equivalent linear model leads to results in better agreement with the experimental results than the linear model. At farther receivers, the results of both methods are different from the experimental results. This might be due to an overestimation of the damping ratios of the deep soil layers (table




1 0 −1 −2 −3 −4

−2

(a)


0 −0.2 −0.4 −0.6 −0.4

(b)

0 −0.2 −0.4 −0.6 −0.4




0.2

(c)

137


0 −0.2 −0.4 −0.6 −0.4

(d)


Figure 5.18: Trace of the particle displacement at (a) {0.1 m, 0, 0}T , (b) {3 m, 0, 0}T , (c) {0.1 m, 0, −4 m}T , and (d) {3 m, 0, −4 m}T due to an impact of the MHB. The motion is shown from the unloaded state (black) to t = 0.50 s (grey). 4.1). The maximum effective shear strain at a depth of 5 m is about 7 × 10−5 . It decreases at larger depths and, according to figure 5.13b, the material damping ratio at shear strains below 10−5 is smaller than the value of 0.04 that has been identified for these layers. In a similar way as in figure 3.12, the effect of the material damping ratio of the deep soil layers on the level of ground vibrations at large distances has been depicted in figure 5.20. In this figure, the Green’s functions of the soil profile in table 4.1 are compared to the Green’s functions of the same soil profile but with a material damping ratio of 0.01 for the third layer and below. It can be observed that the reduction of the material damping ratios of the layers below a depth of 5.9 m does not change the vibration velocity at 5 m, 10 m, and 25 m from the source (figures 5.20a, 5.20b, and 5.20c) while the vibration velocity at larger distances of 57 m and 105 m increases. Another explanation for the underestimation of the response at large distances is the fact that a uniform material damping is assumed for each soil layer in the BE model. At large distances from the source, the amplitude of ground vibrations and, consequently, the shear strains are smaller. Since the material damping ratio

138


Velocity [mm/s/Hz]

20 0 −20 −40

(a) Velocity [mm/s]

0.8

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

0.3 0.25

0 −10

−30

(b)

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

Velocity [mm/s/Hz]

Velocity [mm/s]

2 0 −2 −4

(c)

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

Velocity [mm/s/Hz]

0 −0.5

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

300

0

50


250

300

0

50


250

300

0

50


250

300

0

50


250

300

0.1 0.05

0.06 0.04 0.02

0.03 0.02 0.01 0

2


0.2 0.1 0 −0.1 −0.2

250

0.04

0.5

−1


0.2

0

2

1

(d)

50

0.08

4

−6

0

0.15

0

2

6

Velocity [mm/s]

0.2

20

−20

Velocity [mm/s]

0.4

30

10

(e)

0.6

0

2

Velocity [mm/s/Hz]

Velocity [mm/s]

40

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

2

0.015 0.01 0.005 0

Figure 5.19: Time history (left) and narrow band frequency spectrum (right) of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road due to an impact of the MHB, predicted with the linear (solid black line) and equivalent linear (dashed-dotted line) model and compared with the experimental results (grey line).


139

−7

−7

x 10

1.5

4

Velocity [m/s/Hz]

Velocity [m/s/Hz]

5

3 2 1 0

0


(a)

1

0.5

0

300

Velocity [m/s/Hz]

Velocity [m/s/Hz]

8

2 1

0


300

−9

x 10

3

0

0

(b)

−8

4

x 10

50

(c)


200


100

6 4 2 0

(d)

x 10

0

20


80

100

−9

Velocity [m/s/Hz]

4 3 2 1 0

(e)

x 10

0

20

80

Figure 5.20: Narrow band frequency spectrum of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m due to an impulsive unit load on the soil surface with the profile in table 4.1 (solid line) and the same soil profile with a material damping ratio of 0.01 for the third layer and below (dashed line). depends on the level of shear strain [224], a smaller damping ratio at large distances is expected. The assumption of a uniform material damping ratio results in an underestimation of the response at large distances.

140


Although the equivalent linear method is computationally very appealing, it must be noted that this method has been developed for cyclic loading as occurring during seismic events. The method may therefore not be appropriate for the transient case of impact loading. Moreover, the impacts of the drop hammers cause maximum effective shear strains in the soil larger than 0.01, thereby exceeding the limits up to which the equivalent shear modulus and damping ratio have been defined (figure 5.13). Therefore, a non-linear model that properly accounts for inelastic soil behaviour appears to be more appropriate. This full non-linear soil-structure interaction model is presented in the next chapter.

5.5

Conclusion

In this chapter, ground vibrations generated by pavement breaking are predicted using a linear and an equivalent linear approach. First, the 3D road-soil interaction model developed in section 3.2.1 is coupled to a mass-spring system representing the pavement breaker to study whether the presence of the pavement breaker on the road surface affects ground vibration. The results show that the effect of the mass of the pavement breaker on the ground vibration is negligible. Therefore the 3D model is simplified to an axisymmetric model. In the axisymmetric model, the slab and a part of the soil is modelled with the FE method and the rest of the soil is modelled with the BE method. This model is more efficient than the 3D model because it reduces the dimension of the problem by one. In addition, the BE mesh is shifted from the road-soil interface to further away in the soil domain thus much coarser boundary elements can be used, however, a larger boundary must be meshed. The axisymmetric model has been numerically verified with the Green’s functions of a homogeneous halfspace. The model is used to compute the soil strains due to the impact of the MHB where shear strains as large as a few millistrain are found. At such large strains, the soil behaviour is nonlinear resulting in a poor agreement of the predicted ground vibrations with the experimental results. Therefore, the axisymmetric linear model is modified to an equivalent linear model to approximately account for the non-linear soil behaviour at large shear strains. The equivalent linear method is based on iterative linear analysis where the shear modulus and the material damping ratios are modified after each iteration accounting for the strain level. The degradation curves of sandy soils reported in the literature are used. The results of the equivalent linear model are in a better agreement with the experimental results at small distances while at large distances both the linear and equivalent linear models underestimate the response. This is thought to be caused by the large damping ratios that have been identified for the deep soil layers as well as the assumption of a deformation independent uniform material damping ratio for each soil layer in the BE model. The results

CONCLUSION

141

of the equivalent linear model show that the shear strains in the soil exceed 0.01 which is the upper limit up to which equivalent material properties are defined. Furthermore, a tendency for slab uplifting is observed in the results. Therefore in the following chapter a fully non-linear model is developed.

Chapter 6

Non-linear prediction of ground vibrations due to pavement breaking 6.1

Introduction

The linear and equivalent linear analysis of the road-soil interaction in chapter 5 showed that the deformations in the soil and concrete due to the impacts of the drop hammers are very large. At such large strains, the soil behaves nonlinearly. Therefore, in this chapter a non-linear road-soil interaction model is presented. It takes into account the inelastic soil behaviour as well as the possible slab uplifting. Taking these effects into account requires a time domain solution of the dynamic non-linear road-soil interaction problem. In the literature, several methods have been proposed for this purpose including a FE implementation, a single macro-element approach [49,193], and a hybrid time domain FE-BE scheme [194, 195, 242]. The time domain FE method is a common tool for analysing nonlinear dynamic problems. In case of an unbounded domain, however, difficulties occur due to spurious reflected waves from artificial boundaries. Several techniques such as absorbing boundary conditions (ABC) [167, 257], infinite elements [33, 34], hybrid time-frequency domain FE-BE model [175], perfectly matched layers (PML) [22,142,158] have been proposed to solve the problem. In the following, an efficient two-step approach is presented. In the first step, a FE model of the slab and the soil that includes the region with the inelastic behaviour and a part of the elastic region is analysed in the time domain. The displacements and tractions along a path inside the elastic region of the FE model are subsequently computed and

143

144

NON-LINEAR PREDICTION OF GROUND VIBRATIONS DUE TO PAVEMENT BREAKING

used in a second step where the integral representation theorem is employed to compute the radiated wave field at large distances from the source. This requires a boundary element discretisation. The FE model and the accompanying BE model are verified numerically. In order to predict ground vibrations due to the impacts of the MHB, the critical state soil parameters are estimated from six CPTs. As explained in section 4.5.3, the crack propagation velocity is so high that the slab might be divided into smaller pieces during the course of impact. The non-linear model is used to investigate the effect of two different sizes of the slab on the ground vibrations. The model is further used to investigate the slab and the soil response due to the impacts of the MHB and validated experimentally.

6.2

Model description

The results of the equivalent linear model show that, in a relatively small region near the slab, the soil deformations are beyond the elastic limit (figure 5.14a). Therefore, the slab-soil system can be divided into two subdomains Ω = Ωi ∪ Ωe . Figure 6.1a shows these subdomains and their bounding surfaces and interface. The subdomain Ωi consists of the slab and a part of the soil with inelastic behaviour; it is bounded by the free surface Γiσ and separated from Ωe by the interface Σie . The subdomain Ωe corresponds to the rest of the soil domain with elastic behaviour. The elastic subdomain Ωe is bounded by the free surface Γeσ , the interface Σie , and the outer boundary Γe∞ . A non-linear model is required to analyse the inelastic subdomain Ωi . Since the duration of the impacts of the drop hammers is very short (figure 4.11), the slabsoil system undergoes a non-linear deformation in a very short period of time. Therefore, a FE model with a local absorbing boundary at a sufficiently large distance from the source might be sufficient to avoid spurious reflections from the artificial boundaries during the short impact duration. Figure 6.1b shows the inelastic region and a part of the elastic domain which is denoted as Ω1 and modelled by the FE method. It is bounded by the free surface Γ1σ and the local absorbing boundary Γv where v stands for the viscous boundary conditions. In order to compute the response at large distances from the source, the displacements and tractions along a path Σ inside the elastic region of the FE domain Ω1 are computed. The displacements and tractions along the path Σ are introduced into a second model which consists of a layered elastic halfspace Ωs bounded by the free surface Γsσ and the outer boundary Γs∞ (figure 6.1c). The properties of the halfspace are similar to the elastic subdomain Ωe . The integral representation theorem is used to compute the response of the domain outside the path Σ from the displacements and tractions along the path. The size of the inelastic region Ωi depends on the amplitude, duration, and direction of the applied load as well as on the properties and geometry of the

MODEL DESCRIPTION

(a)

145

(b)

(c)

Figure 6.1: (a) Coupled inelastic and elastic subdomains Ωi ∪ Ωe , (b) the inelastic region and a part of the elastic domain Ω1 coupled to the local absorbing boundary Γv , and (c) the elastic halfspace Ωs . slab and the soil. As a first step, the size of the inelastic region can be estimated from the probable failure surfaces according to Terzaghi’s bearing capacity theory. The location of the local absorbing boundary Γv , and consequently, the size of the FE domain Ω1 , is considered to be sufficiently far from the inelastic region to avoid the effect of spurious reflecting waves on the inelastic region. Therefore, it depends on the duration of the load and the shear and dilatational wave velocities in the soil. The optimised size of the FE domain can be obtained after a few iterations. The path Σ should be as close as possible to the borders of the inelastic region. The FE model is axisymmetric and composed of the slab, a cylinder of soil with a radius Bs and a height Ds , and absorbing boundary conditions at the truncated boundaries. The absorbing boundary is based on the viscous damping of normal and shear stresses along the boundary:   σn + ρ Cp u˙ n = 0 (6.1)  σ + ρ C u˙ =0 t s t

where σn and σt are the normal and shear stresses on the boundary, u˙ n and u˙ t are the normal and tangential velocities, ρ is the soil density, and Cp and Cs are the dilatational and shear wave velocities in the soil. In discretised form, the absorbing boundary is lumped in viscous dampers oriented parallel with and perpendicular to the boundary. Since the energy dissipated in the slab is small (section 4.5), the inelastic behaviour of the slab is disregarded and it is modelled as a linear elastic medium. This reduces the computation time and avoids possible difficulties with convergence due to the localisation of strains in the slab. Inelastic soil behaviour is modelled using the Drucker-Prager yield criterion with a non-associative flow rule. An elastic-perfectly plastic material behaviour is assumed so the yield surface does not change with progressive yielding. The yield

146


surface f is defined as: f (I1 , J2 )

p J2 − k

= αI1 +

(6.2)

where α and k are material constants and I1 is the first stress invariant and J2 is the second deviatoric stress invariant defined in terms of principal stresses σ1 , σ2 , and σ3 as: I1

=

J2

=

σ1 + σ2 + σ3 1 (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 6

(6.3) (6.4)

In the case where the Drucker-Prager yield surface corresponds to the outer apices of the hexagonal Mohr-Coulomb yield surface, the constants α and k are related to the cohesion c and internal friction angle φ as: α =

2 sin φ √ 3(3 − sin φ)

(6.5)

k

6 c cos φ √ 3(3 − sin φ)

(6.6)

=

The flow rule g is defined as: g(I1 , J2 )

= βI1 +

p J2

(6.7)

where β is a material constant related to the dilatancy angle ψ as: β

=

2 sin ψ √ 3(3 − sin ψ)

(6.8)

Since the normal stress has a direct effect on the shear strength of a soil, the self weight of the slab and the soil are considered in the FE model. For this purpose, the analysis is performed in two stages. In the first stage, the vertical displacement of the bottom of the FE domain and the horizontal displacement of the side of the FE domain are temporarily fixed and the self weight is gradually applied. This results in the static stress condition in the domain. Next, the temporary supports are released and the impact load is applied on the slab while the reactions of the temporary supports are applied on the boundary. In this stage, the absorbing boundary conditions are active. This results in an accurate distribution of stresses due to the self weight during the transient part of the analysis.

MODEL DESCRIPTION

147

As shown in figures 5.6a and 5.15a, the slab tends to rebound and lift up from the soil after the impact. This results in unrealistic tensile stresses at the interface and causes difficulties with convergence. Therefore, the slab-soil separation is considered by a contact model. For this purpose, the common contact-target pair concept is used. The soil surface at the slab-soil interface is considered as the contact surface and the bottom side of the slab as the target surface. The surfaceto-surface contact model includes both the normal pressure and frictional stresses and allows sliding and separation of the contacting surfaces. To solve the contact problem, the augmented Lagrangian method is used. This is an iterative penalty method where the contact tractions are augmented at each iteration so that the final penetration is smaller than an allowable tolerance. Compared to the penalty method, the augmented Lagrangian method is less sensitive to the magnitude of the contact stiffness coefficient and usually leads to better conditioning [14, 229]. The FE model is solved in the time domain employing the Newmark time integration method with integration parameters α = 0.2525 and δ = 0.5050. These values satisfy the stability criteria [267]: γ ≥ 0, α ≥ 41 (1+γ)2 , and δ ≥ 21 +γ where γ is an amplitude decay factor. The considered integration parameters guarantee unconditional stability of the solution and provide a small numerical damping that damps high frequency modes. For a large amplitude loading, the main source of energy dissipation in the FE domain is the inelastic behaviour of the soil while small strain material damping does not significantly affect the response inside the FE domain. The hysteretic material damping of the soil under dynamic loads can therefore be approximated by means of a Rayleigh damping formulation. The damping matrix C is thus equal to C = η K, where K is the stiffness matrix and η is a multiplier defined for each soil layer as η = 2β/ω0 with β the material damping ratio of that layer (table 4.1) and ω0 a normalisation frequency. The Rayleigh damping formulation results in a frequency dependent damping that filters the high frequency response. A small normalisation frequency results in a strong attenuation of the high frequency response while a large normalisation frequency causes underestimation of the damping at low frequencies. The later approximation only slightly affects the results as the main source of energy dissipation is the inelastic material behaviour. Therefore, a large normalisation frequency around the maximum frequency of interest is used. After the solution of the FE model under the impact load, the displacements u ˆ p (t) and tractions ˆtp (t) are computed along the path Σ inside the FE domain (figure 6.1b). They are introduced in the integral representation theorem in equation (3.7) to compute the radiated wave field. The mesh size of the BE model is similar to the size of the FE mesh near the path although the nodes are not coincident. The displacements and tractions on the BE mesh are therefore computed by interpolation. Since in the first stage of the FE analysis the self weight has been applied, the displacements and tractions prior to the impact are nonzero. They

148


are also nonzero and different from the initial values at the end of analysis owing to inelastic material behaviour. Computation of the radiated wave field using equation (3.7) requires the initial values of the displacements and tractions to be the same as their final values. Therefore, a time window is applied to gradually change the final values of displacements and tractions to their initial values. The time window has a time history that corresponds to the modulus of the frequency content of a second order Butterworth filter with a low-pass of 0.7T where T is the time period. The proposed combined FE-BE model is verified in the following for two cases of elastic and inelastic material behaviour.

6.3 6.3.1

Numerical verification Linear elastic halfspace

In the following, the proposed FE model is verified for the simplified case of elastic material behaviour. As the material does not yield, the propagating waves are only damped by the material damping so the waves reflected from the absorbing boundary are stronger and the problem is more delicate. The response of a linear model of the soil in table 4.1 under an impact of the MHB (figure 4.11) is computed. The vertical vibration velocity is compared to the results obtained with the direct stiffness method considering the material damping as a linear function of frequency. For this purpose, the material damping ratios are adjusted for each frequency. Several receivers located inside and outside the FE domain are considered. The axisymmetric FE model has a width Bs = 6 m and a depth Ds = 8.1 m (figure 6.2) and material properties according to the first three layers of the soil profile in table 4.1. The impact force is applied to the FE nodes considering an impact footprint with a radius df /2 = 0.065 m. The FE mesh is composed of 8-node quadrilateral elements and has been generated in ANSYS. It is finer near the impact point and gets coarser at larger distances from the source. It provides at least 4 elements (8 DOFs) per wavelength up to a frequency of 500 Hz. To compute the response of the exterior domain, the displacements and tractions along a path Σ at a distance Bb = 4 m and a depth Db = 5 m (figure 6.2) are computed. This will be also the location of the BE mesh in the accompanying BE model. The boundary element discretisation on the path is composed of 3-node line elements with quadratic shape functions. The size of the boundary elements is similar to the size of the finite elements near the path and is shown in figure 5.5b. The material damping - which is considered as a linear function of the frequency - is normalised at ω0 /(2π) = 100 Hz to damp high frequency components and to avoid the necessity of a very fine FE mesh.

