DESIGNOFCABLE-STAYED FOOTBRIDGESUNDER …

DESIGN OF CABLE-STAYED

FOOTBRIDGES UNDER

SERVICEABILITY LOADS

Caterina Ramos-Moreno

Department of Civil and Environmental Engineering,

Imperial College London

September 2015

A thesis submitted for the degree of

Doctor of Philosophy

September 2015

Declaration:

I confirm that this submission is my own work and that any material from published, or

unpublished, work from others is appropriately referred.

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London (the total number of words of the body of the thesis is 91100).

The copyright of this thesis rests with the author and is made available under a Creative

Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to

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of this work.

Caterina Ramos-Moreno

Abstract

During the last decades structural engineers have proposed and developed lighter, slen-

derer and longer footbridges, frequently innovating from the structural and aesthetic

viewpoints, as well as implementing in their design the latest technical developments in

relation to material types, strengths, and light-weights.

As a result, some of these footbridges have prompted serviceability responses with un-

expectedly large and excessive magnitudes (such as the London Millennium Bridge in UK

or the passerelle Leopold Sedar Senghor in Paris, France), which have in turn generated

a response in the structural engineering community, focussing their attention in gaining

understanding about this phenomena. The extensive research that has been published

during the last fifteen years has emphasised the lack of understanding of the nature and

magnitude of the actions transmitted by pedestrians while walking on structures that

may moderately move, such as footbridges. Hence, the inadequate performance of some

footbridges is related to an unrealistic and insufficient representation of the pedestrian

loading and design scenarios and to the simple application of some questionable design

rules available for the design of these footbridges in service. The existing design criteria

for footbridges is not underpinned by sophisticated numerical models which account for

the different phenomena and issues that research in different fields has already identified.

With the aim to address this deficiency, the current research work focuses on the

development of a more accurate and realistic representation of the loads transmitted by

pedestrians while walking, a model capable of accounting for intra- and, inter-subject

variability, as well as pedestrian-structure and pedestrian-pedestrian interaction, and on

its application in order to gain understanding about the structural performance under this

loading. In order to include these characteristics, the model adopts a non-deterministic

approach and combines results and proposals of a wide range of research fields.

The research work of this thesis first investigates the performance of girder bridges and

proposes a simple method that captures the response that would have been obtained with

a more sophisticated model in a very accurate manner. Secondly, the sophisticated loading

model is applied to a set of cable-stayed footbridges which define the structural typology

by means of very comprehensive parametric studies, gaining clear understanding about

the structural behaviour and performance of these bridges under pedestrian loading. This

load model, its implementation in finite element models that represent the cable-stayed

footbridges and the reproduction in the dynamic analyses of the nonlinear nature of the

loads of each pedestrian has been performed combining Abaqus, Matlab and Fortran

software packages.

1

Abstract

Based on this method, the research work generates detailed analyses of the performance

of cable-stayed footbridges that represent this bridge typology to evaluate the sensitivity

of that serviceability performance to multiple modifications of geometrical and structural

characteristics of these bridges (involving every structural element of these footbridges).

According these extensive and accurate analyses, there are several main conclusions

that can be extracted (from the 35 conclusions listed in the conclusion chapter) and

should be considered for the design of footbridges in general and cable-stayed footbridges

in particular:

1. Exclusively load models of lateral loads that include the interaction of pedestri-

ans with the movement of the footbridge can realistically reproduce the effects of

pedestrians on structures in this lateral direction.

2. Codes and guidelines proposing a simplified evaluation method must consider that,

despite the fact that pedestrians do not walk at frequencies above ∼ 2.5 Hz, the

vertical loads of a pedestrian flow have important components well above this fre-

quency.

3. The mass of the deck corresponds to the main parameter that controls the response

of footbridges.

4. The damping ratio is a factor of utmost importance in the performance of footbridges

and designers should seek to increase it.

5. The bearing arrangement of the footbridge has an essential role in its performance

in service, in particular in lateral direction: lateral displacements and rotations of

the deck at support sections must be restrained.

6. The coupling of vertical, lateral and torsional modes leads to a drastic change of the

magnitude of the response. Hence it should be avoided.

7. The reduction of the dynamic deflections is not necessarily related to a reduction

of the accelerations, therefore the assessment of the serviceability performance of

footbridges through dynamic deflections is not reliable.

8. The dynamic events must be used to evaluate the response of different structural

elements of the footbridges (deck, pylons, etc.) as a static analysis with variable

loads of 5kN/m2 does not always describe the largest stresses at these elements.

This point is relevant for safety, as it involves ULS verifications.

2

Acknowledgements

When I first started my PhD research, I could not imagine myself writing the acknowl-

edgements. I always thought that I had a long list of things to do before reaching that

stage. But now the time has come, and I would like to thank all those who have helped

me completing this thesis.

First and foremost, I would like to express my deepest gratitude to my first PhD

supervisor Dr A. M. Ruiz-Teran, for her inestimable dedication, guidance, motivation,

encouragement and patience throughout the different stages of this research work, in

particular for the hardest. I would also like to sincerely thank Dr P. J. Stafford, my

second PhD supervisor, for his invaluable direction, guidance and support. Both Dr

Ruiz-Teran and Dr Stafford have enormously helped me to give shape to my research

work and fulfil the main objective of these last years of my life.

I had my first thoughts of doing a PhD while I was working in Barcelona, at ‘Bridges

Technologies’, under the direction of Prof. A. Aparicio and Prof. G. Ramos. It was with

Prof. Aparicio that I seriously started focusing my attention back to academia. I would

like to thank him for the flexibility, guidance and support during those first stages.

I would also like to acknowledge my gratitude to ”la Caixa” Foundation, that have

provided funding for the first two years of this research work.

I would like as well to thank the Department of Civil and Environmental Engineering

of Imperial College and the library staff for providing facilities and assistance. From

this department, there are many colleagues I would like to kindly thank. Thanks to my

colleagues of room 424 (and others) of Skempton Building: Marianna, Reuben, Shirin,

Fernando, Merih and the rest for your companionship, support and entertainment during

these years. My warmest thanks as well to Ruth for her encouragement and support.

I would like to express my infinite appreciation to my partner Oscar, who decided to

join me in this adventure and moved to London. His help and support during the hard

times (as well as good times) have allowed me to go through this process.

And finally, I would like to put into words my sincerest gratitude to my family. It

is thanks to my parents that I became a civil engineer and that I pursued completing a

PhD.

3

Contents

Abstract 1

Acknowledgements 3

List of Figures 13

List of Tables 29

Nomenclature 33

1 Introduction 37

1.1 Background to the research . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.2 Aims and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1.3 Thesis layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2 State of the art 43

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2 Past, present and future design of footbridges . . . . . . . . . . . . . . . . 44

2.2.1 Historical introduction . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2.2 Unconventional footbridges . . . . . . . . . . . . . . . . . . . . . . . 44

2.2.3 Existing cable-stayed footbridges . . . . . . . . . . . . . . . . . . . 45

2.3 Pedestrian actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.3.1 Pedestrian gait: definition, characteristics and relation to loads . . . 47

2.3.2 Pedestrian characteristics . . . . . . . . . . . . . . . . . . . . . . . 48

2.3.3 Dynamic effects of pedestrians on structures . . . . . . . . . . . . . 49

2.3.4 Interaction pedestrian-structure . . . . . . . . . . . . . . . . . . . . 58

2.3.5 Probabilistic and deterministic approach . . . . . . . . . . . . . . . 62

2.3.6 Current load models: inherent drawbacks . . . . . . . . . . . . . . . 67

2.4 Damping characteristics of footbridge structures . . . . . . . . . . . . . . . 68

2.4.1 Inherent structural damping . . . . . . . . . . . . . . . . . . . . . . 68

2.4.2 Damping devices in footbridges . . . . . . . . . . . . . . . . . . . . 69

2.5 Comfort criteria in structures with pedestrians . . . . . . . . . . . . . . . . 70

2.5.1 Evaluation of vertical movements at footbridges . . . . . . . . . . . 72

2.5.2 Evaluation of lateral movements at footbridges . . . . . . . . . . . . 73

2.6 Failure in service of footbridges . . . . . . . . . . . . . . . . . . . . . . . . 74

5

Contents

2.6.1 Service failure in vertical direction . . . . . . . . . . . . . . . . . . . 74

2.6.2 Service failure in lateral direction . . . . . . . . . . . . . . . . . . . 76

2.7 Design recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.7.1 Guidelines related to serviceability appraisal . . . . . . . . . . . . . 81

2.7.2 Guidelines related to footbridge design . . . . . . . . . . . . . . . . 82

2.8 Footbridge performance analysis . . . . . . . . . . . . . . . . . . . . . . . . 83

2.9 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3 Methodology 85

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.2 Pedestrian loads: model definition . . . . . . . . . . . . . . . . . . . . . . . 86

3.2.1 Definition of the new model . . . . . . . . . . . . . . . . . . . . . . 86

3.2.2 Evaluation of parameters of the proposed load model . . . . . . . . 89

3.2.3 Pedestrian intra-subject variability . . . . . . . . . . . . . . . . . . 94

3.2.4 Representation of inter-subject variability . . . . . . . . . . . . . . 95

3.2.5 Summary of proposed model . . . . . . . . . . . . . . . . . . . . . . 96

3.3 Nondimensional parameters governing the problem . . . . . . . . . . . . . 98

3.4 Comfort criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.4.1 Comfort criteria for walking pedestrians . . . . . . . . . . . . . . . 99

3.4.2 Comfort criteria for standing and sitting pedestrians . . . . . . . . 100

3.5 Footbridges description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.5.1 Girder bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.5.2 Cable-stayed footbridges . . . . . . . . . . . . . . . . . . . . . . . . 103

3.5.3 Evaluated structural schemes . . . . . . . . . . . . . . . . . . . . . 111

3.6 Finite element models: assumptions and representation . . . . . . . . . . . 112

3.6.1 Structure model: finite element model description of girder footbridges112

3.6.2 Structure model: finite element model description of cable-stayed

bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.6.3 Definition of sophisticated user functions within the numerical mod-

els to represent the pedestrian-structure complex interaction . . . . 115

3.6.4 Numerical dynamic analysis . . . . . . . . . . . . . . . . . . . . . . 116

3.6.5 Duration of simulation events . . . . . . . . . . . . . . . . . . . . . 117

3.7 Response analysis and comparison . . . . . . . . . . . . . . . . . . . . . . . 118

3.7.1 Serviceability limit state of vibration . . . . . . . . . . . . . . . . . 118

3.7.2 Serviceability limit state of deflections . . . . . . . . . . . . . . . . 119

3.7.3 Ultimate limit state related to deck normal stresses . . . . . . . . . 119

3.7.4 Ultimate limit state related to shear stresses . . . . . . . . . . . . . 120

3.7.5 Ultimate limit state related to tower stresses . . . . . . . . . . . . . 120

3.7.6 Ultimate limit state of fatigue of cables . . . . . . . . . . . . . . . . 120


6

Contents

4 Relevance of stochastic representation of reality: advantages of the new

load model presented herein 123

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.2 Pedestrian intra-subject variability . . . . . . . . . . . . . . . . . . . . . . 124

4.2.1 Effect of step frequency variability on vertical response . . . . . . . 124

4.2.2 Effect of step frequency variability on lateral response . . . . . . . . 127

4.2.3 Effect of step width variability on lateral response . . . . . . . . . . 129

4.3 Pedestrian inter-subject variability . . . . . . . . . . . . . . . . . . . . . . 130

4.3.1 Variability of vertical load amplitudes . . . . . . . . . . . . . . . . . 130

4.3.2 Variability of weight . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.3.3 Variability of gait characteristics . . . . . . . . . . . . . . . . . . . 133

4.3.4 Variability of step width . . . . . . . . . . . . . . . . . . . . . . . . 135

4.4 Pedestrian flow interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 135


5 Girder footbridge design: evaluation of response in serviceability con-

ditions 139

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.2 Foundations of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.3 Pedestrian loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.4 Vertical and lateral structural frequencies . . . . . . . . . . . . . . . . . . . 140

5.5 Resonance parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.6 Basic vertical and lateral accelerations . . . . . . . . . . . . . . . . . . . . 143

5.7 Maximum vertical and lateral accelerations caused by a single pedestrian . 144

5.7.1 Factor related to the pedestrian mass (φpm) . . . . . . . . . . . . . 145

5.7.2 Factor related to the pedestrian step length (φsl) . . . . . . . . . . 145

5.7.3 Factor related to the pedestrian step width (φsw) . . . . . . . . . . 146

5.7.4 Factor related to the pedestrian height (φph) . . . . . . . . . . . . . 146

5.7.5 Factor related to the structural damping (φd) . . . . . . . . . . . . 146

5.7.6 Factor related to the structural mass (φsm) . . . . . . . . . . . . . . 147

5.8 Vertical and lateral accelerations caused by groups of pedestrians and con-

tinuous streams of pedestrians . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.8.1 Group of pedestrians . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.8.2 Continuous streams of pedestrians . . . . . . . . . . . . . . . . . . . 150

5.9 Verification of the serviceability design appraisal . . . . . . . . . . . . . . . 150

5.9.1 Comparison of the methodology against FEM models . . . . . . . . 150

5.9.2 Comparison of the methodology against real responses . . . . . . . 151

5.10 Evaluation of the serviceability performance in conventional footbridges . . 153


6 Evaluation of the response of a conventional cable-stayed footbridge

under serviceability conditions 157

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7

Contents

6.2 Geometric characteristics of the footbridge representative of the cable-

stayed bridge typology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.3 Fundamental dynamic characteristics of the footbridge representative of

the cable-stayed bridge typology . . . . . . . . . . . . . . . . . . . . . . . . 158

6.4 Characteristics of Pedestrian Traffic . . . . . . . . . . . . . . . . . . . . . . 160

6.5 Response in service of the CSF . . . . . . . . . . . . . . . . . . . . . . . . 161

6.5.1 Structural accelerations predicted by the proposed load model . . . 162

6.5.2 Accelerations felt by users predicted by the proposed load model . . 165

6.6 Structural accelerations estimated by alternative proposals . . . . . . . . . 169

6.7 Comfort appraisal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.8 Serviceability limit state of deflections . . . . . . . . . . . . . . . . . . . . 172

6.9 Deck normal stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.10 Deck shear forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.11 Pylon stresses in serviceability . . . . . . . . . . . . . . . . . . . . . . . . . 180

6.12 Fatigue of cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180


7 Performance of cable-stayed footbridges with a single pylon: parameters

that govern serviceability response 185

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

7.2 Dynamic characteristics of pedestrian loads and the footbridge related to

its performance in service . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

7.3 Strategies to improve the vertical dynamic performance of 1T-CSFs in service187

7.3.1 Articulation of the deck . . . . . . . . . . . . . . . . . . . . . . . . 188

7.3.2 Area of backstay cable . . . . . . . . . . . . . . . . . . . . . . . . . 190

7.3.3 Area of main span stays . . . . . . . . . . . . . . . . . . . . . . . . 192

7.3.4 Material of stays: bars vs strands for the stay cables . . . . . . . . 193

7.3.5 Section of the steel girders . . . . . . . . . . . . . . . . . . . . . . . 194

7.3.6 Concrete slab section . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7.3.7 Transverse section of the pylon . . . . . . . . . . . . . . . . . . . . 197

7.3.8 Pylon height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

7.3.9 Inclination of pylon . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

7.3.10 Shape of the pylon . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

7.3.11 Cable system: anchorage spacing . . . . . . . . . . . . . . . . . . . 203

7.3.12 Cable system: transverse inclination of cables . . . . . . . . . . . . 205

7.3.13 Geometry of deck: deck width . . . . . . . . . . . . . . . . . . . . . 206

7.3.14 Side span length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

7.4 Strategies to improve the lateral dynamic performance of 1T-CSFs in service209


7.4.2 Area of the backstay cable . . . . . . . . . . . . . . . . . . . . . . . 211

7.4.3 Area of the main span stays . . . . . . . . . . . . . . . . . . . . . . 211

7.4.4 Material of stays: bars vs strands for the stay cables . . . . . . . . 212


8

Contents


7.4.7 Transverse section of the pylon . . . . . . . . . . . . . . . . . . . . 214

7.4.8 Pylon height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

7.4.9 Tower longitudinal inclination . . . . . . . . . . . . . . . . . . . . . 216

7.4.10 Pylon shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

7.4.11 Transverse inclination . . . . . . . . . . . . . . . . . . . . . . . . . 218

7.4.12 Cable anchorage distance . . . . . . . . . . . . . . . . . . . . . . . . 219

7.4.13 Geometry of deck: deck width . . . . . . . . . . . . . . . . . . . . . 219

7.4.14 Side span length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

7.5 Cable-stayed footbridges with long main span lengths . . . . . . . . . . . . 220

7.5.1 Geometry of long span cable-stayed footbridges . . . . . . . . . . . 221

7.5.2 Dynamic characteristics of long span cable-stayed footbridges . . . 221

7.5.3 Articulations of the deck . . . . . . . . . . . . . . . . . . . . . . . . 223


7.7 Additional dissipation of the serviceability movements: inherent or external

movement control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230



7.10 Deck shear stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

7.11 Normal stresses at the pylon . . . . . . . . . . . . . . . . . . . . . . . . . . 235

7.12 Performance of the stay cables . . . . . . . . . . . . . . . . . . . . . . . . . 236


8 Performance of cable-stayed footbridges with two pylons: parameters

that govern serviceability response 241

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

8.2 Geometry of conventional cable-stayed footbridges with two pylons . . . . 242

8.3 Dynamic characteristics and response in service of conventional cable-stayed

footbridges with two pylons . . . . . . . . . . . . . . . . . . . . . . . . . . 243

8.4 Principal dynamic characteristics of the pedestrian loads and the footbridge

related to its performance in service . . . . . . . . . . . . . . . . . . . . . . 244

8.5 Strategies to improve the vertical dynamic performance of 1T-CSFs in service245


8.5.2 Area of backstay cable . . . . . . . . . . . . . . . . . . . . . . . . . 246



8.5.5 Concrete slab thickness . . . . . . . . . . . . . . . . . . . . . . . . . 250

8.5.6 Transverse section of the pylons . . . . . . . . . . . . . . . . . . . . 252

8.5.7 Height of pylons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

8.5.8 Longitudinal inclination of the pylon . . . . . . . . . . . . . . . . . 255




9

Contents

8.5.12 Geometry of the deck: deck width . . . . . . . . . . . . . . . . . . . 259

8.5.13 Side span length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

8.6 Strategies to improve the lateral dynamic performance of cable-stayed foot-

bridges with two pylons in service . . . . . . . . . . . . . . . . . . . . . . . 261


8.6.2 Area of backstay cables . . . . . . . . . . . . . . . . . . . . . . . . . 262




8.6.6 Pylon section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

8.6.7 Pylon height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

8.6.8 Longitudinal inclination of the pylon . . . . . . . . . . . . . . . . . 268




8.6.12 Geometry of the deck: deck width . . . . . . . . . . . . . . . . . . . 271

8.6.13 Side span length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

8.7 Cable-stayed footbridges with long main span lengths . . . . . . . . . . . . 273

8.7.1 Geometry of long span cable-stayed footbridges with two pylons . . 273

8.7.2 Dynamic characteristics of long span cable-stayed footbridges . . . 274

8.7.3 Articulations of the deck . . . . . . . . . . . . . . . . . . . . . . . . 276

8.7.4 Dimensions of structural elements . . . . . . . . . . . . . . . . . . . 276

8.7.5 Geometric characteristics of the cable-stayed footbridge . . . . . . . 277


8.9 Additional dissipation of the serviceability movements: inherent or external

movement control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279



8.12 Deck shear stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

8.13 Normal stresses at the pylon . . . . . . . . . . . . . . . . . . . . . . . . . . 284

8.14 Performance of stay cables . . . . . . . . . . . . . . . . . . . . . . . . . . . 285


9 Conclusions and recommendations for future work 291

9.1 Summary of the developed research work . . . . . . . . . . . . . . . . . . . 291

9.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

9.2.1 Conclusions related to the state-of-the-art . . . . . . . . . . . . . . 292

9.2.2 Conclusions related to the methodology for the analysis of the re-

sponse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

9.2.3 Conclusions related to the response of girder footbridges . . . . . . 295

9.2.4 Conclusions related to the response of cable-stayed footbridges . . . 297

9.2.5 Review of the current available design guidelines . . . . . . . . . . . 307

9.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

10

Contents

Annex A 309

Annex B 313

Annex C 319

Annex D 327

Bibliography 335

11

List of Figures

1.1 (a) View of the London Millennium Bridge in the UK, and (b) the passerelle

Leopold Sedar Senghor in Paris, France (Structurae, 2015). . . . . . . . . . 38

1.2 Photo of a group of pedestrians walking, from The University of Arizona,

showing the clear differences between pedestrians which should be consid-

ered by the loading models. . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.1 (a) Tarr steps, Exmoor National Park, UK (Exmoor National Park, 2015);

(b) Glorias Catalanas footbridge, Barcelona, Spain (Structurae, 2015); (c)

Bridge of Aspiration, London, UK (Structurae, 2015). . . . . . . . . . . . . 44

2.2 (a) Plashet School footbridge (Architen Landrell, 2015); (b) Kent Messen-

ger Millennium bridge (Flint & Neill, 2015); (c) Ripshorst bridge (Struc-

turae, 2015); (d) Nesciobrug bridge (2015); (e) Dunajec cable-stayed foot-

bridge (Biliszczuk et al., 2008). . . . . . . . . . . . . . . . . . . . . . . . . 45

2.3 (a) Percentage of footbridges with different number of spans. (b) Main span

length (average, standard deviation, maximum and minimum) for two and

three span cable-stayed footbridges. . . . . . . . . . . . . . . . . . . . . . . 46

2.4 (a) Deck depth and (b) tower height hp according to main span length Lm

of cable-stayed bridges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.5 (a) Human walking cycle with events characterising gait, from Rose et al.

(1994); (b) amplitude of loads as a ratio of the pedestrian weight Wp (Fl is

the lateral, Fld is the longitudinal and Fv is the vertical pedestrian load),

from Nilsson et al. (1987). . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.6 (a) Time amplitude of vertical loads (Vaughan et al., 1987); power spectra

of pedestrian body accelerations while walking of men (b) and women (c)

(Matsumoto et al., 1978). . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.7 (a) Schematic illustration of the parameters defining the vertical load am-

plitude for a single footstep; (b) characterisation of vertical load amplitudes

according to nine parameters and fp (Butz et al., 2008). . . . . . . . . . . . 52

2.8 (a) Relationship between step length and fp of Wheeler (1982); (b) ampli-

tude of force Fourier spectrum obtained by Zivanovic et al. (2007). . . . . . 53

2.9 Time amplitude of lateral force load (Zivanovic et al., 2005). . . . . . . . . 54

2.10 Lateral equilibrium of a pedestrian represented as an inverted pendulum. . 56

13

List of Figures

2.11 Comparison of accelerations caused by a single ’perfect’ resonant pedes-

trian and those cause by real pedestrian (intra-variability) experimentally

observed by Sahnaci et al. (2005). . . . . . . . . . . . . . . . . . . . . . . . 63

2.12 (a) TMDs at LMB, Structurae (2015); (b) Leopold-Sedar-Senghor bridge,

Setra (2006); (c) VFD at LMB, Structurae (2015); (d) TLDs at T-bridge,

Nakamura et al. (2006). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.13 Bridges with large vertical movements in service: (a) Eutinger Waagsteg,

Germany (Butz et al., 2008); (b) Kochenhofsteg bridge, Germany (Butz

et al., 2008); (c) Katzbuckelbrucke bridge, Germany (Butz et al., 2008);

(d) ‘Olga’ Park bridge, Germany (Kasperski, 2006);(e) Erzbahnschwinge

bridge, Germany (Kasperski, 2006). . . . . . . . . . . . . . . . . . . . . . . 76

2.14 Bridges with large lateral movements in service: (a) footbridge over the

Main at Erlach, Switzerland (Franck, 2009); (b) T-bridge, Japan (P. Fujino

et al., 1993); (c) M-bridge, Japan (Nakamura et al., 2006); (d) Leopold-

Sedar-Senghor, France (Setra, 2006); (e) London Millennium Bridge, UK

(Dallard et al., 2001); (f) Lardal footbridge, Norway (Ronnquist et al., 2008). 79

2.15 Bridges with large lateral movements in service: (a) Changi Mezzanine

bridge, Singapore (Brownjohn et al., 2004a); (b) Passerelle Simone de Beau-

voir, France (Hoorpah et al., 2008); (c) Tri-Countries, Germany (Haberle,

2010) ; (d) Pedro e Ines footbridge, Portugal (Adao Da Fonseca et al., 2005). 80

3.1 Normalised ground reaction forces defined using 8th order polynomial func-

tions for different step frequencies (a); comparison of vertical loads defined

with 8th order polynomial functions and three sinusoids, fp = 2.0Hz (b). . 88

3.2 Comparison of vertical accelerations generated by vertical loads defined

with 8th order polynomial functions or three sinusoids (fp = 2.0Hz). . . . . 88

3.3 Definition of the pedestrian gait parameters (step frequency and free ve-

locity according to age and height of the subject). . . . . . . . . . . . . . . 92

3.4 Step frequencies distribution according to density and aim of the journey. . 93

3.5 Correlation between pedestrian velocity and step width ws,t. . . . . . . . . 94

3.6 Summary of the proposed load model (ped represents pedestrian). . . . . . 97

3.7 Comfort criteria for walking pedestrians for (a) vertical and (b) lateral

accelerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.8 (a) Span layout geometries; (b) deck transverse sections and structural

materials considered for girder footbridges. . . . . . . . . . . . . . . . . . . 102

3.9 Displacements and rotations of GFBs. . . . . . . . . . . . . . . . . . . . . 103

3.10 Elevation and transverse sections of benchmark cable-stayed footbridges:

1 Tower (top) and 2 Towers (bottom). . . . . . . . . . . . . . . . . . . . . 104

3.11 Basic an alternative tower shapes: 1) I tower shape, 2) H tower shape, 3)

H shape with a crossing brace and 4) A tower shape. . . . . . . . . . . . . 105

3.12 Depth-to-span length ratios adopted in existing CSFs (black dots) and

ratios of benchmark CSFs (red dots) according to main span length. . . . . 105

3.13 Cable anchorages: (a) bearing socket; (b) fork socket. . . . . . . . . . . . . 107

14

List of Figures

3.14 Geometry of a tensioned stay cable under self-weight. . . . . . . . . . . . . 109

3.15 Wohler-Curves for stay cables: strands and bars (fib Bulletin 30, 2005) . . 109

3.16 Rayleigh damping considered in benchmark CSFs, where ω1 and ω2 are the

undamped natural circular frequencies of modes 2.0 and 6.0 Hz. . . . . . . 110

3.17 Summary of footbridges whose behaviour in serviceability limit state is

thoroughly evaluated in this thesis. . . . . . . . . . . . . . . . . . . . . . . 111

3.18 Discretisation of a GFB structure (elevation, left plot, and transverse sec-

tion, right plot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.19 (a) CSF transverse section and (b) section numerical representation. . . . . 113

3.20 Differences in maximum vertical accelerations (ǫ) at different points of the

deck, according to element mesh size. . . . . . . . . . . . . . . . . . . . . . 114

3.21 Differences in maximum cable stresses (ǫ) at different stayed cables, ac-

cording to element mesh size. . . . . . . . . . . . . . . . . . . . . . . . . . 114

3.22 (a) TMD placed at London Millennium Bridge; (b) numerical representa-

tion of TMD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

3.23 Schematic implementation of UAMP subroutine of Abaqus. . . . . . . . . . 116

3.24 Vertical and lateral RMS accelerations recorded at the deck of a CSF, x =

28.0 and 30.0 m, caused by 5 different pedestrian events with commuters

and density 0.6 ped/m2 (tap describes the time taken by an average pedes-

trian to cross the bridge). . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.25 Recording of accelerations felt by pedestrians; RS describes the right step,

LS the left step, tsfc the time of single foot contact and tdsc the time of

double stance contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.1 Effects of step frequency variability: (a) mean vertical accelerations, (b)

maximum vertical accelerations, (20 simulations of the same event with

fp = 1.8 Hz), (c) detailed description of maximum accelerations around

fs = fp, (d) detailed description of maximum accelerations around fs = 2fpand fs = 3fp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.2 Vertical maximum accelerations generated by loads defined with the pro-

posed model (with σfp = 0.0 Hz) for: (a) fs/fp = 1; or, (b) fs/fp = 2. . . . 126

4.3 Effects of step frequency variability on structural vertical response in sce-

narios with multiple pedestrians: maximum vertical accelerations (50 sim-

ulations of the same event); Ct. corresponds to constant step frequency of

each pedestrian and Var. variable step frequency (µfp = fs and σfp = 0.10

Hz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.4 Maximum midspan lateral accelerations of simply supported structures un-

der single pedestrian loads defined by the new load model with σfp,l = 0.0

(valid for any step frequency). . . . . . . . . . . . . . . . . . . . . . . . . . 128

15

List of Figures

4.5 Effects of step frequency variability on lateral response caused by a single

pedestrian (20 simulations of the same event); the black line correspond to

results of constant step frequency fp, the red line to the maximum accelera-

tions of variable step frequency and the grey line to the mean accelerations

of variable step frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.6 Effects of step frequency variability on lateral response caused multiple

pedestrians scenarios: maximum lateral accelerations (50 simulations), where

Ct. corresponds to constant step frequency of each pedestrian and Var.

variable step frequency (µfpl = fs and σfp = 0.10 Hz). . . . . . . . . . . . . 129

4.7 Effects of step width variability on lateral response caused by a single pedes-

trian (20 simulations of the same event), where the black line correspond

to results of constant step width ws,t, the red line to the maximum acceler-

ations with variable step width and the grey line to the mean accelerations

with variable step width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.8 Sensitivity analysis of vertical load amplitude (φ describes the relative sen-

sitivity of response with respect to each parameter, see Equation 4.3.1)

(parameters defined in Figure 2.7). . . . . . . . . . . . . . . . . . . . . . . 131

4.9 Effects of the variability in the definition of vertical load amplitudes, Ct.

(constant) and Var. (variable), among pedestrians (maximum vertical ac-

celerations at midspan of a simply supported structure). . . . . . . . . . . 133

4.10 Effects of variability of weight, Ct. (constant) and Var. (variable), among

pedestrians (maximum vertical accelerations at midspan of a simply sup-

ported structure). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.11 Effects of variability of step frequency, according to traffic type, among

pedestrians (maximum vertical accelerations at midspan of a simply sup-

ported structure, with fs = 2.0 Hz). . . . . . . . . . . . . . . . . . . . . . . 134

4.12 Effects of step width variability (among pedestrians in a flow) on response

in multiple pedestrian scenarios, where Ct. ws,t represents pedestrian flows

where all pedestrians have the same initial half-step width whereas Var.

ws,t represents the results of pedestrian flows where each pedestrian has a

random half-step width (according to normal distribution). . . . . . . . . . 135

4.13 Effects of collective behaviour simulation, where Setra corresponds a pedes-

trian events characterised according to Setra guideline, N.M. defines pedes-

trian flows where loads are described according to the proposed new load

model, Ct. or Var. refer to constant or variable step intra-subject frequency

and Crowd int. refers to collective behaviour. . . . . . . . . . . . . . . . . 137

5.1 Summary of geometric properties, usual materials (RC and PC stands for

reinforced and prestressed concrete, respectively) and span ranges for dif-

ferent footbridge sections. The slab defining the decking is part of the

structural cross section in sections S.1-S.5, and a non-structural element

for sections S.6-S.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.2 Amplitude of φs,n, according to mode, n, and number of spans. . . . . . . . 143

16

List of Figures

5.3 Values for φsm, according to mode and spans arrangement. . . . . . . . . . 147

5.4 Amplification factor β for lateral response, x = Nφpm. . . . . . . . . . . . . 149

5.5 Comparison of vertical response of two-span bridges, L+ 0.8L. . . . . . . 151

5.6 Comparison of lateral response of simply supported bridges. . . . . . . . . 151

5.7 Evaluation of serviceability of simply supported structures in the vertical

and lateral directions under pedestrian streams of density 0.6 ped/m2 with

commuting or leisure aim of the journey. Section S.6 has non-structural

concrete deck and sections S.7 to S.9 have non-structural wooden decks. . . 153

6.1 (a) Geometric and structural characteristics of the conventional cable-

stayed footbridge; (b) articulation of the footbridge deck (movements re-

stricted by supports). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.2 Modal shapes of the first 16 modes of the conventional cable-stayed foot-

bridge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.3 Distributions of step frequencies adopted by commuters in flows of 0.2, 0.6

or 1.0 ped/m2 (blue bars correspond to simulated events and red lines to

the prediction according to Figure 3.4). . . . . . . . . . . . . . . . . . . . . 162

6.4 Distributions of step frequencies adopted by leisure pedestrians in flows of

0.2, 0.6 or 1.0 ped/m2 (blue bars correspond to simulated events and red

lines to the prediction according to Figure 3.4). . . . . . . . . . . . . . . . 162