NUMERICAL VERIFICATION

149

Figure 6.2: Finite element domain Ω1 and the absorbing boundary conditions Γv with the position of the path Σ. Figure 6.3 compares the vertical velocity predicted using the FE model and the accompanying BE model to the results of the direct stiffness method at four receiver points x = {1 m, 0, 0}T at the soil surface in the FE domain, x = {2 m, 0, −2 m}T inside the FE domain, x = {4 m, 0, −5 m}T at the corner of the assumed path, and x = {10 m, 0, 0}T on the surface outside the FE domain. The response of the first three points is computed using the FE model, while the integral representation theorem (equation (3.7)) is applied to compute the response of the fourth point. Figures 6.3a, 6.3b, and 6.3c show that the agreement between the results of the FE model and the direct stiffness method is very good. At larger distances from the force, the prediction quality slightly declines. This is due to the coarser FE mesh at larger distances as well as the stronger contribution of waves reflected by the absorbing boundary. Figure 6.3d shows the response outside the FE domain at x = {10 m, 0, 0}T which is in good agreement with the result of the direct stiffness method. It must be noted that although the FE model and the accompanying BE model yield satisfactory results for an impulsive load, the quality of the prediction for harmonic loads might not be acceptable, particularly at low frequencies. In the case of harmonic loading, spurious reflections occur that are not sufficiently absorbed by the absorbing boundary. The amplitude of the spurious reflected waves can be further reduced by considering an artificially larger material damping ratio for the region in between the assumed path and the absorbing boundary. This is an improvement to the absorbing layer method proposed by Semblat et al. [226].


4

0.4

2

0.2

Velocity [m/s]

Velocity [m/s]

150

0 −2 −4

0

0.05

−0.2 −0.4

0.15

0.1

0.05

0.05

0 −0.05

(c)

0

0.05

0.1 Time [s]

0.05

0.1

0.15

0.1

0.15

Time [s]

0.1

−0.1

0

(b)

Time [s]

Velocity [m/s]

Velocity [m/s]

(a)

0.1

0

0 −0.05 −0.1

0.15

(d)

0

0.05 Time [s]

Figure 6.3: Time history of the vertical velocity at (a) x = {1 m, 0, 0}T , (b) x = {2 m, 0, −2 m}T , (c) x = {4 m, 0, −5 m}T , and (d) x = {10 m, 0, 0}T due to an impact of the MHB on the soil surface computed with the FE and the accompanying BE model (black line) and the direct stiffness formulation assuming a frequency dependent damping (grey line).

6.3.2

Static elasto-plastic SSI with foundation uplift

In the following, the FE model is verified by means of the results of an example by Gazetas and Apostolou [90] for a static SSI problem including elasto-plastic soil behaviour and soil-structure separation. The response of a 6 m wide rigid strip footing resting on a homogeneous soil layer with a Young’s modulus of 20 MPa and a thickness of 20 m is studied (figure 6.4). Non-linear soil behaviour is modelled with the elasto-plastic Mohr-Coulomb model with cohesion c = 50 kPa and friction angle φ = 30◦ . The Mohr-Coulomb model that is used in the reference is approximated by the Drucker-Prager yield criterion in the FE model. The superstructure is assumed to be a lumped mass with a weight of 1000 kN/m (per unit length) at a point 12 m above the base which is connected to the foundation with a rigid massless element. A displacement-controlled force is applied at the mass centre of the structure, resulting in the loading of the footing by a horizontal force and a moment. The applied displacement is gradually increased until toppling of the structure occurs. Figure 6.5 compares the predicted moment versus tilt angle of the foundation with the result of Gazetas and Apostolou [90], showing a perfect match. At the

CRITICAL STATE SOIL PARAMETERS

151

Figure 6.4: Schematic representation of a rigid structure rocking on a yielding soil (from [90]). 2500

Moment [kNm]

2000 1500 1000 500 0

0

0.05

0.1 0.15 Tilt angle [rad]

0.2

0.25

Figure 6.5: Moment-rotation curve, predicted using the FE model (black line) and the result of Gazetas and Apostolou [90] (grey line). beginning, the overturning moment increases linearly with the tilt angle. Then, the foundation lifts up and the soil begins to yield which results in a gradually softening rocking behaviour. By further increasing the imposed displacement, the P-δ effect causes reduction of the overturning moment.

6.4

Critical state soil parameters

In order to consider inelastic behaviour of the soil in the prediction of ground vibrations, the critical state soil parameters are required. These parameters can be obtained from laboratory or in situ tests of the soil. A parametric study has shown that the level of ground vibration is not very sensitive to the critical state parameters, however. Therefore, the density ρ, critical state friction angle φ′cv , and dilatancy angle ψ of the soil have been estimated from six CPTs next to each other, GEO-00/86-S1 to S6, at 1500 m southeast of the measurement site in Waarschoot (table 6.1). The procedure of obtaining these parameters is explained in appendix B. A cohesion of 20 kPa is considered for the top layer to avoid possible

152


difficulties with convergence and to account for the organic matter content of the surface soil (figure 4.5a). The Young’s modulus E and Poisson’s ratio ν have been obtained from in situ geophysical tests as presented in section 4.3.2 and table 4.1. Some discrepancies are observed between the soil layering in tables 6.1 and 4.1. These arise from the fact that the profile in table 4.1 defines an equivalent soil profile with a dispersion curve that fits the experimental dispersion curve obtained by means of the geophysical tests, while the profile in table 6.1 has been identified from the data of the CPTs at another site near the measurement site. In retrospect, a CPT should have been performed first at the measurement site to identify strength parameters and soil layering. The thicknesses of the layers identified by the CPT could subsequently have been used in the identification of dynamic soil characteristics by inversion of the experimental dispersion curve. In this way, more consistent soil profiles could have been obtained. Layer 1 2 3 4

Thickness [m] 1.6 1.5 3.0 2.3

ρ [kg/m3 ] 1850 2010 1860 2010

φ′cv [°] 33 37 33 39

ψ [°] 9 11 3 9

E [MPa] 141 141 154 342

ν [-] 0.46 0.46 0.49 0.49

Table 6.1: Density ρ, critical state friction angle φ′cv , and dilatancy angle ψ of the soil near the measurement site estimated from the CPT data and Young’s modulus E and Poisson’s ration ν obtained from in situ geophysical tests. In the parametric study, two soil profiles have been considered. The first soil profile corresponds to a dense sand with a friction angle of 37° and a dilatancy angle of 11°. The second profile represents a loose sand with a friction angle of 30° and a zero dilatancy angle. The elastic properties of both soil profiles are the same as the values given in table 4.1. Figure 6.6 compares the vertical velocity predicted using the combined FE and BE model for these soil profiles and five points at the soil surface located at distances of 5, 10, 25, 57, and 105 m from the impact point. It can be observed that the difference between the predicted ground vibrations for the two soil profiles is small. The amplitude of the high frequency components for the loose soil is slightly smaller than for the dense soil. This arises from the fact that in the loose soil material yielding results in more energy dissipation in comparison with the dense soil. The small difference between the response of the two soil profiles indicates that small changes in the critical state soil parameters do not significantly affect the predicted ground vibrations. Therefore, the parameters identified from the available CPT data (table 6.1) are sufficiently accurate for the prediction of ground vibration.


153

Velocity [mm/s/Hz]

20 0 −20 −40

(a) Velocity [mm/s]

0.8

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

0.3 0.25

0 −10

−30

(b)

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

Velocity [mm/s/Hz]

Velocity [mm/s]

2 0 −2 −4

(c)

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

Velocity [mm/s/Hz]

0 −0.5

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

300

0

50


250

300

0

50


250

300

0

50


250

300

0

50


250

300

0.1 0.05

0.06 0.04 0.02

0.03 0.02 0.01 0

2


0.2 0.1 0 −0.1 −0.2

250

0.04

0.5

−1


0.2

0

2

1

(d)

50

0.08

4

−6

0

0.15

0

2

6

Velocity [mm/s]

0.2

20

−20

Velocity [mm/s]

0.4

30

10

(e)

0.6

0

2

Velocity [mm/s/Hz]

Velocity [mm/s]

40

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

2

0.015 0.01 0.005 0

Figure 6.6: Time history (left) and narrow band frequency spectrum (right) of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road due to an impact of the MHB computed with the non-linear model for a dense sand with φ′cv = 37° and ψ = 11° (solid line) and a loose sand with φ′cv = 30° and ψ = 0 (dashed line).

154


As estimated in section 5.3.3, the strain rate in the soil is about 12 s−1 . Whitman and Healy [66, 255] have reported a small effect of loading rate on the friction angle of cohesionless soils. Therefore, the identified soil parameters in table 6.1 are not modified and used to compute the response of the non-linear FE model of the slab-soil system due to the impacts of the MHB. The profile identified by the surface wave test in table 4.1 is used to analyse the linear part of the soil.

6.5

Results obtained with the non-linear model

In the following, the properties of the non-linear FE model are described and the predicted ground vibrations are compared for two sizes of the slab (rr = 0.3 m and rr = 1.425 m). More detailed results are presented afterward for the model with rr = 1.425 m. The radius rr = 1.425 m corresponds to half of the width of the slab in figure 4.4, while rr = 0.3 m is obtained from the size of the impact footprint and assuming a shear plug formation through the thickness of the slab at an angle of about 45°. The FE model has a width Bs = 6 m and a depth Ds = 8.1 m. The FE mesh consists of 8-node quadrilateral elements for the soil and the slab. It is finer near the impact point and coarser at larger distances and provides at least four elements per wavelength below 500 Hz. The damping matrix is approximated by Rayleigh damping with a large normalisation frequency of 500 Hz. The damping coefficients of the viscous absorbing boundary are computed from the shear and dilatational wave velocities of each layer. The FE model is analysed in ANSYS where the soil and the slab are modelled with 8-node quadrilateral plane elements (PLANE183) and the absorbing boundary with viscous dampers (LINK11). The separation of the slab from the soil is modelled using contact elements. The contact elements overlay the top surface of soil describing the boundary of a deformable body that is potentially in contact with a target surface which is the bottom surface of the slab. The target is simply a geometric entity that senses and responds when contact elements move into it. The contact detection points are Gauss integration points and the contact elements are constrained against penetration into the target surface at these points. The contact tractions (pressure and frictional stress) are augmented during equilibrium iterations so that the final penetration is smaller than 1% of the depth of the underlying quadrilateral plane elements. On the slabsoil interface, the bottom surface of the slab is modelled with 3-node 2D target elements (TARGE169) and 3-node 2D contact elements (CONTA172) are used on the soil surface. The limit state soil parameters presented in table 6.1 are used. For the computation of the response outside the FE domain, the integral representation theorem discretised on the BE mesh shown in figure 5.5b is used. Figure 6.7 compares the vertical velocity predicted using the FE and BE models for two sizes of the slab (rr = 1.425 m and rr = 0.3 m) and five points at the soil

RESULTS OBTAINED WITH THE NON-LINEAR MODEL

155

Velocity [mm/s/Hz]

20 0 −20 −40

(a) Velocity [mm/s]

0.8

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

0.3 0.25

0 −10

−30

(b)

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

Velocity [mm/s/Hz]

Velocity [mm/s]

2 0 −2 −4 −6

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

Velocity [mm/s/Hz]

0 −0.5

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

300

0

50


250

300

0

50


250

300

0

50


250

300

0

50


250

300

0.1 0.05

0.06 0.04 0.02

0.03 0.02 0.01 0

2


0.2 0.1 0 −0.1 −0.2

250

0.04

0.5

−1


0.2

0

2

1

(d)

50

0.08

4

(c)

0

0.15

0

2

6

Velocity [mm/s]

0.2

20

−20

Velocity [mm/s]

0.4

30

10

(e)

0.6

0

2

Velocity [mm/s/Hz]

Velocity [mm/s]

40

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

2

0.015 0.01 0.005 0

Figure 6.7: Time history (left) and narrow band frequency spectrum (right) of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road due to an impact of the MHB computed with the non-linear model for slab radii rr = 1.425 m (solid line) and rr = 0.3 m (dashed line).

156


surface located at distances of 5, 10, 25, 57, and 105 m from the impact point. The difference between the predicted ground vibrations for the two sizes of the slab is small. For the small size of the slab, slightly larger peaks of the first arriving P-waves are found. The small difference shows that, even if during the course of impact the cracks divide the slab into smaller parts, the resulting ground vibrations do not change significantly. This is in agreement with the experimental results in figure 4.18 where ground vibrations for eight consecutive impacts of the MHB at four points of a single slab were compared.

−1

−1 z [m]

0.2 0

z [m]

0.2 0

−2

−3 0

(a)

−2

1 x [m]

−3 0

2

1 x [m]

2

(b)

Figure 6.8: Permanent deformation of the soil due to an impact of the MHB for slab radii (a) rr = 1.425 m and (b) rr = 0.3 m. The displacements are scaled by a factor of 20. Figure 6.8 shows the permanent deformation in the soil due to an impact of the MHB for two sizes of the slab. The vertical displacement under the small slab is very large and reaches a value of about 40 mm while for the large slab this value is less than 4 mm. The large deformation predicted for the small slab is far from what was observed in the experiments. The deformed shape of the large slab is more reasonable, however, the hairline cracks on the slab surface are in favour of the idea that the small slab is more rational. This can be explained by the fact that the cracks generated by the MHB are very thin or even do not run through the bottom side of the slab. Thus, the two parts of a broken slab are still interlocking and act as a single slab. This argument is supported by the fact that after the operation of the MHB a heavy roller must pass on the road to complete


157

the breaking operation and to seat the cracked slabs on the soil surface. As the size of the slab does not significantly affect ground vibration at distances larger than 5 m and the deformed shape of the large slab appears to be more realistic, the results for rr = 1.425 m are presented in the following.

−1

−1

−1

−1

−2

−3 0

−2

1 x [m]

(a)

−3 0

2

z [m]

0.2 0

z [m]

0.2 0

z [m]

0.2 0

z [m]

0.2 0

−2

1 x [m]

−3 0

2

(b)

−2

1 x [m]

−3 0

2

(c)

−1

−1

−1

−3 0

(e)

1 x [m]

2

−3 0

(f)

−2

1 x [m]

2

1 x [m]

2

z [m]

−1

z [m]

0.2 0

z [m]

0.2 0

z [m]

0.2 0

−2

2

(d)

0.2 0

−2

1 x [m]

−3 0

(g)

−2

1 x [m]

2

−3 0

(h)

Figure 6.9: Displacement field in the slab and the soil at (a) t = 1.0 s, (b) t = 1.003 s, (c) t = 1.008 s, (d) t = 1.013 s, (e) t = 1.033 s, (f) t = 1.071 s, (g) t = 1.084 s, and (h) t = 1.500 s due to an impact of the MHB. The displacements are scaled by a factor of 100. Figure 6.9a shows the displacement field in the slab and the soil under the self weight while figures 6.9b to 6.9h show the displacement field due to an impact of the MHB. The impact force (figure 4.11) starts 1 ms after the application of the self weight (at t = 1.001 s) and ends at t = 1.003 s. The snap shots are taken at t = 1.000 s (after the application of the self weight), t = 1.003 s (after the impact) and at times t = 1.008 s, 1.013 s, 1.033 s, 1.071 s, 1.084 s, 1.500 s. The time instants t = 1.003 s, 1.008 s, 1.013 s, and 1.033 s correspond to the same time instants as in figures 5.6, 5.7, 5.8, 5.15, 5.16, and 5.17 to simplify comparison. The slab moves downward due to the impact (figure 6.9b) which results in the permanent

158


deformation of the soil underneath and then lifts up from the soil as much as 5.5 mm (figure 6.9e). After some time, the slab returns to the soil surface (figure 6.9f) and rebounds (figure 6.9g). Finally, the edges of the slab rest on the deformed surface of the soil while a void forms under the impact point (figure 6.9h). The permanent vertical displacement of the impact point excluding the displacement due to the self weight is −0.22 mm while the displacement of the soil underneath is −2.99 mm. Figure 6.10 shows the normal stress σzz in the slab and the soil before and after the impact. Before the impact, the vertical stress σzz gradually increases with depth due to the self weight. The impact changes the stress distribution near the slab and residual stresses remain below the slab owing to the inelastic behaviour of the soil. 2

N/m

(a)

2

N/m

4

4

x 10

x 10

0

0

−5

−5

−10

−10

−15

−15

(b)

Figure 6.10: Predicted normal stress σzz in the slab and the soil (a) before and (b) after the impact. Figures 6.11 and 6.12 show snap shots of the horizontal and vertical vibration velocity, respectively, in the FE domain. The snap shots are taken at t = 1.003 s (after the impact) and at later times t = 1.008 s, 1.013 s, 1.033 s, 1.071 s, 1.084 s. Vibration velocities in the horizontal and vertical directions have the same order of magnitude and are larger than 50 mm/s. No waves are reflected from the local absorbing boundary. Figures 6.11e and 6.11f show lower amplitude waves generated by the impact of the slab on the soil surface (figure 6.9f). Figure 6.13 shows the computed traces of the particle displacement at four locations. The traces are shown after the application of the self weight (t = 1.000 s) till t = 1.300 s when the vibrations in the domain have been sufficiently damped


(a)

(b)

159

(c) [mm/s] 100 80 60 40 20 0 −20 −40 −60 −80 −100

(d)

(e)

(f)

Figure 6.11: Horizontal vibration velocity at (a) t = 1.003 s, (b) t = 1.008 s, (c) t = 1.013 s, (d) t = 1.033 s, (e) t = 1.071 s, and (f) t = 1.084 s generated by an impact of the MHB. out. There are rather large permanent deformations at the two points on the surface (figures 6.13a and 6.13b) while the permanent deformation of the two points at a depth of 4 m is small (figures 6.13c and 6.13d). Figure 6.13a shows that during the impact a large permanent deformation occurs. There is also a small elastic motion after the impact. Figure 6.13b shows a large permanent deformation when the shock wave arrives and some recoverable motions afterward. Figure 6.14 shows the region that undergoes plastic deformation due to the impact. The plastic deformation in the soil is mainly concentrated under the slab. It can be observed that the plastic region slightly extends beyond the path Σ where elastic material behaviour has been assumed. The amount of plastic deformation in the region outside the path is very small. The sudden change at the depth of 1.6 m occurs at the interface between layers. Figure 6.15a shows the peak octahedral shear strain as well as its elastic (figure

160


(a)

(b)

(c) [mm/s] 100 80 60 40 20 0 −20 −40 −60 −80 −100

(d)

(e)

(f)

Figure 6.12: Vertical vibration velocity at (a) t = 1.003 s, (b) t = 1.008 s, (c) t = 1.013 s, (d) t = 1.033 s, (e) t = 1.071 s, and (f) t = 1.084 s generated by an impact of the MHB. 6.15b) and plastic (figure 6.15c) part. The shear strain is mostly elastic up to more than 0.2 m under the centre of the slab. This arises from the fact that the confining pressure close to the axis of symmetry is also very large so that the Drucker-Prager soil model results in elastic behaviour (figure 6.16a). At a depth of 0.6 m below the slab as well as near the edge of the slab, the deformations are mostly plastic. The octahedral shear strain in the soil domain is less than 0.007 except under the edge of the slab where stresses and strains are very large. √ Figure 6.16 shows the square root of the second deviatoric stress invariant J2 versus the mean pressure −I1 /3 at 9 points in the soil domain. These points are located at depths z = 0, −0.5 m, and −1.5 m and radial distances x = 0.1 m, 1.0 m, T and 1.5 m. Figure 6.16a shows that √ the mean pressure at x = {0.1 m, 0, 0} reaches a value of 1700 kPa while J2 is about 340 kPa. The ratio of the root of the deviatoric stress to the mean pressure is small which, according to the Drucker-Prager yield criterion in equation (6.2), results in large elastic and small


161



1 0 −1 −2 −3 −4

−2

(a)


0 −0.2 −0.4 −0.6 −0.4

(b)

0 −0.2 −0.4 −0.6 −0.4

(c)



0.2



0 −0.2 −0.4 −0.6 −0.4

(d)


Figure 6.13: Trace of the particle displacement at (a) {0.1 m, 0, 0}T , (b) {3 m, 0, 0}T , (c) {0.1 m, 0, −4 m}T , and (d) {3 m, 0, −4 m}T due to an impact of the MHB. The motion is shown from time t = 1.000 s (black) till t = 1.300 s (grey).