6.5 Peak and 1s-RMS vertical accelerations recorded at the CSF deck generated

by commuter or leisure flows of 0.2, 0.6 or 1.0 ped/m2. The origin for the

abscissa axis is located at the support section of the side span on the

abutment, see Figure 6.1(a). . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.6 Fourier amplitudes [m/s] of the vertical acceleration response of the CSF

at x = 28.0 m under the action of commuter or leisure flows with 0.6 ped/m2.163

6.7 Peak and 1s-RMS lateral accelerations recorded at the CSF deck generated

by commuter or leisure flows of 0.2, 0.6 or 1.0 ped/m2. The origin for

the abscissa axis is located at the support section of the side span on the

abutment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.8 Fourier amplitudes [m/s] of the lateral acceleration response of the CSF at

x = 33.0m under the action of commuter or leisure flows with 0.6 ped/m2. 165

6.9 Relative, compared to amax,P (defined in legend for each scenario), (a) and

absolute (b) maximum vertical accelerations felt by walking pedestrians

vs cumulative number of users that feel maximum vertical acceleration

(according to type and density of flow). . . . . . . . . . . . . . . . . . . . . 166

6.10 Percentage of time and of main span surface for which the maximum ac-

celerations felt by users are larger than the value indicated in the contour

curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.11 Relative (a) and absolute (b) maximum lateral accelerations felt by walking

pedestrians vs cumulative number of users that feel the maximum lateral

acceleration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

17

List of Figures

6.12 Percentage of the time and of the main span surface for which the maximum

lateral accelerations felt by users are larger than the value indicated in the

contour curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.13 Comfort of walking pedestrians due to vertical accelerations according to

traffic scenario and representative acceleration magnitude for the event (‘C’

represents commuter flows and ‘L’ leisure flows and there are two limits for

ranges of unacceptable accelerations, that of Setra and that of NA to BS

EC1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.14 Comfort of walking pedestrians due to lateral accelerations according to

traffic scenario and representative acceleration magnitude for the event. . . 172

6.15 Comfort of standing and sitting pedestrians in the vertical (a) or lateral

direction (b), according to the traffic scenario. . . . . . . . . . . . . . . . . 172

6.16 Dynamic and equivalent static vertical deflections caused by pedestrian

flows with 0.2 ped/m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173


flows with 0.6 ped/m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173


flows with 1.0 ped/m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.19 Dynamic and equivalent static lateral deflections caused by pedestrian flows

with 0.2 ped/m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174


with 0.6 ped/m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174


with 1.0 ped/m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.22 (a) Comparison between peak vertical deflections and peak accelerations

described at the same events, (b) DAFs related to vertical deflections at x =

45.0 and pedestrian flow density causing these dynamic deflections, and (c)

comparison between peak lateral deflections and peak lateral accelerations

at the same events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.23 Dynamic and equivalent static bending moments caused by pedestrian flows

with 0.2 ped/m2 along the length of the deck. . . . . . . . . . . . . . . . . 177





6.26 (a) Comparison between DAFs related to BMs at x = 50.0 m and pedestrian

traffic density causing these dynamic stresses, and (b) similar correlation

considering the weight of the traffic flow compared to the weight of the live

load of ULS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.27 Dynamic and corresponding static shear forces generated by pedestrian

flows with 0.2 ped/m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

18

List of Figures

6.28 Dynamic and corresponding static shear forces generated by pedestrian

flows with 0.6 ped/m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

6.29 Dynamic and equivalent static shear forces generated by pedestrian flows

with 1.0 ped/m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

6.30 (a) Comparison between DAFs related to shear forces at x = 57.5 m and

pedestrian traffic density causing these dynamic stresses, and (b) similar

correlation considering the weight of the traffic flow. . . . . . . . . . . . . . 179

6.31 Maximum dynamic bending moments and axial loads along the height of

the pylon generated by the different traffic scenarios (the intersection of

the pylon with the deck is at 7.5 m high). . . . . . . . . . . . . . . . . . . 180

6.32 Maximum stress variations of the backstay and main span cables generated

by light pedestrian flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181


by medium-density pedestrian flows. . . . . . . . . . . . . . . . . . . . . . 181


by heavy pedestrian flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.1 Energy amplitude [N] of the total vertical loads introduced by the pedes-

trian flow while stepping at the deck between 25 m ≤ x ≤ 30 m. . . . . . . 187

7.2 Plan view of the support configurations of the CSF with LEB bearing

schemes or POT bearing schemes. (a) 2 LEBs and a SK per abutment,

(b) 2 LEBs at each abutment, (c) ‘classical’ POT arrangement and (d)

statically indeterminate POT arrangement. . . . . . . . . . . . . . . . . . . 188

7.3 Peak and 1s-RMS vertical accelerations recorded at the deck of CSFs with

support schemes (a)-(d) according to Figure 7.2. Peak accelerations at the

centre line in scheme (c) have been included for comparison purposes. . . . 189

7.4 Vertical accelerations felt by users amax,P i compared to amax,P . Curves

defined for CSFs with different support conditions (a)-(d). . . . . . . . . . 190

7.5 Static and dynamic behaviour of the CSFs in terms of backstay area ABS

(compared to that of the benchmark CSF ABS,0): (a) main span maximum

static deflections umax (compared to the deflection at the basic CSF umax,0)

and (b) frequencies [Hz] of vertical and torsional modes. . . . . . . . . . . 190

7.6 Vertical service response of the CSF deck according to backstay area ABS:

(a) peak and 1s-RMS vertical accelerations and (b) comparison of maxi-

mum absolute peak and 1s-RMS accelerations to those of the reference case

acc0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

7.7 Vertical accelerations felt by users amax,P i compared to amax,P of the refer-

ence case. Curves defined for CSFs with different backstay areas. . . . . . 191

7.8 Static and dynamic behaviour of the CSF in terms of stay cables area AS

(compared to that of the benchmark CSF AS,0): (a) main span maximum

static deflections umax and (b) frequencies [Hz] of vertical and torsional

modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

19

List of Figures

7.9 Vertical service response of the CSF deck according to area of cables AS: (a)

peak and 1s-RMS vertical accelerations and (b) comparison of maximum

absolute peak and 1s-RMS accelerations to those of the benchmark case

acc 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193


defined for CSFs with different areas of stay cables. . . . . . . . . . . . . . 193

7.11 Static and dynamic behaviour of the CSF in terms of flange girder thick-

ness t bf (compared to that of the benchmark CSF t bf,0): (a) main span

maximum static deflections umax and (b) frequencies [Hz] of vertical and

torsional modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

7.12 Vertical service response of the CSF deck according to bottom flange depth:


mum absolute peak and 1s-RMS accelerations to those of the benchmark

case acc 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195


defined for CSFs with different girder bottom flange. . . . . . . . . . . . . 195

7.14 (a) Transverse section of the deck; (b) dynamic behaviour of the CSF in

terms of slab depth tc: frequencies [Hz] of vertical and torsional modes. . . 196

7.15 Vertical service response of the CSF deck according to depth of the con-

crete slab: (a) peak and 1s-RMS vertical accelerations and (b) comparison

of maximum absolute peak and 1s-RMS accelerations to those of the bench-

mark case acc 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196


defined for CSFs with different slab depths. . . . . . . . . . . . . . . . . . 197

7.17 Dynamic behaviour of the CSF according to diameter of the pylon: fre-

quencies [Hz] of vertical and torsional modes (V2b and T2b are additional

modes). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

7.18 Vertical service response of the CSF deck according to pylon diameter Dt:


mum absolute peak and 1s-RMS accelerations to those of the benchmark

case acc 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

7.19 Fourier spectrum of the vertical acceleration response at x = 28 m of CSF

with pylon diameter 1.7Dt,0 or 2.5Dt,0. . . . . . . . . . . . . . . . . . . . . 198


defined for CSFs with pylon diameters. . . . . . . . . . . . . . . . . . . . . 199

7.21 Static and dynamic behaviour of the CSF in terms of pylon height hp: (a)

main span maximum static deflections umax and (b) frequencies [Hz] of

vertical and torsional modes. . . . . . . . . . . . . . . . . . . . . . . . . . . 199

7.22 Vertical service response of the CSF deck according to pylon height hp: (a)


absolute peak and 1s-RMS accelerations to those of the benchmark case

acc 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

20

List of Figures


defined for CSFs with shorter or higher pylons. . . . . . . . . . . . . . . . 200

7.24 Static and dynamic behaviour of the CSF according to pylon inclination α:

(a) main span maximum static deflections umax and (b) frequencies [Hz] of


7.25 Vertical service response of the CSF deck according to pylon inclination α:


mum absolute peak and 1s-RMS accelerations to those of the reference case

acc0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201


defined for CSFs with pylons inclined towards the side span or main span. 202

7.27 Shapes of CSF pylons: (a) mono-pole pylon, (b) two free-standing poles

pylon, (c) portal shape pylon, (d) ‘A’ shape pylon. . . . . . . . . . . . . . . 202

7.28 Vertical service response of the CSF deck according to pylon shape: (a)


absolute peak and 1s-RMS accelerations to those of the benchmark case acc0.203


defined for CSFs with different pylon shapes. . . . . . . . . . . . . . . . . . 203

7.30 (a) Cable-stayed footbridge geometry according to anchorage of stays and

(b) frequencies [Hz] of vertical and torsional modes. . . . . . . . . . . . . . 204

7.31 Vertical service response of the CSF deck according to cable anchorage dis-

tance: (a) peak and 1s-RMS vertical accelerations and (b) comparison of

maximum absolute peak and 1s-RMS accelerations to those of the bench-

mark case acc0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204


defined for CSFs with alternative stays anchorage spacing. . . . . . . . . . 204

7.33 Maximum static deflections umax at the main span according to lateral

inclination of ‘H’ pylon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

7.34 Vertical service response of the CSF deck according to pylon lateral inclina-

tion α: (a) peak and 1s-RMS vertical accelerations and (b) comparison of


mark case acc0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205


defined for CSFs with stays laterally inclined. . . . . . . . . . . . . . . . . 206

7.36 Dynamic behaviour of the CSF according to deck width: frequencies [Hz]

of vertical and torsional modes. . . . . . . . . . . . . . . . . . . . . . . . . 206

7.37 Vertical service response of the CSF deck according to deck width dimen-

sion: (a) peak and 1s-RMS vertical accelerations and (b) comparison of


mark case acc 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207


defined for CSFs with wider decks. . . . . . . . . . . . . . . . . . . . . . . 207

21

List of Figures

7.39 Static and dynamic behaviour of the CSF according to side span length Ls:

(a) main span maximum static deflections umax and (b) frequencies [Hz] of


7.40 Vertical service response of the CSF deck according to side span length

Ls: (a) peak and 1s-RMS vertical accelerations and (b) comparison of

maximum absolute peak and 1s-RMS accelerations to those of the reference

case acc 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208


defined for CSFs with longer side spans. . . . . . . . . . . . . . . . . . . . 208

7.42 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations

recorded at the deck of CSFs with support schemes (a), (c) and (d); and

(b) lateral accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . 209

7.43 (a) Absolute peak lateral accelerations recorded at the deck of CSFs with

LEBs; (b) time history acceleration at x = 60 m developed at the CSF with

LEBs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210


recorded at the CSF deck according to backstay area and (b) lateral accel-

erations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211


recorded at the CSF deck according to main span stay area and (b) lateral

accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 212


recorded at the CSF deck according to bottom flange thickness and (b)

lateral accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . 213


recorded at the CSF deck according to concrete slab depth and (b) lat-

eral accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . 214


recorded at the CSF deck according to pylon diameter and (b) lateral ac-

celerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215


recorded at the CSF deck according to pylon height and (b) lateral ac-



recorded at the CSF deck according to pylon longitudinal inclination and



recorded at the CSF deck according to pylon shape and (b) lateral ac-



recorded at the CSF deck according to pylon transverse inclination and


22

List of Figures


recorded at the CSF deck according to deck width and (b) lateral acceler-

ations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220


recorded at the CSF deck according to side span length and (b) lateral


7.55 Geometric definition of the representative long span CSFs with transverse

section depth Lm/100 and Lm/200. Dimensions in meters [m]. . . . . . . . 222

7.56 First modal frequencies [Hz] of long span CSF with a deck depth of Lm/100.223

7.57 First modal frequencies [Hz] of long span CSF with a deck depth of Lm/200.223

7.58 Peak vertical (a) and lateral (b) accelerations recorded at the deck of CSFs

with main span length 100 m and support scheme (d). . . . . . . . . . . . 224

7.59 Peak vertical (a) and lateral (b) accelerations recorded at benchmark CSF

or CSF with smaller backstay (0.5ABS,0), with depths Lm/100 and Lm/200. 225


or CSF with larger stays (2.5AS,0), with depths Lm/100 and Lm/200. . . . 226


or CSF with slab depth 2tc,0, with depths Lm/100 and Lm/200. . . . . . . 226


or CSF with shorter pylon (0.25Lm), with depths Lm/100 and Lm/200. . . 227


or CSF with inclined 20o towards the side span, with depths Lm/100 and

Lm/200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227


or CSF with deck width of 5 m, with depths Lm/100 and Lm/200. . . . . . 228

7.65 Comfort assessment of CSF according to the measures implemented to

modify vertical response (where Basic refers to the reference CSF, BC to

deck articulation, BS to backstay, S to main span stays, tf to thickness of

the bottom flange of the steel girder, tc to the thickness of the concrete

slab, hp to the height of the pylon, Inc. to the inclination of the pylon,

‘Pylon’ to its shape, Anch. to the distance between stay anchorage, L. Inc.

to lateral inclination of stays, wd to deck width and Ls to side span length). 228


modify lateral response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

7.67 Comfort assessment of the long span CSF according to the measures im-

plemented to modify vertical and lateral response. . . . . . . . . . . . . . . 229

7.68 Comparison of maximum vertical and lateral movements recorded at the

deck and maximum accelerations felt by 75% of the walking pedestrians. . 229

7.69 (a) Absolute peak vertical and lateral accelerations recorded at medium

span length reference CSF with higher inherent damping ζ = 0.6%, with

TMD located at x = 28 m (D1) or at x = 49 m (D2); (b) accelerations

noticed by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

23

List of Figures

7.70 (a) Static and maximum dynamic vertical deflections generated by medium-

high density pedestrian flows on medium span length CSFs; (b) relationship

between peak vertical (top, y = V) or lateral (bottom, y = L) dynamic

deflections generated by pedestrians and corresponding peak accelerations. 232

7.71 Static bending moments (BM) of the deck produced by the weight of a

flow with 0.6 ped/m2 and dynamic bending moments (and DAFs related

to these) generated by the dynamic actions of this flow at CSFs with al-

ternative dimensions or geometry. . . . . . . . . . . . . . . . . . . . . . . . 233

7.72 (a) Maximum DAFs related to deflections and (b) DAFs related bending

moments according to pedestrian flow density. . . . . . . . . . . . . . . . . 234

7.73 Static shear forces (SF) at the steel girders of the deck generated by the

weight of a flow with 0.6 ped/m2 and dynamic shear forces produced by

the dynamic actions of this flow at CSFs with alternative dimensions or

geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

7.74 Dynamic bending moments and axial forces at critical sections of the pylon

of CSFs with alternative dimensions or geometry. . . . . . . . . . . . . . . 235

7.75 Comparison of accumulated damage at each stay of the CSF produced

at CSFs with geometric and structural characteristics detailed in previous

sections (compared to accumulated damage of stay cables of the benchmark

CSF). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

8.1 Geometry and structural characteristics of CSF with two pylons and trans-

verse section depth Lm/100. Dimensions in meters [m]. . . . . . . . . . . . 242

8.2 Modal frequencies of CSFs with two pylons. . . . . . . . . . . . . . . . . . 243

8.3 Peak vertical (a) and lateral (b) accelerations described at the deck of the

conventional CSF with two pylons. . . . . . . . . . . . . . . . . . . . . . . 244

8.4 Plan view of the support configurations of the CSF with LEB bearing

schemes or POT bearing schemes. (a) ‘classical’ POT arrangement (ar-

rangement of the benchmark 2T-CSF), (b) 2 LEBs at each abutment, (c)

2 LEBs and a SK, (d) statically indeterminate POT arrangement and (e)

POT support scheme with unrestricted longitudinal movements. . . . . . . 246

8.5 (a) Absolute peak vertical, and relative peak and 1s-RMS accelerations

recorded at the 2T-CSFs deck according to support schemes (a)-(e); and

(b) vertical accelerations felt by users. . . . . . . . . . . . . . . . . . . . . 247

8.6 (a) Static and dynamic behaviour of the 2T-CSF in terms of the backstay

area ABS (compared to that of the benchmark CSF ABS,0): (a) main span


torsional modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248


recorded at the 2T-CSFs deck according to backstay area and (b) vertical


24

List of Figures

8.8 (a) Static and dynamic behaviour of the 2T-CSF according to stays area AS

(compared to that of the benchmark CSF AS,0): (a) main span maximum


modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249


recorded at the 2T-CSFs deck according to stays area and (b) vertical


8.10 (a) Static and dynamic behaviour of the 2T-CSF according to thickness of

bottom flange tbf (compared to that of the benchmark CSF tbf,0): (a) main

span maximum static deflections umax and (b) frequencies [Hz] of vertical

and torsional modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250


recorded at the 2T-CSFs deck according to bottom flange of steel girder

and (b) vertical accelerations felt by users. . . . . . . . . . . . . . . . . . . 251

8.12 (a) Transverse section of the deck; (b) frequencies of vertical and torsional

modes (for a slab depth of 1.75tc,0, the sudden change of the frequency of

mode T2 concurs with the coincidence in frequencies of modes V2 and L2). 251


recorded at the 2T-CSFs deck according to concrete slab thickness and (b)

vertical accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . 252

8.14 (a) Static and dynamic behaviour of the 2T-CSF according to pylon di-

ameter Dt (compared to that of the benchmark CSF Dt,0): (a) main span


torsional modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253


recorded at the 2T-CSFs deck according to pylon diameter and (b) vertical


8.16 (a) Static and dynamic behaviour of the 2T-CSF according to pylon height

hp (compared to that of the benchmark CSF): (a) main span maximum


modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254


recorded at the 2T-CSFs deck according to pylon height and (b) vertical


8.18 (a) Static and dynamic behaviour of the 2T-CSF according to pylon longi-

tudinal inclination α (compared to that of the benchmark CSF): (a) main

span maximum static deflections umax and (b) frequencies [Hz] of vertical

and torsional modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255


recorded at the 2T-CSFs deck according to pylon longitudinal inclination


25

List of Figures


recorded at the 2T-CSFs deck according to pylon shape and (b) vertical



recorded at the 2T-CSFs deck according to lateral inclination of stays and

(b) vertical accelerations felt by users. . . . . . . . . . . . . . . . . . . . . 258

8.22 Frequencies [Hz] of vertical and torsional modes of CSFs with anchorages

of stays spaced different distances. . . . . . . . . . . . . . . . . . . . . . . . 258


recorded at the 2T-CSFs deck according to distance of cable anchorages


8.24 Dynamic behaviour of the 2T-CSF according to deck width: frequencies

[Hz] of vertical and torsional modes. . . . . . . . . . . . . . . . . . . . . . . 259


recorded at the 2T-CSFs deck according to deck width and (b) vertical


8.26 (a) Static and dynamic behaviour of the 2T-CSF according to side span

length Ls (compared to that of the benchmark CSF): (a) main span max-

imum static deflections umax and (b) frequencies [Hz] of vertical and tor-

sional modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260


recorded at the 2T-CSFs deck according to side span length and (b) vertical



recorded at the 2T-CSFs deck with alternative support schemes and (b)

lateral accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . 262


recorded at the 2T-CSFs deck according to backstay area and (b) lateral



recorded at the 2T-CSFs deck according to main stays area and (b) lateral



recorded at the 2T-CSFs deck according to bottom flange steel girder thick-

ness and (b) lateral accelerations felt by users. . . . . . . . . . . . . . . . . 265


recorded at the 2T-CSFs deck according to concrete slab thickness and



recorded at the 2T-CSFs deck according to pylon diameter and (b) lat-

eral accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . 267

26

List of Figures


recorded at the 2T-CSFs deck according to pylon height and (b) lateral



recorded at the 2T-CSFs deck according to pylon longitudinal inclination

and (b) lateral accelerations felt by users. . . . . . . . . . . . . . . . . . . . 269


recorded at the 2T-CSFs deck according to pylon shape and (b) lateral



recorded at the 2T-CSFs deck according to lateral pylon inclination and



recorded at the 2T-CSFs deck according to cable anchorage spacing and



recorded at the 2T-CSFs deck according to deck width and (b) lateral



recorded at the 2T-CSFs deck according to side span length and (b) lateral


8.41 Geometric definition of the representative long span CSFs with transverse

section depth Lm/100 and Lm/200. Dimensions in meters [m]. . . . . . . . 274

8.42 First modal frequencies [Hz] of the long span CSF with a deck depth of

Lm/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

8.43 First modal frequencies [Hz] of the long span CSF with a deck depth of

Lm/200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

8.44 Peak vertical (a) and lateral (b) accelerations recorded at CSF with con-

crete slab of 0.3 m and deck depths Lm/100 and Lm/200 (continuous lines). 277


modify vertical accelerations. . . . . . . . . . . . . . . . . . . . . . . . . . . 278


modify lateral accelerations. . . . . . . . . . . . . . . . . . . . . . . . . . . 279

8.47 Comparison of maximum vertical and lateral movements recorded at the

deck and maximum accelerations felt by 75% of the walking pedestrians. . 279

8.48 (a) Static and maximum dynamic vertical deflections generated by medium-

high density pedestrian flows on medium span length CSFs; (b) relationship

between peak vertical dynamic deflections at x = 29.5 m and concomitant

peak vertical accelerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

27

List of Figures

8.49 (a) Comparison of peak lateral dynamic deflections at x = 36.5 m and

concomitant peak lateral accelerations at scenarios with stable lateral re-

sponse; (b) similar comparison including scenarios with unstable lateral

response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

8.50 (a) Static and dynamic moments described at the deck of 2T-CSFs; and

(b) comparison of the peak vertical accelerations recorded at the deck and

DAFs related to hogging and sagging bending moments (HBMs, SBMs) at

x = 26.5 and 43.5 m at corresponding scenarios. . . . . . . . . . . . . . . . 283

8.51 (a) Static and dynamic shear forces (SFs) described at the deck of the 2T-

CSFs; and (b) comparison of the peak vertical accelerations recorded at the

deck and DAFs related to shear forces at x = 0 and 70 m at corresponding

scenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

8.52 Static and dynamic axial forces (N) or bending moments (BMs) at the

pylons of the benchmark 2T-CSFs and dynamic N/BMs at alternative 2T-

CSFs. (a) Static and dynamic BMs, where (*) represents dynamic BMs at

footbridges with larger pylon diameter or side span length; and (b) static

and dynamic N. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

8.53 Comparison of accumulated damage at each stay of the CSF produced CSFs

with geometric and structural characteristics detailed in previous sections

(compared to accumulated damage of stay cables of the benchmark CSF). . 286

28

List of Tables

2.1 Values of the damping ratio according to structural materials of the foot-

bridge, where [1] corresponds to Blanchard et al. (1977), [2] to Bachmann

et al. (1995) and [3] to BSI (2003). . . . . . . . . . . . . . . . . . . . . . . 69

3.1 Median estimates of the parameters of Figure 2.7 in terms of fp (load

amplitudes are normalised to the pedestrian weight Wp and the time terms

are related to the total load time). . . . . . . . . . . . . . . . . . . . . . . . 87

3.2 Values of the factor φj for different journey contexts . . . . . . . . . . . . . 92

3.3 Summary of the geometric characteristics of the cable-stayed footbridges. . 106

3.4 Summary of the characteristics of the concrete employed in the deck of the

cable-stayed footbridges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.5 Summary of the characteristics of the steel employed in the deck longitu-

dinal and transverse beams and tower of the cable-stayed footbridges. . . . 108

3.6 Summary of the characteristics of the steel employed in the strands of

the stayed cables (where the subindex p corresponds to prestressed steel).

(*) Density including mass of stay protection, considered from BBR VL

International Ltd. (2011). . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.7 Summary of the characteristics of the steel employed in the stayed cables

as bars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.8 Summary of service events considered to evaluate the fatigue performance

of the stay cables (C describes commuter events and L leisure events). . . . 121

4.1 Effect of the variability in temporal parameters . . . . . . . . . . . . . . . 132

5.1 Coefficients for obtaining φsl in Equation 5.7.4 for vertical response (y = v),

where x is the ratio between the pedestrian step and the span length, i is

a natural number greater than 2. . . . . . . . . . . . . . . . . . . . . . . . 145

5.2 Coefficients for obtaining φsl in Equation 5.7.4 for horizontal response (y =

l), where x is the ratio between the pedestrian step and the span length, i

is a natural number greater than 1, and j = 2i− 1 . . . . . . . . . . . . . . 145

5.3 Coefficients of φd for Equation 5.7.7 and vertical response. . . . . . . . . . 147

5.4 Coefficients of φd for Equation 5.7.7 and lateral response, where j = 2i− 1

and i is a natural number. . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

29

List of Tables

6.1 Vibration modes and frequencies (Hz) of the CSF, where ‘VN’, ‘TN’ and

‘LN’ denote vertical, torsional and lateral modes with N half-waves in the

main corresponding structural span (i.e., from the pylon to the abutment

support section, for vertical and torsional modes; and between abutment

support sections, for lateral modes), ‘Ld’ longitudinal and ‘P’ pylon modes. 160

6.2 Maximum absolute vertical accelerations [m/s2] at the deck (Max. Deck),

maximum average vertical acceleration felt by walking users (Max. Av.

Ped.) and minimum peak vertical acceleration felt by 50% (aP1), 25%

(aP2) or 5% (aP3) of the users according to the traffic scenario. . . . . . . . 168

6.3 Maximum absolute lateral accelerations [m/s2] at the deck (Max. Deck),

maximum average lateral acceleration felt by walking users (Max. Av.

Ped.) and minimum peak lateral acceleration felt by 50% (aP1), 25% (aP2)

or 5% (aP3) of the users according to the traffic scenario. . . . . . . . . . . 169

6.4 Comparison of the cable-stayed footbridge performance in the vertical di-

rection estimated by alternative proposals. . . . . . . . . . . . . . . . . . . 170

6.5 Comparison of the cable-stayed footbridge performance in the lateral di-

rection estimated by alternative proposals. . . . . . . . . . . . . . . . . . . 171

6.6 Fatigue performance of the stay cables (C describes commuter events and

L leisure events): effects of the density. Values calculated according to

Equation 3.7.3 of Section 3.7.6. . . . . . . . . . . . . . . . . . . . . . . . . 182

6.7 Fatigue performance of the stay cables: effects of the aim of the journey

(see Equation 3.7.3 of Section 3.7.6). . . . . . . . . . . . . . . . . . . . . . 182

6.8 Fatigue performance of cables of bridges with different usages (described

in Table 3.8 of Chapter 3). . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

7.1 Frequencies [Hz] for the vertical and torsional vibration modes of CSFs

for different support arrangements (defined in Figure 7.2), where ‘VN’ and

‘TN’ denote vertical and torsional modes with N half-waves. . . . . . . . . 189

7.2 Frequencies [Hz] for the vertical and torsional vibration modes of CSFs

with strand stay or bar cables. . . . . . . . . . . . . . . . . . . . . . . . . . 194

7.3 Frequencies [Hz] of vertical and torsional modes of the CSF according to

pylon shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

7.4 Frequencies [Hz] of vertical and torsional modes of the CSF according to

pylon lateral inclination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

7.5 First lateral vibration modes [Hz] of CSFs according to support arrangement.210

7.6 Frequencies [Hz] of the vibration modes of long span CSFs according to

their depth magnitude, where ‘VN’, ‘LN’ and ‘TN’ denote vertical, lateral

and torsional modes with N half-waves and ‘P’ denotes modes related to

the pylon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

7.7 Frequencies [Hz] of lateral modes of long span CSFs according to deck

articulation and depth magnitude. . . . . . . . . . . . . . . . . . . . . . . . 224

30

List of Tables

8.1 Frequencies [Hz] for the vertical, lateral and torsional modes of conventional

CSF with two pylons, where ‘VN’, ‘LN’ and ‘TN’ denote vertical, lateral

and torsional modes with N half-waves and ‘P’ denotes modes involving

the pylons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

8.2 Frequencies [Hz] of vertical and torsional vibration modes of CSFs accord-

ing to support arrangement, described in Figure 8.4, where ‘VN’ and ‘TN’

denote vertical and torsional modes with N half-waves. . . . . . . . . . . . 247

8.3 Frequencies [Hz] of the vertical and torsional modes of CSF according to

the shape of pylons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

8.4 Frequencies [Hz] of vertical and torsional modes of CSF according to the

lateral inclination of pylons. . . . . . . . . . . . . . . . . . . . . . . . . . . 257

8.5 Frequencies [Hz] of lateral vibration modes of 2T-CSFs according to sup-

port arrangement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

8.6 Frequencies [Hz] of the vibration modes of long span CSFs with two pylons

according to their depth magnitude, where ‘VN’, ‘LN’ and ‘TN’ denote

vertical, lateral and torsional modes with N half-waves and ‘P’ denotes

modes related to the pylon. . . . . . . . . . . . . . . . . . . . . . . . . . . 275

8.7 First lateral vibration modes [Hz] of long span CSFs according to deck

articulation and depth magnitude (see Figure 8.4). . . . . . . . . . . . . . . 276

9.1 Magnitude of maximum accelerations [m/s2] at medium span length 1T-

CSFs in serviceability events. . . . . . . . . . . . . . . . . . . . . . . . . . 299

9.2 Effect on serviceability response of 1T-CSFs of medium span length of

alternative measures (Part 1), where the subindex ‘0’ refers to the reference

case in Chapter 7, ABS,0 describes the area of the backstay of the reference

CSF, AS,0 the area of the main span stays, tbf,0 the thickness of the steel

girder bottom flange, tc,0 the thickness of the concrete slab, Dt,0 and tt,0 the

diameter or thickness of the pylon, Lm the mains span length, α the pylon

longitudinal or lateral (‘Lat.’) inclination (‘incl.’), wd,0 the deck width and

Ls the side span length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

9.3 Effect on serviceability response of 1T-CSFs of medium span length of

alternative measures (Part 2). . . . . . . . . . . . . . . . . . . . . . . . . . 304

9.4 Effect on the serviceability response of medium span length 2T-CSFs of

alternative measures (Part 1), where the parameters considered are repre-

sented with the names introduced in Chapter 8. . . . . . . . . . . . . . . . 305

9.5 Effect on serviceability response of 1T-CSFs of medium span length of alter-

native measures (Part 2), where the parameters considered are represented

with the names introduced in Chapter 8. . . . . . . . . . . . . . . . . . . . 306

31

Nomenclature

Dc Cable spacing

Lm Cable-stayed footbridge main span length

dh Deck depth

wd Deck width

Ds Distance between abutment and first cable anchorage

fs,h Horizontal structural frequency

Hi Pylon height below deck

hp Pylon height

Ls Side span length

θz Structure lateral rotation

L Total length of a bridge

HT Total pylon height

fs,v Vertical structural frequency

u Acceleration

p(t) Applied loading in time

cc Critical damping ratio

c Damping coefficient

c Damping matrix

Di Dynamic amplification factor of mode i

u Dynamic displacement

33

Nomenclature

Neq Equivalent number of pedestrians on the bridge

C0 Frequency damping coefficient of viscous material

fa Frequency of absorber

us Global lateral acceleration of structure

al Lateral acceleration

pl(t) Lateral applied loading in time

m Mass matrix

m Mass

u0 Maximum dynamic displacement

mi Mode i bridge modal mass

mpi Mode i pedestrian modal mass

Ωi Ratio between pedestrian frequency and structure frequency of mode i

ηv Ratio between the depth of the central kern and the depth of the section

αl Ratio between the horizontal distance from the section centroid to the

lateral extreme fibre and section width

ηl Ratio between the width of the central kern and the width of the struc-

tural section

αv Ratio between vertical distance from the section centroid to the top

extreme fibre and section depth

ζi Rayleigh damping coefficient of ith mode

Il Second moment of area in the lateral direction of the deck section

Iv Second moment of area in the vertical direction of the deck section

k Stiffness matrix

k Stiffness

fs Structure frequency

t Time

Np Total number of pedestrian on a bridge

ωi Undamped natural circular frequency of ith mode

u Velocity

34

Nomenclature

av Vertical acceleration

Eb Bar steel Young’s modulus

fck Characteristic compressive strength of concrete

ρc Concrete density

νc Concrete Poisson’s ratio

αc Concrete thermal expansion coefficient

Ec Concrete Young’s modulus

dh Horizontal length of stay cable

fy,b Maximum steel stress of bars

Nc Number of stress cycles

∆φc,a Stay cable angle of rotation

fy Stay cable maximum steel stress

∆σc,a Stay cable stress variation

ρp Stay cable transverse section density

wc Stay cable weight per unit length

Ep Stay cable Young’s modulus

ρs Steel density

νs Steel Poisson’s ratio

αs Steel thermal expansion coefficient

Es Steel Young’s modulus

Etan Tangent stiffness of stay cable

fs,y Yield strength of structural steel

bn nth Fourier coefficient constant (nth harmonic)

y Acceleration of the CoM of pedestrian

ap Age of pedestrian

Ωp Angular frequency of the lateral oscillations of the pedestrian CoM

35

Nomenclature

ωn Circular frequency of nth harmonic

gN Component of the gravity acceleration normal to the leg

gL Component of the gravity acceleration parallel to the leg

d Density of pedestrian flow

g Earth gravity

φj Effect of aim of journey factor

φd Effect of flow density factor

δpf Final slope of vertical footstep load amplitude

hpd Height of pedestrian

δpi Initial slope of vertical footstep load amplitude

mp Mass of pedestrian

pi Maxima or minimum amplitude of vertical footstep load, i = 1, 2, 3

µfp Mean pedestrian step frequency

Fl Pedestrian foot lateral load

Fld Pedestrian foot longitudinal load

Fv Pedestrian foot vertical load

vf Pedestrian free velocity

ws Pedestrian half-step width

fl,p Pedestrian lateral step frequency

vp Pedestrian speed

fp Pedestrian step frequency

sl Pedestrian step length

Tp Pedestrian step period

Wp Pedestrian weight

φn Phase angle of nth harmonic

ti Time of occurrence of pi of vertical footstep load, i = 1, 2, 3

y Time position of the CoM of pedestrian

tT Total time duration of footstep load

ws,t Total transverse distance between consecutive footsteps of a pedestrian

σfp Variability of pedestrian step frequency

36

Chapter 1Introduction

1.1 Background to the research

Footbridges are structures that have been used since early times in history to shorten

distances and span natural barriers. Their structural design has experienced an enormous

development during the last decades due to the knowledge accumulated by structural

engineers in the field of bridge engineering. This has led engineers to propose and construct

visually striking and technically challenging footbridges.