Figure 6.14: Elastic region (grey) and plastic region (black) in the FE domain due to an impact of the MHB. Superimposed is the position of the path Σ (thick line). plastic deformation (figures 6.15b and 6.15c). Figure 6.16b shows that the point at x = {1 m, 0, 0}T has a similar stress condition as x = {0.1 m, 0, 0}T . Near

162


0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0

(a)

(b)

(c)

oct Figure 6.15: (a) Peak octahedral shear strain γmax due to an impact of the MHB estimated using the non-linear model and its (b) elastic and (c) plastic part.

the edge of the slab at x = {1.5 m, 0, 0}T (figure 6.16c), the deviatoric stress is much larger than the mean pressure because the point is at the free surface of the soil. This results in large plastic strains. Points at the depth z = −0.5 m (figures 6.16d and 6.16e) are close to the failure surface according to Terzaghi’s bearing capacity theory and therefore experience large shear stresses and undergo large plastic strains (figure 6.15c). At the depth z = −1.5 m, the ratio of the root of the deviatoric stress to the mean pressure is small and thus the plastic strains are small. For all points except the point near the edge of the slab at x = {1.5 m, 0, 0}T it can be observed that the mean pressure increases as P-waves arrive while a slight increase of the shear stresses is observed. This causes no yielding. When the Pwave front passes, the mean pressure decreases significantly. Then, the slower S-waves arrive which results in large shear stresses and, consequently, plastic deformation. If a cap model is used, the compressive stresses can also cause yielding. In this case, plastic deformation under the slab as well as near the axis of symmetry would be larger. This results in more concentrated deformations near the slab and, consequently, a smaller non-linear region in comparison with the observation in figure 6.14.

6.6

Comparison of the models

In this section, the results of the non-linear model are compared with the results of the linear and equivalent linear model. Figure 6.17 compares the trace of the particle displacement at several points in the soil domain due to an impact of the MHB computed with the linear,

COMPARISON OF THE MODELS

163

x = 0.1 m

x = 1.0 m

500 400 300 200 100 0

400 800 1200 1600 Mean pressure [kPa]

(a)

100 50

40 20 0


0


80 60 40 20

100 80 60 40 20 0

(e)


0


100

(h)

60 40 20 0


0


0


120 100 80 60 40 20 0

(f) 140

120

80 60 40 20 0

80

(c)

Deviatoric stress [kPa]

100

100

0

140

120

120

140

120

0



60


150

140

z = −1.5 m

80

(b) Deviatoric stress [kPa]


z = −0.5 m

200

(d)

(g)

100

140

250

0

120

0

300

0

140 Deviatoric stress [kPa]


z=0


600

0

x = 1.5 m

140

0


120 100 80 60 40 20 0

(i)

√ Figure 6.16: Square root of the second deviatoric stress invariant J2 versus the mean pressure −I1 /3 at (a) x = {0.1 m, 0, 0}T , (b) x = {1 m, 0, 0}T , (c) x = {1.5 m, 0, 0}T , (d) x = {0.1 m, 0, −0.5 m}T , (e) x = {1 m, 0, −0.5 m}T , (f) x = {1.5 m, 0, −0.5 m}T , (g) x = {0.1 m, 0, −1.5 m}T , (h) x = {1 m, 0, −1.5 m}T , (i) x = {1.5 m, 0, −1.5 m}T due to an impact of the MHB. The stress path is shown from time t = 1.000 s (black) till t = 1.100 s (grey). Superimposed is the yield surface (thick grey line). equivalent linear, and non-linear model. As expected, the non-linear model predicts permanent displacements while the predicted displacements with the two other methods are completely recoverable. The non-linear model also predicts a larger displacement than the linear and equivalent linear model at small distances from the impact point. At large distances, on the other hand, it predicts smaller displacements than the linear model. This arises from the plastic

164


x = 0.1 m

x = 1.5 m

−1 −2 −3



z=0


0

x = 3m

0.2 0 −0.2 −0.4 −0.6


−0.2

(c)

−0.2 −0.4 −0.6 −0.8

0




0.8

−0.1 −0.2 −0.3 −0.4

−1

(d)



z = −4 m

−0.1 −0.2

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Figure 6.17: Trace of the particle displacement at (a) {0.1 m, 0, 0}T , (b) {1.5 m, 0, , 0}T , (c) {3 m, 0, 0}T , (d) {0.1 m, 0, −2 m}T , (e) {1.5 m, 0, −2 m}T , (f) {3 m, 0, −2 m}T , (g) {0.1 m, 0, −4 m}T , (h) {1.5 m, 0, −4 m}T , and (i) {3 m, 0, −4 m}T due to an impact of the MHB computed with the linear (solid line), equivalent linear (dashed-dotted line), and non-linear model (dashed line). deformation of the soil near the slab and, consequently, the concentration of energy dissipation under the slab and attenuation of displacements at larger distances. The displacements at large distances predicted with the equivalent linear model are similar to the results of the non-linear model. Figure 6.18 compares the peak octahedral shear strain due to an impact of the MHB computed with the linear, equivalent linear, and non-linear model. The shear strains far from the slab predicted with the linear model are larger than those predicted by the equivalent linear model and the non-linear model. The

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equivalent linear model predicts the largest shear strains under the slab which decrease rapidly with depth. The non-linear model predicts a concentration of shear strains under the impact point. Leaving out of consideration the soil under the edge of the slab, the maximum octahedral shear strain predicted with the linear, equivalent linear, and non-linear model is 0.006, 0.016, and 0.007, respectively.

0.01

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oct Figure 6.18: Peak octahedral shear strain γmax due to an impact of the MHB computed using (a) the linear, (b) equivalent linear, and (c) non-linear model.

Figure 6.19 compares the vertical ground vibration velocity predicted with the linear, equivalent linear, and non-linear model. The non-linear model predicts larger amplitudes for the P-waves than the linear and equivalent linear model. This arises from the fact that the yielding of the soil under the slab as well as the separation of the edge of the slab from the soil results in a more concentrated transfer of the impact load to the soil. A more concentrated load causes a larger amplitude of the P-waves. In addition, the P-waves do not cause material yielding and are therefore not significantly damped by the non-linear model (figure 6.16). The non-linear model can be further improved by using a cap yield criterion. Figure 6.19a shows that the linear model predicts very large peaks in the time history of the vibration velocity at the closest receiver at x = 5 m. These sharp peaks are drastically reduced in the updated equivalent model. At the farthest receiver at x = 105 m, the results of all methods are more or less similar. To conclude, the results of the linear model are far from the results of the two other methods. The equivalent linear and non-linear models give roughly similar results while the non-linear model is more realistic because it takes into account the plastic behaviour of the soil as well as the slab-soil separation. The non-linear model better represents the characteristics of the problem and thus it is more appropriate for the prediction of ground vibrations due to pavement breaking than the equivalent linear model.

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Figure 6.19: Time history (left) and narrow band frequency spectrum (right) of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road due to an impact of the MHB, predicted with the linear (solid line), equivalent linear (dashed-dotted line), and non-linear model (dashed line).


6.7

167


In this section, ground vibrations predicted using the non-linear model are compared with the experimental results. Figure 6.20 compares the predicted and measured vertical ground vibration velocity. The time histories show that the arrival time of the P-waves as well as the time delay between the P-waves and the surface and shear waves is well predicted. The non-linear model overestimates the amplitude of the P-waves. In the time histories, it can be observed that these P-waves have a higher frequency content than the surface waves. The frequency spectra show that the model overestimates high frequency components. This can arise from the yield criterion assumed in the model. The P-waves generate tension and compression in the soil domain. The tensile stresses are compensated by the self weight. The compressive stresses do not result in failure because the DruckerPrager yield criterion is used. This means that the P-waves are not significantly affected by the inelastic soil behaviour as also observed in figure 6.16. In reality, however, high compressive stresses can also result in the plastic volumetric change of soil, or dilatancy [53]. Using a cap model can therefore give more accurate results. The time histories show that the amplitude of the Rayleigh waves is rather well predicted. As explained in section 5.4.3, the underestimation of the response at the farthest receiver might be due to the overestimation of the material damping ratios of the deep soil layers. To assess this argument, ground vibrations due to an impact of the MHB are computed for the same soil profile but with smaller material damping ratios of the deep layers. Considering the dependency of the material damping ratio on the mean pressure and the shear strain level, a damping ratio of 0.01 for the shear and dilatational waves in the second layer and a value of 0.005 for deeper layers are assumed. Figure 6.21 compares the vertical velocity in the free field predicted with the modified material damping ratios to the experimental results. It can be observed that the smaller damping ratios of the deep layers do not affect the free field response at 5 m and 10 m while the response at farther receivers significantly increases and overestimates the experimental results. The duration of vibrations is, however, still underestimated. These figures illustrate the sensitivity of the predicted response at large distances to the material damping ratio of deep layers. Thus, for an accurate estimation of ground vibrations at far receivers, the soil profile must be identified very accurately.

6.8

Conclusion

In this chapter, a non-linear model for the prediction of ground vibrations generated by pavement breaking is developed. This model is composed of a FE model for the slab and includes a part of the soil that undergoes plastic

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Figure 6.20: Time history (left) and narrow band frequency spectrum (right) of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road due to an impact of the MHB predicted with the non-linear model (dashed line) and compared with the experimental results (grey line).

CONCLUSION

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deformations and an additional bounded elastic region of the soil coupled to absorbing boundary conditions representing the unbounded soil domain with elastic behaviour. To compute the response outside the FE domain, the tractions and displacements along a path inside the FE domain are computed and introduced in the integral representation formulation discretised on a BE mesh that coincides with the path. The non-linear FE model and the accompanying BE model have been numerically verified with two examples. The inelastic soil behaviour is modelled using the Drucker-Prager yield criterion with a non-associative flow rule. Since the soil is mostly composed of sandy soils, a zero value is assumed for the cohesion. The internal friction angle, dilatancy angle, and unit weight have been estimated from the data of six CPTs near the measurement site. The non-linear model has been used to compute the response of the slab and the soil due to an impact of the MHB. First, two different sizes of the slab are considered. It is found that the size of the slabs does not significantly affect the level of ground vibrations which is in agreement with the experimental results presented in section 4.5.4. The deformed shape of the slab with a diameter equal to the width of the rectangular slab is, however, closer to the observations during the experiments. The results of the non-linear model show that the inelastic behaviour of the soil reduces ground borne vibrations especially at small distances, where the agreement between the results of the non-linear model and the experiment is good (and better than the equivalent linear model). At far receivers, the model underestimates the response which is found to be partially due to the overestimation of the material damping ratios of the deep soil layers. The non-linear model tends to overestimate the amplitude of first arriving P-waves. This shortcoming can be improved by adopting a cap yield model for the soil.

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Figure 6.21: Time history (left) and narrow band frequency spectrum (right) of the vertical velocity at (a) 5 m, (b) 10 m, (c) 25 m, (d) 57 m, and (e) 105 m from the road due to an impact of the MHB predicted with the non-linear model for the soil profile presented in table 4.1 but damping ratios of 0.01 and 0.005 for the second and deeper layers, respectively (dashed line), compared with the experimental results (grey line).

Chapter 7

Conclusions and recommendations for further research 7.1

Conclusions

This thesis addresses the problem of predicting ground vibrations generated by road traffic and pavement breaking. The main objectives of the research are threefold: 1) the prediction of the vibration level due to traffic on deteriorated jointed concrete pavements, 2) the quantification of vibration reduction gained by road rehabilitation, and 3) the evaluation of vibrations generated by pavement breaking. The theoretical studies are complemented by several experiments performed during the rehabilitation of the N9 road in Lovendegem and Waarschoot (Belgium). The findings on the prediction of traffic induced vibrations and vibrations due to pavement breaking are recapitulated separately in the following.

7.1.1


The prediction of ground-borne vibrations generated by road traffic is performed in two stages employing the model developed by Lombaert et al. [162]. In the first stage, the dynamic wheel loads due to the passage of a vehicle over road irregularities are estimated. Second, these loads are applied on a road-soil system to compute the radiated wave field in the soil.

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CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH

The dynamic wheel loads are mainly due to local road irregularities and general unevenness in the wavelength range from 0.1 m to 33 m. The APL is found to be able to measure the required range of wavelengths provided an appropriate profiling speed is chosen. The recorded short wavelength irregularities are, however, strongly filtered by the envelopment of the tyres of the travelling vehicles. In order to account for tyre enveloping in the prediction of dynamic vehicle loads, the common rectangular moving average filter and an alternative method proposed by Zegelaar [264] are evaluated. The latter is found to better represent the response of actual tyres. The unevenness of the deteriorated jointed concrete pavement of the N9 road prior to rehabilitation and its new asphalt pavement after rehabilitation are presented. A 3D vehicle model is developed to simultaneously consider the unevenness of the left and right wheel tracks which can excite the axle roll modes of passing vehicles. These roll modes cannot be considered in the common 2D half-car vehicle model. The 3D model of the test truck is calibrated using the results obtained during the measurement on the deteriorated N9. The accelerations of the axles are computed and compared to the experimental results for the passages over the deteriorated and rehabilitated N9. The predicted accelerations agree relatively well with the experimental results. The non-linear behaviour of the vehicle and shortcomings in the recorded road profile - such as lateral tracking errors and the narrow footprint of the APL - are sources of discrepancies between the predicted and measured response. The dynamic vehicle loads during the passage of the truck over the deteriorated and rehabilitated N9 are estimated. The peak dynamic loads on the deteriorated N9 are found to be due to local road irregularities. These peak loads can be as large as the static load. The dynamic loads on the rehabilitated N9 are much smaller and caused by global road roughness. The DLC for the unladen truck driven on the deteriorated concrete road can be as large as 0.3, while it is less than 0.04 for the same truck driven over the new asphalt road. The DLCs computed with the common 2D half-car vehicle model show an error of about 16%. A 3D model for the simulation of the interaction between a jointed concrete pavement and the soil is presented. The model is composed of a FE model for the slab coupled to a BE model for the layered elastic halfspace. The transfer functions of the 3D model are compared to the transfer functions of the beam model of the road developed by Lombaert et al. [162]. It is observed that the free field mobilities computed with both models are very similar in the frequency range of interest so the road structure does not significantly affect the road-soil transfer functions. Finally, the estimated dynamic wheel loads and the computed road-soil transfer functions are used to predict traffic induced ground vibrations along the N9. The predicted ground vibrations are validated where a relatively good agreement is

CONCLUSIONS

173

observed. The vertical velocities are slightly underestimated at small and large distances from the road while they are slightly overestimated at medium distances. This is attributed to inaccuracies in the estimated dynamic vehicle loads as well as errors in the identified soil parameters. Numerical investigations show that an accurate estimation of the material damping ratio in the soil is necessary for an accurate prediction of ground vibrations. The horizontal vibrations have a similar or even larger amplitude than the vertical vibrations. The PPV generally increases with the vehicle speed. The experimental and numerical results show a large variability in the value of the PPV along the jointed concrete pavement of the deteriorated N9. The level of ground vibrations generated by the passages along the rehabilitated N9 is about 5 to 10 times smaller than the level prior to road rehabilitation. Next to the rehabilitated N9, the induced vibrations are barely perceptible at small distances from the road while at large distances, they hardly exceed the level of background noise. The PPV is mainly due to global road unevenness rather than local irregularities, thus its variation along the road is small. The soil strains under the deteriorated N9 near a large faulted crack in the road surface are estimated numerically. Except for small regions near the edges of the road, the shear strains in the soil are found to be small so that the assumption of linear behaviour is valid. The numerical model for the prediction of traffic induced vibrations can be used by road authorities to evaluate the level of ground vibrations along the roads. This can be done by identifying the road and soil parameters and drawing contour plots of the expected PPV along the roads at different levels of perception threshold, annoyance, and other criteria, so the stretches of the roads which might have problems with traffic induced vibrations can be identified. This is similar to a previous study [148] where the level of ground vibrations has been related to the IRI and other road unevenness indices.

7.1.2


The pavement breaking operation generates a high level of ground-borne vibrations which can be potentially damaging to buildings and surface and underground installations next to the road. The amount of energy dissipated by the fracturing of the pavement is found to be only 10% of the impact energy and, therefore, can be disregarded in the prediction of ground vibrations. The major part of energy is transferred into the soil and causes a relatively high level of ground vibrations. The prediction of ground vibrations generated by a falling-weight pavement breaker is performed in two stages: estimation of the impact load due to the blow of the

174


drop hammer on the road surface and prediction of ground vibrations due to the impact load. For predicting the impact load, advantage is taken from the short duration of impact. Since the stress waves can only propagate up to a small distance during the course of impact, the limited size of the road is disregarded so the road is considered as a layer with infinite lateral extension on the soil surface. Therefore, the road-soil system is simplified to a horizontally layered elastic halfspace. The drop hammer-road-soil interaction problem is solved and the dynamic stiffness of the impact point and, subsequently, the impact load is predicted. The predicted impact force is compared to the experimental result where a good agreement is observed. As a first step in the prediction of ground vibrations generated by pavement breaking, the linear 3D road-soil model that was developed earlier for the prediction of ground vibrations due to traffic on jointed concrete pavements is simplified to an axisymmetric model so the dimension of the problem reduces by one. In the axisymmetric model, the slab and a part of the soil are modelled with the FE method and the rest of the soil is modelled with the BE method. The model is used to compute the soil strains due to the impact of the MHB. Shear strains as large as a few millistrain are found. At such large strains, the soil behaviour is non-linear which resulted in a large difference between the predicted ground vibrations and the experimental results. Therefore, in a second step, the axisymmetric linear model is modified to an equivalent linear model to approximately account for the non-linear soil behaviour at large shear strains. The equivalent linear method is based on the iterative linear analysis where the shear modulus and the material damping ratios are modified after each iteration accounting for the strain level. The results of the equivalent linear model are in a better agreement with the experimental results at small distances while at large distances both the linear and equivalent linear models underestimate the response. This is thought to be caused by the large material damping ratios that have been identified for the deep soil layers as well as assuming deformation independent material damping all over a layer. The results of the equivalent linear model show that the shear strains in the soil exceed 0.01 which is the upper limit up to which equivalent material properties are defined. As a final step, a fully non-linear model is developed. It is composed of a time domain non-linear FE model for the slab and a part of the soil coupled to absorbing boundary conditions. To compute the response outside the FE domain, the tractions and displacements along a path inside the FE domain are computed and introduced in the integral representation formulation. The non-linear model has been used to compute ground vibrations due to an impact of the MHB. The results show that the inelastic behaviour of the soil reduces ground vibrations generated by the impacts of the MHB, especially close to the road. The agreement

RECOMMENDATIONS FOR FURTHER RESEARCH

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between the results of the non-linear model and the experiment is good and is better than for the equivalent linear model at small distances. At large distances, the model still underestimates the response which is found to be partially due to the overestimation of the material damping ratios of the deep soil layers. The nonlinear model tends to overestimate the amplitude of first arriving P-waves. This shortcoming can be improved by introducing a yield cap in the soil model.

7.2

Recommendations for further research

From the findings of the present study recommendations for further research on ground vibrations are given in this section. The present source models for the prediction of ground vibrations generated by road traffic and pavement breaking can be employed to perform parametric studies and further completed by additional experiments to develop horizontal and vertical PPV-distance charts for practical applications and support of guidelines. They can also be coupled to a receiver model to study vibration induced damage to structures. Ground-borne vibrations are potentially damaging when they have very large amplitudes or many repetitions or both. Large amplitude ground vibrations generated during construction activities such as pavement breaking, pile driving, dynamic soil compaction, and demolition of structures can cause damage to nearby structures. The present non-linear source model can be adapted and employed to predict ground vibrations due to these large amplitude impulsive sources. It can be further coupled to a receiver model to study the possibility of vibration induced damage to structures and underground installations and pipelines [190]. The receiver model should be an advanced model that includes a damage model for the structure. The source model can be further improved by introducing a yield cap in the soil model. The repeated dynamic loading of the soil due to traffic results in soil compaction and, consequently, the settlement of adjacent buildings. This phenomenon has been numerically and experimentally studied by Fran¸ois et al. [83] and Karg et al. [132] and should be further investigated by long term monitoring of soil strains next to or under roads. Differential settlement can cause architectural and structural damage to old masonry buildings and heritage monuments. This type of damage should be studied using a receiver model (such as [198]) and complemented with long term monitoring of the response of structures. Further theoretical and experimental studies on vibration induced damage will support the definition of limit values in the guidelines and may explain the discrepancies observed between different norms. In addition, the dynamic loading results in soil settlement under concrete roads and formation of voids under the slabs. This can increase the pumping effect and, consequently, accelerate joint faulting. It can further cause cracking of the pavement due to fatigue or an extra large wheel load

176


as also postulated by Pan et al. [192]. The details of this mode of failure should be studied numerically and experimentally. Traffic induced vibrations can be reduced by changing the road structure and implementation of vibration countermeasures in the transmission path and receiver. A simple change in the structure of jointed concrete pavements that can reduce traffic induced vibrations and at the same time may increase the service life of the pavements is constructing the transverse joints skewed so they are not perpendicular to the road axis anymore. In this way, all wheels of an axle do not cross a faulted joint simultaneously and therefore the dynamic wheel loads reduce. This is in accordance with the recommendation of the Federal Highway Administration [129] for undoweled jointed concrete pavements to provide a smoother ride. The performance of skewed joints and their effect on traffic induced vibrations should be studied both numerically and experimentally. Accurate identification of the soil parameters including both the shear modulus and material damping ratio is essential for the correct prediction of ground vibrations. Therefore, developing more advanced soil investigation techniques - like 2D and 3D seismic interferometry approach [81] - is recommended. These techniques should consider anisotropy of the soil, possible non-horizontal layering, and the dependency of the stiffness and damping parameters on the strain level.

Bibliography [1] International experiment to harmonise longitudinal and transverse profile measurement and reporting procedures. Technical Report 01.07.B, PIARC Technical Committee on Surface Characteristics, 2002. [2] Human response to vibration in residential environments. Technical Report NANR172, Department for Environment, Food and Rural Affairs, UK, March 2007. [3] D. Adam, C. Adam, F.-J. Falkner, and I. Paulmichl. Vibration emission induced by Rapid Impact Compaction. In G. De Roeck, G. Degrande, G. Lombaert, and G. M¨ uller, editors, Proceedings of the 8th International Conference on Structural Dynamics EURODYN 2011, pages 914–921, Leuven, Belgium, July 2011. [4] M.M. Ahmadi and P.K. Robertson. Thin layer effects on the CPT q(c) measurement. Canadian Geotechnical Journal, 42(5):1302–1317, 2005. [5] M.O. Al-Hunaidi, W. Guan, and J. Nicks. Building vibrations and dynamic pavement loads induced by transit buses. Soil Dynamics and Earthquake Engineering, 19(6):435–453, 2000. [6] M.O. Al-Hunaidi, J.H. Rainer, and M. Tremblay. Control of traffic induced vibration in buildings using vehicle suspension systems. Soil Dynamics and Earthquake Engineering, 15(4):245–254, 1996. [7] M.O. Al-Hunaidi and M. Tremblay. Traffic induced building vibrations in Montréal. Canadian Journal of Civil Engineering, 24:736–753, 1997. [8] American Concrete Institute. ACI446.1R-91: Fracture Mechanics of Concrete: Concepts, Models and Determination of Material Properties, 1999. [9] American Concrete Institute. ACI446.4R-04: Report on Dynamic Fracture of Concrete, 2004.

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178

BIBLIOGRAPHY

[10] American National Standards Institute. ANSI S3.29-1983: Guide to the evaluation of human exposure to vibration in buildings, 1983. [11] American Society for Testing and Materials. Standard Terminology Relating to Vehicle-Pavement Systems, 2006. [12] W.H. Ames, W. Chow, A. Sequeira, and R. Johnson. Survey of earth-borne vibrations due to highway construction and highway traffic. Technical Report CA-DOT-TL-6391-76-20, Office of Transportation Laboratory, California Department of Transportation, Sacramento, California, April 1976. [13] ANSYS Inc. Basic Analysis Guide, ANSYS Release 11.0, January 2007. [14] ANSYS Inc. Contact Technology Guide, ANSYS Release 11.0, January 2007. [15] L. Areias. Seismic cone test SCPT1. 249/003 Grote Baan (N9), Lovendegem. Technical report, iGeotechnics, 2008. Report no. iG0608. [16] B.L. Atchley and L.F. Howard. Strength and energy absorption capablities of plain concrete under dynamic and static loadings. ACI Journal, 64:745–756, 1967. [17] D. Aubry, D. Clouteau, and G. Bonnet. Modelling of wave propagation due to fixed or mobile dynamic sources. In N. Chouw and G. Schmid, editors, Workshop Wave ’94, Wave propagation and Reduction of Vibrations, pages 109–121, Ruhr Universität Bochum, Germany, December 1994. [18] L. Auersch. Dynamic plate-soil interaction - finite and infinite, flexible and rigid plates on homogeneous, layered or Winkler soil. Soil Dynamics and Earthquake Engineering, 15(1):51–59, 1996. [19] S.A. Badsar, M. Schevenels, W. Haegeman, and G. Degrande. Determination of the damping ratio in the soil from SASW tests using the half-power bandwidth method. Geophysical Journal International, 182(3):1493–1508, 2010. [20] S.A. Badsar, M. Schevenels, M.A. Lak, and G. Degrande. Determination of the dynamic soil properties with the seismic refraction method and the SASW method at a site in Waarschoot. Technical Report BWM-2009-17, Department of Civil Engineering, KU Leuven, August 2009. [21] N. Banthia, S. Mindess, A. Bentur, and M. Pigeon. Impact testing of concrete using a drop-weight impact machine. Experimental Mechanics, 29(1):63–69, 1989. [22] U. Basu. Explicit finite element perfectly matched layer for transient threedimensional elastic waves. International Journal for Numerical Methods in Engineering, 77:151–176, 2009.

BIBLIOGRAPHY

179

[23] M. Bata. Effects on buildings of vibrations caused by traffic. Building Science, 6:221–246, 1971. [24] C.J. Baughan and D.J. Martin. Vibration nuisance from road traffic at fourteen residential sites. Laboratory report 1020, Transport and Road Research Laboratory, 1981. [25] F. Bautista and I. Basheer. Jointed Plain Concrete Pavement (JPCP) preservation and rehabilitation. Design guide, California Department of Transportation, June 2008. [26] Z.P. Baˇzant and E. Becq-Giraudon. Statistical prediction of fracture parameters of concrete and implications for choice of testing standard. Cement and Concrete Research, 32:529–556, 2002. [27] Z.P. Baˇzant and P.A. Pfeiffer. Shear fracture tests of concrete. Matériaux et Constructions, 19(110):111–121, 1986. [28] Z.P. Baˇzant and P.C. Prat. Measurement of mode III fracture energy of concrete. Nuclear Engineering and Design, 106:1–8, 1988. [29] Z.P. Baˇzant, Q. Yu, and G. Zi. Choice of standard fracture test for concrete and its statistical evaluation. International Journal of Fracture, 118:303–337, 2002. [30] W.R. Bellini and E.N. Thrower. A digital computer program to simulate the passage of a vehicle over a road surface. Laboratory report 181, Transport and Road Research Laboratory, 1968. [31] A. Bentur, S. Mindess, and N.P. Banthia. The fracture of reinforced concrete under impact loading. In S. Mindess and S.P. Shah, editors, Materials Research Society Symposia, volume 64, pages 225–234, 1986. [32] N.D. Beskou and D.D. Theodorakopoulos. Dynamic effects of moving loads on road pavements: A review. Soil Dynamics and Earthquake Engineering, 31(4):547 – 567, 2011. [33] P. Bettess. Infinite elements. International Journal for Numerical Methods in Engineering, 11:53–64, 1977. [34] P. Bettess. Infinite elements. Penshaw Press, Sunderland, United Kingdom, 1992. [35] P.H. Bischoff and S.H. Perry. Compressive behavior of concrete at high-strain rates. Materials and Structures, 24(144):425–450, 1991. [36] M.D. Bolton. The strength and dilatancy of sands. Géotechnique, 36(1):65– 78, 1986.

180

BIBLIOGRAPHY

[37] M.D. Bolton and J.M.R. Wilson. Soil stiffness and damping. In Kratzig, W.B. et al., editor, 1st European Conference on Structural Dynamics (EUROPEAN 90), pages 209–216, Bochum, Germany, 1991. [38] J. Borgions and B. Thyssen. Trillingsisolerende schermen. Master’s thesis, Department of Civil Engineering, KU Leuven, 2008. [39] British Standards Institution. BS 6472:1992: Evaluation of human exposure to vibration in buildings (1 Hz to 80 Hz), 1992. [40] BRRC. APL data, N9 Lovendegem, OCW-nr: 15080030, 2008. [41] BRRC. Beproevingsverslag P.V.CO 8288-3/1540, Lovendegem N9, 2008. [42] BRRC. Joints et fissures vue en plan, N9 Lovendegem, 31/03/2008, 2008. [43] G.N. Bycroft. Impact of a rigid body on an elastic half space. Journal of Applied Mechanics, Transactions of the ASME, 44(2):227–230, 1977. [44] Y. Cai, Z. Cao, H. Sun, and C. Xu. Dynamic response of pavements on poroelastic half-space soil medium to a moving traffic load. Computers and Geotechnics, 36(1-2):52–60, 2009. [45] K.M. Captain, A.B. Boghani, and D.N. Wormley. Analytical tire models for dynamic vehicle simulation. Vehicle System Dynamics, 8:1–32, 1979. [46] D. Cebon. Handbook of vehicle-road interaction. Taylor & Francis, 1999. [47] P. Cenek. Vibration Assessment: Waikato Expressway - Cambridge Section. Central laboratories report 10-261647.00, Opus International Consultants, 2010. [48] K. Chatti, J. Lysmer, and C.L. Monismith. Dynamic finite-element analysis of jointed concrete pavements. Transportation Research Record, 1449:79–90, 1994. [49] C.T. Chatzigogos, A. Pecker, and J. Salen¸con. Macroelement modeling of shallow foundations. Soil Dynamics and Earthquake Engineering, 29(5):765– 781, 2009. [50] A.G. Chehab and M.H. El Naggar. Response of block foundations to impact loads. Journal of Sound and Vibration, 276:293–310, 2004. [51] D.H. Chen, Q.L. Huang, and J.M. Ling. Shanghai’s experience on utilizing the rubblization for jointed concrete pavement rehabilitation. Journal of Performance of Constructed Facilities, 22(6):398–407, 2008. [52] J.W. Chen and C.H. Juang. Determination of drained friction angle of sands from CPT. Journal of Geotechnical Engineering, Proceedings of the ASCE, 122(5):374–381, 1996.

BIBLIOGRAPHY

181

[53] W.F. Chen and E. Mizuno. Nonlinear analysis in soil mechanics. Elsevier, 1990. [54] S. Chiostrini, F. Lancieri, A. Marradi, M. Nocentini, and A. Vignoli. Numerical evaluation and experimental measurements of traffic-induced vibrations. In G.M. Carlomagno and C.A. Brebbia, editors, Computational Methods and Experimental Measurements VII, pages 469–476, Italy, 1995. [55] Y.K. Chow, D.M. Yong, K.Y. Yong, and S.L. Lee. Monitoring of dynamic compaction by deceleration measurements. Computers and Geotechnics, 10(3):189–209, 1990. [56] J.-P. Christory and J.-L. Nissoux. Evaluation and maintenance of concrete pavements. Technical Report 07.04.B, PIARC Technical Committee on Concrete Roads, 1992. [57] P. Clemente and D. Rinaldis. Protection of a monumental building against traffic-induced vibrations. Soil Dynamics and Earthquake Engineering, 17(5):289–296, 1998. [58] D. Clouteau and D. Aubry. Computational soil-structure interaction. pages 61–125. Kluwer, 2003. [59] D.J. Cole. Fundamental issues in suspension design for heavy road vehicles. Vehicle System Dynamics, 35(4/5):319–360, 2001. [60] Comité Euro-International du Béton. CEB No 187: Concrete structures under impact and impulsive loading, 1988. [61] Comité Euro-International du Béton. CEB-FIP Model Code, 1998. [62] Vectra Company. APL Continuous measurement of the longitudinal profile. Web page on the Vectra company web site. http://www.vectra.fr/sites/ en/equipments-and-solutions/pavement-monitoring/apl.htmll. Accessed 20 October 2012. [63] P.A. Costa, R. Cal¸cada, A.S. Cardoso, and A. Bodare. Influence of soil non-linearity on the dynamic response of high-speed railway tracks. Soil Dynamics and Earthquake Engineering, 30(4):221–235, 2010. [64] F. Covemaeker and A.-C. Kiekens. Trillingen ten gevolge van verkeer op een wegdek met kasseien. Master’s thesis, Department of Civil Engineering, KU Leuven, 2007. [65] M. Crispino and M. D’Appuzzo. Measurement and prediction of trafficinduced vibrations in a heritage building. Journal of Sound and Vibration, 246(2):319–335, 2001.

182

BIBLIOGRAPHY

[66] B.M. Das and G.V. Ramana. Principles of soil dynamics. Cengage Learning, 2011. [67] A.J. Deeks and M.F. Randolph. Analytical modeling of hammer impact for pile driving. International Journal for Numerical and Analytical Methods in Geomechanics, 17:279–302, 1993. [68] Y. Delanne and P.A.A. Pereira. Advantages and limits of different road roughness profile signal-processing procedures applied in Europe. Transportation Research Record, 1764:254–259, 2001. [69] A. Deraemaeker, I. Babuˇska, and P. Bouillard. Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions. International Journal for Numerical Methods in Engineering, 46(4):471–499, 1999. [70] Deutsches Institut f¨ ur Normung. DIN 4150 Teil 3: Ersch¨ utterungen im Bauwesen, Einwirkungen auf bauliche Anlagen, 1999. [71] R.S. Dickerson and D.G.W. Mace. Dynamic pavement force measurements with a two-axle heavy goods vehicle. Supplementary report 688, Transport and Road Research Laboratory, 1981. [72] C.J. Dodds and J.D. Robson. Simulation of dynamic stresses in vehicles and components. Vehicle System Dynamics, 4(2-3):130–134, 1975. [73] C.H. Dowding. Construction vibrations. Prentice-Hall, New Jersey, second edition, 2000. [74] H.T. Durgunoglu and J.K. Mitchell. Static penetration resistance of soils: I - Analysis. In Proceedings of ASCE Specialty Conference on In-Situ Measurement of Soil Parameters, pages 151–171, New York, 1975. [75] European Committee for Standardization. ENV1992-1-1:1991 Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings, 1991. [76] European Committee for Standardization. EN1997-2:2007 Eurocode 7: Geotechnical design - Part 2: Ground investigation and testing, 2007. [77] P.S. Fancher, R.D. Ervin, C.B. Winkler, and T.D. Gillespie. A factbook of the mechanical properties of the components for single-unit and articulated heavy trucks. Technical Report UMTRI 86-12, University of Michigan Transportation Research Institute, December 1986. [78] R.H. Ferahian and W.D. Hurst. Vibration and possible building damage due to operation of construction machinery. Technical Report 399, National Research Council of Canada, April 1969.