The technological progress that has occurred during the last decades has allowed de-

signers to use new and lighter materials (e.g., glass fibre-reinforced polymers, GFRP)

and structural arrangements with less massive cross sections, with cables as structural

elements, and longer spans. Both parameters, mass of the cross section and span lengths,

are fundamental in the definition of the dynamic behaviour of these footbridges. Foot-

bridges with medium and long spans (with lengths near 50 m and 100 m respectively) and

light decks usually have multiple vertical and lateral vibrational frequencies within the

range 0-5.0 Hz, which experience has shown that, under the passage of pedestrian flows,

it leads to considerably large serviceability accelerations, both in vertical and lateral di-

rections.

Some recent footbridges have been designed and built with the aim of becoming land-

marks, displaying a clear intention of structural and aesthetical innovation. Some of

these footbridges have pushed the boundaries in relation to slenderness, sectional mass,

and span length. As a result, in some cases, such as the London Millennium Bridge in

the UK or the passerelle Leopold Sedar Senghor in Paris, France (both depicted in Fig-

ure 1.1), these new footbridges have displayed unexpectedly large accelerations that have

not been considered acceptable for the urban environment and the expected service load

conditions. Some of these incidents have had significant impact in the media and have

forced the profession to seek explanations for such phenomena.

However, this progression in structural design is not entirely the single cause of the

serviceability problems in some of these footbridges. Detailed studies of these events

have proved that loading assumptions used to design and to assess their serviceability

performance were not sufficiently realistic, both in terms of nature and magnitudes.

37

1. Introduction

(a) (b)

Figure 1.1: (a) View of the London Millennium Bridge in the UK, and (b) the passerelleLeopold Sedar Senghor in Paris, France (Structurae, 2015).

Regarding their nature, it has been observed that humans are sensitive to the move-

ment of the platform where they walk showing a nonlinear pedestrian behaviour. In lateral

direction, the nonlinear nature of these loads has led to a large number of research publi-

cations focused on the simulation of these lateral loads in different scenarios (e.g., those

cited in Ingolfsson et al., 2012a) and to a conservative design of pedestrian bridges in this

direction to ensure an adequate serviceability response. In relation to the magnitude of

vertical loads introduced by pedestrians, first appraisals for the assessment of footbridge

response were initially published at the end of 1970s and beginning of 1980s. However,

with the aim of improving these magnitude descriptions, multiple research groups have

developed studies focused on the loads transmitted by pedestrians while walking during

the last fifteen years (e.g., Butz et al., 2008). Moreover, apart from an accurate individ-

ual description of vertical and horizontal loading, researchers have explored the effects of

crowd flows (e.g., Venuti et al., 2009, or Carroll et al., 2012).

Despite these advances, the research developed so far has not produced more accu-

rate and meticulous models to be applied in design. There is not a methodology for the

analysis of footbridge structures that encloses these progressions yet (accurate individual

load modelling, accounting for the uncertainties involved in the description of these loads,

Figure 1.2, including the nonlinearity of lateral loads and a detailed flow description).

Instead, the analyses of the vibration serviceability limit state of structures subjected

to multiple pedestrians are based on widely accepted models that simulate these pedes-

trian flow scenarios with easy to implement in practice but far too simple and inaccurate

methods.

This lack of knowledge, in terms of realistic and reliable pedestrian load descriptions,

affects design considerations of these structures, both from the economical (in terms

of over-design to avoid inadequate serviceability response) and aesthetic point of view

(by disregarding geometrical proportions that have deficient performance under these

simplified load models).

Accordingly, a more accurate model of pedestrian loading, considering variables that

enable a better and more precise description of the phenomenon, will allow a better under-

standing and reasonable prediction of the response developed by different structural ar-

38

1. Introduction

Figure 1.2: Photo of a group of pedestrians walking, from The University of Arizona,showing the clear differences between pedestrians which should be considered by the loadingmodels.

rangements that can be adopted when designing such structures. Based on this improved

response prediction, a design guideline that provides a better understanding about the

structural response, and guidance about the most apropriate configuration of the struc-

tural elements, to be used under design circumstances, can prove to be a necessary and

useful instrument for developing successful footbridge projects.

1.2 Aims and objectives

The main aims of the current research work are to define a realistic and accurate

methodology to be applied in the study of the structural response of footbridges under

the action of pedestrians during service situations, to investigate the structural response

of cable-stayed footbridges (including classical and innovative layouts) under the action

of pedestrians by applying the previously mentioned methodology, and to develop a gen-

eralised and robust set of design criteria applicable for the design of these cable-stayed

footbridges.

The main aims of this research work are fulfilled through the development of the

following objectives:

(a) A definition of a new methodology inside a non-deterministic framework with a re-

alistic and accurate description of the pedestrian loading on the basis of combining

existing but disconnected multidisciplinary research outcomes. The definition of this

methodology includes:

• The identification of the variables involved in pedestrian actions and selection

of those that are essential for their description. Based on these parameters,

establishment of a realistic description of the characteristics of the population

that crosses the footbridges (using probabilistic and deterministic relationships).

• The description of the pedestrian movement and actions, identifying key pa-

rameters and developing a model that reproduces such descriptions (using both

deterministic and probabilistic relationships as well). This corresponds to the

load model proposed in this thesis.

39

1. Introduction

• A vast and accurate revision of the state-of-the-art of pedestrian actions described

while walking, including outcomes and representations proposed by researchers

and existing models considered in codes and design guidelines, from both a de-

terministic and probabilistic point of view. Representations for vertical and hor-

izontal loading found to be defining most appropriately the actions are compiled

and selected to develop the proposed new unified load model for vertical and

lateral loading.

• A literature review of the different criteria related to users comfort to be consid-

ered to assess whether or not the serviceability limit state of footbridge vibration

is fulfilled under pedestrian actions.

(b) The comparison of the performance predicted by this new methodology with existing

methods for the assessment of footbridges, and the comparison with experimental

data.

(c) The development of nonlinear finite element models in ABAQUS, as well as pre-

processing, post-processing, and user (those that interact with ABAQUS to simulate

the pedestrian-structure interaction) FORTRAN subroutines, used to implement this

stochastic approach in the structural schemes studied in this thesis.

(d) The application of a simplified version of this complex methodology to analyse the

structural performance of beam girders (which are the most commonly used struc-

tural type) with different structural sections and materials, and the development of a

simplified assessment method that allows the designer to obtain an accurate response

of these bridges by performing very simple calculations.

(e) The development of a data base of existing footbridges with a cable-stayed structural

scheme and the definition of the average, and a range of variation, for the different

structural parameters that define the structural typology.

(f) The implementation of this complex methodology to a set of cable-stayed footbridges

that are representative of this structural typology by means of a set of parametric

analyses with the aim of providing understanding about the structural response of

these bridges under pedestrian loading.

(g) The development of a comprehensive set of design criteria for cable-stayed footbridges.

1.3 Thesis layout

The thesis comprises nine chapters with the following content:

Chapter 1, Introduction (this chapter), which corresponds to the introduction to

the thesis.

Chapter 2, State of the art, provides a general review of the geometry and main

structural characteristics of footbridges with a cable-stayed structural scheme and sum-

marises the main outcomes of all the research work that has been published during the

40

1. Introduction

last years in relation to this field, i.e., the magnitudes and nature of the pedestrian loads,

the stochastic character of these loads and of the representation of events with multiple

pedestrians, the research related to the criteria applicable for the assessment of the foot-

bridge performance in service, and a summary of several footbridges where vertical or

lateral accelerations were excessive in service and the solutions considered in each case.

The chapter presents as well a section describing the main characteristics related to the

devices usually implemented to reduce the serviceability accelerations of footbridges and,

finally, the last sections provide a series of recommendations that are currently available

to address the design of footbridges to ensure an adequate response in service.

Chapter 3, Methodology, presents the characteristics of the methodology adopted

to fulfill the aims and objectives of this research work. It outlines the derivation and fea-

tures of the developed load model representing the actions of pedestrians in a traffic flow

(it highlights as well its advantages in comparison to other load models). Furthermore, it

enumerates the criteria selected to appraise the validity of the accelerations generated by

the modelled pedestrians in the serviceability events. Finally, it summarises the character-

istics of the girder footbridges and cable-stayed footbridges (including geometry, material

properties, numerical representation and dynamic analysis characteristics) and parame-

ters used to assess the serviceability response (SLS of vibration, SLS of deflections, ULS

related to normal and shear stresses in the deck and the pylons and ULS of fatigue at the

stay cables).

Chapter 4, Relevance of stochastic representation of reality: advantages

of the new load model presented herein, highlights the importance of the repre-

sentation of the stochastic nature of pedestrian loads (including pedestrian intra-subject

variability, inter-subject variability and collective behaviour) when predicting the service-

ability response of footbridges and substantiates the proposed load model to assess the

response in service of footbridges.

Chapter 5, Girder footbridges design: evaluation of response in serviceabil-

ity conditions, presents the application of a simpler version of the complex methodology

to analyse the structural response of beam girders with a range of different structural

sections and materials. The methodology is founded on a series of nondimensional pa-

rameters. Based on this evaluation, the chapter provides a simple and reliable tool to

assess the performance of footbridges in service without the need of any computational

tool. The chapter compares as well some predictions of the load model with experimental

data.

Chapter 6, Evaluation of the response of a conventional cable-stayed foot-

bridge under serviceability conditions, provides a detailed description of the service-

ability response of the cable-stayed footbridge with most common geometric characteris-

tics. The serviceability response is obtained for different pedestrian traffic conditions and

is compared to the predictions given by currently available codes and guidelines. Further

analyses of the results include an assessment of the magnitudes of deflections, normal and

shear stresses of the deck and the pylon as well as the normal stresses at the stays and a

comparison of these to the values described at the corresponding ULS.

41

1. Introduction

Chapter 7, Performance of cable-stayed footbridges with a single pylon,

presents the structural response of cable-stayed footbridges with a single pylon and it

highlights what characteristics have the largest impact on this serviceability response

(either considering the vertical and lateral accelerations at the deck, stresses at the deck,

the pylons and the stay cables), what parameters improve or worsen this response, and

emphasises the set of characteristics to consider to enhance their comfort in service. These

results are founded on a comprehensive parametric analysis that explores how the response

of these cable-stayed footbridges changes in relation to these parameters.

Chapter 8, Performance of cable-stayed footbridges with two pylons, de-

scribes the structural response of cable-stayed footbridges with two pylons. This response

is based on an extensive and comprehensive parametric analysis that assesses the response

of these cable-stayed footbridges in relation to the parameters considered. The chapter

highlights what characteristics have the largest impact on this serviceability response (ei-

ther considering the vertical or lateral accelerations at the deck, stresses at the deck,

the pylons and the stay cables), what parameters improve or worsen this response, and

emphasises the set of characteristics to consider to enhance their comfort in service.

Finally, Chapter 9, Conclusions and general recommendations for future

work, closes the thesis by outlining the main conclusions of the work and suggesting

further work that can be conducted in this field.

42

Chapter2State of the art

2.1 Introduction

For the last decades, engineers and architects have frequently proposed and constructed

footbridges with innovative structural solutions characterised by lighter and longer spans.

These proposals are the consequence of vast structural knowledge that these designers have

accumulated and their disposition to provide pedestrians with a pleasant and memorable

experience.

Some of these footbridges have experienced unusually large movements under the

action of pedestrian traffic flows, which are considered uncomfortable. Although lighter

and longer span footbridges are more prone to vibrate, experience has shown that these

innovative designs are not the sole reason that triggers these large responses.

These events have originated numerous research publications that attempt to propose

a more realistic description of these loads and to adequately predict the response of

these bridges in service. Despite this vast research, current design models do not include

some of the published advances yet. Furthermore, this has not caused a modification or

specific proposal of sound design guidelines for these structures. In fact, the number of

publications related to this topic in the last years is scarce.

In this context, the current chapter revises the multiple topics involved in the design

of innovative and conventional footbridges. Section 2.2 emphasises one of the typologies

that allows designers to develop longer and lighter structures, i.e. cable-stayed foot-

bridges; Section 2.3 illustrates the principles and recent advances of research in relation

to pedestrian load models; Section 2.4 details one of the most important characteristics

in dynamic response of footbridges, i.e. their energy dissipation capacity; Section 2.5

describes the comfort criteria usually considered to assess the serviceability response of

these bridges; Section 2.6 enumerates footbridges that have had problems in serviceabil-

ity; Section 2.7 identifies the available proposals for footbridge designers and Section 2.8

emphasises considerations for the numerical evaluation of the serviceability response of

footbridges.

43

2. State of the art

2.2 Past, present and future design of footbridges

2.2.1 Historical introduction

From time immemorial, humanity has sought to shorten distances while avoiding nat-

ural obstacles. Between the oldest structures of stones and ropes and current ones using

stay cables or polymer materials, human knowledge of bridge construction has experienced

an immense transformation.

The development of vehicles such as trains and cars in the 19th century and beginning

of the 20th demanded the construction of bridges capable of resisting heavy traffic flows,

a technical progress in bridge construction. This technical evolution, in particular for

bridges with cables as structural elements, occurred parallelly to the development of

the automobile. This was due to the focus of engineers, in particular in Germany, who

considered cable-supported structures and cable-stayed bridges as aesthetically appealing

and economically advantageous.

This improvement in design of cable-stayed structures led to the development of iconic

road bridges such as the Maracaibo bridge (1962) in Venezuela or the Marianski bridge

(1998) in the Czech Republic.

During the last decades of the 20th century and beginning of the 21st, engineers have

focused on pedestrian bridges and the experience of their users while crossing.

The technical knowledge developed in rail and road bridges together with the tendency

of civil engineers and architects to innovate in the design of pedestrian bridges, has led to

the proposal and construction of visually striking and technically challenging footbridges.

In these, the desire of designers is to improve pedestrians’ experience in rural or urban

areas, with cost not always regarded as a design determinant factor.

The following two sections emphasise the features of some of these innovative foot-

bridge designs as well as the characteristics of more conventional footbridges.

(a) (b) (c)

Figure 2.1: (a) Tarr steps, Exmoor National Park, UK (Exmoor National Park, 2015); (b)Glorias Catalanas footbridge, Barcelona, Spain (Structurae, 2015); (c) Bridge of Aspiration,London, UK (Structurae, 2015).

2.2.2 Unconventional footbridges

There are multiple compendia of innovative footbridges proposed and designed during

the last decades, e.g., Strasky (1995), Baus et al. (2007) and Idelberger (2011). These

summaries show the atypical and original solutions adopted by designers for footbridge

typologies such as girders, arches, trusses, suspension bridges or cable-stayed bridges.

44

2. State of the art

These can be exemplified by bridges such as the Plashet School footbridge girder, located

in London and built in 2001 (Figure 2.2(a)); the Kent Messenger Millennium stress ribbon

bridge located in Maidstone and finished in 2001 (Figure 2.2(b)); the Ripshorst arch bridge

in Oberhausen completed in 1997 (Figure 2.2(c)); the Nesciobrug suspension bridge in

Amsterdam finished in 2006 (Figure 2.2(d)); and the Dunajec cable-stayed footbridge in

Sromowce Nizne, Poland, finished in 2006 (Figure 2.2(e)).

Engineers have not only proposed innovative structural arrangements but have also

started using new and lighter materials such as fibre-reinforced polymers (e.g., a footbridge

over the River Flaz in Switzerland, or the Aberfeldy cable-stayed bridge in the UK),

with fibre reinforced concrete (e.g., a footbridge over the River Dollnitz in Germany, or

the footbridge in front of the Guggenheim in Bilbao, Spain), weathering steel (e.g., a

footbridge over the Onyar in Girona, Spain), aluminium (e.g., the Lockmeadow bridge

in Maidstone, UK), granite (e.g., the Sackler Crossing bridge in Kew Gardens, London,

UK), teflon (used in the Plashet Shool bridge) or laminated wood (e.g., the Dunajec

cable-stayed footbridge) among others.

Further details of these designs can be found in publications such as Strasky (1995),

Baus et al. (2007), and Idelberger (2011) or publications of the Footbridge conferences

that take place every three years, e.g., Debell et al. (2014).

(a) (b) (c)

(d) (e)

Figure 2.2: (a) Plashet School footbridge (Architen Landrell, 2015); (b) Kent Messen-ger Millennium bridge (Flint & Neill, 2015); (c) Ripshorst bridge (Structurae, 2015); (d)Nesciobrug bridge (2015); (e) Dunajec cable-stayed footbridge (Biliszczuk et al., 2008).

2.2.3 Existing cable-stayed footbridges

Girder footbridges are a conventional solution adopted for the design of footbridges

due to their, in general, modest cost and simple design. Alternatively, designers frequently

develop footbridge projects with cables as structural elements, as cable-stayed bridges and

suspension bridges.

In this thesis, the performance of the cable-stayed footbridges under the effect of

45

2. State of the art

pedestrian traffic loads is evaluated in detail in order to propose sound design criteria.

In order to establish design guidelines, the geometrical and material characteristics of

conventional cable-stayed bridges are described hereunder.

Figure 2.3 represents the number of spans of cable-stayed footbridges and their span

lengths. These and other characteristics are extracted from a data set with 38 cable-

stayed footbridges found in literature (further details are summarised in Annex A). The

plots of this figure highlight that most common cable-stayed footbridges have two and

three spans. Their average main span lengths are approximately 50 and 125 m for two

and three spans, respectively.

Num

be

r of

str

uctu

res [

%]

10

20

30

40

1 2 3 4 5 7Number of spans

Main

spa

n length

[m

]50

100

150

2 3Number of spans

(a) (b)

Figure 2.3: (a) Percentage of footbridges with different number of spans. (b) Main spanlength (average, standard deviation, maximum and minimum) for two and three span cable-stayed footbridges.

Further analysis of the data reveals that the deck materials correspond to steel box

girders, steel girders or trusses with deck slabs of wood, aluminium, etc. (not concrete),

steel girders with concrete slab and concrete girders. Each of the first three material

alternatives is used on 30% of the bridges whereas concrete girders are implemented in a

more reduced number of occasions.

Regarding the cable system, designers place two planes of cables in a modified fan

disposition (fan arrangement with each cable anchored at different anchor points) instead

of harp disposition. The distance between anchorages at the deck is not related to the

span length and its median value corresponds to 7 m approximately.

The most favoured pylon design is that of a single vertical steel mast. Considerably

less used are tower sections in ’A’, ’Y’ or ’H’.

The deck materials selected for the research work of this thesis are concrete (at the

slab) with steel girders. In relation to these bridges, the usual span lengths, deck depths

and tower heights are represented in Figure 2.4.

These properties define the geometric and material characteristics of the cable-stayed

bridges whose behaviour in serviceability is thoroughly assessed in posterior chapters of

this thesis.

2.3 Pedestrian actions

Pedestrians introduce dynamic loads on the surfaces where they walk. Despite their

modest magnitudes, these loads are very relevant in the design of footbridges because

46

2. State of the art

Main span length Lm [m]

Deck d

epth

[m

]

50.0 100.0 150.00.0

1.0

2.0

Main span length Lm [m]

hp /

Lm

50.0 100.0 150.00.0

0.2

0.6

0.4

(a) (b)

Figure 2.4: (a) Deck depth and (b) tower height hp according to main span length Lm ofcable-stayed bridges.

of the small masses and low frequencies of these structures. Additionally, pedestrians

modify the dynamic properties of the bridges (modal mass and damping ratio). These

modifications are consequence as well of the reduced masses and small damping ratios of

footbridges. However, the definition of loads is not clear and there is limited knowledge

regarding the effects on mass and damping of bridges caused by pedestrians. Consequently

codes and guidelines have not fully incorporated them.

The following sections describe the most relevant research works and conclusions on

the representation of these pedestrian loads, on the effects of pedestrians on bridges modal

mass, and on damping ratio.

2.3.1 Pedestrian gait: definition, characteristics and relation to loads

When crossing a bridge, pedestrians may walk, jog or run; human locomotions that

are mainly differentiated by the movement speed. Other users sitting, standing, jumping

or even deliberately exciting the structure may be found. All these individuals transmit

loads onto the deck of the bridge that depend on their gait characteristics.

For the analysis of footbridge performance in service, the actions generated by runners

or vandals are usually not considered, although exceptions can be found, as in BSI (2008).

In relation to runners, this assumption is based on the observation that common flows of

pedestrians using a bridge will rarely include a runner and even on fewer occasions several

runners will coincide on the bridge. Regarding vandals (users jumping, bobbing, etc., as

mentioned by Caetano et al., 2011), the scenario corresponds to an even lower occurrence

event and it is not consistent with a serviceability scenario as their purpose is to disturb

normal service of the structure (accidental action).

Human walking is a repetitive activity consisting in the consecutive placement of

feet on the floor that overlap in time. Due to this repetitiveness, this locomotion is

characterised by the time interval between the occurrence of the same event (Tp), or its

frequency (fp = 1/Tp). The events that take place within that cycle are described in

Figure 2.5(a): stance and swing phase (individually for each foot) and single limb stance

and double support phase (when considering both feet).

This walking cycle, designated by step frequency and speed, is related to the anthro-

pometric characteristics of the individual (Nilsson et al., 1987; Rose et al., 1994), to the

movement described by his centre of mass, CoM, (Gard et al., 2001) and to the actions

47

2. State of the art

Foot 1 strike Foot 2 toe-off

Foot 2 strike

Foot 1 toe-off

% Cycle

Cycle events

Foot 1 strike

0% 50% 100%

Double

support

Double

support

Single limb

stance

Single limb

stance

Foot 1 eventsStance phase Swing phase

a b c

a: Initial Swing b: Mid Swing c: Terminal Swing

(a)

Vertical

Longitudinal

Lateral

Foot 1 Foot 2

1.0

0.3

0.15Fl

Wp0.00

0.0

0.0

Fld

Wp

Fv

Wp

Gait cycle (%)10050

(b)

Figure 2.5: (a) Human walking cycle with events characterising gait, from Rose et al.(1994); (b) amplitude of loads as a ratio of the pedestrian weight Wp (Fl is the lateral, Fld

is the longitudinal and Fv is the vertical pedestrian load), from Nilsson et al. (1987).

transmitted by this pedestrian.

In the vertical and the longitudinal directions this relation between CoM movement

and gait is intuitive, as it can be seen from the comparison of vertical and longitudinal

loads of Figure 2.5(b) with the phases of the gait cycle in Figure 2.5(a).

In the lateral direction, the movement of the CoM of an individual while walking

cannot merely be attributed to characteristics of the gait such as walking velocity, step

frequency or step length. However, several researchers have noticed a link between the

lateral movement of the CoM and the position of the foot in the lateral direction (lateral

step width), e.g., Mackinnon et al. (1993) or Townsend (1985). The first authors observed

experimentally that the equilibrium of the human body in the lateral direction was ensured

by the correct positioning of the foot in relation to the CoM and a similar observation

was stated earlier by the second, on the basis of a theoretical study of human movement

in the lateral direction making use of an inverted pendulum model.

Of the two parameters that describe the gait of a pedestrian, step frequency and

speed (or step length), the first is involved in the representation of the individual loads

(see sections below) whereas the second is commonly used to illustrate the movement of

that user (and to describe the position of his loads). Hence, pedestrian loads require the

assessment of anthropometric characteristics, the link between step frequency and load

amplitudes and the relation between step frequency and walking velocity (Nilsson et al.,

1987; Rose et al., 1994).

2.3.2 Pedestrian characteristics

As argued in the previous section, actions transmitted by pedestrians depend on their

anthropometric characteristics. One of the most important is the user weight Wp. A

thorough description of this magnitude of each pedestrian in terms of other anthropomet-

ric characteristics can be found (e.g., for the UK population, in Health and Social Care

Information Centre and Office for National Statistics). This magnitude may be related

to the gait adopted by the user (as cited in the previous section), however, codes and

guidelines in use consider this factor as uniform among pedestrians (e.g., Setra, 2006,

48

2. State of the art

considers 700 N).

In relation to other anthropometric characteristics, Weidmann (1993) observed that

the speed adopted by humans depends upon characteristics such as height and their age

and additional circumstances (such as the aim of the journey and the density of the

pedestrian flow, among others). Regarding the first factors, Weidmann (1993) claimed

that when there are no external constraints upon the pedestrian (no particular aim for the

journey and no external crowd pressure), pedestrians will choose their velocity according

to their anthropometric characteristics and their age, so-called free speed. In relation to

the other factors, the author suggested describing the gait speed by linearly scaling this

free speed with factors that account for these external constraints.

Regarding the step frequency, multiple researchers have considered relationships be-

tween the gait speed and this factor to adequately reproduce its magnitude (due to the

impact that it has on the pedestrian loads definition, see sections below), e.g., Venuti et al.

(2007a), and Ingolfsson et al. (2012a). Nonetheless such evaluation requires a fine descrip-

tion of the walking speed of pedestrians to be representative. An accurate description is

reported by Weidmann (1993), and a similar alternative is that of Venuti et al. (2007a),

however these have not been used for the representation of pedestrians on structures.

Alternatively, other researchers have simplified the definition of this factor (step fre-

quency) and described its magnitude from observations in particular events (Matsumoto

et al., 1978; Bachmann et al., 1987; Mouring et al., 1994; Bachmann et al., 1995; Zivanovic

et al., 2005; Pachi et al., 2005; Pedersen et al., 2010). Due to the variability among users,

this factor is described by a normal distribution and further explanation on this account

is given in Section 2.3.5. However, the most important characteristic of these descriptions

is the large variation between different observations.

2.3.3 Dynamic effects of pedestrians on structures

The description of the movement in time of a dynamic system is given by the equilib-

rium of the forces acting on the system as formulated by d’Alembert’s principle (Clough

et al., 1993). According to this principle, the external loads p(t) acting on the system are

resisted by the inertial forces (related to the mass of the system), the damping forces (re-

lated to the viscous dissipation) and the elastic forces (dependent on the system stiffness),

first to third terms at the left side of Equation 2.3.1.

mu(t) + c u(t) + k u(t) = p(t) (2.3.1)

It has usually been considered that pedestrians introduce dynamic loads p(t) and

increase the total mass of the system. Nonetheless, experimental evaluations point out

towards an additional effect of pedestrians: the modification of the dissipation capacity

of the structure. Furthermore, research has shown that the loads considered in design

are not fully well characterised yet (in particular lateral loads) and that the assumptions

in relation to mass are not clear either. Following sections report the theoretical and

experimental proposals published in relation to loads, mass and damping introduced by

walking pedestrians on structures.

49

2. State of the art

2.3.3.1 Vertical loads

Vertical loads generated by pedestrians have a time amplitude with a high and sharp

peak at the beginning (heel strike) and two main peaks separated by a lower amplitude

(saddle shape) that coincide with the initial and terminal swing of the locomotion cycle

(Figure 2.6(a)). These main peaks are flat for low and medium speeds, as the CoM

vertical displacements are small and smooth, and sharper for higher speeds, coinciding

with a larger movement of the CoM (Vaughan et al., 1987; Rose et al., 1994; Kerr, 1998).

Nilsson et al. (1989a) observed increments of the first main peak magnitude from 1.0 to

1.5Wp and decrements of the minimum from 0.9 to 0.4Wp when increasing the walking

speed from slow to fast.

1.0

2.0

100%50%

Time (%Stance phase)

0%

Fast

Normal

Slow

2.0 4.0 6.0 8.0 10.0[Hz]

1.0

S(f

) [g

2 s

ec]

0.5

man

(a) (b) (c)

2.0 4.0 6.0 8.0 10.0[Hz]

1.0

S(f

) [g

2 s

ec]

0.5

woman

Fv

Wp

Figure 2.6: (a) Time amplitude of vertical loads (Vaughan et al., 1987); power spectra ofpedestrian body accelerations while walking of men (b) and women (c) (Matsumoto et al.,1978).

The first evaluations of vertical pedestrian loads were developed in the field of medicine

(as described in Wheeler, 1982). The first mathematical representations of these loads cor-

responded to Fourier series (Equation 2.3.2). The resulting Fourier series model described

the total load transmitted by both feet in time p(t) and highlighted the fact that the gait

parameter with largest influence on loads (and CoM movement) is the step frequency. In

1978, Matsumoto et al. observed such clear influence from the power spectra of the body

accelerations of multiple pedestrians while walking (as described in Figures 2.6(b,c)).

p(t) = Wp

[

1 +∞∑

n=1

bn sin(ωnt− φn)

]

(2.3.2)

This representation of the vertical loads generated by a subject is very similar to that

proposed by Blanchard et al. in 1977, which was one of the first of such representations

to be used for the evaluation of the structural response of footbridges, and where the

authors considered a heavily truncated series, with a single harmonic bn. It should be

emphasised that the form of this representation is at least partly dictated by the available

computational methods of the time. This first proposal of Blanchard et al. was adopted

in design codes for vertical loads, e.g., the BS 5400: Part 2 (1978), and versions of this

formulation continued to be used well into the beginning of the 21st century (BSI, 2006a).

As a result of the adoption of this formulation in codes, a number of researchers

50

2. State of the art

focused their attention on the estimation of the harmonic parameters (bn, ωn and φn)

used in this definition. It should be highlighted that researchers put their efforts in the

evaluation of bn and ωn rather than φn, although the importance of this last parameter

in load reproduction has been highlighted (e.g., for jumping loads, Ji et al., 2001) as it is

related to the shape of the footloads of a pedestrian.

Apart from that of Blanchard et al. (1977), other initial proposals include those of

Ellingwood et al. (1984) (who highlighted the magnitude of the third harmonic), Rainer

et al. (1986) (who proposed the use of the first three harmonics) or Bachmann et al.

(1995).

Efforts in this vein continued. For example, the work of Kerr (1998) presented results

from more than 40 subjects and 1000 individual vertical foot traces (generated with step

frequencies prompted by a metronome). The author described the first six harmonics

and emphasized the relationship between the magnitude of the first and the weight and

height of the individual (observed as well by Sahnaci et al., 2005), as opposed to the rest

of harmonics, where no clear relationship was observed.

Later on, the work of Butz et al. (2008) estimated the Fourier parameters for vertical

loads from 60 different subjects and more than a thousand individual footsteps. In addi-

tion, this study also described both vertical and lateral loads in terms of their temporal

variation. For vertical loads, the author characterised this temporal variation with nine

parameters: peaks p1 and p3, local minimum p2, initial and final slopes δpi and δpf , and

the times of the peaks t1, t3, the local minimum t2 and the total time tT (as seen in Fig-

ure 2.7). This empirical dataset developed by Butz et al. (2008) suggested that all these

parameters tend to have a clear dependence upon the step frequency fp (as described by

Figure 2.7).

These researchers (Butz et al., 2008) evaluated as well the effects of the platform

movement on these vertical loads (load amplitudes were appraised when the platform was

moving with amplitudes ranging from 2 to 10 mm and movement frequencies of 1.55,

1.9 or 2.1 Hz). Based on these observations, they reached the conclusion that differences

between those obtained on a still or a moving platform are minimal.

An alternative representation of the vertical loads to that of Fourier series is the

definition of the real time amplitude of each foot load individually. This approach was

adopted by Wheeler (1982), who considered amplitudes according to the speeds chosen by

the represented pedestrian (similar to those plotted in Figure 2.6) placed on the structure

according to step lengths whose magnitude was related to the speed (Figure 2.8(a)). More

recently, Butz et al. (2008) adopted a similar approach to propose a simplified method to

assess the vertical response of simple span bridges.

One of the latest advances of these models describing vertical loads is the considera-

tion of the inability of individual subjects to repeat monotonous activities with constant

features such as step frequency (already mentioned by Pavic et al., 2002). In the light

of the differences between the movements predicted by load models and those recorded

experimentally in footbridges, several researchers (Zivanovic et al., 2007; Ricciardelli et

al., 2007) attempted to include such phenomenon in models representing vertical loads

51

2. State of the art

p1 p3

p2

dpi dpf

tT

t3t2t1

Time [s]

Fv(t)Wp

1.4 1.6 1.8 2.0 2.2 2.4

10

20

30

40

[

]d

pf

fp [Hz]

0.3

0.5

0.7

0.9

1.1

1.3

t T [

s]

1.4 1.6 1.8 2.0 2.2 2.4

fp [Hz]

0.4

0.8

1.2

1.6

2.0

p1 /

Wp [

]

1.4 1.6 1.8 2.0 2.2 2.4

fp [Hz]

0.2

0.4

0.6

0.8

t 1 /

tT [

]

1.4 1.6 1.8 2.0 2.2 2.4

fp [Hz]

1.4 1.6 1.8 2.0 2.2 2.4

0.4

0.8

1.2

1.6

2.0

p3 /

W p [

]

fp [Hz]

0.2

0.4

0.6

0.8

t 3 /

tT [

]

1.4 1.6 1.8 2.0 2.2 2.4

1.0

fp [Hz]

0.4

0.8

1.2

1.6

2.0

p2 /

W p [

]

1.4 1.6 1.8 2.0 2.2 2.4

fp [Hz]

1.4 1.6 1.8 2.0 2.2 2.4

fp [Hz]

20

40

60

80

[

]

1.4 1.6 1.8 2.0 2.2 2.4

dpi

0.2

0.4

0.6

0.8

t 2 /

tT [

]

(a) (b)

Figure 2.7: (a) Schematic illustration of the parameters defining the vertical load ampli-tude for a single footstep; (b) characterisation of vertical load amplitudes according to nineparameters and fp (Butz et al., 2008).

through Fourier series. These authors proposed alternative amplitudes of the dynamic

load factors or suggested the introduction of additional terms in the Fourier series (named

by researchers as subharmonics, described in Figure 2.8(b)) to account for the effects of

intra-variability that had been experimentally recorded. Based on the definition of the

load amplitude in time, other research proposals included this phenomenon (Racic et al.,

52

2. State of the art

frequency / Walking frequency

1.0 2.0 3.0 4.0 5.0

0.05

0.10

0.15

0.20

Fouri

er

am

plit

ude /

weig

ht

Walking frequency

1.0 2.0 3.0 4.0 5.0

0.5

1.0

1.5

2.0

Str

ide

length

[m

]

forward speed

Walk

Jog

Run

[m/s] Subha

rmon

ic 1

Subha

rmon

ic 2

Subha

rmon

ic 3

Subha

rmon

ic 4

Subha

rmon

ic 5

Harm

onic

1

Harm

onic

2

Harm

onic

3

Harm

onic

4

Harm

onic

5

(b)(a)

Figure 2.8: (a) Relationship between step length and fp of Wheeler (1982); (b) amplitudeof force Fourier spectrum obtained by Zivanovic et al. (2007).

2011). As the authors adopted a stochastic approach, this model is further discussed in

Section 2.3.5.

Both the first and second representations of vertical pedestrian loads require a fine

description of the step frequency of the user fp (apart from the load amplitude according to

this step frequency fp). Therefore, equally large efforts have been devoted to realistically

predict this value. Multiple researchers have measured this parameter in unconstrained

events, as already enumerated in Section 2.3.2, where average step frequencies range from

1.8 to 2.2 Hz although they can be as low as 1.6 Hz (as described in SCI, 1989, for indoor

floors).

The wide ranges of the mean value of fp can be attributed to differences in the an-

thropometric characteristics of the observed populations as well as the density of the flow,

aim of the journey, etc., as highlighted in Section 2.3.2. Due to this stochastic character,

those authors have usually described the magnitude of fp through normal distributions.

However the question remains on how to select a representative value of such factor for a

pedestrian, a group, or a flow of pedestrians. This selection is linked to the type of repre-

sentation of the population adopted (deterministic or probabilistic), discussed in Section

2.3.5. Nonetheless, it must be emphasized that the validity of the load model is linked to

the accuracy of the value of this parameter as well.

2.3.3.2 Lateral loads

Lateral loads generated by pedestrians while walking have a considerably smaller mag-

nitude than those in the vertical direction (see Figure 2.9, as measured by Nilsson et al.,

1989a). Due to the load opposite sign of consecutive steps, the frequency of lateral loads

is half that of vertical loads: fl,p = fp/2.

In relation to this, the large movements recorded at the T-bridge in Japan (P. Fu-

jino et al., 1993) or later at the London Millennium Bridge (Dallard et al., 2001) in the

UK highlighted: a) the importance of these loads despite their small magnitude, and b)

the existence of an underlying mechanism by which pedestrians may become engaged to

the structure movement and hence introduce larger loads and trigger larger structural

movements. The underlying mechanisms that explain such effect are still not fully under-

stood, therefore researchers need to adopt assumptions in order to consider them in the

numerical models.

53

2. State of the art

0.06

0.03

0.03

0.06

Time

Fl

Wp

Figure 2.9: Time amplitude of lateral force load (Zivanovic et al., 2005).

Hereunder there is a description of the lateral loads models with an experimental

basis. Models with theoretical assumptions regarding this interaction of pedestrians or

predictions of the conditions that trigger such phenomenon are not included hereunder

but are separately described in Section 2.3.4.

Similarly to vertical loads, there are two main groups of models for lateral loads pl(t):

one mathematically represented through Fourier series (Equation 2.3.3) and another based

on the description of the movement of the CoM.

pl(t) = Wp

[

∞∑

n=1

bn sin(ωnt− φn)

]

(2.3.3)

The first quantification of the Fourier series harmonics of these loads for structures is

that published by Bachmann et al. in 1987, where the largest harmonics were the first and

the third b1 = b3 = 0.04 (later on the same authors, Bachmann et al., 1995, recommended

slightly larger values, i.e., b1 = b3 = 0.10).

In 1993, P. Fujino et al. published a detailed description of the large lateral movements

recorded at the T-bridge in Japan (see Section 2.6). By comparing the movements of

pedestrians on the bridge with those of the structure, the authors reached the conclusion

that 20% of the users walked with a step frequency coinciding with that of the lateral

movement. Furthermore, it was estimated that the load introduced by these was larger

than that proposed by Bachmann et al. (1987) (maximum amplitude 0.06 instead of

0.04). Hence, authors highlighted that pedestrians could become synchronised with the

structure and that, in that case, lateral loads were larger than those of a non-synchronised

pedestrian.

After that event, multiple researchers evaluated the actions of pedestrians on a moving

platform. Based on multiple experimental tests on a concrete slab, Charles et al. (2005)

observed that the lateral loads have maximum amplitudes ranging from 20 to 100 N and

proposed an average harmonic b1 of amplitude 35 N (which corresponds to b1 = 0.05

for a pedestrian weight of 700 N), valid as long as the acceleration of the platform is

below 0.1-0.15 m/s2. Ronnquist et al. (2007) performed single pedestrian tests on a

laboratory platform moving at lateral frequencies between 0.75 and 1.14 Hz (obtaining

more than 1000 samples) and proposed magnitudes of the first harmonic that depended

on the distance between the lateral step frequency fl,p and structure frequency fs (defined

by Equation 2.3.4, where u describes the structure movement and u is its second time

54

2. State of the art

derivative, i.e., the platform lateral acceleration).

b1 = 0.145− 0.1 exp

[

−(

0.45 + 1.5 exp

[

−1

2

(

fp,l − fs0.07

)2])

u1.35

]

(2.3.4)

A simpler evaluation of the first harmonic of the lateral loads was that introduced by

Sun et al. (2008), according to which the magnitude depends exclusively on the amplitude

u of the lateral structural movement (defined by Equation 2.3.5).

b1 = 0.05 + 1.18u (2.3.5)

Butz et al. (2008) observed that lateral loads were larger with the lateral acceleration

of the platform (as Ronnquist et al., 2007) and that their amplitudes with time had a

rectangular shape. Due to the large scatter of their results, they proposed load ampli-

tudes according to different ranges of the lateral movement: for pedestrians walking on

a fixed platform the amplitude of the rectangle is 0.04Wp; if the lateral acceleration of

the platform is between 0 and 0.5 m/s2, the load amplitude is 0.055Wp and, above that

acceleration, the amplitude is 0.075Wp.

A more sophisticated load model based on Fourier series is that presented by Pizzimenti

et al. (2005) and later by Ricciardelli et al. (2007). Based on experimental tests, the

authors theorized that lateral loads are the sum of an action equal to that that would have

been generated by the user while walking on a still floor pl,fp(t) and a second term centred

at the frequency of the movement of the structure pl,fs(t) (see Equation 2.3.6). They

proposed values for the first five harmonics of the first term that did not depend on the step

frequency (the 95% characteristic values are b1 = 0.04, b2 = 0.008, b3 = 0.023, b4 = 0.004

and b5 = 0.011) and divided the second into two different terms (see Equation 2.3.6).

The components of the second term may be regarded as additional damping and inertia

forces.

pl(t) = pl,fp(t) + pl,fs(t) = pl,fp(t) + pin sin(2πfst) + pout cos(2πfst) (2.3.6)

Continuing the work of Pizzimenti et al. (2005), Ingolfsson et al. (2011) performed

more experimental tests (with 71 individuals) on a static or laterally moving platform

(with frequencies between 0.33 and 1.07 Hz and amplitudes ranging from 4.5 to 48 mm).

For the loads transmitted to a static platform, the authors proposed values for the first

five harmonics, the amplitudes of which were obtained considering a broad-band around

the frequency of the harmonic (instead of a narrow-band, as Pizzimenti et al., 2005). In

relation to the loads at a moving platform, the authors suggested values for the equivalent

damping and inertial forces (second and third terms of Equation 2.3.7, terms that depend

on the movement of the platform and maximum lateral displacement u0). Due to the large

scatter of results, the authors described these forces with a stochastic representation.

pl(t) = pl,fp(t) + cp(fs/fl, u0)u+mpp(fs/fl, u0)u (2.3.7)

55

2. State of the art

Results of Ingolfsson et al. (2011) showed that: a) pedestrians damp out the movement

of the moving platform they walk on irrespective of its vibration frequency (valid for those

frequencies tested in experiments); and b) at lower movement frequencies, pedestrians

decrease the modal mass whereas at higher frequencies they increase the modal mass.

The second group of lateral load models corresponds to that based on the movement of

the CoM of the user. The appraisal of this movement is based on the representation of the

transverse movement of a walking human as an inverted pendulum (IP), see Figure 2.10

(where y and y are the acceleration and position of the CoM in local coordinates, Ωp =√

g/Leq represents the natural angular frequency of the lateral oscillations of the CoM,

Leq the length of the inverted pendulum, ws half of the step width or the position of each

foot relative to the equilibrium position of the CoM, mp the mass of the pedestrian, us the

global lateral acceleration of the structure, gL the component of the gravity acceleration

parallel to the leg, and gN to the component normal to the first). Through the equilibrium

of forces in the frontal plane, given a known previous foot position and the movement of

the bridge, it allows one to define the position in time of both the next step and the CoM

(equations are given in Section 3.2.1).

This IP model was developed by Townsend (1985) and investigated by Mackinnon

et al. (1993) or Bauby et al. (2000) (from the field of biomechanics, who commented that

the most important parameter in the equilibrium is the lateral foot position in relation

to the CoM). This movement can be directly related to characteristics of the gait, step

width, anthropometric characteristics and the lateral movement of the walking surface.

us

Global

axes

CoMus

y

Local

axes gN

Leq

Fl

g

Fl

gL

Structure ws

Figure 2.10: Lateral equilibrium of a pedestrian represented as an inverted pendulum.

A very similar model was first applied by Barker (2002), who observed that the model

predicted an increment of the step width when stepping on a moving floor, and later by

Macdonald (2009), who observed that this pedestrian load model generated loads centred

on the walking frequency and load terms centred on the frequency of the structure (as

seen by Pizzimenti et al., 2005, or Ingolfsson et al., 2011). Furthermore, Macdonald

(2009) pointed out that under certain combinations the load model was equivalent to

negative damping and mass (which matches what was seen in LMB). This load model

was implemented as well by Bocian et al. (2012) or, with some variations, by Elricher et

al. (2010), who used a modified Van der Pol/Rayleigh oscillator to represent pedestrians

56

2. State of the art

(with parameters fitting results of tests performed by 6 pedestrians).

Carroll et al. (2013a) used this model as well, after experimentally observing lateral

loads with components as those of Pizzimenti et al. (2005) or Ingolfsson et al. (2011).

With the implementation of this model, the authors (Carroll et al., 2013a) were able to

predict an event similar to that recorded at Clifton suspension bridge.

Nonetheless, it should be highlighted that lateral loads defined with this simplified

theoretical model have one particular shortcoming in that they cannot reproduce the

loading corresponding to the case that both feet of a subject are in contact with the

ground (so-called double stance). However, this is not a significant shortcoming and has

very little impact upon the results given that the accelerations of the CoM are very small

during this double-stance phase.

2.3.3.3 Mass

The equivalent mass introduced by a pedestrian on a vibrating structure has been

thoroughly researched in areas related to human response to vibrations (e.g., in transport

systems such as trains, etc.) and, in a smaller degree, for the design of structures where a

large number of pedestrians can be gathered (e.g., stadia). In both cases, pedestrians are

mainly considered to be standing or sitting. However, this research has not been extended

to the study of the effects of walking pedestrians (of interest for dynamic footbridge

analysis).

In the area of pedestrian comfort, researchers have observed that numerical represen-

tations of humans through spring-damper elements with a lumped mass (Matsumoto et

al., 2003; Zheng et al., 2001) adequately predict the mass added by the pedestrian to the

vibrating system. This numerical model has been used as well by some researchers in the

analysis of the vertical movement caused by a single pedestrian on a footbridge (Fanning

et al., 2005), finding good agreements with experimental results. However, these numeri-

cal models are not still fully developed and have several drawbacks: they have not been

used to assess the equivalent masses of humans when subjected to lateral movements and

a large computational effort would be required in the representation of a large number of

pedestrians.

In relation to observations of experimental tests in stadia, results are not conclusive:

Ellis et al. (1994) observed that standing or sitting pedestrians increased the vertical

frequencies whereas Ellis et al. (1997) or Brownjohn et al. (2004a) observed that standing

and sitting pedestrians only added mass vertically. In the lateral direction, Brownjohn et

al. (2004a) noticed that standing subjects did not change the modal mass of the structure.

In a general situation, Agu et al. (2010) discussed that these effects depended on the

frequency of the structure: if the frequency of the structure is considerably smaller, similar

or larger than that of the human body (4.5-5.0 Hz approximately), standing pedestrians

either add mass to the overall vibrating system, they add mass and dissipation of the

movement or exclusively act as damping dissipators respectively.

Exclusively for footbridges, there are multiple observations of the effects of pedestrians

on distinct structures. Zivanovic et al. (2010) measured the response of the Reykjavik

City footbridge and found a decrement of the first vertical frequency due to the presence

57

2. State of the art

of walking pedestrians on the bridge. Similar effects in the vertical direction were seen by

Pimentel (1997) or Zuo et al. (2012). In the lateral direction, there are more discrepancies:

Zuo et al. (2012) and Carroll et al. (2012) considered added mass whereas Brownjohn et

al. (2004a) did not observe this modification.

Hence, in general, researchers point towards the consideration of pedestrian mass in

the vertical direction whereas in the lateral direction it is not clear yet.

2.3.3.4 Damping

Research and experimental evaluations at real footbridges and other structures such

as stadia suggest that pedestrians increase the energy dissipation capacity.

This is very clear for standing or sitting pedestrians on vertical vibration movements

(Ebrahimpour et al., 1991; Ellis et al., 1994; Ellis et al., 1997; Willford, 2002; Sachse et

al., 2002; Setra; Barker et al., 2008). Although in few cases researchers disregarded this

effect (Brownjohn et al., 2004a). This observation is well founded for walking pedestrians

as well (Ebrahimpour et al., 1991; Willford, 2002; Brownjohn et al., 2004a; Barker et al.,

2008; Zivanovic et al., 2010), although some others have not observed such effect (Ellis

et al., 1997; Zivanovic et al., 2010).

In the lateral direction researchers have not found such beneficial effect: Brownjohn

et al. (2004a) observed that in the lateral direction damping was little affected by standing

pedestrians in the Changi Mezzannine bridge and Ellis et al. (1997) highlighted the need

to study the effects of human body in the lateral direction.

Consequently, several researchers (Barker et al., 2008) point towards ignoring such

effect both in the vertical and lateral movements.

2.3.4 Interaction pedestrian-structure

Multiple research works have assessed the effect that vertical or lateral movements

of bridges have on the gait of pedestrians (and loads transmitted) while crossing them

(named as synchronisation by P. Fujino et al., 1993, ‘lock-in’ or synchronous lateral exci-

tation – SLE – by Dallard et al., 2001). In the vertical direction, there are few proposals

evaluating this effect. However, lateral movements and their effects on pedestrians have

received the largest attention of researchers, mainly due to the large movements observed

in footbridges such as the London Millennium bridge (Dallard et al., 2001) or the Solferino

bridge (Danbon et al., 2005).

In the vertical direction, researchers have estimated that this occurrence is related to

vertical movements of large magnitudes. However, proposals such as that of Smith (1969),

as explained by Willford (2002), in relation to accelerations, or Bachmann (2002), in

relation to displacements, described limiting movements that were below current comfort

limits (see Section 2.5). Additionally, it should be highlighted that this phenomenon has

not been recorded or observed in real footbridges.

In the lateral direction, multiple researchers have studied this phenomenon. There is

much evidence that points towards the generation of such interaction between pedestrians

and the moving structure as a result of the important effect that lateral movements have

on their lateral stability. Publications related to this phenomenon are mainly focused

58

2. State of the art

on modelling pedestrians capable of reproducing such effect or on the proposal of simple

expressions for predicting whether this phenomenon will be developed or not.

Research focused on the first aim generally takes into account that this phenomenon

is a two-stage process: with small structural movements pedestrian actions are inde-

pendent of these structural movements and, when these are noticeable, some become

synchronised and cause the structure to develop lateral movements at a faster pace. For

this second stage, many publications considered that synchronisation involved adopting

a step frequency similar to that of the movement, however latest experimental analyses

have shown that synchronisation may not involve changing the step frequency but merely

adopting a wider step while walking (Carroll et al., 2013b) and leave this change of step

frequency as a last resort that may occur in some occasions.

One of the first proposals related to this event is that of P. Fujino et al. (1993), who

proposed a model to reproduce the movements recorded at the T-bridge (see Section 2.6).

This model was based on the assumption that 20% of the users where synchronised with

the structure and that these users produced lateral loads that were larger than those of

unsynchronised pedestrians (see Section 2.3.3.2).

Similarly to P. Fujino et al. (1993), other researchers explored models based on the

number of synchronised pedestrians and used them to predict movements recorded in

real structures, e.g., Nakamura (2004) and Nakamura et al. (2006) for the M-bridge in

Japan or Danbon et al. (2005) for the Solferino bridge in Paris. Nakamura (2004) and

Nakamura et al. (2006) proposed a model that intended to describe the beginning of the

synchronisation and related it to the movement of the structure (see Equation 2.3.8):

F (t) = b1k2 H[u(t)]G(fB)mp g (2.3.8)

where F (t) is the total lateral modal force caused by pedestrians, b1 is the amplitude of

the first harmonic of lateral loads (assumed to adopt a value of 0.04), k2 is the proportion

of synchronised pedestrians, u(t) is the girder modal velocity, H[u(t)] is a function that

describes the process of synchronisation according to the movement of the bridge which

is proposed by authors and G(fB) is a function describing how pedestrians synchronise to

the movement (assumed to have a value of 1.0). Posterior evaluations of this model led

the authors Nakamura et al. (2008) to associate synchronisation to the frequency of the

lateral movement (50% for lateral movements of frequency magnitude 1.0 Hz and 20% for

0.87 Hz movements).

Danbon et al. (2005) related the synchronisation to the amplitude of the lateral move-

ments, instead of their speed, through a proposed function φ(u) (Equation 2.3.9) where

G is the amplitude of the first harmonic and bρφ(u) is the number of synchronised pedes-

trians. In relation to φ(u), the authors considered that 100% of the pedestrians were

synchronised if the amplitude of the lateral movements was larger than 6 mm.

F (x, t) = Gbρφ(u) cos(2πfst) (2.3.9)

There are several other models (Carroll et al., 2012; Bodgi et al., 2007; Venuti et al.,

59

2. State of the art

2005) similar to that of Nakamura (2004). Venuti et al. (2005) developed a model where

the number of synchronised pedestrians and the magnitude of the first harmonic of their

lateral loads b1, were related to the velocity of the deck lateral movement:

b1 = b1,s + b1,d u(t) (2.3.10)

where b1 is the total amplitude of the first harmonic of synchronised pedestrians, with b1,sas the first harmonic amplitude of the lateral loads on a static platform and b1,d as the

first harmonic amplitude dependent on the deck lateral movement u(t). Later on, these

authors changed the function that predicted the synchronisation factor and the lateral

loads of the synchronised pedestrians (Venuti et al., 2007b).

Later proposals have implemented more sophisticated models. These represent syn-

chronised pedestrians through a function that changes their step frequency (of the Fourier

series loads) instead of adopting assumptions to predict the number of synchronised pedes-

trians according to the movement of the deck. This is the basis of models such as that

of Bodgi et al. (2008) or Marcheggiani et al. (2010), which used proposals similar to the

Kuramoto model (Strogatz, 2000).

All these models established some degree of synchronisation (change of step frequency)

between pedestrians and lateral movements, however there is a limited number of experi-

mental analyses of this phenomenon. One of the few is that performed at Imperial College

(Willford, 2002). However, it should be highlighted that those results were obtained on a

platform where pedestrians were able to perform a limited number of steps.

Previous models are based on the use of Fourier series to describe the lateral loads.

However, the proposals based on the use of inverted pendulum (IP) load models are

capable of reproducing an interaction of pedestrians with a platform moving laterally

without considering assumptions related to proportions of synchronised pedestrians. This

is the case of models such as that of Morbiato et al. (2005) or Morbiato et al. (2011),

where the authors included as well a second stage of synchronisation, i.e., the adoption

of a more convenient step for some pedestrians (through a function predicting a shift of

the phase angle between pedestrians and the structure), or those of Bocian et al. (2012)

and Carroll et al. (2013a).

Regarding the models focused on the prediction of the conditions that trigger large

lateral movements (second group of publications focused on pedestrian-structure inter-

action), one of the first proposals is that triggered by the phenomenon recorded at the

London Millennium bridge, LMB, (Dallard et al., 2001). Based on multiple experimental

tests at that structure, Dallard et al. (2001) proposed an expression (Equation 2.3.11) to

predict the number of real pedestrians that could trigger large lateral movements. This

criterion is based on results of those tests according to which lateral loads of pedestrians

were linearly related to the speed of the lateral movement (the proportion magnitude is

cp = 300 Ns/m).

Ncr =4πζfsm

cp1L

∫ L

0[Φ(x)]2dx

(2.3.11)

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2. State of the art

In Equation 2.3.11, ζ is the damping ratio of the lateral mode, fs its frequency, m its

modal mass, Φ(x) its modal shape and L the total length of the bridge. Nonetheless, it

should be highlighted that other research works have obtained alternative values of cp,

e.g., Strobl et al. (2007).

Similarly to this proposal, other authors have presented alternative formulae to predict

the susceptibility of any structure to develop large lateral movements. The formulation of

Charles et al. (2005) is based on experimental research conducted at the Solferino bridge.

Based on that, they concluded that in general 10% of the users may be considered to

be engaged with the movement and that, if the lateral acceleration (predicted with the

model defined in Section 2.3.3.2) is larger than 0.1-0.15 m/s2, instability can take place.

Apart from these expressions based on experimental results, other authors evaluated

this phenomenon and published alternative theoretical expressions. This is the case of

Roberts (2005), who introduced an expression derived considering lateral loads defined

by Fourier series and a proportion of synchronised pedestrians. According to the author,

if the lateral response is larger than the magnitude given by Equation 2.3.12, the number

of pedestrians that trigger SLE will be described by Equation 2.3.13:

ui =LFi

Npmp,iω2i

(2.3.12)

Np

L=

mi

mp,iΩ2iDi

(2.3.13)

where in the first equation ui is the critical lateral displacement of mode i, L is the bridge

length, Fi is the pedestrian modal lateral force, Np is the total number of pedestrians

on the bridge, mp,i the pedestrian modal mass, and ωi is the modal angular frequency;

and where in the second equation mi is the bridge modal mass, Ωi is the ratio between

the lateral step frequency of users and the structure lateral modal frequency and Di a

dynamic amplification factor.

A similar alternative is that of Newland (2004), based on the movement of the CoM of

pedestrians and the proportion of synchronised pedestrians in relation to the total number

of pedestrians on the bridge β (Newland, 2004, suggested considering β = 0.40 for lateral

movements up to 10 mm), see Equation 2.3.14, where α is the ratio of the amplitude of

the CoM movement to the structure movement amplitude, mp is the pedestrian modal

mass per unit length, and m is the bridge modal mass per unit length. This expression is

very similar to that of McRobie et al. (2003), although this second proposal was defined

for vertical movements.

ζ >1

2αβmp/m (2.3.14)

Piccardo et al. (2008) suggested a similar expression which considered the fact that load

amplitudes of synchronised pedestrians changed with the lateral movement (harmonics of

lateral loads were defined as in Venuti et al., 2005).

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2. State of the art

One of the latest proposals is that of Ronnquist et al. (2007) who, based on observa-

tions from experimental tests, fitted an expression to describe the number of equivalent

synchronised pedestrians generating large lateral responses:

Neq = 35− 34 exp

[

−(

Np

60

)1.6]

(2.3.15)

where Np is the number of pedestrians on the footbridge and Neq is a fictitious number of

equivalent pedestrians that generate the same response as the real number of pedestrians

on the footbridge.

2.3.5 Probabilistic and deterministic approach

A satisfactory prediction of the response of footbridges under serviceability conditions

relies, to a large extent, upon the assumptions made regarding the imposed human ac-

tions. However, the parameters that are commonly used to define anthropogenic loads

are associated with a significant degree of inherent variability. This variability arises from

a combination of the wide range of anthropometric characteristics that exist within any

typical sample of the human population (inter-subject variability) and the inability of

individual subjects to repeat monotonous activities with constant features such as step

length or step frequency (intra-subject variability), as described by Giakas et al. (1977).

With the representation of traffic flows (with multiple pedestrians), not only intra and

inter-variability have an important role in the definition of the overall loads transmitted

to the structure, but also their arrival in time to the bridge and their movement among

others (pedestrian interactions).

In order to accurately assess the dynamic footbridge response, it is important to cap-

ture these effects in the load models used for such evaluation. Due to the stochastic

nature of these parameters, multiple research proposals include the use of probabilistic

tools. However, in order to simplify these, equally numerous formulae attempting to

replicate this stochastic character in a deterministic manner have been published.

The following sections enumerate different published proposals attempting to include,

through probabilistic or deterministic models, the intra-variability of the loads generated

by one pedestrian, the different parameters used to describe loads of different subjects

(inter-variability), such weight, step frequency, step length or speed and load amplitudes,

and the representation of pedestrian flows.

Finally, the last section enumerates as well evaluations quantifying the response of

footbridges stochastically, which aim to reflect on results the variability of the pedestrian

characteristics and the traffic flow using a footbridge.

2.3.5.1 Pedestrian intra-variability

Most of the load models described in Section 2.3.3 have as main assumption the consid-

eration of an average step frequency (and speed or step length). However, it is intuitively

obvious that, even in the absence of other pedestrians, an individual will vary the char-

acteristics of his walking from step-to-step. When observed over a period of time that

individual may have well-defined average characteristics, but there will also be an element

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2. State of the art

of inherent variability that is naturally propagated through to the imparted loads. Fol-

lowing there is a description of research proposals considering such characteristic through

stochastic definitions or deterministic simplifications.

The effect of intra-variability was first included in loads transmitted while jumping

(Tuan et al., 1985; Ellis et al., 1994). These authors observed the difficulty of maintaining

a constant frequency while jumping and described the peak amplitudes of these loads

through Gaussian or Rayleigh distributions or suggested modelling the contact times of

consecutive steps through normal distributions.

Researchers considered the effects of intra-variability while walking slightly later. One

of the first proposals is that of Ebrahimpour et al. (1996), who proposed a normal dis-

tribution to replicate the double-stance times of consecutive steps. However, not until

several years later researchers emphasised this phenomenon and tried to replicate it.

Authors such as Sahnaci et al. (2005) observed experimentally the effects of this vari-

ability in the response of different structures caused by these variable loads, (see Fig-

ure 2.11).

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

Absolu

te m

axim

um

accele

ration

[m/s2]

Constant step frequency

Variable step frequency

fp = 2.1Hz

2.0 4.0 6.0 8.0 10.0

Fundamental frequency [Hz]

Figure 2.11: Comparison of accelerations caused by a single ’perfect’ resonant pedestrianand those cause by real pedestrian (intra-variability) experimentally observed by Sahnaciet al. (2005).

For both vertical and lateral actions, there are proposals that have tried to capture

the variability of the consecutive loads transmitted by a pedestrian by describing the

power-spectral density around the first harmonics through functions fitting these densities,

e.g., Brownjohn et al. (2004b). Some of these proposals have attempted to convey this

variability through deterministic models, by defining the amplitudes of the first harmonics

of the Fourier series by means of those fitted functions (although these functions convey

the stochastic character of these loads). This is the case of Zivanovic et al. (2007) for

vertical loads or Ricciardelli et al. (2007) and Ingolfsson et al. (2011) for lateral loads. In

the vertical direction the authors included additional terms in the Fourier series defining

the time loads (subharmonics or components of the power-spectral density between integer

harmonics) that attributed to the intra-variability (due to the fact that loads of each step

correspond to Fourier series with frequency half that of the loads of both feet). It should

be highlighted that in both cases authors did not provide phase angles for these harmonics

63

2. State of the art

(Zivanovic et al., 2007, suggest random values).

A more sophisticated and fully stochastic proposal is that of Racic et al. (2011) who,

from experimental tests on a treadmill, observed that consecutive step frequencies were

best described by the auto-spectral density function of this random process. Together

with this representation of consecutive step frequencies, the total impulse of a load step

(that authors related to the step time interval) and the scaling of real shape loads to match

this impulse, authors reproduced stochastically this intra-variability with a time-domain

model.

2.3.5.2 Pedestrian inter-variability: anthropometric characteristics and gait

Pedestrians adopt alternative gait characteristics (speed, step frequency and step

length) due to their different anthropometric characteristics (height, weight, etc.) and

to the effects caused by their individual aim of the journey or the number of pedestrians

around them in a traffic flow. The introduction of these differences among pedestrians

in numerical models was soon explored after the first proposals for pedestrian loads were

published (e.g., Wheeler, 1982, and Matsumoto et al., 1978).

As emphasized in Section 2.3.3, this variability is very evident in the magnitude of the

step frequency adopted by pedestrians. Due to the importance of the estimation of this

parameter in the representation of loads, multiple researchers have put major efforts in

adequately representing this parameter. For this reason there have been many proposals

such as those enumerated in Section 2.3.2, that have represented the inter-variability

through the definition of these frequencies with normal probability density functions. In

general, these proposals indicate that usual step frequencies have a mean value between

1.7 and 2.2 Hz.

In order to account for this variability, proposals that evaluate the dynamic response

of footbridges consider the worst case scenario, i.e., the same likelihood for pedestrians

to adopt a mean step frequency between 1.8-2.0 Hz (BSI, 2008) or between 1.7-2.1 Hz as

defined by Setra (for lateral loads this guideline proposes a range of frequencies 0.5-1.1 Hz).

The adoption of these wide ranges is justified by the importance of this parameter in the

response prediction (evaluated by Pedersen et al., 2010), although such assumptions may

lead to large overestimations.

Another parameter that reflects the population variability in load models is the pedes-

trian weight. This parameter is represented in guidelines or codes by a single value, al-

though it is clear that pedestrians with particular anthropometric characteristics (height

and weight) will tend to adopt a gait according to those (as observed by Kerr, 1998).

Researchers have included the weight variability by randomly assigning this value to

the simulated subjects from a normal distribution (e.g., Mouring et al., 1994, or Pedersen

et al., 2010). Results from these evaluations have yielded that the impact of the variability

of this factor is moderate.

Apart from the step frequency and weight, the variability of the step length used by

different individuals has been introduced as well in load models (values extracted from

a normal distribution). However results highlighted the minor effect of this variability

(Pedersen et al., 2010).

64

2. State of the art

2.3.5.3 Pedestrian inter-variability: actions

Differences among pedestrians are not only constrained to gait and anthropometric

characteristics but affect as well their loads: two pedestrians with the same weight walk-

ing at equal speeds and step frequency may not transmit the same loads (i.e., their CoM

movements may be different). This effect in the vertical loads can be observed in Fig-

ure 2.7.

One of the first research studies recognising this issue is that of Rainer et al. (1986),

who proposed values of the first three harmonics of the Fourier series describing vertical

loads as an average of those obtained for three pedestrians.

More recently, based on vertical loads of multiple pedestrians, Kerr (1998) appraised

this variability through the description of polynomial functions including 95% of the values

of the first harmonic captured in his experimental tests. Based on these values as well,

Willford et al. (2006) proposed values of the first harmonic to be used in design (with a

25% chance of exceedance). However, numerical evaluations performed by Pedersen et al.

(2010) including this characteristic led the authors to conclude that it had a moderate

impact in results prediction.

For lateral loads, researchers have included this effect as well. Ronnquist et al. (2007)

and Ingolfsson et al. (2011) proposed harmonics of these loads that corresponded to the

mean values of those recorded, whereas Ricciardelli et al. (2007) and Ingolfsson et al.

(2011) suggested as characteristic values of these harmonics those with a 95% probability

of non-exceedance.