BIBLIOGRAPHY

183

[79] A. Forsén. Road-induced longitudinal wheel forces in heavy vehicles. SAE Transactions, 1997. Paper No. 973260. [80] A. Forsén. Heavy vehicle ride and endurance - modelling and model validation. PhD thesis, Department of Vehicle Engineering, Royal Institute of Technology, Sweden, 1999. [81] S. Foti, S. Parolai, D. Albarello, and M. Picozzi. Application of surface-wave methods for seismic site characterization. Surveys in Geophysics, 32(6):777– 825, 2011. [82] S. Fran¸cois. Nonlinear modelling of the response of structures due to ground vibrations. PhD thesis, Department of Civil Engineering, KU Leuven, 2008. [83] S. Fran¸cois, C. Karg, W. Haegeman, and G. Degrande. A numerical model for foundation settlements due to deformation accumulation in granular soils under repeated small amplitude dynamic loading. International Journal for Numerical and Analytical Methods in Geomechanics, 34(3):273–296, 2010. [84] S. Fran¸cois, M. Schevenels, and G. Degrande. BEMFUN: MATLAB toolbox for boundary elements in elastodynamics. Version 2.1 Build 16. User’s guide BWM-2009-26, Department of Civil Engineering, KU Leuven, December 2009. [85] S. Fran¸cois, M. Schevenels, G. Lombaert, P. Galv´ın, and G. Degrande. A 2.5D coupled FE-BE methodology for the dynamic interaction between longitudinally invariant structures and a layered halfspace. Computer Methods in Applied Mechanics and Engineering, 199(23-24):1536–1548, 2010. [86] T.E. Freeman. Evaluation of concrete slab fracturing techniques in mitigating reflective cracking through asphalt overlays. Technical Report VTRC 03-R3, Virginia Transportation Research Council, September 2002. [87] F. Fuchs. Achieving and maintaining the evenness of concrete pavements. Technical Report 07.03.B, PIARC Technical Committee on Concrete Roads, 1991. [88] Fugro Roadware Company. ARAN Subsystems. Web page on the Fugro Roadware web site. http://www.roadware.com/products/ARAN-Subsystems/. Accessed 19 April 2012. [89] L. Gavrić. Finite element computation of dispersion properties of thin-walled waveguides. Journal of Sound and Vibration, 173(1):113–124, 1994. [90] G. Gazetas and M. Apostolou. Nonlinear soil-structure interaction: Foundation uplifting and soil yielding. In M. Celebi, M.I. Todorovska, I. Okawa, and M. Iiba, editors, Third UJNR Workshop on Soil-Structure Interaction, California, USA, 2004.

184

BIBLIOGRAPHY

[91] G. Gazetas and E.T Selig, editors. Vibration problems in geotechnical engineering. ASCE, New York, USA, 1985. [92] T.D. Gillespie and S.M. Karamihas. Simplified models for truck dynamic response to road input. International Journal of Vehicle Design, Heavy Vehicle Systems, 7(2-3):231–247, 2000. [93] T.D. Gillespie, S.M. Karamihas, D. Cebon, M.W. Sayers, M.A. Nasim, W. Hansen, and N. Ehsan. Effects of heavy vehicle characteristics on pavement response and performance. Technical report, University of Michigan Transportation Research Institute, 1992. Project 1-25(1). [94] T.D. Gillespie and M.W. Sayers. The effect of suspension system nonlinearities on heavy truck vibration. Vehicle System Dynamics, 10(23):97–102, 1981. [95] T.D. Gillespie, M.W. Sayers, and L. Segel. Calibration and correlation of response-type road roughness measuring systems. Technical report, Highway Safety Research Institute, The University of Michigan, September 1980. UMHSRI-80-30. [96] C.G. Gordon. Generic criteria for vibration-sensitive equipment. In Proceedings of the SPIE Conference on Vibration Control and Metrology, volume 1619, pages 71–85, San Jose, California, USA, November 1991. [97] Z.K. Guo, A.S. Kobayashi, and N.M. Hawkins. Dynamic mixed mode fracture of concrete. International Journal of Solids and Structures, 32(17):2591–2607, 1995. [98] J.J. Hajek, C.T. Blaney, and D.K. Hein. Mitigation of highway trafficinduced vibration. In Annual Conference of the Transportation Association of Canada, Charlottetown, 2006. [99] A.A. Halabian and M.H. El Naggar. Effect of non-linear soil-structure interaction on seismic response of tall slender structures. Soil Dynamics and Earthquake Engineering, 22:639–658, 2002. [100] K. Hall, D. Dawood, S. Vanikar, and et al. Long-life concrete pavements in Europe and Canada. Technical Report FHWA-PL-07-027, U.S. Department of Transportation, Federal Highway Administration, AASHTO, and NCHRP, August 2007. [101] T. Hanazato. Ground vibrations produced during construction of pile foundations. In Y.M. Chen and H. Takemiya, editors, International Seminar on Environmental Vibration: Prediction, Monitoring and Evaluation, pages 186–201, Hangzhou, China, 2003.

BIBLIOGRAPHY

185

[102] T. Hanazato, K. Ugai, M. Mori, and R. Sakaguchi. Three-dimensional analysis of traffic-induced ground vibrations. Journal of Geotechnical Engineering, Proceedings of the ASCE, 117(8):1133–1151, 1991. [103] H. Hao and T.C. Ang. Analytical modelling of traffic-induced ground vibrations. Journal of the Engineering Mechanics Division, Proceedings of the ASCE, 124(8):921–928, 1998. [104] H. Hao, N. Chouw, and A.J. Deeks. Theoretical prediction of traffic-induced ground vibration with road filtering effect. In Y.B. Yang and J.D. Yau, editors, International Symposium on Environment Vibrations: Prediction, Monitoring, Mitigation and Evaluation. ISEV 2007, pages 115–122, Taiwan, 2007. [105] H. Hao, A.J. Deeks, and T.W.N. Yau. Measurement and analysis of construction activity-induced ground vibration around Perth western Australia. In Y.B. Yang and J.D. Yau, editors, International Symposium on Environment Vibrations: Prediction, Monitoring, Mitigation and Evaluation. ISEV 2007, pages 492–497, Taiwan, 2007. [106] M.S.A. Hardy and D. Cebon. Response of continuous pavements to moving dynamic loads. Journal of the Engineering Mechanics Division, Proceedings of the ASCE, 119(9):1762–1780, 1993. [107] Y.A. Hegazy and P.W. Mayne. Statistical correlations between Vs and CPT data for different soil types. In International symposium on Penetration Testing, CPT95, volume 2, pages 173–178, Linköping, Sweden, 1995. [108] R. Hendriks. Transportation related earthborne vibrations (Caltrans experiences). Technical Report TAV-04-01-R0201, California Department of Transportation, Division of Environmental Analysis, Office of Noise and Hazardous Waste Management, January 2004. [109] A. Hillerborg. Concrete fracture energy tests performed by 9 laboratories according to a draft RILEM recommendation. Technical report, Lund Institute of Technology, 1983. Report TVBM-3015. [110] S. H¨ oller and H. Kotze. Results of the survey on maintenance of concrete roads. Technical report, PIARC Technical Committee D.2.c Concrete Roads, 2011. Issue 2: Improved Maintenance Methods. [111] G.T. Houlsby. How the dilatancy of soils affects their behaviour. In 10th European Conference on Soil Mechanics and Foundation Engineering, volume 4, pages 1189–1202, Florence, Italy, May 1991. [112] M.E. House. Traffic induced vibrations in buildings. The Highway Engineer, February:6–15, 1973.

186

BIBLIOGRAPHY

[113] B.P. Hughes and H. Al-Dafiry. Impact energy absorption at contact zone and supports of reinforced plain and fibrous concrete beams. Construction and Building Materials, 9(4):239–244, 1995. [114] J.L. Humar. Dynamics of structures. Taylor & Francis, 2002. [115] H.-H. Hung and Y.-B. Yang. Elastic waves in visco-elastic half-space generated by various vehicle loads. Soil Dynamics and Earthquake Engineering, 21:1–17, 2001. [116] H.E.M. Hunt. Stochastic modelling of traffic-induced ground vibration. Journal of Sound and Vibration, 144(1):53–70, 1991. [117] J.H. Hwang and T.Y. Tu. Ground vibration due to dynamic compaction. Soil Dynamics and Earthquake Engineering, 26:337–346, 2006. [118] R.N. Hwang and J. Lysmer. Response of buried structures to traveling waves. Journal of Geotechnical and Geoenvironmental Engineering, Proceedings of the ASCE, 107(GT2):183–200, 1981. [119] J.H. Hyde and H.R. Lintern. The vibrations of roads and structures. In Minutes of proceedings of the institution of civil engineers, pages 187–243, 1929. [120] F. Ihlenburg and I. Babuˇska. Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation. International Journal for Numerical Methods in Engineering, 38(22):3745–3774, 1995. [121] Antigo Construction Incorporation. T8600 Badger Breaker®. Web page on the Antigo Construction web site. http://www.antigoconstruction.com/ specs-t8600.html. Accessed 26 March 2012. [122] International Organization for Standardization. ISO 2631-1:1985: Evaluation of human body exposure to whole-body vibration - Part 1: General requirements, 1985. [123] International Organization for Standardization. ISO 2631-2:1989: Evaluation of human body exposure to whole-body vibration - Part 2: Continuous and shock-induced vibration in buildings (1 to 80 Hz), 1989. [124] International Organization for Standardization. ISO 4866:1990: Mechanical vibration and shock - Vibration of buildings - Guidelines for the measurement of vibrations and evaluation of their effects on buildings, 1990. [125] International Organization for Standardization. ISO 8608:1991: Mechanical vibration - Road surface profiles - Reporting of measured data, 1991.

BIBLIOGRAPHY

187

[126] M.B. Jaksa, M.C. Griffith, and R.W. Grounds. Ground vibrations associated with installing enlarged-base driven cast-in-situ piles. Australian Geomechanics, 37:67–73, 2002. [127] M. Jamiolkowski, C.C. Ladd, J.T. Germaine, and R. Lancellotta. New developments in field and laboratory testing of soils. In Eleventh International Conference of Soil Mechanics and Foundation Engineering ICSMFE, volume 1, pages 57–153, San Francisco, 1985. [128] S.-H. Ju. Finite element investigation of traffic induced vibrations. Journal of Sound and Vibration, 321(3–5):837–853, 2009. [129] A.R. Kane. Concrete pavement joints. Technical Report Technical Advisory 5040.30, Federal Highway Administration, November 1990. [130] S.M. Karamihas. Critical profiler accuracy requirements. Technical report, University of Michigan Transportation Research Institute, August 2005. [131] S.M. Karamihas, T.D. Gillespie, and S.M. Riley. Axle tramp contributions to the dynamic wheel loads of a heavy truck. In C.B. Winkler, editor, Proceedings of the Fourth International Symposium on Heavy Vehicle Weights and Dimensions, pages 425–434, Ann Arbor, MI, 1995. [132] C. Karg, S. Fran¸cois, W. Haegeman, and G. Degrande. Elasto-plastic longterm behavior of granular soils: modeling and experimental validation. Soil Dynamics and Earthquake Engineering, 30:635–646, 2010. [133] L. Karl. Dynamic soil properties out of SCPT and bender element tests with emphasis on material damping. PhD thesis, Universiteit Gent, 2005. [134] E. Kausel. Fundamental solutions in elastodynamics: Cambridge University Press, New York, 2006.

a compendium.

[135] E. Kausel and J.M. Roësset. Stiffness matrices for layered soils. Bulletin of the Seismological Society of America, 71(6):1743–1761, 1981. [136] D.S. Kim and J.S. Lee. Propagation and attenuation characteristics of various ground vibrations. Soil Dynamics and Earthquake Engineering, 19:115–126, 2000. [137] S.-M. Kim, J.M. Roësset, and K.H. Stokoe. Numerical simulation of rolling dynamic deflectometer tests. Journal of Transportation Engineering, Proceedings of the ASCE, 125(2):85–92, 1999. [138] W. Konon and J.R. Schuring. Vibration criteria for historic buildings. Journal of Construction Engineering and Management, 111(3):208–215, 1985.

188

BIBLIOGRAPHY

[139] V.V. Krylov. Computation of ground vibrations generated by accelerating and braking vehicles. Journal of Vibration and Control, 2:299–321, 1996. [140] V.V. Krylov. Generation of ground elastic waves by road vehicles. Journal of Computational Acoustics, 9(3):919–933, 2001. [141] K. Ksaibati, W. Miley, and J. Armaghani. Rubbilization of concrete pavements. Technical Report FL/DOT/SMO/98-426, Florida Deparment of Transportation, August 1998. [142] S. Kucukcoban and L.F. Kallivokas. A mixed perfectly-matched-layer for transient wave simulations in axisymmetric elastic media. Computer Modeling in Engineering and Sciences, 64(2):109–145, 2010. [143] R.L. Kuhlemeyer and J. Lysmer. Finite element method accuracy for wave propagation problems. Journal of the Soil Mechanics and Foundation Division, Proceedings of the ASCE, 99(SM5):421–427, 1973. [144] A.R. Kukreti, M.R. Taheri, and R.H. Ledesma. Generation of ground elastic waves by road vehicles. Journal of Transportation Engineering, Proceedings of the ASCE, 118(3):341–360, 1992. [145] F.H. Kulhawy and P.W. Mayne. Manual on estimating soil properties for foundation design. Technical Report EL-6800, Electric Power Research Institute, August 1990. [146] S. Kunimatsu and Y. Hirao. Vibration characteristics of house in Japan by 1/3 oct band frequency to environmental vibration estimation. In H. Xia and H. Takemiya, editors, Environment Vibrations: Prediction, Monitoring, Mitigation and Evaluation, pages 134–140, Beijing, China, 2009. [147] M.A. Lak, G. Degrande, and G. Lombaert. Free field vibrations due to traffic and the operation of a multi-head breaker on a concrete road. In H. Xia and H. Takemiya, editors, Environmental vibrations: Prediction, Monitoring, Mitigation and Evaluation, Volume I, pages 47–55, Beijing, China, October 2009. Science Press. [148] M.A. Lak, G. Degrande, and G. Lombaert. The effect of road unevenness on the dynamic vehicle response and ground-borne vibrations due to road traffic. Soil Dynamics and Earthquake Engineering, 31(10):1357–1377, 2011. [149] M.A. Lak, S. Fran¸cois, G. Degrande, and G. Lombaert. Experimental validation of a numerical model for the prediction of traffic induced ground vibrations. Report BWM-2010-10, Department of Civil Engineering, KU Leuven, July 2010. IWT-Project VIS-CO 060884.

BIBLIOGRAPHY

189

[150] M.A. Lak, G. Lombaert, S.A. Badsar, E. Reynders, and G. Degrande. Simultaneous vehicle and free field response measurement during the passage of a truck before and after road renovation. Report BWM-2008-15, Department of Civil Engineering, KU Leuven, October 2008. IWT-Project VIS-CO 060884. [151] M.A. Lak, G. Lombaert, S.A. Badsar, M. Schevenels, and G. Degrande. Determination of the dynamic soil characteristics with the SASW method at a site in Lovendegem. Report BWM-2008-11, Department of Civil Engineering, KU Leuven, September 2008. IWT-Project VIS-CO 060884. [152] M.A. Lak, G. Lombaert, H. Verbraken, and G. Degrande. Vibration measurements during the operation of a multi-head breaker on a concrete road at a site in Waarschoot. Technical Report BWM-2009-14, Department of Civil Engineering, KU Leuven, August 2009. [153] M.A. Lak, E. Reynders, G. Lombaert, and G. Degrande. Vibration measurements during the cracking of concrete slabs by means of a multihead breaker. Report BWM-2008-16, Department of Civil Engineering, KU Leuven, October 2008. IWT-Project VIS-CO 060884. [154] D.E. Lambert and C.A. Ross. Strain rate effects on dynamic fracture and strength. International Journal of Impact Engineering, 24:985–998, 2000. [155] M. Larcher. Development of discrete cracks in concrete loaded by shock waves. International Journal of Impact Engineering, 36:700–710, 2009. [156] J. Lee, J. Eun, K. Lee, Y. Park, and M. Kim. In-situ evaluation of strength and dilatancy of sands based on CPT results. Soils and Foundations, 48(2):255–265, 2008. [157] Q.M. Li, S.R. Reid, and A.M. Ahmad-Zaidi. Critical impact energies for scabbing and perforation of concrete target. Nuclear Engineering and Design, 236:1140–1148, 2006. [158] Q.H. Liu. Perfectly matched layers for elastic waves in cylindrical and spherical coordinates. Journal of the Acoustical Society of America, 105(4):2075–2084, 1999. [159] G. Lombaert. Development and experimental validation of a numerical model for the free field vibrations induced by road traffic. PhD thesis, Department of Civil Engineering, KU Leuven, 2001. [160] G. Lombaert and G. Degrande. Experimental validation of a numerical prediction model for free field traffic induced vibrations by in situ experiments. Soil Dynamics and Earthquake Engineering, 21(6):485–497, 2001.