2.3.5.4 Pedestrian flows

The first research work appraising the effects caused by multiple pedestrians was put

forward by Blanchard et al. (1977), who suggested that a single pedestrian inducing a

resonant response could represent the worst case scenario for serviceability. This very

simplified approach naturally circumvented the need to consider simulations of crowd

movement (computationally demanding at that time). This assumption was subsequently

adopted in codes such as the BS 5400-2:1978 and by several researchers such as Wheeler

(1982) and Ellingwood et al. (1984).

However, this proposal was quickly modified. Through experimental observation of

traffic flows crossing bridges, several researchers had already observed that the arrival

of pedestrians to a bridge matched a Poisson distribution (Kajikawa et al., 1977; Mat-

sumoto et al., 1978) and suggested assumptions and formulae to replicate the stochastic

response caused by these pedestrian flows. According to these authors, if the effect of

multiple pedestrians is equivalent to the superposition of individual effects, the crowd

flow is equivalent to√

Np equal pedestrians (where Np is the number of pedestrians on

the bridge at any time). Despite the advantage of such simple deterministic approach to

represent a pedestrian flow, other researchers have discussed its validity, e.g., Bachmann

(2002) and Brownjohn et al. (2004a).

After those first simplifications, other researchers put forward similar deterministic

expressions to represent the effects of pedestrian flows, e.g., Grundmann et al. (1993),

model that was included in Eurocode 5 (BSI, 2004), or more recently in Setra or the UK

65

2. State of the art

NA to BS EC1 (BSI, 2008). Setra suggested representations of the flows that depended

on the damping capacity of the structure and the flow density (obtained from multiple

Monte Carlo flow simulations). The UK NA to BS EC1 proposed factors that represented

pedestrian flows or groups mathematically derived (Barker et al., 2005a).

Despite these simplifications, other researchers adopted numerical simulations to eval-

uate the effects of these pedestrian flows. This is the case of Ebrahimpour et al. (1996),

who included pedestrian loads with stochastic simulation of phase lags, or the Monte

Carlo simulations adopted by Kerr (1998) for pedestrians using stairs or Willford (2002)

and Ingolfsson et al. (2007) for pedestrians on bridges.

These proposals were obtained for the evaluation of vertical movements and in many

cases are not valid for lateral movements, as argued in Section 2.3.4. In order to include

the pedestrian interaction to the equivalent effects of a flow in the lateral direction, there

are multiple deterministic evaluations such as those enumerated in Section 2.3.4. Alterna-

tively, there are equally numerous proposals based on numerical simulations that include

the stochastic character of pedestrians in a flow, e.g., Carroll et al. (2012).

It should be stressed that multiple of these proposals have been obtained exclu-

sively considering that each pedestrian has a different mean step frequency (normally

distributed) and random arrival time at the bridge (in particular for simulations focused

on vertical movements of the bridge).

Apart from the random arrival of pedestrians to the bridge, several researchers have

included in the flow representation the effects that pedestrians have on other pedestrians’

gait (named as interaction pedestrian-pedestrian). This phenomenon has obviously a

random nature as it depends on the simulation of local concentrations of pedestrians

(i.e., the coincidence of slower and faster than average pedestrians at the same area of the

bridge). According to some researchers (Brownjohn et al., 2004b; Piccardo et al., 2012)

it may involve the adoption of the same step frequency and phase (pedestrian-pedestrian

synchronisation), although this effect has not been explored experimentally in a rigorous

manner.

One of the few proposals representing this stochastic event through a deterministic

approach is that of Kramer (1998) (described by fib Bulletin 32, 2006), who proposed

a coefficient assessing this effect in flows. Many other proposals included this effect

in stochastic models (representing pedestrian events with macroscopic models -fluid- or

microscopic models -pedestrians as particles-). The first model was used by Bodgi et al.

(2007) and Venuti et al. (2007b) (these authors related density to speed of pedestrians

and in some cases included pedestrian-pedestrian synchronisation), and the second by

Carroll et al. (2012).

Simpler stochastic models, including the effect caused by pedestrians in others’ gait,

proposed the use of normal distributions describing step frequencies with a standard

deviation that depended on the density of the flow (Butz et al., 2008).

2.3.5.5 Structural movements prediction

Due to the importance of the stochastic effects in the prediction of traffic flows, few

authors have recognised and considered this variability to predict the peak structural ac-

66

2. State of the art

celerations with methods simpler than those exposed in previous sections. The approaches

are based on the assumption that the peak responses of a bridge are a random event.

Matsumoto et al. (1978) considered that the deflections of a bridge could be described

with a normal distribution with a mean µ = 0. From numerical simulations, Ingolfsson

et al. (2007) fitted a Gumbel and a Generalized Extreme Value (GEV) distribution to

the peaks obtained in the simulation. Later on, Ingolfsson et al. (2012b) fitted these two

distributions to describe the peak accelerations recorded at a real footbridge.

A relatively similar process was followed by Georgakis et al. (2008) or Butz et al.

(2008). The first authors obtained a representative peak acceleration response spectrum.

This was developed considering a reference structure and pedestrian traffic, and additional

factors allowed changing these to describe other scenarios. The method predicted the peak

acceleration associated to a desired return period.

Butz et al. (2008) defined a spectral load model (for vertical and lateral loads of

pedestrians crossing a simply supported bridge), and based on these they described the

variance of the acceleration response. This variance allowed the authors to propose a

prediction for the peak acceleration (50% exceedance probability) and the characteristic

peak acceleration (5% exceedance probability).

2.3.6 Current load models: inherent drawbacks

Despite the availability of the research previously cited, current models used for ser-

viceability evaluation have introduced very few of these developments, or have introduced

amendments that do not really help to improve matters. Examples of the shortcomings

with the approaches that continue to be used are:

a) The description of vertical and lateral loads through truncated Fourier series does not

provide a rigorous description of the energy introduced by individual footsteps.

b) Lateral loads described by Fourier series are unable to capture the phenomenon of

pedestrian-structure synchronisation, which experimental recordings prove to be of

utmost importance. This effect appears to arise from a two-stage process: a first stage

including pedestrians that produce noticeable lateral response without changing their

step frequency; and a second stage where some users may further influence the lateral

response by changing their step frequency to one more comfortable given the structural

movement. However, Fourier series are unable to capture the first and can reproduce

the second only through assumptions that have very weak empirical support. Recent

proposals show that a plausible alternative is that based on the movement of the CoM,

which would be able to consider the first without including substantial conjectures.

c) Some proposals suggest that intra-subject variability may have an important impact

in response prediction. Nonetheless, the most commonly used load model (Fourier

series) is not the most adequate method to define intra-subject variability of pedestrian

loads: considering few harmonics with larger amplitudes than recorded to include

contributions around these or attributing random phases to these harmonics may be

reproducing entirely different loads compared to those intended to be defined.

67

2. State of the art

d) The assumption of deterministic load models (e.g. Wp or bn in Equation 2.3.2) to

describe all the population should be thoroughly assessed to ensure its validity. This

evaluation should account for the characteristics of the actual flow crossing the struc-

ture (thorough description of speed, pedestrian weight and step frequency) and include

possible relationships between these (unless statistical tests show these to be uncorre-

lated).

e) The representation of pedestrian flows is of extreme importance in response prediction

(both for vertical and lateral response). Monte Carlo simulations may be realistic

for flows with low densities, where interactions (such overtaking or reducing speed)

among pedestrians may be unusual, however these cannot be disregarded for flows

with higher densities. Accordingly, evaluations using more sophisticated methods may

prove extremely valuable for a correct response assessment.

2.4 Damping characteristics of footbridge structures

The damping ratio is related to the inherent capacity of structures to dissipate the

energy of their movement. The mechanisms involved in such effect are multiple and not

fully understood, therefore its estimation for a numerical representation of structures will

always comprise a large degree of uncertainty and only experimental testing can provide

an adequate and realistic estimate.

Due to this large uncertainty, multiple experimental works have attempted to assess

this parameter reporting ranges instead of single values. Nonetheless, for footbridges,

it is usually recommended to make allowances in their design for the implementation of

damping devices that increase this dissipation capacity (Setra, 2006).

The following paragraphs describe the common values of damping ratios considered

in the design of footbridges as well as the characteristics of the most frequent damping

devices placed in footbridges.

2.4.1 Inherent structural damping

The damping ratio depends on the materials of the transverse section, the amplitude

of the movements (larger movements are related to higher dissipation, as mentioned by

Tilly et al., 1984), and even the structural scheme (as pointed out by fib Bulletin 32,

2006). Some authors have suggested as well that it was related to the magnitude of the

structure fundamental frequency (the higher the modal frequency the larger the damping

ratio, Wheeler, 1982). However others have argued the lack of evidence supporting that

statement (Tilly et al., 1984).

One of the first summaries of damping ratios based on experimental observations of

multiple bridges is that of Blanchard et al. (1977), which reported these values according

to the materials of the transverse section. This proposal has been largely used since then,

e.g., Wheeler (1982), BSI (1978), Tilly et al. (1984) and Setra (2006).

These values coincide with the minimum magnitudes recorded by Matsumoto et al.

(1978) (steel and prestressed concrete bridges), they are similar or larger than the mean

values reported by Bachmann et al. (1995) and similar or smaller than those reported in

68

2. State of the art

Table 2.1: Values of the damping ratio according to structural materials of the footbridge,where [1] corresponds to Blanchard et al. (1977), [2] to Bachmann et al. (1995) and [3] toBSI (2003).

Material [1] [2] (mean value) [3]

Steel, with asphalt or epoxy surfacing 0.55% 0.4% 0.5%

Composite (concrete-steel) 0.64% 0.6% 0.5%

Prestressed concrete 0.8% 1.0% 1.0%

Reinforced concrete 0.8% 1.3% 1.5%

BSI (2003) (see Table 2.1), which highlight the difficulty of estimating such value. For

cable-stayed bridges specifically, BSI (2003) suggested reducing the damping ratio by a

factor of 0.75.

An alternative evaluation of this damping ratio was given by fib Bulletin 32 (2006),

that proposed obtaining the total damping ratio as a sum of that provided by the material,

the structural scheme (differentiating cable-stayed bridges or suspension bridges among

others) and the bearing conditions (in agreement to comments of Tilly et al., 1984). The

values assigned to each alternative were experimentally obtained, as mentioned in the fib

Bulletin 32.

The proposals of Blanchard et al. (1977) or fib Bulletin 32 were obtained from ex-

perimental assessments at bridges not exclusively used by pedestrians, which may have

an impact as suggested by Tilly et al. (1984) due to the magnitudes of the movements

that can be given in footbridges in comparison to road bridges. Nonetheless, the few

observations recorded in pedestrian bridges (Matsumoto et al., 1978; Bachmann et al.,

1995) described values similar to those reported in Table 2.1.

2.4.2 Damping devices in footbridges

There are multiple devices capable of performing a control and reduction of the dy-

namic movements of footbridges. These have different principles upon which they dissi-

pate energy of the movement and, according to these, they can be classified as passive,

active, hybrid or semi-active (further details and examples in civil engineering structures

can be found in Housner et al., 1997, or Soong et al., 2002).

In footbridges, the most common control devices are passive. These enhance damping

and stiffness of the structure where they are located regardless the movements that are

developed at the structure. The most widely used passive dampers are Tuned Mass

Dampers (TMDs), (for both vertical and lateral accelerations), and in a smaller degree

Viscous Fluid Dampers (VFDs) or Tuned Liquid Dampers (TLDs) (as described by Setra

and fib Bulletin 32).

TMDs are devices that have a mass with spring and damping elements tuned to a

single mode, idea that was already explored in the middle of the past century (Den

Hartog, 1984). These can be found in the London Millennium bridge (Dallard et al.,

2001), Solferino bridge (Setra) (Figure 2.12(a,b)) or multiple smaller bridges, as reported

by Zivanovic et al. (2005) or by fib Bulletin 32. The characteristics of TMDs can be

69

2. State of the art

derived from the formulation proposed by Den Hartog (1984), given by Equations 2.4.1

and 2.4.2, or that of Ioi et al. (1978). These equations define the optimum tuning and

damping of the device (where µ corresponds to the ratio between absorber and structure

modal mass, fa and fs the frequencies of the absorber and structure, and (c/cc)opt the

optimum damping of the absorber).

fa =1

1 + µfs (2.4.1)

(

c

cc

)

opt

=

√

3

8

µ

(1 + µ)3(2.4.2)

Structures such as the London Millennium bridge (Dallard et al., 2001) also have

Viscous Fluid Dampers (VFDs) controlling lateral movements (Figure 2.12(c)). The dis-

sipation of these devices is based on the movement of a fluid through orifices caused by a

piston moved by the bridge. For movements at frequencies below 4.0 Hz, the force trans-

mitted by this device can be considered proportional to the velocity of the movement

(Soong, 1997). In this case the force is described by Equation 2.4.3 (C0 is the frequency

damping coefficient of the material at zero-frequency and can be evaluated from Soong,

1997).

f(t) = C0dx(t)

dt(2.4.3)

Even fewer bridges have Tuned Liquid Dampers (TLDs) regulating lateral movements

caused by pedestrians. This is the case of the T-bridge (P. Fujino et al., 1993; Nakamura

et al., 2006) (Figure 2.12(d)). These devices consist of a liquid in a container that re-

duces movements of the structure by means of wave breaking or sloshing and the viscous

characteristics of the fluid. These have a low cost and maintenance although they have a

highly nonlinear response (Soong et al., 2002). An analysis of their dimensioning can be

found in Y. Fujino (1993).

2.5 Comfort criteria in structures with pedestrians

The serviceability of a structure includes aspects such as durability, function and

appearance (Menn, 1990). From those, function is the factor that constraints the design

of footbridges: function is fulfilled if users consider the vibrations of the structure as

comfortable.

According to multiple researchers (Wheeler, 1982; Ellingwood et al., 1984; Corbridge

et al., 1986; Bachmann et al., 1987; Gierke et al., 1988), human comfort or response to

vibration depends on multiple factors such as its characteristics (magnitude, frequency,

direction in relation to the subject, duration and decay duration), user characteristics

(posture – sitting, standing, lying –, activity – standing, walking, running – and age),

simultaneous presence of other users, etc.

Due to the large number of parameters influencing the user perception, it is difficult to

establish a single representative boundary for comfort. In this sense, evaluations such as

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2. State of the art

(a) (b)

(c) (d)

Figure 2.12: (a) TMDs at LMB, Structurae (2015); (b) Leopold-Sedar-Senghor bridge,Setra (2006); (c) VFD at LMB, Structurae (2015); (d) TLDs at T-bridge, Nakamura et al.(2006).

that of Reiher et al. (1931) or Goldmann (1948) classified the magnitudes of movements

in four or six levels of perception respectively. More recently, Kasperski (2006) observed

a ratio of 4.0 between movements considered unpleasant by 95% and 5% of the users.

Research works on human comfort such as those of Gierke et al. (1988), Reiher et al.

(1931), and Goldmann (1948) were developed within the analysis areas of physiological

response of the human body and psychological response of humans in vehicles (trains, cars

or aircrafts), in buildings or in working areas (machinery, hand-tools, etc.) among others.

Consequently, their criteria were developed considering standing or sitting individuals

and showed minimum comfort at movement frequencies of 5.0 or 1.5 Hz (for vertical and

lateral movements respectively), which is due to the resonance of the human body under

vibrations.

These research works, together with those of Bachmann et al. (1987), Irwin (1978),

and Griffin (1998), represent the basis upon which codes and guidelines (e.g., ISO 2631,

2003, or ISO 10137, 2005) propose tolerance criteria for vibration exposure of humans in

these analysis areas. One of the main characteristics of these proposals is the evaluation

of comfort according to a representative acceleration value of the movement, which can

be: peak, root-mean-squared (RMS) acceleration or more recently vibration dose value

(VDV) described by Equations 2.5.1 and 2.5.2:

RMS =

√

∫ t2

t1a2(t) dt

t2 − t1(2.5.1)

VDV =

[∫ t2

t1

a4(t) dt

]

14

(2.5.2)

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2. State of the art

where a(t) is the time acceleration, and t1 and t2 describe the weighting time interval.

The acceleration is selected as the magnitude representing the vibration of an event

based on observations of Bachmann et al. (1987) or SCI (1989). The first stated that

pedestrian perception is proportional to acceleration for movements at frequencies ranging

from 1.0 to 10.0 Hz and proportional to velocity for movements at frequencies above

10.0 Hz. The second author stated that, at very low frequency movements, human reaction

is related to the rate of adjustment of inertia forces on the body. The accelerations

weighted with time (RMS or VDV) are considered by several codes or guidelines (e.g.,

ISO 10137, according to Pantak et al., 2012) to emphasise the response throughout the

event instead of that of a single instant.

For footbridges in particular, multiple proposals (BSI, 2008; Butz et al., 2008; Setra,

2006) assessed the comfort according to the magnitude of the peak acceleration recorded

at any point of the deck. However, this comparison does not take into account whether

this peak is recurrent in time or not. Hence, in order to overcome this deficiency, many

other researchers have argued that magnitudes such as the RMS acceleration or the VDV

values should be used (Barker et al., 2005a; Ingolfsson et al., 2012b).

2.5.1 Evaluation of vertical movements at footbridges

Comfort of footbridge users is influenced by more factors than those mentioned above:

among others, there is the aspect of the structure (pedestrians expect light bridges to

vibrate and robust bridges to remain still), previous user experience at the same bridge

or height of the structure above the ground (Wheeler, 1982).

In relation to comfort limits, several proposals for vertical movements are based on

experimental data that reproduce conditions given in real bridges, e.g., Smith (1969),

Leonard (1966) and Kobori et al. (1974). The first study considered both standing and

walking pedestrians whereas the other two focused on evaluating comfort for standing

pedestrians exclusively.

Results of Leonard (1966) and Smith (1969) were used by Blanchard et al. (1977)

to propose a comfort limit for pedestrians on a bridge (0.5√fs, where fs is the vertical

mode of vibration excited by the users), that was adopted by the first British code con-

sidering the serviceability of footbridges (BSI, 1978). This code has been largely used

until recently (BSI, 2006a). The work published by Kobori et al. (1974) (where unpleas-

ant structural movements were related to the peak dynamic deflection u0 according to

u0 2πf > 24 mm/s) was very similar to the limit proposed in another of the first codes

for footbridges, the Ontario Bridge Code ONT83 (described in the fib Bulletin 32), which

corresponded to 0.25f 0.78s .

Due to the mentioned variability inherent to the evaluation of comfort, just after the

publication of these codes, several proposals were already suggesting different limits (Tilly

et al., 1984; Bachmann et al., 1987; BSI, 2004). The first author suggested increasing the

limit of acceptable vertical accelerations of the first British code (BSI, 1978) to 1.0√fs if

fs was below 1.7 or above 2.2 Hz, the second a limit between 0.5 and 1.0 m/s2 and the

Eurocode 5 a limit of 0.7 m/s2. Some of these alternative assessments proposed comfort

limits independent of the frequency of the movement, which was justified by the fact that

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2. State of the art

the movement of a bridge may be caused by multiple frequencies instead of one.

The latest published comfort limits for footbridge users have proposed comfort ranges

independent of the vibration frequency as well and, additionally, they have introduced

factors that adapt the comfort limits to consider their aspect, their height over the ground,

whether there is an alternative route, etc. This is the case of the UK NA to BS EC1 (2008),

based on the evaluations of Mackenzie et al. (2005), the Synpex (Butz et al., 2008) and

Setra guidelines (2006), based on the opinions of users at several bridges, or that of Butz

et al. (2007), which is similar to that of Mackenzie et al. (2005). According to some of

these proposals, peak vertical accelerations provide a maximum comfort if they are below

0.5 m/s2, medium if they are between 0.5 and 1.0 m/s2, minimum at the range between

1.0 and 2.0-2.5 m/s2 and are not acceptable beyond 2.5 m/s2 (2.0 m/s2 according to the

UK NA to BS EC1).

From these recent proposals, it should be highlighted that their ranges are valid for

walking pedestrians (although it is not explicitly stated).

In relation to the comfort of standing or sitting pedestrians there are even fewer

proposals. ISO 10137 (2005) suggested limits (0.3 m/s2) that were valid for both standing

and sitting users (in agreement to the observations of Thuong et al., 2002) that were half

of those for walking users. Nonetheless, several researchers considered the limits defined

in ISO 10137 (2005) with caution due to their origin and experimental results pointing

towards drastically different limits (Kasperski, 2006).

2.5.2 Evaluation of lateral movements at footbridges

The assessment of comfort of pedestrians under lateral movements has been evaluated

by an even more reduced number of researchers.

Leonard (1966) proposed the first limit to ensure comfort in this direction (the comfort

limit corresponded to that for standing users enduring vertical movements), although

the author recognised that further evaluations should be performed to establish a sound

proposal.

In design codes, one of the first comfort assessments is that of EC5 (BSI, 2004) (where

the limit 0.2 m/s2 was proposed), although Willford (2002) commented that this was not

valid for walking users since it had been described by standing pedestrians. The same au-

thor observed that in high-rise buildings lateral accelerations of magnitude 0.25 m/s2 (and

frequency around 1.0 Hz) still allowed pedestrians to walk normally and that accelerations

of 0.7 m/s2 (frequency around 0.5 Hz) caused pedestrians to walk difficultly.

One of the few comfort evaluations conducted in a footbridge is that detailed by Naka-

mura (2003) who, based on observations at the M-bridge and the T-bridge, extracted some

interesting conclusions: some users found it was uncomfortable to walk with a movement

amplitude of 10 mm (equivalent to a lateral acceleration of 0.3 m/s2); some had difficulties

in walking at normal pace when the movement amplitude was 25 mm (0.75 m/s2); and

with amplitudes of 45 mm (1.35 m/s2) people would loose balance and stop walking (in

comparison, the London Millennium Bridge registered lateral accelerations of 2.1 m/s2

with a maximum amplitude of 70 mm).

Based on experimental tests of Charles et al. (2005) and the results of Nakamura

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2. State of the art

(2003), the Setra guideline (2006) proposes limiting the lateral accelerations to 0.10 m/s2

to ensure maximum comfort. The same guideline relaxes this limit to values of 0.3 and

0.8 m/s2 if less restrictive movements are permitted.

ISO 10137 (2005) describes a comfort limit for walking, standing and sitting pedes-

trians on a footbridge where the user perceives lateral movements. In this case the same

limit applies to all these users (0.2 m/s2), despite the comments of Thuong et al. (2002)

pointing towards the opposite effect.

2.6 Failure in service of footbridges

There are multiple footbridges that have developed dynamic responses considered by

users as excessive (once they were built). These have been described in relevant literature

and some of these are summarised hereunder.

It should be emphasised that in some of these structure, despite the large movements,

owners did not adopt further measures. Furthermore, when comparing the magnitudes of

unacceptable movements at different footbridges, the importance of factors that modify

comfort perception of users (enumerated in Section 2.5) becomes apparent (e.g., lateral

movements of the Changi Mezzannine bridge compared to acceptable magnitudes at the

M-bridge).

2.6.1 Service failure in vertical direction

2.6.1.1 Shibuya footbridges, Tokyo, Japan

On an exhaustive study performed by Matsumoto et al. in 1978, the authors evalu-

ated the performance of 5 existing footbridges in Tokyo. This assessment was based on

the structural movements caused by one pedestrian, the characteristics of the traffic flow

(arrival of pedestrians) and the prediction of the response under a pedestrian flow (stochas-

tic superposition). According to these analyses, the authors found that two structures,

Shibuya East and West Exits (steel girders of span lengths 40.3 and 48.5 m and vertical

frequencies 2.51 and 2.09 Hz respectively) could exhibit vertical accelerations larger than

1.0 m/s2, the considered serviceability limit, under flows of 1.0 ped/m/s (1.2 and 2.3 m/s2

respectively). Accordingly, the owners of the bridges undertook modifications to damp

the vibrations of the bridges.

2.6.1.2 Eutinger Waagsteg bridge in Pforzheim, Germany

The Eutinger Waagsteg bridge (Butz et al., 2008) (completed in 1992, see Figure 2.13(a))

is a stress ribbon bridge with a main span of 50 m that crosses the lake of the Enz river.

The main span has a length of 50 m and a deck width of 2.88 m. A modal analysis iden-

tified several modes between 1.0 and 2.0 Hz as well as near 4.0 Hz whereas experimental

tests with groups of 4 to 6 uncoordinated pedestrians developed vertical accelerations with

peak movements of 0.8 m/s2. Despite these movements being larger than the considered

limit of serviceability pedestrians have not complained about the large movements.

2.6.1.3 Kochenhofsteg footbridge, Stuttgart, Germany

The Kochenhofsteg footbridge (Butz et al., 2008) (completed in 1992, see Figure 2.13(b))

is a suspension bridge with a main span of 42.5 m and deck width of 3.0 m that crosses

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2. State of the art

a street linking a car park and the fair of Stuttgart. The bridge is back anchored at one

side (where there are masts) and anchored to the abutment at the other. Under the deck,

there are inverted prestressed cables providing additional lateral stiffness.

The structure has vertical vibration modes with frequencies near 1.0 and 2.0 Hz. Tests

with pedestrians described how the structure could develop very large vertical movements

when crossed by a group of 4 pedestrians walking in step or a flow of uncoordinated

pedestrians. Despite the likelihood of being used by groups of pedestrians walking fast,

no additional measures have been implemented.

2.6.1.4 Katzbuckelbruecke, Duisburg, Germany

This suspension bridge (Butz et al., 2008) (completed in 1999, see Figure 2.13(c)) is a

moving structure that spans a distance of 73.7 m with a deck made of hinged plates.

During a multi-pedestrian event, data was collected to appraise its dynamic properties

and responses developed. Results showed a vertical mode at a frequency of 0.72 Hz

with a damping ratio of 1.4%. With pedestrians, tests showed that the structure could

move considerably in the vertical direction under the effect of 34 pedestrians (2.3 m/s2),

although movements were lower with continuous flows of 0.3 ped/m2. Therefore, no

additional measures have been implemented.

2.6.1.5 ‘Olga’ Park bridge, Oberhausen, Germany

As part of a broader study (see Figure 2.13(d)), several researchers (Kasperski, 2006;

Kasperski et al., 2007) recorded the response of a cable-stayed pedestrian bridge under

uncontrolled and controlled flow conditions. The structure (steel girders and concrete

slab) connects a park to a public transport station and has a total length of 66 m and a

deck of width of 4.5 m.

The first vertical frequency modes have magnitudes of 1.8 and 3.9 Hz and associated

critical damping ratios of 0.5%. Under 9 pedestrians the structure described peak vertical

accelerations of 1.1 m/s2 and under uncontrolled flows of pedestrians (3951 users crossed

the bridge during 3 hours after a festival) the peak accelerations had a magnitude of

1.35 m/s2 (many users were alarmed by the movements). According to the authors these

indicated the need of measures to reduce the movement.

2.6.1.6 Erzbahnschwinge bridge, Bochum, Germany

Another structure evaluated dynamically by Kasperski (2006) is the Erzbahnschwinge

bridge (see Figure 2.13(e)), a mono-cable suspension bridge opened in 2003 with curved

deck (steel truss with concrete slab). A numerical analysis of the structure showed that

the bridge has two vertical modes at frequencies 1.80 and 1.85 Hz with very low damp-

ing associated (0.34 and 0.22% respectively). Experimental tests with two unprompted

pedestrians demonstrated that they could trigger accelerations near 1.0 m/s2. However,

no alternative measures have been applied to reduce this response.

2.6.1.7 Other reported cases

Among others, bridges where large vertical movements were detected and alternative

solutions were implemented to avoid them are: the Britzer Damm in Berlin; the Schwedter

Strasse bridge in Berlin; the Mjomnesun det bridge, in Norway; the Belloagio to Bally’s

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2. State of the art

footbridge in Las Vegas; the Forchheim footbridge in Germany, or the Minden Footbridge

in Germany (fib Bulletin 32, 2006). Some bridges where large lateral movements were

reported (see following section) presented as well large vertical movements. Other cases

with large lateral movements can be found in the Proceedings of the Footbridge Conference

that takes place every three years since 2002.

(a) (b)

(c) (d) (e)

Figure 2.13: Bridges with large vertical movements in service: (a) Eutinger Waagsteg,Germany (Butz et al., 2008); (b) Kochenhofsteg bridge, Germany (Butz et al., 2008); (c)Katzbuckelbrucke bridge, Germany (Butz et al., 2008); (d) ‘Olga’ Park bridge, Germany(Kasperski, 2006);(e) Erzbahnschwinge bridge, Germany (Kasperski, 2006).

2.6.2 Service failure in lateral direction

2.6.2.1 Footbridge over the Main, Erlach, Switzerland

The footbridge over the Main, located at Erlach (Franck, 2009) (see Figure 2.14(a)),

was opened in 1972 and consists of an arch supporting a steel box girder deck with a

width of 3.0 m that spans a length of 110 m. During its opening day, flows with a

maximum number of coincident pedestrians on the bridge ranging from 300 to 400 crossed

the bridge and caused strong lateral accelerations (Bachmann, 1992) (maximum lateral

displacements of 25 mm). In order to avoid further serviceability problems, several tuned

vibration absorbers were installed to act in horizontal direction.

2.6.2.2 Toda Park bridge (T-bridge), Japan

The T-bridge (opened in 1989, see Figure 2.14(b)) as mentioned by P. Fujino et al.,

1993) and Nakamura (2004) is a cable-stayed bridge that crosses a river and connects a

sports stadium with a bus terminal. The structure has two planes of cables and a deck

consisting of a steel box girder with effective width of 5.25 m spanning a total distance

of 180 m. Dynamically, the bridge presents vertical and lateral modes with frequencies

below 1.0 Hz.

On days with sport events, the bridge would receive dense traffic flows, with 2000 users

at a time on the bridge. During these episodes large vertical and lateral responses were

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2. State of the art

recorded: P. Fujino et al. (1993) reported lateral movements with maximum amplitudes of

10 mm and some pedestrians had their walking affected. In order to suppress the excessive

lateral movements, tuned liquid dampers were installed in the box girder (Nakamura et

al., 2006).

2.6.2.3 Maple valley suspension bridge (M-bridge), Nasu Shiobara, Japan

The M-bridge (built in 1999, see Figure 2.14(c)) (Nakamura et al., 2006) is a suspension

bridge that spans a dam lake. The main span of the structure has a length of 320 m and

the deck consists of two steel beams and sway bracings that support a deck of wooden

plates and steel grating slabs. Additionally, in a horizontal plane the deck is stiffened by

wind-ropes.

Dynamic tests showed that the structure has several lateral modes below and near

1.0 Hz. Under normal pedestrian flows (part of the tests), the structure developed peak

lateral movements of 45 mm amplitude (1.35 m/s2) and some users were seen to loose

balance and stop walking. However, movements of these magnitudes are considered to be

the serviceability limits in the lateral direction by the owner, hence countermeasures to

reduce movements would be applied only if larger events were observed (Nakamura et al.,

2006).

2.6.2.4 Leopold-Sedar-Senghor bridge (Solferino), Paris, France

The Leopold-Sedar-Senghor bridge is a steel arch bridge with total length of 140 m and

width 12.0-14.8 m that crosses the Seine river in Paris (Setra, 2006), see Figure 2.14(d).

On its opening day, in December of 1999, large and unexpected lateral accelerations were

registered.

Dynamic analyses of the structure showed that the bridge has a lateral mode with

frequency 0.81 Hz and vertical and torsional modes with frequencies near 2.0 Hz. Tests

with pedestrians demonstrated that 16 pedestrians walking or running on the structure

could induce peak lateral accelerations of 2.0 m/s2 and vertical accelerations of 2.5 m/s2

respectively. In order to avoid such large movements, six tuned mass dampers for swing

movements (pendular systems supporting masses) and eight tuned mass damper (for

torsional and swing movements) were placed at the structure (Setra, 2006).

2.6.2.5 London Millennium Bridge, London, United Kingdom

The London Millennium Bridge (Dallard et al., 2001) is a shallow suspension bridge

with three spans of 81, 144 and 108 m that crosses the Thames river (see Figure 2.14(e)).

Transverse steel box sections hold the deck from the cables every 8 m. The 4S m wide deck

consists of two lateral steel edge tubes that support the aluminium surface and transmit

the loads to the transverse arms.

During the opening day, in 2000, crowd flows of maximum density between 1.3 and

1.5 ped/m2 crossed the bridge. Under this intense flow, the bridge developed large lateral

vibrations. The movements at the lateral span near the southbound had a frequency

near 0.8 Hz, those of the central span around 0.5 Hz and those of the northbound span

a frequency around 1.0 Hz. From recordings of the event it was estimated that the

movements had reached peak lateral accelerations between 2.0 and 2.5 m/s2. In order to

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2. State of the art

avoid further movements of such magnitude during serviceability, the bridge was reopened

after the installation of 37 viscous dampers (damping out horizontal movements) and 26

pairs of tuned mass dampers.

2.6.2.6 Lardal footbridge, Norway

The Lardal bridge (2001) (Ronnquist et al., 2008) is a shallow arch/truss bridge located

near Lardal, in Norway, that crosses the Lagen River (see Figure 2.14(f)). The deck has

a main span of 91 m and consists of timber longitudinal beams connected with transverse

steel girders with under-deck steel cable reinforcements at mid-span.

During the opening day, the bridge supported the crossing of numerous pedestrians

simultaneously and exhibited large lateral movements at a frequency of 0.83 Hz. Experi-

mental tests with pedestrian flows showed that 40 pedestrians could cause movements with

lateral peak accelerations of 1.0 m/s2 (Ronnquist et al., 2008), despite the large damping

ratio associated to that mode (2.5%). Due to this large damping ratio, Ronnquist et al.

(2008) recommended stiffening the deck instead of placing damping devices.

2.6.2.7 Changi Mezzanine bridge, Singapore Airport, Singapore

The Changi Mezzanine bridge (2002) (Brownjohn et al., 2004a) is a shallow arch bridge

with a main span of 140 m and side spans of 30 m that connects two terminals within

the airport (see Figure 2.15(a)). The structure consists of a steel truss girder, of variable

depth and width, with glass cladding supported by pins.

Prior to the opening, experimental tests were conducted. These reported lateral,

torsional and vertical frequencies below 2.0 Hz. The experimental tests with pedestrians

showed that 150 pedestrians could cause large lateral movements (pedestrians stopped

walking when peak accelerations were 0.17 m/s2 approximately). In the vertical direction,

three users walking at a prompted step frequency were able to generate peak vertical

accelerations of 0.1 m/s2. Due to these results, a pair of tuned mass dampers was placed

on the structure.

2.6.2.8 Passerelle Simone de Beauvoir, Paris, France

The Simone de Beauvoir footbridge (Hoorpah et al., 2008) (inaugurated in 2006, see

Figure 2.15(b)) has a structure that combines a stress ribbon and a shallow arch walkways.

This bridge spans a distance of 200 m over the Seine river in Paris.

Numerical analyses of the structure showed that the design had several transverse

modes with frequencies near or below 1.1 Hz and multiple vertical modes between 1.4 and

2.1 Hz (Cespedes et al., 2005). These frequencies were confirmed through vibration tests

performed at the structure. Uncoordinated flows of 100 pedestrians produced lateral

movements with peak amplitudes of 30 mm which could be considerably larger if 60

pedestrians walked in a synchronised manner on the bridge. Despite these observations, no

further measures have been implemented and response in service has remained acceptable.

2.6.2.9 Pedro e Ines footbridge, Coimbra, Portugal

The Pedro e Ines footbridge (Adao Da Fonseca et al., 2005) (see Figure 2.15(d))

consists of a central arch and two lateral half arches that crosses the Mondego river at

Coimbra and opened in 2006. The bridge spans a total distance of 274.5 m and has a

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2. State of the art

(c)

(a) (b)

(f)

(d)

(e)

Figure 2.14: Bridges with large lateral movements in service: (a) footbridge over the Mainat Erlach, Switzerland (Franck, 2009); (b) T-bridge, Japan (P. Fujino et al., 1993); (c) M-bridge, Japan (Nakamura et al., 2006); (d) Leopold-Sedar-Senghor, France (Setra, 2006);(e) London Millennium Bridge, UK (Dallard et al., 2001); (f) Lardal footbridge, Norway(Ronnquist et al., 2008).

deck with a composite steel-concrete box section and 4 m width.

In plan, the bridge is anti-symmetrical in the longitudinal direction, which provides

larger lateral stiffness. Nonetheless ambient and free vibration tests showed that the

structure had a lateral vibration mode at a frequency of 0.91 Hz and an average damping

ratio associated to that mode between 0.56 and 0.89% (Caetano et al., 2010). Tests with

pedestrians showed that flows with 145 pedestrians could trigger lateral accelerations with

a maximum magnitude of 1.2 m/s2 and peak displacements of 80 mm. Following these

results, it was decided to place six lateral tuned mass dampers (Caetano et al., 2008).

2.6.2.10 Tri-Countries, Weil am Rhein, Germany

The Tri-Countries bridge (see Figure 2.15(c)) (Haberle, 2010) is an arch bridge that

crosses the Rhine River and connects Germany with France near the border of that country

with Switzerland at the cities of Weil am Rhein (Germany) and Huningue (France). The

bridge has two arches that span a distance of 230 m from which the deck (5.0 m clear

width) is suspended (deck that leaves a vertical clearance over the river of 7.80 m).

Before the opening in 2007, modal tests identified three lateral modes with frequencies

0.90, 0.95 and 1.00 Hz (Ingolfsson et al., 2012a), whereas tests with traffic flows showed

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2. State of the art

that only more than 500 people walking on the bridge at a speed above 1.61 m/s would

generate very large lateral movements. Since this situation was deemed to be very unlikely,

dampers have not been installed.

2.6.2.11 Other reported cases

Other bridges where excessive lateral movements have been observed include: the

Clifton Suspension bridge in Bristol, the Auckland Harbour bridge, a footbridge at the

Geneva Airport, the Link bridge from National Exhibition Centre to Railway Station in

Birmingham, the Groves Suspension bridge in Chester (Dallard et al., 2001), the Alexan-

dra road bridge in Ottawa, or the Brooklyn bridge, in New York (Franck, 2009).

(c)(a) (b)

(d)

Figure 2.15: Bridges with large lateral movements in service: (a) Changi Mezzanine bridge,Singapore (Brownjohn et al., 2004a); (b) Passerelle Simone de Beauvoir, France (Hoorpahet al., 2008); (c) Tri-Countries, Germany (Haberle, 2010) ; (d) Pedro e Ines footbridge,Portugal (Adao Da Fonseca et al., 2005).

2.7 Design recommendations

Numerous research proposals introduce methods related to the serviceability analysis

focused on: a) the avoidance or prediction, during design stages, of dynamic responses

such as those described in Section 2.6, or b) the numerical representation of pedestrian

scenarios that cause these responses. The dynamic response predicted by these proposals

may not fulfil the comfort criteria established by codes (described in Section 2.5). In

this case designers are advised to modify the structural design in order to avoid large

amplitudes of these movements. Regarding this design optimization of footbridges, there

are several suggestions based on experience of designers or on theoretical proposals that

have been developed using research described in previous sections. Following sections

enumerate recommendations published in relation to these topics, which constitute the

main design rules available for footbridge designers.

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2. State of the art

2.7.1 Guidelines related to serviceability appraisal

Due to the random nature of pedestrian loads, the vibration SLS evaluation was in-

troduced in codes together with the publication of simple methods appraising this SLS.

This was the case of the proposals of Blanchard et al. (1977) (introduced in the code BS

5400:1978 Part 2, BSI, 1978), of Rainer et al. (1988) or of Grundmann et al. (1993), three

simple methods that evaluated the peak response on a simply supported structure (the

first method due a single resonant pedestrian and the other two with multiple pedestri-

ans), without the need of numerical simulations. The last two methods were based on a

relationship between the dynamic peak accelerations and the dynamic peak deflections,

and between the dynamic peak deflections and the static deflections caused by the static

loads introduced by the pedestrian traffic flows.

However, it was soon seen that these proposals were not adequately representing the

pedestrian serviceability scenarios. Considering this fact and the random nature of loads,

guidelines and codes have proposed load models instead of simple assessments of the

movements. These load models predict the dynamic response by means of FE models.

Examples of these can be found in codes such as the prenorm of Eurocode 1, Part 1 (fib

Bulletin 32, 2006), the current British Standards for footbridges (the UK NA to BS EC1,

BSI, 2008) or the Setra guideline (2006).

The UK NA to BS EC1 (BSI, 2008) comprehends the work commissioned by the UK

Highway Agency to two consultancy companies to evaluate the serviceability response of

footbridges (Barker et al., 2005a; Mackenzie et al., 2005; Barker et al., 2005b). This work

attempted to overcome the deficiencies of prior design procedures such as the population

representation (model used was far too simplified) or the comfort evaluation beyond the

peak response magnitudes (bridges with similar peak responses may not be equivalently

regarded by users). The standard provides a load model (with largest amplitude for pedes-

trian step frequencies between 1.8 and 2.0 Hz) to evaluate the vertical responses whereas

in the lateral direction it describes ranges of structural characteristics that ensure an ad-

equate lateral response (which the code points out that could occur for lateral structural

frequencies above 1.5 Hz). The vertical movements predicted by the standard correspond

to peak responses representing events that will be surpassed on few occasions, despite the

fact that the work of Barker et al. (2005a) proposed using accelerations averaged with

time (e.g., RMS accelerations). The prediction of the vertical accelerations of a foot-

bridge is based on static loads (and variable time amplitude) that represent continuous or

discontinuous pedestrian flows (similar to other methods such as that of Setra guideline,

2006).

The Setra guideline (2006) describes the effects of multiple pedestrian events through

a simpler static load with variable time amplitude that exclusively depends on the pedes-

trian density and the footbridge damping ratio (the density of these traffic flows is limited

to 1.0 ped/m2). In relation to the movements, the guideline explicitly mentions that the

accelerations describe events that may be surpassed on 5% of the occasions.

The latest design proposals that have been published favour the prediction of dynamic

movements instead of the representation of traffic flows and, for particular structures

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2. State of the art

and considering simplified load models, they propose a straightforward method to predict

vertical accelerations, e.g, that of Georgakis et al. (2008), or vertical and lateral accel-

erations, as in Synpex (Butz et al., 2008) (which was developed for simply supported

footbridges). The approach of Georgakis et al. (2008) is based on a response spectrum

method and stems from Monte Carlo simulations performed on simply supported bridges

crossed by a specific pedestrian flow. The method requires designers to make an assump-

tion in relation to the mean step frequency of the users. The proposal of Synpex, which

was obtained using a response spectrum method as well, predicts vertical and lateral ac-

celerations of simply supported footbridges under the passage of three pedestrian flows

considered representative of a large number of events (0.5, 1.0 and 1.5 ped/m2).

2.7.2 Guidelines related to footbridge design

A second group of design recommendations includes proposals focused on the improve-

ment of the design of footbridges in vibration SLS.

The main focus of these recommendations involve the avoidance of critical ranges of

structural frequencies, vertically and laterally, e.g., BSI (1978), Bachmann et al. (1987),

Bachmann et al. (1995), and BSI (2006a). In order to achieve this design characteristic,

researchers have mentioned the modification of the bridge stiffness and mass, e.g., Tilly

et al. (1984) or Bachmann et al. (1995).

In relation to the stiffness, researchers have indicated measures to increase this pa-

rameter without substantially modifying the mass such as: the thickness of steel flanges,

the depth in truss girder bridges, the span arrangement or the material Young’s modu-

lus. Regarding the mass, some proposals include the use of heavy decks or lightweight

materials (fib Bulletin 32 and Setra).

In the lateral direction, the main modifications that have an impact on the lateral

vibration SLS consist in increasing the width or adding lateral cables to stiffen the deck

(these are effective regardless the footbridge structural type).

Alternatively, the improvement of the damping capacity or the implementation of

dampers are mentioned as measures to reduce the magnitude of the movements. The

damping ratio can be increased by changing the bearings or by applying an asphalt surface

to the deck.

For cable-stayed footbridges, researchers have highlighted the better performance of

cable fan arrangements compared to harp dispositions, the positive effect of increasing

the cable areas and the advantage of using taller pylons. These measures increase the

vertical stiffness of the deck cable stayed system. The use of two individual lateral pylons

is not recommended as it increases torsional flexibility (fib Bulletin 32 and Setra).

For suspension footbridges, Setra highlights the lack of effects that an increment of

cable areas has on their performance. Additionally, several research groups have evaluated

the performance of suspension bridges with alternative cable arrangements using relatively

simplified pedestrian load models, e.g., Huang et al. (2005), Huang et al. (2007), Bruno

et al. (2012), and Faridani et al. (2012).

82

2. State of the art

2.8 Footbridge performance analysis

The dynamic performance of a structure can only be predicted through an adequate

consideration and representation of the essential physical parameters that intervene in this

performance: mass, damping and stiffness. These three parameters are involved in the

resistance of the movement caused by external loads applied to the structure (as described

by d’Alembert’s Principle, Equation 2.3.1, described in Clough et al., 1993).

When assessing the dynamic response of a bridge, this equilibrium equation is consid-

ered for selected n points of that structure, i.e. for an equivalent discrete system. The

equations that describe the dynamic motion of these points or multiple-degree-of-freedom

(MDOF) system is an extension of Equation 2.3.1 (mass, damping and stiffness are given

by matrices and movements and loads by vectors).

There are two methods available to resolve this dynamic assessment. If the system

can be easily represented by one or few nodes (e.g., girder bridges), the movements are

evaluated by resolving these equations (considering certain simplifications). Otherwise,

the system is represented, and these equations resolved, with FE methods.

In relation to the first alternative, it is usually assumed that damping of this struc-

ture is viscous and that the response in time of each of these n nodes can be described

considering modal coordinate superposition. It must be highlighted that this approach is

valid as long as the system is linear during the response (not affected by changes in the

properties, such as yielding of materials or geometrical nonlinearities). Since this is the

usual case for many footbridges, this method has been largely used by many researchers.

This is the case of Blanchard et al. (1977), Matsumoto et al. (1978), Wheeler (1982),

Ellingwood et al. (1984), Rainer et al. (1988), P. Fujino et al. (1993), or more recently

Ronnquist et al. (2007), Georgakis et al. (2008) and Pedersen et al. (2010).

However, when some of the assumptions adopted to simplify those equations are not

valid (nonlinearities, axial forces, etc.) or a large number of points is needed to represent

accurately the dynamic response of the structure, designers usually appraise the dynamic

response through FE models. This option has been considered by many designers and

it has several advantages in the simulation of lateral and torsional movements and in

the prediction of modes of structures with considerable axial forces (this is the case of

cable-stayed footbridges).

In these FE simulations, the model can be more or less sophisticated depending on the

elements used to represent the structure. When simulating footbridges, multiple authors

have suggested the use of shell elements for the representation of the deck, since these

improve the response prediction in comparison to beam elements. The shell elements

representing the deck allow an accurate representation of the distribution of masses, which

is related to: a) an adequate representation of the deformations of the deck in torsion, b)

a fine introduction of dynamic loads of pedestrians, and c) a precise description of stresses

(shear, bending moments and axial forces) at the deck.

In relation to the first point, Park et al. (2014) and Daniell et al. (2007) highlighted

the more precise description of torsional movements obtained with shell elements. In this

context, Kanok-Nukulchai et al. (1992) pointed out that cable-stayed bridges in service

83

2. State of the art

have high level of torsional movements, which require this type of numerical representation

to reproduce deformations generated by the effects of flexure, torsion and axial forces.

Furthermore, Brownjohn et al. (1994) obtained a more accurate prediction of lateral

modes when modelling concrete panels in a suspension bridge as plate elements instead of

beam elements. Gardner-Mores (1993) found a better prediction of modes (compared to

experimental results) when modelling a timber deck with shell elements instead of adding

the mass to beam elements.

In relation to axial forces, several authors have highlighted the need of a previous

geometric nonlinear step to account for these axial forces affecting the vibration modes.

Regardless the option chosen to obtain the dynamic response of a bridge, multiple

authors have pinpointed other issues that should be considered in these numerical rep-

resentations of a bridge: material properties, such as Youngs’ modulus (in particular

of concrete), as commented by Tilly et al. (1984) and Brownjohn et al. (1994); non-

structural masses, including non-structural elements such as anchorages or web stiffeners,

and pedestrian mass (as considered by Setra and Daniell et al., 2007, and commented in

Section 2.3.3.3); non-structural stiffness, depending on the handrail it may be considerable

(Brownjohn et al., 1994, and Zivanovic et al., 2005); and bearing conditions, which should

be as realistic as possible, including soil characteristics and bearing stiffness (Brownjohn

et al., 1994, and Butz et al., 2008).

2.9 Concluding remarks

Design of pedestrian footbridges has evolved rapidly during the last years. In the course

of the last decades, engineers and architects have constructed and proposed alternative

designs of footbridges using innovative materials, with elaborate and unconventional plans

and elevations, with long spans and slender depths, with suspension or cable-stayed decks.

These design modifications have led to the larger proneness of these bridges to vibrate

under the passage of pedestrians (see Section 2.6), despite the moderate magnitudes of

their loads (see Section 2.3.3).

Footbridges with large amplitude movements in serviceability such as those described

in Section 2.6 have prompted the publication of countless research works focused on the

evaluation of pedestrian loads (see Section 2.3.3), the study of the pedestrian perception of

these movements (see Section 2.6) and the design and implementation of devices capable

of mitigating their effects (see Section 2.4).

The first codes including the evaluation of the vibration SLS of footbridges were based

on simple appraisal methods. In comparison, the last codes and guidelines are more

sophisticated. Nonetheless, these do not include numerous research outcomes that have

been published in relation to load description and users comfort.

Furthermore, these sophisticated and realistic pedestrian load models have not been

used to develop accurate design guidelines for specific footbridge typologies (see Sec-

tion 2.7). The design guidelines for structures such as cable-stayed footbridges are limited

and the existing ones seem to be founded on static evaluations or very simplified dynamic

pedestrian load models.

84

Chapter3Methodology

3.1 Introduction

Footbridges are increasingly characterised as lightweight and slender structures dic-

tating that their design tends to be governed by their performance under serviceability

conditions. In particular, the adequate performance of footbridges is very strongly linked

to the satisfaction of the limit-state conditions corresponding to vibration. This is well-

known as a result of a number of documented cases in the literature (e.g., Bachmann

et al., 1987; Bachmann, 1992; P. Fujino et al., 1993; Dallard et al., 2001; Bachmann,

2002; Charles et al., 2005).

These and other events of large dynamic movements in service are assumed to not to

pose a threat to the structural integrity of these footbridges, although their resolution

may involve considerable additional costs. During the last fifteen years, these events have

captured the attention of multiple researchers, which has led to the publication of numer-

ous advances in this field. However design criteria of footbridges have not experienced a

substantial modification.

In order to attempt a realistic evaluation of the response of footbridges under pedes-

trian loads and to establish design criteria, it is of utmost importance to establish a real-

istic model of analysis, an accurate representation of the footbridges and of the dynamic

response, clearly stating the assumptions and simplifications adopted.

All these details are introduced in this chapter, where there is a description of: the

proposed load model used to represent pedestrian loads (Section 3.2); the proposal of

a method to evaluate the impact of each of the characteristics of this load model on

the service response of footbridges (Section 3.3); the criteria upon which the structural

response is assessed in service (Section 3.4); the geometric and structural characteristics,

material assumptions and actions considered in serviceability analyses of cable-stayed

and girder footbridges subjected to analysis (Section 3.5); the numerical representation

and simplifications adopted for cable-stayed and girder footbridges (Section 3.6); and

the criteria adopted to evaluate and compare different aspects related to the dynamic

performance of the structures in service (Section 3.7).

85

3. Methodology: modelling and basic assumptions

3.2 Pedestrian loads: model definition

A satisfactory prediction of the response of footbridges under serviceability condi-

tions relies, to a large extent, upon the assumptions made regarding the imposed human

actions. The parameters that are commonly used to define anthropogenic loads are asso-

ciated with a significant degree of inherent variability. As thoroughly detailed in Sections

2.3.2, 2.3.3 and 2.3.5, this variability arises from a combination of the wide range of

anthropometric characteristics that exist within any typical sample of the human popula-

tion (inter-subject variability), the inability of individual subjects to repeat monotonous

activities with constant features (intra-subject variability), and the different constraints

that arise from collective behaviour scenarios (interactions between subjects).

In recent years, research related to more sophisticated load-models has appeared in

the literature with the aim of addressing some over-simplifications or inconsistencies of

previous proposals. However, very few of these advances have been included in load

models for structural design (as detailed in Sections 2.3 and 2.7).

In order to overcome these disagreements and to accurately predict the response of

structures subjected to these pedestrian actions, the structural analyses developed in this

thesis are based on the implementation of a novel pedestrian load model (developed as part

of this thesis and based on a realistic and rigorous basis) that is capable of representing

the previously mentioned components of variability that have an important impact in

response.

Following sections provide a description of this load model whereas Chapter 4 demon-

strates and quantifies the impact that the stochastic representation adopted in this model

has on structural response of footbridges compared to available proposals.

3.2.1 Definition of the new model

The proposed pedestrian load model is based on a temporal definition of the evolution

of the load amplitudes (both vertical and lateral) associated with individual steps as func-

tions of the gait characteristics (pedestrian speed and step frequency of the footsteps), the

subject properties (e.g., pedestrian mass, age and height), and the walkway movements

(lateral accelerations). This description of individual loads in time reproduces the interac-

tion among them by changing their speed, direction of movement, etc. This proposal has

been developed following a meta-analysis of the state-of-the-art multidisciplinary research

that has recently been published.

3.2.1.1 Vertical Load Model

3.2.1.1.1 Correlation between gait and load amplitudes

As just mentioned, the load model associates the gait characteristics of pedestrians with

the vertical load amplitudes they impart on the bridge. This correlation is based on

the relationship between anthropogenic characteristics, gait and vertical movement of the

CoM (or loads) described in Section 2.3.

3.2.1.1.2 Vertical load amplitudes (Fv)

A comprehensive empirical dataset useful for defining the relationship between gait

86


characteristics and vertical load amplitudes is provided by Butz et al. (2008). The authors

of the dataset describe the temporal variation of these forces normalised by the weight of

the subject (Wp) through nine parameters as shown in Figure 2.7 that depend upon the

step frequency fp.

For any given step frequency, fp, median estimates of the 9 parameters shown in

Figure 2.7 can be obtained (these are defined in the Table 3.1). These values can be used

to determine the coefficients of an 8th order polynomial, defined as:

Fv(t)/Wp =8∑

i=0

aixi (3.2.1)

that satisfies the constraints provided by the nine parameters.

Table 3.1: Median estimates of the parameters of Figure 2.7 in terms of fp (load amplitudesare normalised to the pedestrian weight Wp and the time terms are related to the total loadtime).

Parameter Median

p1 0.58 + 0.38fpp2 1.62− 0.51fpp3 0.74 + 0.24fpt1 (0.46− 0.11fp)tTt2 (0.41 + 0.06fp)tTt3 (0.64 + 0.06fp)tTtT 1.55− 0.46fpδpi 400.16− 670.98fp + 362.55f2

p − 60.41f3p

δpf 4.54− 9.15fp

Vertical step loads described using these polynomial expressions have a time varia-

tion that depends upon the step frequency as shown in Figure 3.1(a). The load shapes

predicted by these polynomial expressions are corroborated by the observations of many

previous researchers, e.g., Keller et al. (1996).

Alternatively, the form of these ground reaction forces can be approximated through

the use of three half-sine waves. The first half-sinusoid models the heel strike (capturing

p1 and t1), while the third reflects the foot pushing off the ground as the foot lifts off

(defined considering p3, t3 and tT ). The second sinusoid is used for modelling the transition

between these phases (representing p2 and t2). This approximation (shown schematically

in Figure 3.1(b)) is adopted considering the unsubstantial effect of two parameters, δpiand δpf , (evaluation discussed in Section 4.3) .

This simplification provides very similar results to those expressed in terms of 8th or-

der polynomial functions (comparisons such as those depicted in Figure 3.2 have been

performed for multiple values of fp to evaluate the simplification introduced by the three

sinusoids load amplitudes) and is computationally faster, especially when multiple pedes-

trian loads are defined. The vertical pedestrian loads of the proposed load model are

defined through this three half-sine waves model.

87


Time [s]

1.0

Fv(t)

Wp

Time [s]

Equivalent load:

sum of sinusoids8th order

polynomial function

1st half-sinusoid

3rd half-sinusoid

2nd half-sinusoid

Fv(t)

Wp

a) b)

fp = 1.6 Hzfp = 1.9 Hzfp = 2.2 Hzfp = 2.4 Hz

Figure 3.1: Normalised ground reaction forces defined using 8th order polynomial func-tions for different step frequencies (a); comparison of vertical loads defined with 8th orderpolynomial functions and three sinusoids, fp = 2.0Hz (b).

fs / fp

0.5 1.0 3.0 5.0 7.0 9.0

0.002

0.004

0.006

0.008Acceleration [m/s2]

8th order polynomial

Sum of sinusoids

Figure 3.2: Comparison of vertical accelerations generated by vertical loads defined with8th order polynomial functions or three sinusoids (fp = 2.0Hz).

3.2.1.2 Lateral Load Model

3.2.1.2.1 Correlation between gait and loads

As opposed to studies linking the vertical movement of the CoM to vertical loads,

lateral loads are related to gait characteristics as well as the position of the foot in the

lateral direction (in relation to the CoM), as argued in Section 2.3.

Amplitudes of pedestrian-induced lateral loads have been recorded experimentally by

several researchers (e.g., Butz et al., 2008, or Nilsson et al., 1989b). These studies describe

lateral loads with a large variability but do not provide relationships between these loads

and gait characteristics. However, the work of Townsend (1985), Macdonald (2009) and

Carroll et al. (2012) formulate a theoretical basis for describing how movements of the

CoM relate to modifications of gait characteristics as well as the positioning of steps

among other parameters.

3.2.1.2.2 Lateral load magnitudes (Fl)

The work developed by Townsend (1985) and later implemented by Macdonald (2009)

and Carroll et al. (2012) provides the theoretical basis upon which lateral loads are de-

scribed in this work. These proposals describe the lateral position of a pedestrian’s CoM

in time by representing the subject as an inverted pendulum (IP) (see Figure 2.10). This

movement can be directly related to characteristics of the gait, step width, anthropo-

88


metric characteristics and the lateral acceleration of the walking surface. The model has

been used extensively in the field of biomechanics. However, its application in structural

engineering has only occurred recently. The development of the model is based upon the

equilibrium of forces in the frontal plane and, given a known previous foot position and

the acceleration of the bridge, allows one to define the position in time of both the next

step and the CoM.

The displacement of the CoM in time and the resulting lateral load (Fl) are defined

in Equations 3.2.2, 3.2.3 and 3.2.4 respectively. In these equations, y and y are the

acceleration and position of the CoM in local coordinates and gL and gN the parallel and

normal components of the gravity acceleration to the pedestrian legs.

gN =g

L(ws − y) (3.2.2)

us + y = −gN = − g

L(ws − y) (3.2.3)

Fl = −mp(us + y) = mpΩ2p(ws − y) (3.2.4)

According to these equations, the basic magnitudes that are required to define the

model are the pedestrian mass mp, the half step width ws (which relates the foot position

in relation to the equilibrium position of the CoM), the lateral step frequency fp,l, and

anthropometric characteristics that govern the geometry of the inverted pendulum (Ωp =√

g/Leq, where Leq is the length or height of the CoM). The lateral step frequency fp,lcorresponds to half of the vertical step frequency fp (fp,l = 0.5fp).

Lateral loads defined with this simplified theoretical model correspond to the most

accurate prediction of real actions due to its ability to include feedback between the

movement of the platform and the load amplitudes, despite their shortcoming in that

they cannot reproduce the loading corresponding to the case that both feet of a subject

are in contact with the ground (see Section 2.3), although the loads induced over this

short period of time are minimal.

3.2.2 Evaluation of parameters of the proposed load model

In order to ensure that one can obtain a robust estimate of the structural response

under the action of multiple pedestrians, the conceptual framework related to the load

model must be sound. Moreover, the actual parameters of this model must also be defined

in a rigorous manner. The present section provides the calibration models of these relevant

parameters as well as a detailed explanation of their definition. Vertical loads require the

description of step frequency of the subjects as well as their mass, whereas the lateral

loads require the specification of the pedestrian height and step width in addition to the

step frequency and mass.

3.2.2.1 Parameter description: basis data

A rigorous definition of the parameters used to define the load model can only be ob-

tained on the basis of sound and consistent data. This data set is developed through data

of multiple and diverse experimental recordings (mainly focused on physical medicine).

In general, data from each study is reported in clusters or bins that represent groups

89


or subgroups of the multiple subjects that have been used to perform the experimental

analyses (instead of providing data of each individual for each parameter). Hence, graph-

ical representation of this data will provide one or more points (depending on the number

of subgroups) with error bars that correspond to the mean and standard error associated

with that mean value.

Data used to describe the definition of step frequency is originally from studies that

include pedestrians from Japan, USA and Western Europe (studies are enumerated in

following sections). Almost all these studies are considered for the derivation of the

characterisation of the velocity. These, together with research using pedestrians from

Canada and Central Europe describe the database for this second parameter (these latter

studies are considered to be consistent with the rest since they describe populations

with very similar anthropometric characteristics). Finally, the third derived relationship

(predicting the step width) is obtained using several experimental works used in previous

relationships as well as 13 others that describe results for populations of countries located

in Western Europe or the USA (studies are enumerated in following sections). Results

derived for the step width are considered to be consistent with the other relationships

given the common regional origin of the underlying data (they intend to describe the

same population and age ranges).

3.2.2.2 Step frequency

Generally speaking, the parameter that most readily describes the movement of an

individual behaving in a particular condition, such as during an early morning commute,

or walking in a light or heavy crowd, etc. is the pedestrian speed. However, for the load

model presented previously the key parameter is the step frequency. Therefore, in order

to calibrate the model parameters it is important to propose a robust relationship relating

the step frequency adopted by a pedestrian to the typical speed of walking for a variety

of loading scenarios.

Despite the multiple attempts to quantify such relationship (see Section 2.3), for the

present study, a meta-analysis of a number of studies (Kirtley et al., 1985; Himann et

al., 1988; Oberg et al., 1993; Sekiya et al., 1997; Sekiya et al., 1998; Stolze et al., 2000;

Boyer et al., 2012) that have considered the correlation between step frequency and speed

for a number of different subsets of the global population is undertaken. Collectively,

these studies provide information related to 909 different subjects which are represented

in Figure 3.3a in 33 bins (data is gathered according to the study it came from as well as

in age intervals), where vertical error bars describe the standard error of the mean of the

dependent variable and horizontal error bars reflect the standard error of the independent

variable in its own bin.

Figure 3.3a depicts the complete dataset considered for the definition of the step

frequency in terms of the velocity along with the median prediction of the proposed

model.

The model for the expected step frequency, fp, is best described as a quadratic function

of the speed, vp (Equation 3.2.5, where fp is expressed in Hz and vp in m/s). The model was

fitted using a weighted regression analysis in which the weights are inversely proportional

90


to the variances of the binned data. The residual standard error of this model is 0.178 and

together with the expected value of fp defines a normal distribution of step frequencies in

[Hz] for a given pedestrian speed vp in [m/s].

fp = α0 + α1vp + α2v2p

= 0.11 + 2.11vp − 0.47v2p(3.2.5)

3.2.2.3 Pedestrian velocity

Given that the new model for the step frequency is a function of the pedestrian velocity,

naturally it is also needed to develop a model that enables this velocity to be established

under a variety of loading scenarios. The model for the pedestrian velocity is based upon

the research of Weidmann (1993) (who describes this pedestrian velocity in terms of a

velocity in unrestricted conditions, vf , and effects of the flow density or the purpose of

the journey among others, further detailed in Section 2.3). Adopting a procedure similar

to that of Weidmann (1993), the pedestrian’s velocity, vp, is defined as a function of their

free velocity, vf , a factor representing the effect of the aim of the journey, φj, and another

capturing the effects of the flow density, φd (Eq. 3.2.6).

vp = vfφjφd (3.2.6)

To develop the model for the velocity first a model for the free velocity vf (in m/s)

is obtained. Again, a meta-analysis approach is adopted based on compiled dataset with

information regarding age, height, mass as well as sociological factors absorbed within

the country of origin. The dataset is compiled from the work of Cunningham et al.

(1982), Pearce et al. (1983), Himann et al. (1988), Oberg et al. (1993), Boonstra et al.

(1993), Sekiya et al. (1998), Stolze et al. (2000), Brach et al. (2001), Grabiner et al.

(2001), Helbostad et al. (2003), Fiser et al. (2010), Boyer et al. (2012) and Alcock et al.

(2013). This dataset comprises attributes from 1492 different pedestrians from 9 different

countries (mainly European countries, in addition to the USA and Japan).

From analysis of the dataset a model relating the free velocity to both pedestrian age

(ap in years), and height (hpd in metres), with a quadratic dependence upon age ap and

a linear dependence upon height hpd is developed. The developed model is shown along

with the underlying data in Figure 3.3b. The data depicted here is binned according to

the research study where it is defined and age intervals of the subjects. The final model

is presented in Equation 3.2.7. The standard deviation of this model is 0.087.

vf = β0 + β1ap + β2a2p + β3hpd

= 0.22 + 1.28× 10−2ap

− 1.71× 10−4a2p + 0.55hpd

(3.2.7)

Naturally, the use of this model requires information regarding the age of the pedestri-

ans and their height. However, this information is usually available for any given country.

For example, reports defining these characteristics for the UK population, which have

been used for this thesis, can be found in Health and Social Care Information Centre

91


0.5 1.0 1.5 2.0 2.5

1.0

1.5

2.0

2.5

Pedestrian Velocity [m/s]

Ste

p f

requency [

Hz]

(a) Correlation between free pedestrian velocityand step frequency.

10 20 30 40 50 60 70 80

0.9

1.1

1.3

1.5

Age [years]

Pedestr

ian F

ree V

elo

city [

m/s

]

(b) Correlation between subject age and height andfree pedestrian velocity.

Figure 3.3: Definition of the pedestrian gait parameters (step frequency and free velocityaccording to age and height of the subject).

(2013) and Office for National Statistics (2013).

The effect of the flow density φd on the mean speed adopted by pedestrians within

the pedestrian flow is quantified by an expression suggested by Weidmann (1993) (Equa-

tion 3.2.8), which is corroborated by empirical data of different traffic conditions found

in Ped-net.org (2013).