190

BIBLIOGRAPHY

[161] G. Lombaert and G. Degrande. The experimental validation of a numerical model for the prediction of the vibrations in the free field produced by road traffic. Journal of Sound and Vibration, 262:309–331, 2003. [162] G. Lombaert, G. Degrande, and D. Clouteau. Numerical modelling of free field traffic induced vibrations. Soil Dynamics and Earthquake Engineering, 19(7):473–488, 2000. [163] G. Lombaert, G. Degrande, and D. Clouteau. The influence of the soil stratification on free field traffic induced vibrations. Archive of Applied Mechanics, 71(10):661–678, 2001. [164] G. Lombaert, M. Schevenels, G. Degrande, F. Covemaekers, and A.-C. Kiekens. Invloed van de langsvlakheid van de weg op de dynamische voertuigrespons en de trillingen in de omgeving. OCW Mededelingen, (73):18–20, 2007. [165] J. Lucas and A. Viano. Systematic measurement of evenness on the road network: High output longitudinal profile analyser. Technical Report 101, Laboratoire Central des Ponts et Chaussees, 1979. [166] P. Lugner, H. Pacejka, and M. Plöchl. Recent advances in tyre models and testing procedures. Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 43(6):413–426, 2005. [167] J. Lysmer and R.L. Kuhlemeyer. Finite dynamic model for infinite media. Journal of the Engineering Mechanics Division, Proceedings of the ASCE, 95(EM4):859–877, 1969. [168] Resonant Machines. Rubblizing. Web page on the Resonant Machines web site. http://www.resonantmachines.com/Rubblizing/tabid/207/Default.aspx. Accessed 26 March 2012. [169] M.S. Mamlouk and T.G. Davies. Elasto-dynamic analysis of pavement deflections. Journal of Transportation Engineering, Proceedings of the ASCE, 110(6):536–550, 1984. [170] V.J. Marks and C. Anderson. Crack and seat PCC pavement prior to resurfacing, US 59 - Shelby county. Final Report Project HR-527, Iowa Department of Transportation, Highway division, Office of materials, 1993. [171] D.J. Martin. Ground vibrations caused by road construction operations. Supplementary report 328, Transport and Road Research Laboratory, 1977. [172] C.J. Martins, J.A.M. Carrer, W.J. Mansur, and F.C. Ara´ ujo. On the use of pseudo-forces to consider initial conditions in 3D time- and frequencydomain acoustic analysis. Computer Methods in Applied Mechanics and Engineering, 195:4371–4382, 2006.

BIBLIOGRAPHY

191

[173] H.R. Masoumi. Numerical modeling of free field vibrations due to pile driving. PhD thesis, Department of Civil Engineering, KU Leuven, 2008. [174] H.R. Masoumi, G. Degrande, and G. Lombaert. Prediction of free field vibrations due to pile driving using a dynamic soil-structure interaction formulation. Soil Dynamics and Earthquake Engineering, 27(2):126–143, 2007. [175] H.R. Masoumi, S. Fran¸cois, and G. Degrande. A non-linear coupled finite element-boundary element model for the prediction of vibrations due to vibratory and impact pile driving. International Journal for Numerical and Analytical Methods in Geomechanics, 33(2):245–274, 2009. [176] K.R. Massarsch. Ground vibrations caused by soil compaction. In N. Chouw and G. Schmid, editors, Proceedings of the 3rd International Workshop Wave 2002, Wave propagation, Moving load, Vibration reduction, pages 25–37, Okayama University, Japan, September 2002. A.A. Balkema. [177] The MathWorks. MATLAB Optimization Toolbox User’s Guide, 2005. [178] P.W. Mayne. Cone penetration testing state-of-practice. Technical report, Transportation Research Board Synthesis Study, February 2007. NCHRP Project 20-05. [179] P.W. Mayne, J.S. Jones, and J.C. Dumas. Ground response to dynamic compaction. Journal of Geotechnical Engineering, Proceedings of the ASCE, 110(6):757–774, 1984. [180] G.G. Meyerhof. The ultimate bearing capacity of wedge-shaped foundations. In Fifth International Conference of Soil Mechanics and Foundation Engineering ICSMFE, volume 2, pages 103–109, Paris, 1961. [181] M. Mhanna, M. Sadek, and I. Shahrour. Numerical modeling of trafficinduced ground vibration. Computers and Geotechnics, 39:116–123, 2012. [182] S. Mindess and A. Bentur. A preliminary study of the fracture of concrete beams under impact loading, using high speed photography. Cement and Concrete Research, 15(3):474–484, 1985. [183] C.G.B. Mitchell and L. Gyenes. Dynamic pavement loads measured for a variety of truck suspensions. In Proceedings of 2nd International Conference on Heavy Vehicle Weights and Dimensions, Kelowna, British Columbia, 1989. [184] V. Muluka. Optimal suspension damping and axle vibration absorber for reduction of dynamic tire loads. Master’s thesis, Department of Mechanical Engineering, Concordia University, Montréal, Quebec, Canada, 1998.

192

BIBLIOGRAPHY

[185] M.A. Nasim, S.M. Karamihas, T.D. Gillespie, W. Hansen, and D. Cebon. The behaviour of a rigid pavement under moving dynamic loads. Transportation Research Record TRB, 1307:129–135, 1991. [186] B.M. New. Ground vibration caused by civil engineering works. Research report 53, Transport and Road Research Laboratory, 1986. [187] B.M. New. Ground vibration caused by construction works. Tunnelling and Underground Space Technology, 5(3):179–190, 1990. [188] Databank Ondergrond Vlaanderen. Geotechniek. Web page on the Database of the Subsoil of Flanders web site. https://dov.vlaanderen.be/dovweb/html/ geotechniek.html. Accessed April 2010. [189] G.A. Ordón ˜ ez. SHAKE2000. A computer program for the 1-D analysis of geotechnical earthquake engineering problems. User’s manual. EERC, University of California, Berkeley, 2008. [190] Z. Ozdemir, P. Coulier, M.A. Lak, S. Fran¸cois, G. Lombaert, and G. Degrande. Numerical evaluation of the dynamic response of pipelines to vibrations induced by the operation of a pavement breaker. Soil Dynamics and Earthquake Engineering, 44:153–167, 2013. [191] H.B. Pacejka. Tyre and vehicle dynamics. Society of Automotive Engineers Inc, 2002. [192] G. Pan, H. Okada, and S.N. Atluri. Nonlinear transient dynamic analysis of soil-pavement interaction under moving load: a coupled BEM-FEM approach. Engineering Analysis with Boundary Elements, 14(1):99–112, 1994. [193] R. Paolucci and R. Figini. Non-linear dynamic soil-structure interaction: application to seismic analysis of structures on shallow foundations. In G. De Roeck, G. Degrande, G. Lombaert, and G. M¨ uller, editors, Proceedings of the 8th International Conference on Structural Dynamics EURODYN 2011, pages 164–169, Leuven, Belgium, July 2011. [194] G.D. Pavlatos and D.E. Beskos. Dynamic elastoplastic analysis by BEM/FEM. Engineering Analysis with Boundary Elements, 14(1):51–63, 1994. [195] G.D. Pavlatos and D.E. Beskos. Dynamic inelastic soil-structure interaction using a hybrid BEM/FEM scheme. In H.A. Mang and F.G. Rammerstorfer, editors, IUTAM Symposium on Discretization Methods in Structural Mechanics, pages 233–240, Dordrecht, 1999. Kluwer Academic Publishers.

BIBLIOGRAPHY

193

[196] M.G. Pottinger, K.D. Marshall, J.M. Lawther, and D.B. Thrasher. A review of tire/pavement interaction induced noise and vibration. In M.G. Pottinger and T.J. Yager, editors, The tire pavement interface, ASTP STP 929, pages 183–287, Philadelphia, 1986. American Society for Testing and Materials. [197] L. Pyl. Development and experimental validation of a numerical model for traffic induced vibrations in buildings. PhD thesis, Department of Civil Engineering, KU Leuven, 2004. [198] L. Pyl, G. Degrande, and D. Clouteau. Validation of a source-receiver model for road traffic induced vibrations in buildings. II: Receiver model. ASCE Journal of Engineering Mechanics, 130(12):1394–1406, 2004. [199] L. Pyl, G. Degrande, G. Lombaert, and W. Haegeman. Validation of a sourcereceiver model for road traffic induced vibrations in buildings. I: Source model. ASCE Journal of Engineering Mechanics, 130(12):1377–1393, 2004. [200] J.H. Rainer. Effect of vibrations on historic buildings: an overview. Technical Report 1091, National Research Council Canada, 1982. [201] S. Rao, H.T. Yu, and M.I. Darter. The longevity and performance of diamond-ground pavements. PCA Research and Development Bulletin RD118, 1999. [202] P.K. Robertson. Soil classification using the cone penetration test. Canadian Geotechnical Journal, 27(1):151–158, 1990. [203] P.K. Robertson. Interpretation of cone penetration tests – a unified approach. Canadian Geotechnical Journal, 46(11):1337–1355, 2009. [204] P.K. Robertson and K.L. Cabal (Robertson). Guide to Cone Penetration Testing for Geotechnical Engineering. Technical report, Gregg Drilling & Testing Inc., July 2010. [205] P.K. Robertson and R.G. Campanella. Interpretation of cone penetration tests - Part I (Sand). Canadian Geotechnical Journal, 20(4):718–733, 1983. [206] J.M. Roësset, E. Kausel, V. Cuellar, J.L. Monte, and J. Valerio. Impact of weight falling onto the ground. Journal of Geotechnical Engineering, Proceedings of the ASCE, 120(8):1394–1412, 1994. [207] C.A. Ross, J.W. Tedesco, and S.T. Kuennen. Effects of strain-rate on concrete strength. ACI Materials Journal, 92:37–47, 1995. [208] P. Rossi. A physical phenomenon which can explain the mechanical-behavior of concrete under high-strain rates. Materials and Structures, 24:422–424, 1991.

194

BIBLIOGRAPHY

[209] V. Rouillard, R.A. Sek, and T. Perry. Analysis and simulation of road profiles. Journal of Transportation Engineering, 122(3):241–245, 1996. [210] F.F. Rudder. Engineering guidelines for the analysis of traffic-induced vibration. Technical Report FHWA-RD-78-166, U.S. Department of Transportation, Washington, 1978. [211] G. Ruiz, X.X. Zhang, R.C. Yu, E. Poveda, R. Porras, and J. Del Viso. Effect of loading rate on the fracture behaviour of high-strength concrete. Applied Mechanics and Materials, 24:179–185, 2010. [212] M.W. Sayers. Dynamic terrain inputs to predict structural integrity of ground vehicles. Technical Report UMTRI 88-16, University of Michigan Transportation Research Institute, April 1988. [213] M.W. Sayers and T.D. Gillespie. The Ann Arbor Road Profilometer Meeting. Technical Report UMTRI 86-19, University of Michigan Transportation Research Institute, April 1986. [214] M.W. Sayers, T.D. Gillespie, and C.A.V. Queiroz. The international road roughness experiment. Technical Report 45, World Bank, 1986. [215] M.W. Sayers and S.M. Karamihas. Interpretation of road roughness profile data. Technical Report UMTRI 96-19, University of Michigan Transportation Research Institute, June 1996. [216] M.W. Sayers and S.M. Karamihas. The little book of profiling: basic information about measuring and interpreting road profiles. University of Michigan Transportation Research Institute, 1998. [217] H.A. Schenck. Improved integral formulation for acoustic radiation problems. Journal of the Acoustical Society of America, 44:41–58, 1968. [218] M. Schevenels, S.A. Badsar, G. Lombaert, and G. Degrande. The robustness of ground vibration predictions based on SASW tests. In Proceedings of Leuven Symposium on Applied Mechanics in Engineering, Leuven, Belgium, March 2008. CD-ROM. [219] M. Schevenels, G. Degrande, and S. Fran¸cois. EDT: Elastodynamics Toolbox for MATLAB. Version 2.2 Build 20. User’s guide BWM-2010-11, Department of Civil Engineering, KU Leuven, August 2010. [220] M. Schevenels, G. Degrande, and G. Lombaert. The influence of the depth of the ground water table on free field road traffic induced vibrations. International Journal for Numerical and Analytical Methods in Geomechanics, 28(5):395–419, 2004.

BIBLIOGRAPHY

195

[221] M. Schevenels, S. Fran¸cois, and G. Degrande. EDT: An ElastoDynamics Toolbox for MATLAB. Computers & Geosciences, 35(8):1752–1754, 2009. [222] A.M. Schutzbach. The crack and seat method of pavement rehabilitation. Technical Report 104, Illinois Department of Transportation, 1988. [223] H.B. Seed and I.M. Idriss. Soil moduli and damping factors for dynamic response analyses. Report No. EERC-70/10, Earthquake Engineering Research Center, University of California at Berkeley, 1970. [224] H.B. Seed, R.T. Wong, I.M. Idriss, and K. Tokimatsu. Moduli and damping factors for dynamic analyses of cohesionless soils. Journal of the Soil Mechanics and Foundation Division, Proceedings of the ASCE, 112(11):1016– 1032, 1986. [225] J.-F. Semblat and J.-J. Brioist. Efficiency of higher order finite elements for the analysis of seismic wave propagation. Journal of Sound and Vibration, 231(2):460–467, 2000. [226] J.-F. Semblat, L. Lenti, and A. Gandomzadeh. A simple multi-directional absorbing layer method to simulate elastic wave propagation in unbounded domains. International Journal for Numerical Methods in Engineering, 85(12):1543–1563, 2011. [227] J.-F. Semblat and A. Pecker. Waves and vibrations in soils: earthquakes, traffic, shocks, construction works. IUSS Press, 2009. [228] S.P. Shah and R. John. Rate-sensitivity of mode I and mode II fracture of concrete. In S. Mindess and S.P. Shah, editors, Materials Research Society Symposia, volume 64, pages 21–38, 1986. [229] J.C. Simo and T.A. Laursen. An augmented Lagrangian treatment of contact problems involving friction. Computers and Structures, 42(1):97–116, 1992. [230] B.O. Skipp, editor. Ground dynamics and man made processes. Thomas Telford, 1998. [231] M. Srbulov. Ground vibration engineering. Springer, 2010. [232] Stichting Bouwresearch. SBR deel A: Schade aan gebouwen door trillingen: meet- en beoordelingsrichtlijn, 2002. [233] Stichting Bouwresearch. SBR deel B: Hinder voor personen in gebouwen door trillingen: meet- en beoordelingsrichtlijn, 2002. [234] Stichting Bouwresearch. SBR deel C: Storing aan apparatuur door trillingen: meet- en beoordelingsrichtlijn, 2002.

196

BIBLIOGRAPHY

[235] Jones & Stokes. Transportation- and construction-induced vibration guidance manual. Technical report, 2004. June. (J&S 02-039.) Sacramento, CA. Prepared for California Department of Transportation, Noise, Vibration, and Hazardous Waste Management Office, Sacramento, CA. [236] A.J. Sutherland and P.D. Cenek. Vibration assessment: Christchurch southern motorway project. Central laboratories report 07 5c1124.06, Opus International Consultants, 2007. [237] M.R. Svinkin. Impact of weight falling onto the ground. Journal of Geotechnical Engineering, Proceedings of the ASCE, 122(5):414–415, 1996. [238] J.J. Takeda and H. Tachikawa. Mechanical properties of several kinds of concrete at compressive, tensile and flexural tests in high rates of loading. Transactions of Architect Institute of Japan, 77:1–6, 1962. [239] E. Taniguchi and S. Okada. Reduction of ground vibrations by improving soft ground. Soils and Foundations, 21(2):99–113, 1981. [240] S. Thilakasiri, G. Mullins, P. Stinnette, and M. Gunaratne. Impact of weight falling onto the ground - Discussion. Journal of Geotechnical Engineering, Proceedings of the ASCE, 122(5):415–416, 1996. [241] R.T. Uil. Non-lagging effect of motorcycle tyres. Technical Report DCT 2006.046, Department of Mechanical Engineering, Eindhoven University of Technology, 2006. [242] O. Von Estorff and M. Firuziaan. Coupled BEM/FEM approach for nonlinear soil/structure interaction. Engineering Analysis with Boundary Elements, 24:605–621, 2000. [243] D. Watstein. Effect of strain rate on the compressive strength and elastic properties of concrete. Journal of the American Concrete Institute, 49:729– 744, 1953. [244] G.R. Watts. Vibration nuisance from road traffic, results of a 50 site survey. Laboratory report 1119, Transport and Road Research Laboratory, 1984. [245] G.R. Watts. Traffic-induced ground-borne vibrations in dwellings. Research report 102, Transport and Road Research Laboratory, 1987. [246] G.R. Watts. Case studies of the effects of traffic-induced vibrations on heritage buildings. Research report 156, Transport and Road Research Laboratory, 1988. [247] G.R. Watts. Traffic vibration and building damage, TRRL papers presented at acoustics ’87. Research report 146, Transport and Road Research Laboratory, 1988.

BIBLIOGRAPHY

197

[248] G.R. Watts. The effects of traffic-induced vibrations on heritage buildings, further case studies. Research report 207, Transport and Road Research Laboratory, 1989. [249] G.R. Watts. Groundborne vibrations generated by HGVs - effects of vehicle, road and ground parameters. Proceedings of the Institute of Acoustics, 11(5):99–109, 1989. [250] G.R. Watts. Traffic induced vibrations in buildings. Research report 246, Transport and Road Research Laboratory, 1990. [251] G.R. Watts. The generation and propagation of vibration in various soils produced by the dynamic loading of road pavements. Journal of Sound and Vibration, 156(2):191–206, 1992. [252] G.R. Watts and V.V. Krylov. Ground-borne vibrations generated by vehicles crossing road humps and speed control cushions. Applied Acoustics, 59:221– 236, 2000. [253] P.D. Welch. The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms. IEEE Transactions on Audio and Electroacoustics, 15(2):70– 73, 1967. [254] H.M. Westergaard. Computation of stresses in concrete roads. Highway Research Board Proceedings, 5:90–112, 1926. [255] R.V. Whitman and K.E. Healy. Shear strength of sands during rapid loading. Trans. ASCE, 128:1553–1594, 1963. [256] J.F. Wiss. Construction vibrations: state-of-the-art. Journal of Geotechnical Engineering, Proceedings of the ASCE, 107(GT2):167–181, 1981. [257] J.P. Wolf. Soil-structure-interaction analysis in the time domain. PrenticeHall, Englewood Cliffs, New Jersey, 1988. [258] G. Wu and W.D.L. Finn. Dynamic nonlinear analysis of pile foundations using finite element method in the time domain. Canadian Geotechnical Journal, 34:44–52, 1997. [259] H. Xia, P.B. Wei, Y.M. Cao, and G. De Roeck. Traffic induced vibrations of ground environments and buildings. In H. Takemiya, editor, Proceedings of the 2nd International Symposium on Environmental Vibrations: Prediction, Monitoring, Mitigation and Evaluation. ISEV 2005, pages 529–539, Okayama, Japan, September 2005. A.A. Balkema, Rotterdam.