φd = 1− exp

[

−1.913

(

1

d− 1

5.4

)]

(3.2.8)

Here, d describes the density of the flow in units of pedestrians per metre square,

ped/m2, φd is the factor applied to the free velocity (as detailed in Section 2.3) to ac-

count for the effects of the pedestrian density, and the values 1.913 and 5.4 result from

the calibration of the function against data gathered by Weidmann (1993) (including 25

different field studies).

The data available from Ped-net.org (2013) also allow one to infer how observed speeds

are influenced by the purpose of the journey φj (Section 2.3). Weidmann (1993) had pre-

viously recognised this effect (distinguishing purposes related to ‘Business’, ‘Commuting’

and ‘Leisure’) and proposed the values for the modifier φj shown in Table 3.2.

Table 3.2: Values of the factor φj for different journey contexts

Activity φj

Business 1.20

Commuting 1.11

Leisure 0.86

Here, ‘Business’ represents flows of pedestrians that perform journeys related to work

excluding those to and from home (pedestrians have an age ranging from 20 to 65 years

old approximately). ‘Commuting’ describes traffic flows where pedestrians perform the

same journey repeatedly, including people moving between home and study/working place

92


(which may include not only people at working age but also younger and older pedestrians,

i.e., between 10 and 80 years old). The final class of ‘Leisure’ corresponds to pedestrians

performing leisure activities such as strolling or shopping.

The overall definition of the step frequency to be used to define pedestrian loads

therefore depends upon the attributes of the population using the structure (in terms

of age and height, which define the free-speed) and the activity and density of the flow

(modifying the free speed). As an example, the distributions of step frequencies that

are obtained through application of this model for conditions relevant to the Western

European population are shown in Figure 3.4.

Flow density [pedestrians/m2]2

0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8

Business Commuters Leisure

1.2

1.6

2.0

2.4

2.8

1.2

1.6

2.0

2.4

2.8

1.2

1.6

2.0

2.4

2.8

fp [Hz]

Figure 3.4: Step frequencies distribution according to density and aim of the journey.

3.2.2.4 Transverse step width

The step width, ws,t, defined as the total transverse distance between consecutive

footsteps (ws,t = 2ws), is estimated based upon gait characteristics. Data examining the

magnitude of this parameter for multiple pedestrians is obtained from research works

that describe the step width (ws,t) as well as characteristics such as age, height, mass

and pedestrian velocity. Studies considered include Hageman et al. (1986), Blanke et

al. (1989), Sekiya et al. (1997), Stolze et al. (2000), Brach et al. (2001), Donelan et al.

(2001), Donelan et al. (2002), Helbostad et al. (2003), Donelan et al. (2004), Orendurff

et al. (2004), Owings et al. (2004), Browning et al. (2007), Dean et al. (2007), Hof et al.

(2007), Schrager et al. (2008), Hurt et al. (2010), Rosenblatt et al. (2010) and Alcock

et al. (2013) (which define data for 294 different subjects). Nonetheless, some data of

these may have been considered outliers (those with Cook’s distance above unity have

been disregarded for the derivation of the relationship).

A model is described here to enable the prediction of the step width as a function

of the velocity and height of the pedestrian. Contrary to what has been reported in the

literature by some authors (e.g., Collins et al., 2003, as reported by Ortega et al., 2008),

it is not possible to observe any significant dependence of the step width upon age. The

model is fitted using a weighted regression analysis and is represented by Equation 3.2.9,

where the units of the step width is [cm], those of the velocity [m/s] and height [m]

(comparison of the model and the underlying data is represented in Figure 3.5). The

residual standard error of the model is 1.642, and this corresponds to a larger coefficient

93


0.8 1.0 1.2 1.4 1.6

68

10

12

14

Pedestrian Velocity [m/s]

Pedestr

ian s

tep w

idth

[cm

]

Figure 3.5: Correlation between pedestrian velocity and step width ws,t.

of variation than has been found in the earlier relationships. The step width parameter

may therefore be influenced by other factors not currently considered in the new model (or

not measured in experimental studies), or this larger variation may simply reflect greater

aleatory variability.

ws,t = δ0 + δ1v + δ2v2 + δ3hpd

= 26.86− 37.74v + 13.37v2 + 4.92hpd

(3.2.9)

3.2.3 Pedestrian intra-subject variability

As described in Section 2.3, over a period of time an individual may have well-defined

average gait characteristics (speed, step frequency or step width), but these have an

inherent variability that is naturally propagated through to the imparted loads. Following

sections describe how this is introduced in the load model.

3.2.3.1 Step frequency variability

In order to reproduce this step frequency variability for an individual, two strong

assumptions are made: a whole sequence of consecutive steps performed by a pedestrian

during an event are well-described by a normal distribution (as pointed out in studies

such Maruyama et al., 1992, and Butz et al., 2008); and, the properties of each step

depend upon the characteristics of the previous step, i.e., pedestrians do not use very

large steps after short steps and vice versa but, rather, subconsciously define these with

a smooth transition exhibiting a degree of autocorrelation (several studies identify this

subconscious choice, e.g., Hausdorff et al., 1995, or Jordan et al., 2009, and suggest

that it would disappear under controlled gait conditions). The first assumption implies

a long-term relationship of the step characteristics and the second implies a short-term

relationship featuring local temporal correlation.

Based on the previous considerations, the variability of the step frequency is captured

using the Metropolis-Hastings algorithm, a Markov chain Monte Carlo (MCMC) method

that generates synthetic step lengths (given a certain speed) according to long and short-

term relationships. The long-term relationship is defined by ensuring that all the steps of

94


an event describe a normal distribution, N(µ, σfp), and the short-term is considered by

sampling each step from a conditional distribution (corresponding to a normal distribution

with a standard deviation smaller than that of the long-term relationship) that depends

upon the previous step.

3.2.3.2 Step width variability

Intra-variability of consecutive step widths performed by an individual is not included

in the proposed load model (Section 4.3 describes results supporting this conclusion). In-

stead, this parameter is described according to the model predicting the lateral movement

of the CoM.

3.2.4 Representation of inter-subject variability

The following sections outline how inter-subject variability is taken into account to

characterise subjects of multi-pedestrian scenarios.

3.2.4.1 Probabilistic definition of vertical load magnitudes

The functions that describe the parameters upon which vertical load amplitudes are

described (which are functions of fp), represent the mean values of those observed by

Butz et al. (2008). However, there is a significant degree of variability in these various

parameters for any given step frequency fp.

Despite this variability in the definition of these amplitude parameters, inter-variability

of the vertical load amplitudes is not included when representing events with multiple

pedestrians (further description of the analyses performed demonstrating the validity of

this conclusion can be found in Section 4.3).

3.2.4.2 Probabilistic description of pedestrian weight

Codes and guidelines currently in use consider flows of pedestrians with a uniform

weight of 700 N. However, this pedestrian characteristic differs considerably among sub-

jects according to gender, age and other factors. In this proposed methodology a uniform

weight of 780 N among represented users is considered, despite the previous observation.

This uniform magnitude is considered as a result of the observations made when comparing

results of simulations representing individuals of a flow stochastically or deterministically

(see Section 4.3).

3.2.4.3 Probabilistic description of gait characteristics

Gait characteristics (speed and step frequency) of each individual in a flow are stochas-

tically generated considering the flow density and the aim of the journey. Distributions

such those depicted in Figure 3.4 show that step frequencies of pedestrians can be sub-

stantially different from those considered as critical in guidelines and codes (with step

frequency mean values around 1.8-2.0 Hz).

3.2.4.4 Probabilistic description of step width

The step width adopted by different pedestrians within a flow is described stochasti-

cally according to Equation 3.2.9.

95


3.2.4.5 Collective behaviour

Collective behaviour, with characteristics such those proposed by Helbing et al. (2000),

is considered in the proposed model. This collective behaviour consists in the representa-

tion of the alteration of the trajectory and walking velocity of each pedestrian according

to the proximity of other pedestrians, structural barriers (e.g., balustrades) or whether

they walk in groups or not. This model of collective behaviour permits the simulation of

the two-dimensional movement of subjects on a structure to obtain the gait parameters

of each subject crossing structures under different initial conditions (density and targeted

velocity that is related to the aim of the journey, as suggested in previous sections).

Further details of the simulation model can be obtained in Helbing et al. (2000).

Results in Section 4.4 support the introduction of this model, in particular for medium

and heavily crowded flows.

3.2.5 Summary of proposed model

The characteristics of the model used to develop analyses of the serviceability response

of footbridges in this thesis are presented in terms of the following steps (see scheme

represented in Figure 3.6):

1. According to the design crowd flow density and anticipated mode of traffic for this

density, the first step of the method consists in statistically generating the anthropo-

metric characteristics of the pedestrians of the flow: age, gender (which is related to

height and mass), height and mass. The mass of each pedestrian could be considered

as the mean mass of the population being described.

2. Based on the sampled age and height, as well as the average density of the flow and

type of journey: the second step consists in determining the desired mean walking

speed of each pedestrian for the particular type of journey (considering Equations

3.2.6 - 3.2.8 and the distribution around this equation, Table 3.2).

3. With the desired speed, the third step is the derivation of the mean step frequency

of each pedestrian (using Equation 3.2.5).

4. Using the desired speed and height of each pedestrian, the initial half-step width

ws,t is computed from Equation 3.2.9.

5. The next step is the simulation to account for crowd interactions, using an appro-

priate algorithm such as that proposed by Helbing et al. (2000):

• Each pedestrian starts crossing the bridge at a time generated according to a

Poisson arrival process and at a lateral position along the deck width (assigned

according to a uniform distribution).

• The simulation then defines the position (longitudinal and transversal) and

velocity of each pedestrian at any time while crossing the structure, accounting

for collective behaviour. Note that this simulation traces the general path of

pedestrians, but not the locations of each footstep of pedestrians in their general

path.

96


• The simulation can account for the arrival of groups of pedestrians (two or three)

by introducing a distribution for group size. The treatment of groups in the

algorithm is treated by assigning similar desired speeds to the pedestrians in a

group and forcing them to remain within a particular fixed transverse distance.

The desired speed of pedestrians in a group corresponds to the speed of the

slowest pedestrian affected by a factor described in Willis et al. (2004).

6. Using the results of the previous simulation, the initial step position (sampled from

a uniform distribution over [0.0 - 0.75], where 0.75 m corresponds to an upper value

of step length) and the initial step frequency of each pedestrian (equal to the mean

step frequency of the pedestrian, previously obtained), the positions of subsequent

footsteps are based upon the instantaneous velocity from the above simulation:

If the velocity is equal to that of the previous step (not affected by other pedes-

trians or structural elements), the step frequency for the next step will be defined

using MCMC sampling. This frequency will define the time at which the next foot

will touch the bridge. This MCMC sampling method describes the intra-variability

of each pedestrian.

If the velocity is not equal to that of the previous step (due to the effect of other

users on the desired path of the pedestrian), the step frequency (and the time of

contact of the following step) is defined using Equation 3.2.5.

7. The instantaneous step width of each pedestrian depends on the initial step width

(assigned according to Equation 3.2.9), the acceleration felt by each pedestrian dur-

ing previous instants and on the equations of the inverted pendulum model that

captures the lateral movement of the pedestrians.

Density- Leisure

- Commuter

- Business

Traffic type

Pedestrian characteristics:

age / height / gender / mass

Desired speed, vd

Mean step frequency fp

Crowd

simulation

Initia

l flow

data

Pedestr

ian

data

Gait

data

Half-step width ws,t

- Speedped n,ti

- fp ped n,ti

- x ped n,ti

- y ped n,ti

ped

instant i

Figure 3.6: Summary of the proposed load model (ped represents pedestrian).

For the assessment of the dynamic serviceability responses of footbridges developed

in this research work, the previous methodology is considered to represent traffics of

commuters and leisure pedestrians with densities of 0.2, 0.6 or 1.0 ped/m2:

• These three density values are chosen in accordance to comments in BSI (2008) and

Weidmann (1993). These values are representative of events likely to occur on a

large number of occasions at different footbridges.

97


• The journey purposes considered in this thesis correspond to leisure and commuting.

These are deemed to be more likely to occur in comparison to exclusively ‘business’

flows.

3.3 Nondimensional parameters governing the problem

When evaluating the serviceability response of any footbridge, there are several param-

eters that affect the magnitude of the accelerations caused by a single pedestrian at those

footbridges, e.g., the number of steps of the individual on the bridge, his mass in relation

to that of the structure, the mean step frequency in relation to that of the structure, etc.

The individual effect of one of these parameters (e.g., pedestrian step frequency) can

only be grasped when comparing results at different bridges if the rest of the parameters

remain constant. This observation is accounted for in the actual appraisal of a bridge

response under pedestrian actions taking into consideration the Pi Theorem of Dimen-

sional Analysis (Buckingham, 1914). This theorem ensures that a function that describes

a physical problem f(x) = f(x1, x2, ..., xn) = 0, with n arguments that are defined with

respect to q fundamental units U1, U2, ..., Uq, can be represented as g(π1, π2, ..., πk) = 0,

where k < n. In this second function k = n − r, and r is the rank of the dimensional

matrix n× q. The π terms are independent and dimensionless products formed from the

original n variables x1, ..., xn.

For the dynamic response of a footbridge under pedestrian actions, f(x) = 0 is the

function that predicts the structural response, which arises from the equation of motion of

a dynamic system. For the case of vertical accelerations in a simply-supported bridge, this

is a function of both the structural properties (the structural mass, the span length, the

material damping ratio, and the flexural stiffness) and the pedestrian characteristics (the

pedestrian mass, their step frequency, and their step length) and provides a prediction

of the expected maximum acceleration of the bridge. Therefore, there are seven input

variables used to predict one response variable (eight variables in total in implicit function

f(x), i.e., n = 8). However, when the lateral accelerations are considered two additional

parameters are required and these are the pedestrian step width and the height of the

pedestrian, meaning that in the lateral case n = 10.

This physical problem involves the three fundamental units of mass, length and time

(which are defined as U1, U2 and U3). To identify the required number of nondimensional

parameters it is necessary to represent each of the 8 (or 10) variables in terms of these

three basic units through an n × q matrix. The difference between the total number of

variables and the rank of this matrix then defines the required number of nondimensional

parameters. In the present case, with the fundamental units defined as above, the rank

of this matrix will be r = 3 and will imply that five (or seven) π terms are required. Four

(or six) of these terms relate to the input variables while one term is associated with the

response, and is simply the normalised acceleration a/g, with g being the gravitational

acceleration of the Earth - considered to be a constant.

The dimensionless parameters are then chosen to be the ratio of the frequency of the

structure to the pedestrian step frequency, π1 = fs/fp, the ratio of the structural mass

98


to the pedestrian mass, π2 = ms/mp, the ratio of the pedestrian step length and the

span length π3 = sl/L, and the damping ratio, π4 = ζ. For the lateral case π5 and

π6 are also used and are set as pedestrian step widths and heights normalized as step

width/pedestrian height and step width/span length.

Considering a range of values of each dimensionless term at a time and specific repre-

sentative values for the remaining terms allows the appraisal of the effect of that dimen-

sionless term on the structural response.

This procedure is applied in following chapters to evaluate the impact of each of

the variables considered in the proposed load model as well as those that have been

disregarded.

3.4 Comfort criteria

As previously remarked, one of the principal aims of this thesis is to establish design

criteria of cable-stayed footbridges based on their performance in service (under the action

of pedestrian traffics). The fulfilment of this aim is partly based on a consistent evaluation

of the comfort attributed by users to the movements they perceive while using those

structures.

Since bridge users may be walking, standing or sitting, movements of the deck bridges

will be considered to appraise the comfort of each of these users. The criteria considered

in each case are based on comfort evaluation proposals detailed in Sections 2.5.1 and 2.5.2

for individuals that could be placed at any location of the footpaths. For walking pedes-

trians, apart from the comfort appraisal based on the comparison of deck accelerations

with proposals of those sections, the comfort is assessed as well by comparing the actual

movements felt by pedestrians while walking (see Section 3.7.1) against the comfort ranges

used to assess the magnitudes of the response recorded at the deck.

This second assessment is justified by the fact that the first is based on the assumption

that the maximum deck movements are felt by users, regardless the duration of this event,

the number of times that it is repeated during a whole event and its location at the deck,

and which may not always be valid.

3.4.1 Comfort criteria for walking pedestrians

On the basis of the maximum movements recorded at the deck of the bridge, the

serviceability limit state of footbridges is evaluated as:

• In the vertical direction, maximum vertical accelerations of the deck are compared

to criteria proposed by the NA to BS EN 1991-2:2003 and the Setra (2006) and

Synpex guidelines (Butz et al., 2008).

• In horizontal direction, maximum lateral accelerations are compared to comfort lim-

its of Setra and a limit value between the two suggested by Nakamura (2003) (see

thresholds in Figure 3.7).

Limits of Setra are used to appraise longitudinal movements as well, although these

should be considered with caution since: a) the reaction of walking pedestrians to

99


the effects of longitudinal movements of the platform has not been analysed, and b)

multi-disciplinary research has highlighted the significant sensitivity of pedestrians

to lateral movements when walking, as opposed to longitudinal movements.

Considering the movements perceived by users while crossing the footbridge, the ser-

viceability limit state of footbridges is evaluated as:

• The vertical movements noticed by pedestrians are compared to a) a comfort limit of

0.5 m/s2, upper threshold of the limits proposed by Leonard (1966) and ISO (2005)

for walking pedestrians, and b) the limits enumerated for the assessment of vertical

movements recorded at the deck, see Figure 3.7.

• In horizontal direction, additionally to the thresholds proposed for the assessment

of the lateral accelerations recorded at the deck, perceived horizontal movements

will be compared to the limit proposed in ISO (2005) for walkers on a horizontally

moving bridge, see Figure 3.7.

ah [m/s2]

1.4

1.0

0.8

0.6

0.4

0.2

0.0

1.2

2.0 4.0 6.0 8.0 10.0fs,v fs,h

(*)

av [m/s2]

3.0

2.5

2.0

1.5

1.0

0.5

0.02.0 4.0 6.0 8.0 10.0

Leonard (1966)

ISO 10137 (2005) Nakamura (2004)

(a) (b)

Setra: max

Setra: mean

Setra: min

BS

Envelope

Setra: max

Setra: mean

Setra: min

Nakamura (2004)

[ISO 10137

(* )Extended

Figure 3.7: Comfort criteria for walking pedestrians for (a) vertical and (b) lateral accel-erations.

3.4.2 Comfort criteria for standing and sitting pedestrians

For standing pedestrians, vertical accelerations recorded at the deck are compared to

an envelope value of the limits proposed by Leonard (1966) and ISO (2005), 0.3 m/s2. In

horizontal direction, the limit proposed by ISO (2005) is used as benchmark value, 0.2

m/s2. Since these limits have not been derived under real conditions (most have been

obtained in laboratory environments), the ranges of the Setra guideline are included.

For sitting pedestrians, the degree of comfort is appraised with the same limits as those

for standing pedestrians. In the vertical direction, according to Thuong et al. (2002) (who

performed experimental tests), these are considered reasonably adequate. In the lateral

direction, despite the disagreements found by some researchers (Thuong et al., 2002),

these are adopted as well since they correspond to the best available assessment.

100


3.5 Footbridges description

Footbridge typologies with cables, and in particular cable-stayed bridges, are pro-

posed by designers considering economic, structural efficiency and aesthetic points of

view, although the last usually prevails due to their outstanding and appealing appear-

ance. Alternatively, a different balance of these three main criteria usually leads designers

to propose and develop pedestrian girder bridges with a single or multiple spans crossing

similar distances.

The dynamic characteristics of bridges with cables as structural elements such cable-

stayed bridges (which usually comprises light decks and long span lengths) and more

recently of girder footbridges (due to the use of lighter materials and longer spans) cause

these structures to present multiple modes of vibration in the range of the pedestrian

actions, therefore becoming susceptible of vibrating considerably in normal serviceability

conditions.

This effect in the dynamic response in service of footbridges raised the awareness of

designers when anticipating their response at design stages from the 1980’s and particu-

larly from the beginning of the 21st century. This has originated the publication of several

proposals since then (see Section 2.7).

However, these currently available predictions are not based on the latest state-of-the-

art load descriptions. Additionally, there is a substantial need of analysis of serviceability

for these structures in the lateral direction (see Section 2.3) as well as design proposals

to improve their dynamic performance under these pedestrian loads (see Section 2.7).

In order to obtain conclusions in relation to these topics, analyses are performed

considering girder footbridges with geometry and material characteristics described in

Section 3.5.1 and cable-stayed footbridges with structural and material features given in

Section 3.5.2.

3.5.1 Girder bridges

3.5.1.1 Geometric description

Among the numerous options involved in the design of girder footbridges (GFBs),

characteristics such as the span layout, the structural materials, the structural trans-

verse section and mass of non-structural elements can be distinguished as some of the

parameters with largest impact in the dynamic response of these structures.

In order to include the largest number of possible cases, the analysis of the dynamic

response in service of these footbridges has been developed considering the widest feasible

ranges of these parameters. In terms of span layout, the assessment is valid for bridges

of single, two or three spans: bridges with two spans include those with a smaller span of

length 0.2 to 1.0 times the longest span (see Figure 3.8(a)); bridges with three spans com-

prise those of side spans of length equal or smaller than that of the main span (proportion

between 0.2 and 1.0, see Figure 3.8(a)).

In relation to the transverse section, alternatives such as slab sections, box girders

or longitudinal girders with a slab (which may or may not provide structural resistance)

have been examined (see Figure 3.8(b)).

101


L

0.2 < < 1.0

Section

Material

Figure 3.8: (a) Span layout geometries; (b) deck transverse sections and structural mate-rials considered for girder footbridges.

3.5.1.2 Material properties

The analysis of the dynamic performance of these girder bridges under pedestrian loads

is performed for structural materials that are considered traditional in the design of these

structures as well as those lately introduced by designers (Firth et al., 2002). Among the

first, reinforced (RC) and prestressed concrete (PC), composite (concrete and steel) and

timber bridges have been assessed. Among the second group, aluminium and glass fibre

reinforced polymers (GFRP) have been included.

The characterisation of these materials can be found in the pertinent Eurocodes and

guidelines: EC2 (BSI, 2011) for concrete, EC3 (BSI, 2010a) for steel, EC5 (BSI, 2009; BSI,

2010c) for timber, EC9 (BSI, 2010b) for aluminium and Eurocomp Handbook (Clarke,

1996) for GFRP. The concrete grades considered for reinforced concrete sections are

C20/25 to C30/37, those for prestressed concrete sections are C45/55 to C60/75. The

steel grades considered for composite or steel sections corresponds to S275 and S375. In

relation to aluminium, it has been considered that the properties are those of an Alloy

EN AW 6082 (with Young’s modulus of 70000 MPa and weight of 27000 N/m3). The tim-

ber grades considered have a Young’s modulus of 12000 MPa and weight of 7000 N/m3

and that of the GFRP corresponds to a Young’s modulus of 17200 MPa and weight of

25600 N/m3.

In relation to the dissipation capacity of these footbridges and materials, as detailed in

Section 2.4, it is a difficult parameter to predict. Characteristics such as bearings, bolted

unions, balustrades and deck finishings have an impact on this parameter, hence it can

only be determined once experimental measurements can be performed.

Despite the number of uncertainties in relation to this parameter, the performance in

service of GFBs has been evaluated considering that the magnitude of this factor for the

considered materials and footbridges can range from ζ = 0.2% to 2.5% (according to the

multiple proposals that can be found such as BSI (2008), Setra (2006), Butz et al. (2008)

and Bachmann et al. (1995), with an average value corresponding to ζ = 0.5%.

102


3.5.1.3 Boundary conditions

Abutments and intermediate piers only restrain vertical movements uz of the bridges

at those sections. Horizontally, deck sections supported on abutments and piers rotate

with respect to a vertical axis located at the middle of the deck section, θz (see Figure 3.9).

GFB elevation GFB transverse sectionz

yx

Vertical displacement uz

GFB plan

Horizontal rotation z

z

yx

z

y

x

Figure 3.9: Displacements and rotations of GFBs.

3.5.1.4 Design actions

The actions taken into account for the appraisal of the dynamic response of GFBs

with one to three spans are: self-weight, dead load from non-structural elements of the

transverse section, balustrades, surface finishings, etc. (which in analyses is considered to

adopt a value including 6 cm of asphalt layer and 200 kg for the balustrades and the mass

of the non-structural decks for some sections) and the dynamic vertical and lateral loads

generated by pedestrians. The mass of pedestrian flows (as discussed in Section 2.3.3.3)

is included as non-structural mass.

3.5.2 Cable-stayed footbridges

3.5.2.1 Geometric description

As seen in Section 2.2, there is a wide variety of geometric characteristics (involving

the deck, the cable system, the tower, etc.) that can be considered when developing the

design of cable-stayed footbridges (CSFs). The importance of these design characteristics

on the static behaviour of the bridge is well known (Strasky, 1995; Gimsing et al., 2012),

as opposed to their dynamic response under pedestrian loads.

In order to assess their impact on the dynamic response in service of these footbridges,

the response caused by different traffic scenarios on two CSFs considered benchmark or

representative are compared to those of CSFs with alternative geometrical parameters

(such as tower height, tower shape and tower inclination, cable layout, deck transverse

section, etc.).

The main features of these two representative CSFs and the magnitudes of alterna-

tive geometrical proportions whose impact in dynamic response is assessed, are based on

observations of current designs (a summary of which can be found in Section 2.2). For

103


Dp

Lm

Hi

hp

HT

Ha

Dc

Ds

Ls

hp = 0.36 Lm

Scale: Scale:

1

2

1

1

hp = 0.20 Lm

Hi

hpHT

Dp Dc Dc

LmLs Ls

Ha

Figure 3.10: Elevation and transverse sections of benchmark cable-stayed footbridges: 1Tower (top) and 2 Towers (bottom).

these benchmark cases, the following features can be highlighted:

• These footbridges generally present main spans that cover distances between 50 and

100 m (span distances of 150 m are considerably less usual). Hence, the benchmark

footbridges adopt a main span lengths of 50 m with one or two towers (geometry

represented in Figure 3.10). As an alternative, results obtained with span lengths of

100 m are obtained in both cases.

• Usual shapes of towers of CSFs present a single mast and elevation over the deck

as depicted in Figure 3.10. Alternatively, designers may adopt tower shapes (as

those depicted in Figure 3.11), elevations (shorter or higher than 0.36Lm or 0.20Lm)

or longitudinal inclinations (towards the side span or main span) that differ from

the most used geometry and dimensions. The performance of these alternatives in

service are compared to that of the benchmark cases to discern which case provides

the best behaviour.

The magnitude of the tower height below the deck Hi corresponds to 7.5 m. This

magnitude represents a median value of those adopted in constructed CSFs and is

justified by their typical locations: crossing streets, highways, rivers, etc..

• The deck depth of these benchmark bridges is equivalent to a ratio with the span

length of 1/100, ratio selected in accordance to geometries adopted by designers

(Section 2.2), as represented in Figure 3.12. The effect of smaller ratios of 1/200 for

lengths Lm of 100 m are explored as well.

• The cable system of the benchmark bridges is arranged as a semi-fan configuration

104


(1) (2) (3) (4)

Figure 3.11: Basic an alternative tower shapes: 1) I tower shape, 2) H tower shape, 3) Hshape with a crossing brace and 4) A tower shape.

Main span length [m]

Deck d

epth

[m

]

50.0 100.0 150.00.0

1.0

2.0

1/100

1/200

Inferior limit (0.5 m)

Figure 3.12: Depth-to-span length ratios adopted in existing CSFs (black dots) and ratiosof benchmark CSFs (red dots) according to main span length.

with two planes of cables (see Section 2.2). The presence of two planes of cables al-

lows the deck transverse section to be an open section, consisting of two longitudinal

steel girders holding a concrete slab (geometry and materials selected in agreement

to observations of Section 2.2).

The distance between consecutive cables Dc at the benchmark cases is 7.0 m (se-

lected on the basis of magnitudes implemented in constructed footbridges), although

alternative values have been assessed in order to consider the large spread of this

magnitude in actual footbridges.

• The magnitude of the deck width is generally chosen according to the size of the

pedestrian streams expected to use the footbridge. Existing CSFs present an average

deck width of 4 m (Section 2.2), which is the magnitude adopted in the benchmark

footbridges. This total width, extracting a region occupied by balustrades and a

width to ensure pedestrian comfort near cables, corresponds to an effective width

for pedestrians of 3.4 m. This characteristic of the cable-stayed footbridge and others

previously described are summarised in Table 3.3.

The dimensions of each structural element have been adopted considering their per-

formance under permanent and live loads. As permanent loads, their self-weight has been

included as well as those representing balustrades, etc. (equivalent to 100 kg/m). In

relation to the live loads, those corresponding to traffic, wind and temperature have been

included. The loads representing the traffic correspond to a vertical UDL of 5 kN/m2, a

concentrated load of 10 kN and a lateral load equivalent to 20% of the vertical load. The

105


Structural Parameter Geometry

Main span length Lm 50 m, 100 mTower height above the deck hp 0.36 Lm, 0.20 Lm

Tower height below the deck Hi 7.5 mDeck depth d Lm/100, Lm/200Deck width wd 4 mCable system semi-fanDistance cable anchorages Dc 7 m

Table 3.3: Summary of the geometric characteristics of the cable-stayed footbridges.

wind load has been derived considering a location 125 m above the sea, at 7.5 m above

the ground and near London. In relation to the temperature, both a minimum shade and

maximum shade temperatures of -10 and 35 have been included. These loads have been

combined to represent different load scenarios.

The area of each cable has been adopted in accordance to its performance under

the loads described in the previous paragraph. The cable technology considered for the

benchmark footbridges corresponds to stays, hence the number of strands are obtained

from ensuring that the maximum axial stress is limited to 45% of the ultimate tensile

strength of the steel and that 200 MPa stress ranges are resisted during 2·106 cycles. A

pedestrian traffic of 5 kN/m2 is adopted for the design of the sections since the structures

may stand heavy streams of pedestrians during their lifetime (BSI, 2008).

Similarly, the dimensions of the thickness of the tower section and deck girders have

been adopted to fulfil the ULS of normal and shear stresses (considering load scenarios

with alternative predominant live loads). In all cases the dimensions of these transverse

sections (deck and tower) remain constant throughout their entire length.

Boundary conditions of the structures represent an important factor for structural

response both in the vertical and the lateral direction. There are multiple supporting

conditions that can be adopted by designers and not one is more favoured than others:

(a) In relation to the movements restricted at the embankments, the response of multiple

options has been assessed and that providing the best performance in service (together

with considerations related to temperature effects and economical criteria) has been

adopted as benchmark case.

(b) In relation to the movements constrained at the tower support, it has been assumed

that the tower has joint displacements with the deck (rotations are not transmitted).

(c) Another geometric aspect that affects the boundary restraints corresponds to the

length of the side spans. The chosen dimensions of these have a magnitude Ls =

0.20Lm for the benchmark cases, although the effects of shorter and longer side spans

are appraised in Chapters 7 and 8.

In relation to the technology of the cable anchor, stayed cables are anchored to the deck

through bearing sockets (see Figure 3.13), although multiple cable-stayed footbridges with

106


fork sockets have been found. The main difference in performance is the rotation allowance

that the second type has in comparison to the first. This is taken into account when using

bearing sockets by locally increasing the cable stresses according to the rotation angles

that the cable anchorage endures during service (magnitude described by Equation 3.5.1,

given in Gimsing et al., 2012, where σc,a is the cable stress at the anchorage due to its

rotation φc,a, Ep is the cable Young’s modulus, Ac and dc its transverse section area and

diameter and Tc its tension).

∆σc,a = 1/2∆φc,a

√

16EpTc

Acd2c(3.5.1)

Figure 3.13: Cable anchorages: (a) bearing socket; (b) fork socket.

The benchmark structures do not present devices of additional dissipation energy, al-

though their beneficial effect on response is evaluated as well (Chapters 7 and 8). These

correspond to Tuned Mass Dampers (TMD), which are the most common devices imple-

mented in these structures. A characterisation of their response under dynamic loads is

given in Section 2.4 and their numerical representation is detailed in Section 3.6.2.

3.5.2.2 Properties of materials

The decks correspond to composite concrete-steel sections, with steel girders connected

to a reinforced concrete slab, the towers are made of steel (or steel filled with concrete)

and the cables of steel strands (bar cables have been contemplated as an alternative).

In all cases, it is assumed that the response of these materials is linear elastic with

this elastic behaviour infinitely extended in tension and compression. The assumption

of material linearity is considered adequate in the serviceability analysis to be performed

since deformations and material stresses that the structure endures in service (dynamic

effects of streams of pedestrians) are of a lower order of magnitude in comparison to those

that it would sustain under ULS events (static effects caused by the worst case scenario,

i.e., the heaviest possible pedestrian event).

• The concrete employed in the slab of the deck transverse section corresponds to class

C40/50. The main characteristics of this material are adopted from BS EN 1992-1-1:2004

(BSI, 2011) and summarised in Table 3.4.

• The structural steel used in the longitudinal girders, transverse beams and section of

the tower is that of a steel grade S355 with elastic behaviour as well and characteristics

as summarised in Table 3.5 (BSI, 2010a).

• The majority of bridges have been developed considering that the stays correspond

to stranded cables. As an alternative parameter, stays have been characterised as bars.