198

BIBLIOGRAPHY

[260] Y.L. Xu and X.J. Hong. Stochastic modelling of traffic-induced building vibration. Journal of Sound and Vibration, 313(1-2):149–170, 2008. [261] W. Yang, M.F.M. Hussein, C. Cox, and A. Marshall. Physical and numerical modelling of ground-borne vibration from a surface source. In G. De Roeck, G. Degrande, G. Lombaert, and G. M¨ uller, editors, Proceedings of the 8th International Conference on Structural Dynamics EURODYN 2011, pages 748–753, Leuven, Belgium, July 2011. [262] Y.B. Yang and H.H. Hung. A 2.5D finite-infinite element approach for modelling visco-elastic bodies subjected to moving loads. International Journal for Numerical Methods in Engineering, 51:1317–1336, 2001. [263] N. Yoshida, S. Kobayashi, I. Suetomi, and K. Miura. Equivalent linear method considering frequency dependent characteristics of stiffness and damping. Soil Dynamics and Earthquake Engineering, 22(3):205–222, 2002. [264] P.W.A. Zegelaar. The dynamic response of tyres to brake torque variations and road unevennesses. PhD thesis, Delft University of Technology, 1998. [265] W. Zhai, H. Takemiya, G. De Roeck, and E. Tutumluer, editors. Proceedings of the 5th International Symposium on Environmental vibrations: Prediction, Monitoring, Mitigation and Evaluation ISEV 2011, Chengdu, China, October 2011. Science Press. [266] X.X. Zhang, C.Y. Rena, G. Ruiz, M. Tarifa, and M.A. Camara. Effect of loading rate on crack velocities in HSC. International Journal of Impact Engineering, 37:359–370, 2010. [267] O.C. Zienkiewicz and R.L. Taylor. The finite element method, Volume 1: the basis. Butterworth-Heinemann, Oxford, United Kingdom, fifth edition, 2000.

Appendix A

Mass, stiffness, and damping matrices of the 3D vehicle model The mass, stiffness, and damping matrices of the 3D vehicle model shown in figure 2.7 and repeated in figure A.1 are given in this appendix.

Figure A.1: Three-dimensional model of a two-axle truck.

199

200

A.1

MASS, STIFFNESS, AND DAMPING MATRICES OF THE 3D VEHICLE MODEL

Mass matrix

The mass matrix in equation (2.1) is: 

Mvv

A.2

     =     

mb 0 0 0 0 0 0

0 Ibx 0 0 0 0 0

0 0 Iby 0 0 0 0

0 0 0 ma1 0 0 0

0 0 0 0 Ia1 0 0

0 0 0 0 0 ma2 0

0 0 0 0 0 0 Ia2

           

(A.1)

Stiffness matrix

The submatrices Kvv , Kvc , and Kcc of the stiffness matrix are: 

Kvv

kp1 + kp2  k l +k l p2 2  p1 1   0  = 2 −kp1   0    −kp2 0

0 0 −kp1 wp wak1 0 2 kp1 wak1 + kt1 wt2 0 0

kp1 l1 + kp2 l2 kp1 l12 + kp2 l22 0 −kp1 l1 0 −kp2 l2 0 −kp2 −kp2 l2 0 0 0 kp2 + kt2 0

0 0 (kp1 + kp2 )wp2 0 −kp1 wp wak1 0 −kp2 wp wak2

0 0 −kp2 wp wak2 0 0 0 2 kp2 wak2 + kt2 wt2



      ,     

−kp1 −kp1 l1 0 kp1 + kt1 0 0 0

STIFFNESS MATRIX

201



Kvc = KT cv

Kcc



0 0 0 0  0 0 0 0    0 0 0 0  = 0 0 −k −k t1 t1   0 0  −kt1 wt kt1 wt   0 0 −kt2 −kt2 0 0 −kt2 wt kt2 wt

kt1 0 0 0  0 k 0 0  t1 =  0 0 kt2 0 0 0 0 kt2

    



      ,     

(A.2)

Parameters kpi and kti represent the stiffness of the primary suspension and the tyre(s) on each side of axle i, respectively. The scalar li is the position of axle i in the y-direction relative to the centre of gravity of the vehicle body and it therefore has a negative sign for the rear axle. The distance between the chassis beams (where the springs and dampers are connected to the body) is denoted by 2wp and is the same for the rear and front side. 2waki is the distance between the springs on axle i which can be different from 2wp . The wheel track of the vehicle is represented by 2 wt and is usually approximately similar for the rear and front wheels.

202

A.3

MASS, STIFFNESS, AND DAMPING MATRICES OF THE 3D VEHICLE MODEL

Damping matrix

The submatrices Cvv , Cvc , and Ccc of the damping matrix have a similar structure as the stiffness submatrices and are equal to: 

Cvv

cp 1 l 1 + cp 2 l 2 cp1 l12 + cp2 l22 0 −cp1 l1 0 −cp2 l2 0

cp 1 + cp 2  c l +c l p2 2  p1 1   0  = 2 −cp1   0    −cp2 0

0 0 −cp1 wp wac1 0 2 cp1 wac1 + ct1 wt2 0 0

−cp2 −cp2 l2 0 0 0 cp 2 + ct 2 0



Cvc = CT cv

Ccc



0 0 (cp1 + cp2 )wp2 0 −cp1 wp wac1 0 −cp2 wp wac2

0 0 −cp2 wp wac2 0 0 0 2 cp2 wac2 + ct2 wt2

0 0 0 0  0 0 0 0    0 0 0 0  = 0 0 −c −c t t 1 1   0 0  −ct1 wt ct1 wt   0 0 −ct2 −ct2 0 0 −ct2 wt ct2 wt

ct 1 0 0 0  0 c 0  t1 0 =  0 0 ct 2 0 0 0 0 ct 2

    

−cp1 −cp1 l1 0 cp 1 + ct 1 0 0 0



      ,     



      ,     

(A.3)

Parameters cpi and cti represent the damping coefficients of the primary suspension and the tyre(s) on each side of axle i, respectively. 2waci is the distance between the dampers on axle i which can be different from 2waki .

Appendix B

Critical state soil parameters B.1

Introduction

0

0

2

2

2

4

4

4

6

6

8

10

(a)

Depth [m]

0

Depth [m]

Depth [m]

The Database of the Subsoil of Flanders [188] contains data of six CPTs next to each other, GEO-00/86-S1 to S6, at 1500 m southeast of the measurement site in Waarschoot and 700 m east of the measurement site in Lovendegem. These CPTs show that the groundwater table is located at about 0.5 m below the surface. Figure B.1 shows the cone resistance qc , the sleeve friction fs , and the friction ratio Rf = fs /qc obtained from these CPTs as a function of depth. These data are used to estimate the geotechnical soil properties in the following.

6

8

0

10 20 30 40 Cone resistance qc [MPa]

10

(b)

8

0

0.1 0.2 0.3 Sleeve friction fs [MPa]

10

(c)

0

1 2 3 4 Friction ratio Rf [%]

5

Figure B.1: (a) Cone resistance qc , (b) sleeve friction fs , and (c) friction ratio Rf of the soil near the measurement sites as a function of depth, obtained from six CPTs. Superimposed are the average values (thick lines).

203

204


Table B.1 presents the recommendation of the Database of the Subsoil of Flanders for soil classification and estimation of the unit weight γ, effective friction angle φ′ , drained cohesion c′ , and undrained cohesion cu from the cone resistance qc and the friction ratio Rf . Soil type gravel

Admixture

loamy or clayey

sand loamy or clayey loam -

sandy

clay -

sandy

peat

Density condition

qc (MPa)

Rf (%)

moderate 0-20 dense >20 moderate 0-20 dense >20 loose 2-4 moderate 4-10 dense >10 loose 2-4 moderate 4-10 dense >10 less firm 0.4-1 moderate firm 1-2 rather firm 2-4 firm >4 less firm 0.4-1 moderate firm 1-2 rather firm 2-4 firm >4 less firm 0.4-1 moderate firm 1-2 rather firm 2-4 firm >4 less firm 0.4-1 moderate firm 1-2 rather firm 2-4 firm >4 less firm 0.2-0.5 moderate firm 0.5-1 firm >1

<1% 1-2% <1%

1-2%

2-4%

1-3%

3-6%

2-5%

>6%

γ γ above w.t. below w.t. (kN/m3 ) (kN/m3 ) 18 20 19 21 19 21 20 22 16 18 17 19 18 20 16 18 17 19 18 20 17 17 18 18 19 19 20 20 17 17 18 18 19 19 20 20 16 16 17 17 18 18 19 19 16 16 17 17 18 18 19 19 10 10 12 12 14 14

φ′ c′ cu ◦ ( ) (kPa) (kPa) 35 40 32 37 27 30 35 25 27 30 22 22 22 22 25 25 25 25 20 20 20 20 22 22 22 22 15 15 15

0 0 0 0 0 0 0 0 0 0 0 2 4 8 0 2 4 8 2 4 8 15 2 4 8 15 2 5 10

10 25 50 100 10 25 50 100 20 50 100 200 20 50 100 200 10 20 40

Table B.1: Soil classification and geotechnical soil properties based on CPT data (proposal of the Database of the Subsoil of Flanders [188] for the national annex of Eurocode 7 - part 2).

It must be noted that the recommended strength parameters are for practical engineering design purposes and therefore rather conservative. Hence in the

UNIT WEIGHT

205

following, more realistic values especially for the friction angle φ and the dilatancy angle ψ are estimated.

B.2

Unit weight

The shear strength of cohesionless soils directly depends on the effective vertical stress σv′ . The effective vertical stress, in turn, depends on the depth d, unit weight γ of the soil, and the level dw of groundwater table as: σv′ = d γ − (d − dw )γw

(B.1)

where γw is the unit weight of water. The soil unit weight has not been measured on site and is estimated from the CPT data. Robertson and Cabal [204] suggest the following formula for the estimation of soil unit weight γ: γ/γw = 0.27 log Rf + 0.36 log(qt /pa ) + 1.236

(B.2)

where qt is the corrected total cone resistance which can be considered equal to qc for sands [203] and pa is the atmospheric pressure. Figure B.2a shows the unit weight as a function of depth estimated using equation (B.2) as well as the values proposed by the DOV in table B.1. The agreement between equation (B.2) and the values recommended by the DOV is relatively good. Herein, the values recommended by the DOV are considered because the presence of the groundwater table is accounted for. Figure B.2b shows the effective vertical stress σv′ as a function of depth. Above and below the groundwater table, the effective vertical stress increases almost linearly with depth.

B.3

Relative density

The density state of the soil determines its volume change behaviour during shear which can be dilative or contractive (e.g. for a very dense or a very loose cohesionless soil, respectively). This dilatation or contraction appears in the form of strain softening or hardening in the non-linear behaviour of the soil and must be provided in the potential flow rule of a plastic model. In the following, the CPT data are used to estimate the relative density of the soil which will be used subsequently to determine the dilatancy angle. Kulhawy and Mayne [145] suggest the following formula for the relative density ID based on the results of 24 sets of calibration chamber tests on sands. qc /pa 2 ID = /QF (B.3) (σv′ /pa )0.5


0

0

2

2

4

4

Depth [m]

Depth [m]

206

6

8

8

10 15 16 17 18 19 20 21 22

(a)

6

Unit weight γ [kN/m3]

10

0

20

40

60

80

100

(b) Effecetive vertical stress σ’v [kPa]

Figure B.2: (a) Unit weight of the soil as a function of depth, estimated from six CPTs and (b) the computed effective vertical stress. Superimposed on figure (a) is the average value (thick line) and the values recommended by the DOV (dots). where QF is the limit value of possible particle crushing and depends on the compressibility of the sand. For low compressibility sands which correspond to clean quartz sands QF = 332, for medium compressibility QF = 292, and for high compressibility sands which have more fines, mica, and other compressible minerals QF = 280. Jamiolkowski et al. [127] suggest the following formula for the relative density, based on calibration chamber tests on five different normally consolidated sands: ! ! qc −1 (B.4) ID = 68 log p pa σv′

To correlate the field and chamber values of qc , Jamiolkowski, et al. [127] recommend dividing the field value of qc by Kq = 1 + (ID − 30)/300 before entering it into equation (B.4). This process requires one iteration because the value of ID is not known beforehand.

Figure B.3 shows the relative density estimated by equations (B.3) and (B.4). The difference between the relative densities estimated using equation (B.3) for different compressibility levels (QF values) is small, so only the result for high compressibility sand (QF = 280) is shown. The agreement between the two methods is relatively good except for very high relative densities where equation (B.3) passes the limit value of possible particle crushing.

INTERNAL FRICTION ANGLE

207

0

Depth [m]

2

4

6

8

10

0

20 40 60 80 100 Relative density ID [%]

Figure B.3: Relative density of the soil as a function of depth estimated using the method suggested by Kulhawy and Mayne [145] (dashed line) and Jamiolkowski et al. [127] (solid line).

B.4

Internal friction angle

The internal friction angle of a cohesionless soil can be defined as the peak friction angle φ′p or the critical state (critical void ratio, ultimate) friction angle φ′cv depending on the strain level (figure B.4). Development of the peak friction angle also depends on soil density state. For dense cohesionless soils which exhibit dilative behaviour during shear, the peak friction angle φ′p is high and develops at very small strains typically on the order of a few percent. On the other hand, loose cohesionless soils exhibit contractive behaviour with a lower peak friction angle φ′p that develops at large strains typically upwards of 10 − 20% [145].

Figure B.4: Definition of the peak friction angle φ′p and the critical state friction angle φ′cv (from [145]).

208


A significant amount of literature exists on the estimation of the friction angle of cohesionless soils from CPT data. Meyerhof [180] recommends an empirical correlation between qc and φ based on tests carried out in France and Germany. In this correlation, qc is the limiting static cone resistance, i.e. the maximum value of qc obtained at the ‘critical depth’ below which the penetration resistance shows little or no increase with continued penetration. Eurocode 7 - part 2 [76] suggests the following equation for the estimation of the effective friction angle φ′ for poorly-graded sands above groundwater and cone penetration resistances in the range 5 MPa ≤ qc ≤ 28 MPa: φ′ = 13.5 log qc + 23

(B.5)

where qc is input in unit of MPa. The standard states that this correlation should be considered as conservative estimate. Although not explicitly stated, the friction angles estimated from this method and Meyerhof’s method probably correspond to the critical state friction angle φ′cv . Figure B.5 compares the estimated φ′cv as a function of depth using the aforementioned methods and the DOV recommendations according to table B.1. The presented graphs are the average values of six estimations from the available CPT data. Although Eurocode 7 suggests that equation (B.5) should be used for the soil above the groundwater, its agreement with Meyerhof’s method is very good even below the water table. The recommended values by the DOV are conservative and are smaller than the results of the other methods, as expected. In the following, the critical state friction angle estimated by Meyerhof’s method is used. 0

Depth [m]

2

4

6

8

10 20 25 30 35 40 45 50 Critical state friction angle φcv [deg]

Figure B.5: Critical state friction angle φ′cv of the soil as a function of depth estimated using the methods suggested by Meyerhof [180] (solid line), Eurocode 7 [76] (plus signs), and the DOV (dots).

INTERNAL FRICTION ANGLE

209

Kulhawy and Mayne [145] calibrated the following equation for the drained peak friction angle φ′p from 20 data sets obtained in calibration chambers: φ′p

= 17.6 + 11.0 log

qc /pa ′ (σv /pa )0.5

(B.6)

Robertson and Campanella [205] established an empirical correlation between the cone resistance qc and drained peak friction angle φ′p based on CPT tests on normally consolidated sands of medium compressibility. This correlation was presented in the form of a design chart to which Chen and Juang [52] have fit the following equation. 1 qc /σv′ ′ tan(φp ) = (B.7) ln C1 C2 where σv′ is the vertical effective stress at the depth of the cone tip and the regression coefficients are C1 = 6.820 and C2 = 0.266. Durgunoglu and Mitchell [74] suggested a method for the determination of φ′p based on the bearing capacity theory. Chen and Juang [52] have also fit equation (B.7) to Durgunoglu and Mitchell’s design chart which yielded C1 = 7.629 and C2 = 0.194. Robertson and Campanella [205] and Durgunoglu and Mitchell [74] use bearing capacity theories neglecting the curvature of the failure envelope [52]. This curvature is attributed to soil dilatancy [145]. Thus, the estimated friction angles are peak friction angles which also contain the dilatancy component. Penetration of the cone in sand can generate very large stresses, therefore Chen and Juang [52] recommend that the curvature of the failure envelope should be considered and the predicted φ′p from the cone resistance qc using the bearing capacity theories should be reduced. Based on the work of Meigh (1987) a rule of thumb is suggested for the reduction ∆φ′ of the drained peak friction angle φ′p : 3 2 ∆φ′ = 9.310ID − 5.122ID + 3.753ID − 0.013

(B.8)

where ID is the relative density in decimal value. Figure B.6a compares the estimated φ′p as a function of depth using the aforementioned methods. The presented graphs are the average values of six estimations from the available CPT data. The agreement between the methods is relatively good except at shallow depths. This arises from the fact that equations (B.6) and (B.7) are very sensitive to small values of the effective vertical stress σv′ as found near the surface. Figure B.6b presents the similar data as in figure B.6a except that the reduction in φ′p according to equation (B.8) has been considered. It can be observed that the suggested reduction reduces very large values of the


0

0

2

2

4

4

6

Depth [m]

Depth [m]

210

6

8

10 20 25 30 35 40 45 50 Peak friction angle φp [deg]

(a)

8

10 20 25 30 35 40 45 50 Peak friction angle φp [deg]

(b)

Figure B.6: Peak friction angle φ′p of the soil as a function of depth estimated using the methods suggested by Robertson and Campanella [205] (solid line), Durgunoglu and Mitchell [74] (dashed line), and Kulhawy and Mayne [145] (dotted line) (a) without and (b) with the reduction according to equation (B.8). peak friction angle which coincide with the peaks of the relative density (figure B.3). Since the estimated values of φ′p at shallow depths seem to be unreliable, the type of the soil near the location of the CPTs is further investigated. Borehole sample kb13d39e-B277 with a depth of 3.0 m at 300 m west is the closest borehole to the CPTs. It reveals a layer of grey sandy loam with a thickness of 0.80 m overlaying yellow loose sand with a thickness of at least 2.3 m. Considering that this borehole sample and the one close to the measurement site (kb13d39e-B278) describe the top 3.0 m of the soil as sandy loam and loose sand, the estimated peak friction angles for the shallow depths in figure B.6 seem to be erroneous. In addition, figure B.1a shows that the cone resistance for the top 1.5 m of the soil is between 2 MPa to 3 MPa which is very low and according to table B.1 corresponds to loose sand. Considering that the difference between the ground level at the location of the borehole samples and the CPTs is in the range of ±1 m, it is concluded that the loose soil at the location of the CPTs is only at the top 1.5 m and there is a dense layer underneath. Another explanation for this discrepancy between the borehole samples and the CPTs is the fact that the cone tip resistance is influenced by the soil ahead and behind the cone tip [205]. This effect is limited to a range of ±0.5 m as the cone can sense a soil interface up to 15 times the cone diameter ahead and behind, depending on the strength and stiffness of the soil and the in situ effective stresses [4]. The peak friction angle estimated using Kulhawy and Mayne’s method appears to be more reasonable and will be used in the rest of this study.

DILATANCY ANGLE

B.5

211

Dilatancy angle

There are several methods to estimate the dilatancy angle from the CPT data using the relative density of the soil. Kulhawy and Mayne [145] based on the work of Bolton [36] suggest the following formula for the estimation of the dilatancy angle for triaxial compression ψ = 3IRD

(B.9)

where IRD is a relative density index, given by: IRD = ID [Q − ln(100 p¯f /pa )] − R

(0 < IRD < 4)

(B.10)

Q is a coefficient depending on the soil mineralogy and compressibility (Q = 10 for quartz and feldspar, 8 for limestone, 7 for anthracite, 5.5 for chalk), p¯f = (σ1′ + σ2′ + σ3′ )f /3 is mean principal effective stress at failure which can be assumed to be p¯f ≈ 2 σv′ , and R = 1 is a fitting coefficient. Herein, a value Q = 10 is considered. Lee et al. [156] suggest the same relation as equation (B.9) for the estimation of the dilatancy angle and a semi-empirical equation for the relative density index IRD : IRD = ID [QCPT − ln(100 qc/pa )] − RCPT

(B.11)

where QCPT and RCPT are functions of silt content and can be obtained from table B.2. Silt content [%] QCPT RCPT

0 14 1.0

2 15.4 -0.12

5 14.0 -0.12

10 14.3 -0.01

15 11.8 0.01

20 12.1 0.12

Table B.2: Intrinsic soil variables QCPT and RCPT proposed by Lee et al. [156]. Since equation (B.11) is based on the work of Bolton [36], the same limits as in equation (B.10) are considered for IRD , although Lee et al. [156] do not mention any limit. The dilatancy angle might also be estimated from relations between the peak and the critical state friction angles. There is no consistency on this matter in the literature [111]. The more widely accepted relation suggested by Bolton [36] is considered here as ψ = φ′p − φ′cv /0.8.

Figure B.7 compares the estimated dilatancy angle based on Kulhawy and Mayne [145], Lee et al. [156] considering 0 and 10% of silt content, and Bolton’s relation

212


0

Depth [m]

2

4

6

8

10

0

2 4 6 8 10 12 Dilatancy angle ψ [deg]

Figure B.7: Dilatancy angle of the soil as a function of depth estimated according to Kulhawy and Mayne [145] (thick line), Lee et al. [156] with 0% (thin solid line) and 10% of silt content (dashed line), and Bolton’s relation [36] (dotted line). [36]. In the estimation of the dilatancy angle, the relative density has been computed following equation (B.4) proposed by Jamiolkowski et al. [127]. The agreement between the estimated dilatancy angles is not good. There is also a large difference in the estimated dilatancy angles with the method of Lee et al. for different values of fines content. Herein, the values estimated by Kulhawy and Mayne’s method are considered as they slightly better correspond to the identified soil type.

B.6

Shear wave velocity

The region near the measurement sites in Waarschoot and Lovendegem is quite flat. Two deep boreholes that previously discussed in section 4.3.2 also show that it is reasonable to assume the soil is horizontally layered. In order to verify whether the soil profile at the location of the CPTs is similar to the soil at the measurement site, the shear wave velocity of the soil is estimated from the CPT data and compared with what has been identified from the surface wave test in Waarschoot [20] and the SASW test [151] and the SCPT [15] in Lovendegem. Robertson [203] suggests the following formula for the estimation of the shear wave velocity Cs from CPT data. p (B.12) Cs = αcs (qt − σv )pa

where αcs is the shear wave velocity cone factor and estimated as: αcs = 10(0.55Ic +1.68)

(B.13)

COHESION

213

in which Ic is the soil behaviour type index, defined as: 0.5 Ic = (3.47 − log Qt1 )2 + (log Fr + 1.22)2

(B.14)

Qt1 = (qt − σv )/σv′

(B.15)

Fr = [fs /(qt − σv )] 100%

(B.16)

Hegazy and Mayne [107] derived the following relationship between the in situ shear wave velocity and CPT data from a database that included sands, silts, clays, as well as mixed soil types. Cs = (10.1 log qt − 11.4)1.67 (100 fs/qt )0.3

(B.17)

where qt and fs are in kPa and Cs is in m/s. From a similar database of well-documented experimental sites in saturated clays, silts, and sands, Mayne [178] showed that: Cs = 118.8 log fs + 18.5

(B.18)

where fs is in kPa and Cs in m/s. Figure B.8 compares the shear wave velocities estimated from the CPT data with the velocity identified from the surface wave test, the SASW test, and the SCPT. A relatively good agreement is found between the estimated shear wave velocities and the measured values. This reveals that the soil profiles are similar at the measurement sites and the location of the CPTs.

B.7

Cohesion

Since the borehole sample data as well as the soil classification based on the CPT data [202] reveal the presence of mostly sandy soils, a zero value is assumed for the cohesion.

B.8

Summary of the estimated soil parameters

As explained earlier, the area around the measurement sites is very flat and the topographical maps show a level difference of at most ±2 m between the measurement sites and the locations of the boreholes and CPTs. From the CPT data and the borehole samples, the soil profile is identified as a layer of loose sand

214


0

Depth [m]

2

4

6

8

10

0

100 200 300 400 Shear wave velocity [m/s]

Figure B.8: Shear wave velocity as a function of depth measured with the SCPT in Lovendegem [15] (thick solid line), surface wave test in Waarschoot [20] (thick dashed line), and SASW test in Lovendegem [151] (thick dotted line) and estimated from the CPT data using the method suggested by Robertson [203] (thin solid line), Hegazy and Mayne [107] (thin dashed line), and Mayne [178] (thin dotted line). with a thickness of 1.6 m overlaying a 1.5 m dense sand, a 3.0 m moderately firm sandy loam, a 2.3 m dense sand, on top of rather firm sandy loam. The presence of a 1.5 m stiff layer at depths between 1.6 m and 3.1 m is not identified by the surface wave tests (figure 4.5b and table 4.1). Figure B.9a shows the experimental dispersion curve obtained in theses tests. The experimental dispersion curve has been transformed to an initial shear wave velocity profile (figure B.9b) by a scaling operation where the wavelength λ is transformed to a depth z according to a rule of thumb z = λ/2.5 [20]. Figure B.9b shows an inverse layering with a stiff layer at a depth about 2 m [20]. This stiff layer was then smeared out in the adjacent layers during the optimisation process. The presence of this stiff layer is also apparent in the shear wave velocity profile identified from the SCPT (figures 3.9b and B.8). The unit weight, critical state friction angle, and dilatancy angle of the soil have been averaged over the thickness of each layer and are shown in Figure B.10 and table B.3. It is assumed that the top loose layer at the measurement sites has been compacted by road traffic over the years and is not loose anymore. Therefore the relatively high value of the dilatancy angle that has been estimated from the CPT data is not reduced.

SUMMARY OF THE ESTIMATED SOIL PARAMETERS

215

0

0

5 Depth [m]

Wavelength [m]

10

20

30

15

40

50 0

(a)

10

100 200 300 400 Phase velocity [m/s]

(b)

20 0 100 200 300 400 Shear wave velocity [m/s]

0

0

2

2

2

4

4

4

6

6

8

6

8

10 15 16 17 18 19 20 21 22

(a)

Depth [m]

0

Depth [m]

Depth [m]

Figure B.9: (a) Experimental dispersion curve determined from the active and passive surface wave tests at the site in Waarschoot in terms of the wavelength and (b) initial shear wave velocity profile (from [20]).

Unit weight γ [kN/m3]

8

10 30 33 36 39 42 45 Critical state fiction angle φcv [deg]

(b)

10

0

(c)

2 4 6 8 10 12 Dilatancy angle ψ [deg]

Figure B.10: (a) Unit weight, (b) critical state friction angle, and (c) dilatancy angle of the soil near the measurement sites as a function of depth estimated from the CPT data. Superimposed are the averaged values over the thickness of each layer (thick lines). Layer 1 2 3 4

Thickness [m] 1.6 1.5 3.0 2.3

γ [kN/m3 ] 18.1 19.7 18.2 19.7

φ′cv [°] 33 37 33 39

ψ [°] 9 11 3 9

Table B.3: Unit weight, critical state friction angle, and dilatancy angle of the soil near the measurement sites estimated from the CPT data.

Curriculum vitae Mohammad Amin Lak ◦ 6 September 1978, Hamedan (Iran)

Education 2007-2013 PhD in Engineering, Department of Civil Engineering, KU Leuven, Belgium 2000-2002 MSc in Structural Engineering, University of Tehran, Iran 1996-2000 BSc in Civil Engineering, University of Tehran, Iran 1992-1996 Secondary education, Math-Physics, Allameh Helli, Hamedan, Iran

Work 2007-2013 Research assistant, Department of Civil Engineering, KU Leuven, Belgium 2005-2007 Lecturer, Azad University and UCATR, Hamedan, Iran 2004-2007 Structural engineer and designer, freelancer, Tehran and Hamedan, Iran 2002-2003 Civil engineer, Sazeh & Energy, Tehran, Iran 2000-2002 Structural engineer, Panehsazan, Tehran, Iran 217

218

CURRICULUM VITAE

Publications International journal papers [1] Z. Ozdemir, P. Coulier, M.A. Lak, S. Fran¸cois, G. Lombaert, and G. Degrande. Numerical evaluation of the dynamic response of pipelines to vibrations induced by the operation of a pavement breaker. Soil Dynamics and Earthquake Engineering, 44:153–167, 2013. [2] M.A. Lak, G. Degrande, and G. Lombaert. The effect of road unevenness on the dynamic vehicle response and ground-borne vibrations due to road traffic. Soil Dynamics and Earthquake Engineering, 31(10):1357–1377, 2011. International conference papers [1] Z. Ozdemir, M.A. Lak, P. Coulier, S. Fran¸cois, G. Lombaert, and G. Degrande. Numerical evaluation of the dynamic response of pipelines to vibrations induced by the operation of a pavement breaker. In W. Zhai, H. Takemiya, G. De Roeck, and E. Tutumluer, editors, Advances in Environmental Vibration. Proceedings of the 5th International Symposium on Environmental Vibration. ISEV 2011., pages 808–815, Chengdu, China, October 2011. Science Press. [2] Z. Ozdemir, M.A. Lak, S. Fran¸cois, G. Degrande, and G. Lombaert. A 2.5d coupled be-fe model for the prediction of the dynamic response of lifelines to ground vibrations. In M. Papadrakakis, M. Fragiadakis, and V. Plevris, editors, Proceedings of COMPDYN 2011, 3rd International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Corfu, Greece, May 2011. CD-ROM. [3] M.A. Lak, G. Degrande, and G. Lombaert. The influence of the pavement type on ground-borne vibrations due to road traffic. In G. De Roeck, G. Degrande, G. Lombaert, and G. M¨ uller, editors, Proceedings of the 8th International Conference on Structural Dynamics EURODYN 2011, pages 777–784, Leuven, Belgium, July 2011. [4] Z. Ozdemir, M.A. Lak, S. Fran¸cois, P. Coulier, G. Degrande, and G. Lombaert. A numerical model for the prediction of the response of pipelines due to vibrations induced by the operation of a pavement breaker. In G. De Roeck, G. Degrande, G. Lombaert, and G. M¨ uller, editors, Proceedings of the 8th International Conference on Structural Dynamics EURODYN 2011, pages 928– 935, Leuven, Belgium, July 2011. CD-ROM. [5] M.A. Lak, G. Degrande, and G. Lombaert. Free field vibrations due to traffic and the operation of a multi-head breaker on a concrete road. In H. Xia

CURRICULUM VITAE

219

and H. Takemiya, editors, Environmental vibrations: Prediction, Monitoring, Mitigation and Evaluation, Volume I, pages 47–55, Beijing, China, October 2009. Science Press. National conference papers [1] M.A. Lak, S. Fran¸cois, G. Degrande, and G. Lombaert. Prediction of ground vibrations generated by a falling-weight pavement breaker. In Proceedings of the 9th National Congress on Theoretical and Applied Mechanics NCTAM 2012, Brussels, Belgium, May 2012. [2] G. Lombaert, M.A. Lak, S. Fran¸cois, and G. Degrande. Oorzaken, gevolgen en voorspelling van trillingen ten gevolge van verkeer op wegen uit betonplaten. OCW-K.U.Leuven studiedag, Trillingsgecontroleerd stabiliseren van betonplaten voor duurzame asfaltverhardingen met scheurremmende lagen, December 2011. [3] G. Lombaert, M.A. Lak, Z. Ozdemir, P. Coulier, S. Fran¸cois, and G. Degrande. Trillingen ten gevolge van het beuken van wegen uit betonplaten. OCWK.U.Leuven studiedag, Trillingsgecontroleerd stabiliseren van betonplaten voor duurzame asfaltverhardingen met scheurremmende lagen, December 2011. [4] G. Lombaert, S. Fran¸cois, M.A. Lak, and G. Degrande. Trillingen in de omgeving ten gevolge van wegverkeer. Belgisch Wegencongres, Gent, Belgium, September 2009. [5] A. Vanelstraete, J. Maeck, A. Beeldens, J. De Visscher, F. Vervaecke, C. Van Geem, G. Lombaert, G. Degrande, and M.A. Lak. Trillingsgecontroleerd stabiliseren van betonplaten voor duurzame asfaltoverlagingen met scheurremmende lagen. Belgisch Wegencongres, Gent, Belgium, September 2009. [6] M.A. Lak, G. Lombaert, and G. Degrande. Free field vibrations due to traffic and construction activities on a concrete road. In Proceedings of NCTAM 2009, 8th National Congress on Theoretical and Applied Mechanics, pages 263–270, Brussels, Belgium, May 2009. CD-ROM. [7] G. Lombaert, M.A. Lak, S. Fran¸cois, and G. Degrande. IWT VIS-CO project: Trillingsgecontroleerd stabiliseren van betonplaten voor duurzame asfaltoverlagingen met scheurremmende lagen. Kennisdag AWV, March 2009. Internal reports [1] M.A. Lak, S. Fran¸cois, M. Schevenels, G. Degrande, and G. Lombaert. Prediction of ground vibrations generated by a falling-weight pavement

220

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breaker. Report BWM-2012-06, Department of Civil Engineering, KU Leuven, July 2012. IWT-Project VIS-CO 060884. [2] M.A. Lak, J. Houbrechts, G. Degrande, and G. Lombaert. Measurement of ground vibrations during the passage of a truck on a concrete road in Overijse. Report BWM-2012-02, Department of Civil Engineering, KU Leuven, January 2012. IWT-Project VIS-CO 060884. [3] Z. Ozdemir, M.A. Lak, P. Coulier, S. Fran¸cois, G. Lombaert, and G. Degrande. Numerical evaluation of the dynamic response of pipelines to vibrations induced by the operation of a pavement breaker. Report BWM-2011-04, Department of Civil Engineering, KU Leuven, June 2011. [4] M.A. Lak, S. Fran¸cois, G. Degrande, and G. Lombaert. Experimental validation of a numerical model for the prediction of traffic induced ground vibrations. Report BWM-2010-10, Department of Civil Engineering, KU Leuven, July 2010. IWT-Project VIS-CO 060884. [5] S.A. Badsar, M. Schevenels, M.A. Lak, and G. Degrande. Determination of the dynamic soil properties with the seismic refraction method and the SASW method at a site in Haren. Technical Report BWM-2009-18, Department of Civil Engineering, KU Leuven, August 2009. [6] S.A. Badsar, M. Schevenels, M.A. Lak, and G. Degrande. Determination of the dynamic soil properties with the seismic refraction method and the SASW method at a site in Waarschoot. Technical Report BWM-2009-17, Department of Civil Engineering, KU Leuven, August 2009. [7] M.A. Lak, G. Lombaert, H. Verbraken, and G. Degrande. Vibration measurements during the operation of a multi-head breaker on a concrete road at a site in Waarschoot. Technical Report BWM-2009-14, Department of Civil Engineering, KU Leuven, August 2009. [8] M.A. Lak, E. Reynders, G. Lombaert, and G. Degrande. Vibration measurements during the cracking of concrete slabs by means of a multihead breaker. Report BWM-2008-16, Department of Civil Engineering, KU Leuven, October 2008. IWT-Project VIS-CO 060884. [9] M.A. Lak, G. Lombaert, S.A. Badsar, E. Reynders, and G. Degrande. Simultaneous vehicle and free field response measurement during the passage of a truck before and after road renovation. Report BWM-2008-15, Department of Civil Engineering, KU Leuven, October 2008. IWT-Project VIS-CO 060884. [10] M.A. Lak, G. Lombaert, S.A. Badsar, M. Schevenels, and G. Degrande. Determination of the dynamic soil characteristics with the SASW method at a site in Lovendegem. Report BWM-2008-11, Department of Civil Engineering, KU Leuven, September 2008. IWT-Project VIS-CO 060884.

Arenberg Doctoral School of Science, Engineering & Technology Faculty of Engineering Department of Civil Engineering Research group: Structural Mechanics Kasteelpark Arengerg 40, Box 2448, BE-3001 Leuven, Belgium

Numerical Prediction of Ground Vibrations Generated by Road Traffic

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