For the first case, the steel of the stayed cables corresponds to Grade 1860 MPa with

properties as detailed in Table 3.6 and strands consisting of seven twisted wires of total

107


Compressive strength fck = 40 MPaDensity ρc 2500 kg/m3

Poisson’s ratio νc 0.2Young’s modulus Ec 35 GPaCoefficient of thermal expansion αc 10−5 K−1

Table 3.4: Summary of the characteristics of the concrete employed in the deck of thecable-stayed footbridges.

Yield strength fs,y = 355 MPaDensity ρs 7850 kg/m3

Poisson’s ratio νs 0.3Young’s modulus Es 210 GPaCoefficient of thermal expansion αs 1.2 · 10−5 K−1

Table 3.5: Summary of the characteristics of the steel employed in the deck longitudinaland transverse beams and tower of the cable-stayed footbridges.

nominal area 150 mm2. The decreased Young’s modulus of the stay cable in relation to

that of steel is caused by the helical disposition of wires in strands. For the second case,

the properties of the considered bars are described in Table 3.7.

Maximum stress fy = 1860 MPa

Steel density ρs 7850 kg/m3

Cable density(*) ρp 8870 kg/m3

Material Young’s modulus Ep 195 GPa

Coefficient of thermal expansion αs 1.2 · 10−5 K−1

Table 3.6: Summary of the characteristics of the steel employed in the strands of the stayedcables (where the subindex p corresponds to prestressed steel). (*) Density including massof stay protection, considered from BBR VL International Ltd. (2011).

Maximum stress fy,b = 1030 MPa

Steel density ρs 7850 kg/m3

Young’s modulus Eb 205 GPa

Coefficient of thermal expansion αs 1.2 · 10−5 K−1

Table 3.7: Summary of the characteristics of the steel employed in the stayed cables asbars.

Despite the stiffness of cables represented in Table 3.6, the relationship between force

and deformation of cables is not linear due to the shape that these present in reality. The

cable dead weight causes cables to describe a catenary shape with a sag that depends

on the magnitude of their axial tension (see Figure 3.14). The axial tension and the

equivalent horizontal length of the stay cable determine the equivalent tangent stiffness of

the cable Etan. This tangent modulus is described by Equation 3.5.2 according to Ernst

108


(1965), where Ep is the material Young’s modulus, wc the weight of the stay cable per

unit length, dh the equivalent horizontal length of the cable (see Figure 3.14), Ac the cable

cross-sectional area and Tc the tension of the cable.

Tc

TcSag of Cable

Real Inclined Cable,

catenary shape (Ac, wc)

Theoretical Inclined Cable,

straight line shape

dh

Figure 3.14: Geometry of a tensioned stay cable under self-weight.

Etan =Ep

1 +Epw2

cd2hAc

12T 3c

(3.5.2)

Substituting the corresponding values in Equation 3.5.2, it has been seen that the

reduction of the stiffness of the cables of bridges in Figure 3.14 is practically negligible,

fact mainly explained by the short lengths of the cables and relatively high axial ten-

sion. Hence, the behaviour of cables has been considered linear in all cases with stiffness

equivalent to that of straight cables Ep.

Due to the dynamic nature of the service loads, which cause repeated loading and

unloading of the stayed cables, the fatigue strength of the stayed cable materials is of

utmost importance. The fatigue life of these materials can be characterised by an S-N

curve or Wohler-Curve, which describes the maximum magnitude of the stress ranges

capable to be endured according to the number of cycles Nc of repetition. Figure 3.15

represents the performance curves considered for strands and bars to assess their fatigue

endurance (fib Bulletin 30, 2005).

s

105 106 107

2106

K1

K2200

[N/mm2]Stress Range

Nc

Number of cycles

s

105 106 107

2106

K1

K2110

[N/mm2]

Nc

Strands Bars

K1 K2

Bars

Strands 5 6

4 6

Figure 3.15: Wohler-Curves for stay cables: strands and bars (fib Bulletin 30, 2005)

• As indicated for GFBs, the capacity of energy dissipation of CSFs depends on mech-

anisms that are not fully understood and probably are more complex than an energy-loss

proportional to the movement velocity (as considered for GFBs where linear modal su-

perposition is adopted for the dynamic analyses, Clough et al., 1993).

For CSFs, the time response is obtained according to step-by-step analyses instead

of simple linear methods (reasons for such adoption are given in Section 3.6.4). Hence,

damping cannot be represented by viscous damping. However, in order to preserve the

109


orthogonality of the mode shapes, damping can be represented by an explicit damping

matrix defined by Rayleigh (Clough et al., 1993). This Rayleigh damping matrix is

proportional to mass and stiffness matrices (see Equation 3.5.3, given in Clough et al.,

1993), where c, m and k are the damping, mass and stiffness matrices respectively and

a0 and a1 are constants). Therefore the damping coefficient of each mode is given by

Equation 3.5.4 (where ζi is the damping coefficient of ith mode and ωi is the undamped

natural circular frequency of the same mode).

c = a0m+ a1k (3.5.3)

ζi =a02ωi

+a1ωi

2(3.5.4)

According to Bachmann et al. (1995), codes, guidelines and diverse research works

(Section 2.4) propose representative ranges of damping values according to the structural

material and structural scheme. For composite sections there is no distinction between

dissipation capacity of each material and rather an homogeneous damping coefficient is

associated to both.

From available coefficients for cable-stayed composite bridges, an average low value

has been chosen for the benchmark cases: ζ = 0.4%. This value is adopted as an average

damping dissipation coefficient for modes with frequencies in the range 2.0-6.0 Hz (see

Figure 3.16). This distinction is made in order to avoid attributing lower dissipation

capacity to modes in this region. The impact of alternative values of this dissipation

factor is assessed as well in posterior chapters.

1 2

1

2 = 0.4%

= a0

2

= a1

2

Figure 3.16: Rayleigh damping considered in benchmark CSFs, where ω1 and ω2 are theundamped natural circular frequencies of modes 2.0 and 6.0 Hz.

3.5.2.3 Design actions

The serviceability limit state of CSFs has been evaluated considering the following

permanent loads:

1. The self-weight of the structural elements (these are determined by the volume of

the structural elements and densities previously described, as ρg, where g = 9.81

m/s2 is the earth gravitational acceleration and ρ is the density of each material).

2. Permanent loads corresponding to balustrades, anchorages and other services located

at the edges of the deck (with a total equivalent mass of 100 kg/m per side of the

110


deck) and the mass of the pedestrian flows as discussed in Section 2.3.3.3 (on average

it is considered that each pedestrian has a mass of 79.5 kg, which is added as a

uniform constant mass at the deck, considering the density of the pedestrian flow).

In relation to live loads, serviceability analyses include vertical and lateral loads of

pedestrians defined as detailed in Section 3.2.5 and numerically implemented as explained

in Section 3.6.3.

3.5.3 Evaluated structural schemes

Figure 3.17 represents a summary of the footbridge structural schemes evaluated at

serviceability limit state as well as the traffic, structural and geometric characteristics

considered for their evaluation.

1 S

pan

2 S

pans

3 S

pans

Traffic

Structural

variables

Single pedestrian

Groups

Streams

Commuting/Leisure

Transverse section

Material

Span layout

Damping ratio

1 T

ow

er

2 T

ow

ers

0.2 ped/m2 streams

0.6 ped/m2 streams

1.0 ped/m2 streams

groups of pedestrians

Commuting/Leisure

Lm = 50 / 100 m

Lm = 50 / 100m

medium/high

pedestrian

commuter flows

medium/high

pedestrian

commuter flows

Girder

footbridges

Cable stayed

footbridges

Chapter

5

Chapter

6

Chapter

7

Chapter

8

Boundary conditions

Deck depth

Deck trans. section

Deck width

Tower transv. section

Tower height

Tower long. inclination

Tower shape

Cable dimensions

Cable arrangement

Energy dissipation

Chapter

Figure 3.17: Summary of footbridges whose behaviour in serviceability limit state is thor-oughly evaluated in this thesis.

111


3.6 Finite element models: assumptions and representation

In order to predict the dynamic structural response of girder footbridges (GFBs) and

cable-stayed footbridges (CSFs) under pedestrian actions, finite element models are used

to characterise, in a simplified but representative and accurate manner, these structures

and additional factors that intervene in their dynamic behaviour. The accuracy of the

results is related to the structural geometry represented, to the mechanical behaviour

adopted for the elements representing this geometry as well as to the analysis chosen to

predict the time response.

Some of these characteristics are adopted according to established and largely consid-

ered criteria (e.g., behaviour of concrete and steel) whereas others are assessed and chosen

in terms of their impact on the prediction of the dynamic response (e.g., numerical ele-

ments, structure nodal definition, time analysis).

3.6.1 Structure model: finite element model description of girder footbridges

The response of GFBs under the effects of pedestrian loads has been obtained on the

basis of mode displacement superposition. This procedure is justified by the overall linear

load-deformation relationships of these structures under the design loads.

For the resolution of this dynamic response in service, the GFBs have been represented

by mathematical models that simplify the structure as finite elements connecting nodes

with one degree of freedom (vertical or lateral displacements, see Figure 3.18). From this

model, mode shapes and frequencies are obtained. The resolution of the dynamic response

is based on these modes of vibration, which remain uncoupled due to the representation

of the energy dissipation as viscous damping (damping ratio adopts a constant value for

all the vibration modes).

Nodal representation

of GFB

1 2 i n

Degree of freedom

of node i

Figure 3.18: Discretisation of a GFB structure (elevation, left plot, and transverse section,right plot).

A detailed description of the procedure developed in the numerical computing software

Matlab to obtain the nodal representation of the GFBs and the dynamic response is given

in Annex B.

3.6.2 Structure model: finite element model description of cable-stayed bridge

Cable-stayed footbridges are numerically represented through Finite Element Models

(FEM) in order to predict their dynamic response in service as accurately as possible.

These FEM models are developed with Abaqus (ABAQUS, 2013), a commercial soft-

ware widely used in research analyses in the civil engineering field. Following, there is a

112


description of the elements and size used to represent the CSFs in this software.

3.6.2.1 Discretization of the deck

The deck of the cable-stayed bridges consists of a composite section, with structural

elements of different materials: a concrete slab and longitudinal and transverse steel

girders. These have been represented individually: the slab is characterised through shell

elements of constant thickness, the longitudinal girders are modelled with shell elements

as well with thickness according to their dimensions of web and flanges and the transverse

steel girders are represented with beam elements (see Figure 3.19). Shell elements are

chosen to represent the structure in agreement to observations of Section 2.8.

The shell elements used to represent the concrete slab and the longitudinal steel girders

correspond to S4R, shell elements with 4 nodes and first order interpolation that use thick

shell theory as thickness increases and considers thin shell theory and shear deformation

as it decreases. The thickness of these shell elements is represented by 5 nodes (top and

bottom and three intermediate nodes). The transverse steel girders are represented as

space beam elements with linear interpolation (B31). The different elements are connected

through constraints that adjust rotations and maintain the original offset between these.

The offset corresponds in each case to the distance between centers of gravity of the

represented sections (see Figure 3.19(b)).

Longitudinal

steel girders

Transverse

steel girders

Concrete slab Balustrades

Constraint

Longitudinal steel girders

(Shell elements)

Transverse steel girders

(Beam elements)

Concrete slab

(shell elements)

Balustrade

mass

(a) (b)

Figure 3.19: (a) CSF transverse section and (b) section numerical representation.

3.6.2.2 Deck mesh sensitivity

Several authors pinpoint the importance of a thorough representation of the struc-

ture, with numerous nodes, to numerically define accurate responses (e.g., Brownjohn

et al., 2000, and Daniell et al., 2007). Nonetheless, such detailed analysis entails large

computational costs.

Accordingly, the size of the shell elements used to represent the deck of the CSFs has

been calibrated comparing the computation time and the effects on response of a particular

pedestrian event on a cable-stayed bridge where the deck is comprised of elements of

alternative sizes. The different element sizes considered have an average element size of

0.5, 0.4, 0.3 and 0.2 m2 and correspond to a representation of the concrete slab transverse

section with 10, 12, 14 and 18 elements respectively.

In each case, the point loads generated by the feet of each pedestrian are converted

into nodal loads in proportion to the distances to the closest four nodes (two nodes if the

113


point load is located at a point where two nodes have the same x or y coordinate, x being

the longitudinal coordinate and y the transversal coordinate).

Figures 3.20 and 3.21 describe the differences (with respect to results of the finest

mesh) in the magnitudes of accelerations recorded at different points of the deck and stress

variations at stayed cables when the deck is represented with the previously mentioned

deck meshes. These figures depict how, except for the mesh 0.3 m2 (where some elements

have a rectangular shape), smaller elements yield results that are closer to those of the

model with smallest mesh size, and the largest mesh (0.5 m2 elements) predicts results

that are approximately ±5% those of the smallest mesh. Therefore considering differences

yielded by models with size elements of the deck mesh between 0.2 and 0.5 m2, a mesh

with elements of 0.5 m2 is adopted for the dynamic analyses in Abaqus.

0.3 0.50

2.55

7.510

ε [%

]

x = 5.0m

0.3 0.50

2.55

7.510

ε [%

]

x = 21.0m

0.3 0.50

2.55

7.510

ε [%

]

x = 28.0m

0.3 0.50

2.55

7.510

Err

or

ε [%

]

x = 35.0m

0.3 0.50

2.55

7.510

ε [%

]

x = 42.0m

0.3 0.50

2.55

7.510

ε [%

]

x = 49.0mǫ[%] =

Respi −Resp0.2

Resp0.2

Element size

Figure 3.20: Differences in maximum vertical accelerations (ǫ) at different points of thedeck, according to element mesh size.

0.3 0.50

2.55

7.510

ε [%

]

Backstay

0.3 0.50

2.55

7.510

ε [%

]

CB1

0.3 0.50

2.55

7.510

Err

or

ε [%

]

CB2

0.3 0.50

2.55

7.510

ε [%

]

CB3

0.3 0.50

2.55

7.510

ε [%

]

CB4 (longest cable)

0.3 0.50

2.55

7.510

ε [%

]

CB0 (shortest cable)

ǫ[%] =Stressi − Stress0.2

Stress0.2

Element size

Figure 3.21: Differences in maximum cable stresses (ǫ) at different stayed cables, accordingto element mesh size.

3.6.2.3 Discretization of the cables

Cables are represented in the numerical models by means of truss elements (elements

that do not have bending stiffness). In relation to the number of elements of each cable, as

argued by Abdel-Ghaffar et al. (1991), the representation of each stay cable with multiple

elements (multiple element cable stay, MECS) instead of one element (one element cable

stay, OECS) allows the simulation of numerous vibration modes that otherwise would

have not appeared. These modes would correspond to cable vibration modes as well as

modes in which deck and cable motions are coupled (for lateral and torsional movements).

Nonetheless, models with MECS involve larger computational effort in comparison to

114


those with OECS (Daniell et al., 2007).

In order to choose an adequate and cost-effective (in terms of time) representation

of the cables, the impact caused by considering bridges with OECS instead of MECS

is measured by comparing the dynamic results caused by the same pedestrian scenario

on cable-stayed footbridges of main span length 50 m (with one or two towers) where

cables are discretized with one or multiple elements per cable (same number regardless

the length of each cable). This comparison shows that the maximum differences in the

predicted accelerations and cable stresses are approximately 5% and hence negligible.

Therefore, for all the numerical simulations the cables of the CSFs have been represented

with a single truss element.

3.6.2.4 Discretization of the tower

As opposed to other structural elements, towers do not constitute the most difficult

elements to be represented numerically with accuracy. For the numerical models developed

in this work, the discretisation of the tower is done by means of space beam elements

(with linear interpolation) located at the center of gravity of the section with a maximum

element length of 0.5 m.

3.6.2.5 Other elements

Additionally to the structural components, other elements located at the bridge have

an impact on the dynamic performance of the structure. These correspond to components

that modify the mass of the structure, its stiffness or the energy dissipation capacity:

• In relation to the first, balustrades located at both edges of the deck add mass to

the structure. This mass has been included in the FEM as point masses located at

their real position in relation to the deck section (see Figure 3.19).

• Regarding stiffness, boundary conditions corresponding to Laminated Elastomeric

Bearings (LEBs) have been simulated by linear springs with stiffness according to

their dimensions (BSI, 2006b).

• In relation to components modifying the damping ratio, these correspond to tuned

mass dampers (TMD) passive damping devices. A TMD consists of a spring in

parallel with a dashpot element linking a node of the bridge to a point mass element

(as represented in Figure 3.22).

3.6.3 Definition of sophisticated user functions within the numerical models

to represent the pedestrian-structure complex interaction

The pedestrian load model described in Section 3.2 corresponds to the most realistic,

accurate and up-to-date representation of the actions transmitted by pedestrian streams

on the surface where they walk that can be found.

As it has been detailed in Section 3.2, the velocity and trajectory of pedestrians are

not affected by the movements registered at the deck of the bridge they cross. However,

due to the sensitivity of pedestrians to their lateral equilibrium and to the large impact

115


M (mass)

c

(damping coefficient) K

(stiffness)

Bridge surface

(a) (b)

Figure 3.22: (a) TMD placed at London Millennium Bridge; (b) numerical representationof TMD.

of this lateral equilibrium on the lateral loads they transmit, their step widths and lateral

loads are modified by the lateral movements they sense in prior steps.

These prior movements correspond to the lateral accelerations recorded at the location

where each pedestrian places the previous foot, i.e., the closest mesh node to that location.

In time, the movements that affect the following step are those felt with the other foot,

i.e., recorded between the prior foot strike and 0.1 seconds before the strike of that step.

Numerically, this interaction between dynamic lateral response and the definition of

lateral load amplitudes can only be implemented in Abaqus through the use of UAMP

(User Amplitude) subroutine, a Fortran code that defines load amplitudes according to

results developed during the same analysis within the previous time steps. A schematic

operation of this subroutine is described in Figure 3.23 and a summary of the main

characteristics of the subroutine and its implementation in Abaqus models can be found

in Annex C.

Pedestrian i

step j

step j

characteristics

structure

movement

Pedestrian i

step j+1

UAMP Subroutine

step j+1 step j

Abaqus: numerical analysis

New subroutine input file:

Figure 3.23: Schematic implementation of UAMP subroutine of Abaqus.

3.6.4 Numerical dynamic analysis

Many authors (Fleming et al., 1980; Daniell et al., 2007) observe that a more accurate

dynamic response of cable-stayed bridges is obtained if a nonlinear static step precedes

the dynamic assessment. According to Fleming et al. (1980) this effect is explained by the

load-deformation relationships, which are nonlinear for cable stays, towers and girders.

Similar statements are described by Daniell et al. (2007), who explain as well that an

accurate prediction of vertical and torsional modes is affected by this nonlinear static

analysis (in particular for torsion modes).

116


In relation to the dynamic analysis, the same authors highlight the fact that a nonlinear

analysis with stiffness matrix recalculated at each time step produces very similar dynamic

results as those of a linear dynamic analysis. However, the use of UAMP subroutines in

Abaqus does not support modal superposition procedures.

Hence, in agreement to these observations, the serviceability analyses of cable-stayed

footbridges in Abaqus are calculated through a nonlinear static step followed by a direct-

solution dynamic analysis.

Due to the length of the dynamic event (time taken by a pedestrian flow to cross

the bridge) and the computational cost of this direct-solution analysis, this dynamic

event is divided in Abaqus into small dynamic steps of shorter total duration (by using

restart analyses). The duration of these shorter steps is chosen in order to minimise the

calculation time of Abaqus. This calculation time is related to the number of times that

Abaqus calls the subroutine (which depends on the number of pedestrians and step loads

of each pedestrian during that interval of time) and the time taken by Abaqus to restart

the dynamic analysis. At each dynamic restart step, nodes have as initial conditions

(displacement, velocity and acceleration) the final conditions of the prior step.

The static nonlinear step is represented in Abaqus by a general static step with acti-

vated large-displacement formulation. The dynamic steps are represented in Abaqus by

dynamic implicit analyses with fully nonlinear direct integration based on Hilber-Hughes-

Taylor integration method. These have a maximum time step of 0.01 sec. to consider

modes with a total modal mass participation ratio larger than 95%.

3.6.5 Duration of simulation events

The evaluations of the responses of girder footbridges are developed considering deter-

ministic definitions of the pedestrian loads. However, in cable-stayed footbridges, charac-

teristics and arrival of pedestrians are stochastically described, as detailed in Section 3.2.

In this case, loads defined by pedestrians describe a random excitation process.

Once the footbridge develops steady state response (after the bridge deck has been fully

loaded for a while), it can be considered that the response of the structure corresponds

to a stationary ergodic process. In relation to stationarity, the response is expected to

present a mean value independent of time. In relation to ergodicity, it implies that results

obtained from sampling at instants from different events or from a single event during a

time length should be similar.

Based on these statistical characteristics, the analysis of the serviceability response

of CSFs will be performed on the basis of a single long event from which results at

different instants will be used to appraise the serviceability response. It is clear that due

to the limited time length of this event, the peak response will not represent a magnitude

only trespassed on 5% of the occasions when the bridge is fully loaded (as in Setra,

2006). Nonetheless, this event will provide results that will faithfully describe an average

serviceability response of the footbridge under the particular type of traffic.

In relation to the length of this single event, the analysis of multiple events of a

particular traffic (0.6 ped/m2 of commuters) describes how average response developed

during a time length equivalent to that used by an average pedestrian to cross the bridge

117


tap three times is the same as that developed during longer events (see Figure 3.24). This

figure represents vertical and lateral RMS accelerations at different points of the deck

of a CSF obtained during five different pedestrian events of time length equal to 3tapor 5tap. Based on these results, it can be inferred that the response of both events is

similar regardless the length of the event. Therefore, for the comparison of the dynamic

performance of CSFs with different structural characteristics in this thesis, the shorter

time event (3tap) will be implemented to evaluate the dynamic response.

0 2 40

0.5

1

1.5

2

RM

S V

ert

acc.

[m/s

2]

0 2 40

0.5

1

1.5

2

0 2 40

0.5

1

1.5

2

0 2 40

0.5

1

1.5

2

0 2 40

0.5

1

1.5

2

0 2 40

0.2

0.4

Time [s]

RM

S L

at

acc.

[m/s

2]

0 2 40

0.2

0.4

Time [s]

0 2 40

0.2

0.4

Time [s]

0 2 40

0.2

0.4

Time [s]

0 2 40

0.2

0.4

Time [s]

Time: 3 tap

Time: 5 tap

x = 30.0m

x = 28.0m

Figure 3.24: Vertical and lateral RMS accelerations recorded at the deck of a CSF, x =28.0 and 30.0 m, caused by 5 different pedestrian events with commuters and density 0.6ped/m2 (tap describes the time taken by an average pedestrian to cross the bridge).

3.7 Response analysis and comparison

3.7.1 Serviceability limit state of vibration

The main purpose of verifying the serviceability limit state of vibration at footbridges

is ensuring an acceptable degree of comfort to users when using such structure. Based

on the considerations outlined in Section 3.4.1, it is proposed in this thesis to assess the

comfort of pedestrians walking, standing and sitting on a footbridge.

In relation to walking users this is performed through two comparisons. Regarding the

first, limits given in Section 3.4.1 are compared to the magnitudes of the peak and RMS

(expression defined in Section 2.5) accelerations recorded at the deck at three different

nodes of sections every 2.0-3.0 m along the deck. This comparison is based on the assump-

tion that these values are equivalent to the movements noticed by users. Peak values are

representatives of a particular event (with similar pedestrian traffics these may be slightly

smaller or larger) whereas RMS values, which weight response with time, describe the

average response caused by the traffic.

The limits of the second method (Section 3.4.1) are compared to the largest peak

accelerations noticed by more than 50, 25 or 5% of the users, as well as the average

maximum acceleration felt by all pedestrians while crossing (magnitudes of accelerations

accepted by 50% of the users are generally considered to set comfort limits, as detailed

in Section 2.5).

118


The movements felt by walking pedestrians are described by the accelerations recorded

at the soles of their feet while crossing. This is performed by registering, for each pedes-

trian, the accelerations at the node closest to the position of their feet when touching the

deck. During the walking phase of double stance (both feet in contact to the ground),

it is assumed that the acceleration felt by the pedestrian corresponds to that felt by the

foot last placed on the floor (see Figure 3.25).

Mesh nodei i+1

LS

RS

LS

RS

Feet contact timeLS

RStsfc tsfc

tdsc

Acceleration felt by

pedestrian

(recorded movement)

aleft aright aleft aright

Figure 3.25: Recording of accelerations felt by pedestrians; RS describes the right step, LSthe left step, tsfc the time of single foot contact and tdsc the time of double stance contact

Comfort of standing and sitting users is assessed according to movements recorded at

the deck. These values are compared to comfort values of Section 3.4.2.

3.7.2 Serviceability limit state of deflections

The maximum dynamic deflection caused by a pedestrian traffic is compared to the

static deflection caused by the equivalent static forces of these traffic scenarios.

In the vertical direction, the dynamic deflections are compared to the maximum static

displacements caused by the weight of the pedestrian streams. In the lateral direction,

the dynamic deflections are compared to the static lateral deflections caused by horizontal

loads that result from the average lateral load of all the step loads of each pedestrian that

crosses the bridge, Amplat, described by Equation 3.7.1 (where Ampi,j,lat corresponds to

the lateral load amplitude of the step i of pedestrian j and Kj is the total steps on the

bridge of pedestrian j).

Amplat =N∑

ped,j=1

Kj∑

step,i=1

∣

∣Ampi,j,lat

∣

∣

∑

Kj

(3.7.1)

3.7.3 Ultimate limit state related to deck normal stresses

The time history dynamic bending moments of the deck (hogging and sagging) and

peak values are obtained at sections every 2.0-3.0 m along the deck length and compared

to: a) the envelope of static bending moments caused by the equivalent static weight of

the pedestrian flow and b) the envelope of the static bending moments generated by the

ULS uniformly distributed load (5 kN/m2).

Bending moments are obtained considering the stresses at multiple points of the deck

119


transverse section (concrete slab and steel girders).

3.7.4 Ultimate limit state related to shear stresses

The time history dynamic shear stresses and peak values described at the web of the

steel girders are described at sections located every 2.0-3.0 m along the deck length and

compared to: a) the static shear stresses caused by the equivalent static weight of the

pedestrian flow and b) the static shear stresses caused by the ULS uniformly distributed

load (5 kN/m2).

3.7.5 Ultimate limit state related to tower stresses

The variation in time and peak magnitudes of the normal and shear stresses of the

tower at multiple sections are compared to normal and shear stresses caused by the

static equivalent weight of the pedestrian traffics and those caused by the ULS uniformly

distributed load (5 kN/m2).

3.7.6 Ultimate limit state of fatigue of cables

The fatigue of a cable is related to the accumulated damage caused by the successive

stress cycles of different amplitudes endured by this cable during its lifetime (generated

by the passage of pedestrians).

In order to evaluate the damage caused by a particular traffic event at a cable, the

effect of the stress cycles is assessed considering the cable fatigue resistance (presented in

Section 3.5.2) and the Palmgren-Miner linear damage hypothesis given by Equation 3.7.2.

k∑

i=1

ni

Ni

= Damage ≤ 1.0 (3.7.2)

According to this rule, the number of cycles of each stress amplitude ni is compared

to the maximum number of cycles of that amplitude that the cable is capable to resist Ni

(given in Section 3.5.2). The effects of each stress range are accumulated and, in order to

ensure that failure does not occur, this total effect (Damage of Equation 3.7.2) is limited

to 1.0.

Based on this fatigue assessment, two damage evaluations of cables are proposed: one

involving the overall performance of a cable during its lifetime and another comparing

that performance to the behaviour of the same cable of bridges with different structural

or geometrical characteristics. For the first assessment, the total damage accumulated

during the lifetime (50 years) of each cable of a footbridge is described using Equation

3.7.2 considering the traffic events described in Table 3.8. For the second assessment,

the fatigue performance is evaluated by comparing the accumulated damage at the cable

in each bridge as described by Equation 3.7.3 (where the numerator describes the total

damage of the cable at the footbridge with an alternative structural parameter and the

denominator describes that of the benchmark footbridge).

Damage Comparison =(∑k

i=1ni

Ni)alt

(∑k

i=1ni

Ni)bas

(3.7.3)

120


Table 3.8: Summary of service events considered to evaluate the fatigue performance ofthe stay cables (C describes commuter events and L leisure events).

Bridge usage 0.2 ped/m2 0.6 ped/m2 1.0 ped/m2

Seldom5 hours/weekdays: C4 hours/weekend: L

Regular6 hours/weekdays: C4 hours/weekend: L

5 hours/weekdays: C4 hours/weekend: L

Heavy6 hours/weekdays: C4 hours/weekend: L

5 hours/weekdays: C4 hours/weekend: L


This chapter summarises the fundamental characteristics considered in this thesis re-

lated to load models, footbridges and parameters considered for their numerical represen-

tation and serviceability evaluation.

The first section of the chapter proposes a model for the vertical and lateral loads trans-

mitted by pedestrians while walking over footbridges. The proposal aims to be realistic

by including the inherent intra-subject variability associated with human movement, the

inter-subject variability among different pedestrians, and the variations in flow movement

produced when individuals interact in crowds.

The key features of the proposed model are the definition of vertical and lateral loads

induced by pedestrians through individual footsteps that accurately capture the energy

transmitted by these steps, the feedback between bridge response and the pedestrian

movement considered for lateral loads and the relation of these loads to the gait charac-

teristics of the user, which in turn are related to the situation considered (type of flow

and density). The parameters of the model are represented in a probabilistic manner that

adds realism to the underlying components and assumptions of the model.

The second section of the chapter highlights the most adequate ranges of comfort

limits upon which to assess the serviceability of footbridges from the point of view of users

(considering that these may be walking, standing as well as sitting). The third section

of the chapter provides a detailed outline of the characteristics of girder and cable-stayed

footbridges henceforth considered in this work. The fourth section of the chapter describes

and reasons the assumptions considered for the numerical representation of the bridges as

well as their dynamic response. Finally the last section introduces the criteria that will

be used in following chapters to assess and compare the dynamic performance of different

footbridges.

121

Chapter4Relevance of stochastic

representation of reality: advantages

of the new load model presented

herein

4.1 Introduction

The load model developed for the assessment of the dynamic response of footbridges

under the effect of pedestrian actions (Section 3.2) has a stochastic basis that can be easily

grasped in reality: the inability of pedestrians to maintain a constant step frequency while

walking at a particular speed, the gait differences among pedestrians (selected speed is

affected by anthropometric characteristics, aim of the journey and density of the pedes-

trian flow where they walk), the different step width of pedestrians as well as the effects

of pedestrians on others while walking (collective behaviour).

Despite the consideration of these variable parameters, there are other factors of

stochastic nature that have not been included in the model, e.g., the variability of the step

width at consecutive steps of a user or the different load amplitudes and weight of indi-

viduals. In order to substantiate these decisions, following sections appraise the impact of

each of these measures on the dynamic response of structures and compare them as well

to the predictions of movements for the same cases generated by available deterministic

load models.

The dynamic assessments are conducted with traffic events including one or multi-

ple pedestrians. The appraisal of the effects of intra-variability factors is performed by

comparing dynamic movements caused by single pedestrians at different structures or

comparing movements generated by multiple pedestrians to those of similar traffics de-

scribed deterministically. In the analysis of inter-variability characteristics, comparisons

include multiple pedestrian events exclusively.

The dynamic movements caused by a single pedestrian are obtained at structures with

123

4. Relevance of stochastic representation of reality

vibration frequency coincident with the mean step frequency as well as others where these

magnitudes are different. The comparison of these results is performed on the basis of

observations and concepts detailed in Section 3.3.

Section 4.2 presents results related to intra-variable characteristics (step frequency and

lateral step width), Section 4.3 describes the results caused by populations described with

different characteristics (inter-variability), and Section 4.4 those related to flow interac-

tions.

4.2 Pedestrian intra-subject variability

One of the aspects of pedestrian loads that requires a stochastic definition to include it

in a pedestrian load model is the intra-subject variability. This phenomenon corresponds

to the inability of pedestrians to walk at a constant pace, with identical gait characteristics

(Sections 2.3.5 and 3.2.3), and it involves parameters of the human gait such as the step

frequency or the lateral step width. The step frequency variability has been experimentally

evaluated in research works of the biomechanics field (focused on the study of the human

gait) and lately of the civil engineering field. The intra-subject variability related to

the magnitude of the lateral step width at consecutive steps has only been studied and

characterised in few studies.

The following sections assess the importance of the intra-subject variability related

to step frequency and lateral step width on the movements caused at footbridges in the

vertical and the lateral direction.

4.2.1 Effect of step frequency variability on vertical response

As just mentioned, several experimental works characterise the step frequency intra-

subject variability (Maruyama et al., 1992; Butz et al., 2008). These researchers define

the intra-variability of fp using normal distributions with standard deviations σfp with

values in the range from 0.037 to 0.207 Hz (Maruyama et al., 1992) and 0.09 Hz (Butz

et al., 2008). Some of these studies associate larger values of σfp with higher speeds and

others with lower speeds. Nonetheless, due to the insufficient information in this regard,

in the work developed herein the standard deviation of this distribution is homoskedastic

with respect to the speed of the pedestrian.

As proposed in Section 3.2.3, the intra-variability of σfp is represented using the

Metropolis-Hastings algorithm where the normal distribution describing the long-term re-

lationship between steps has a standard deviation similar to those proposed by Maruyama

et al. (1992) and Butz et al. (2008). The impact of this step frequency variability is in-

vestigated for pedestrians using different mean step frequencies.

Underpinning the importance of the introduction of such characteristic in a model of

pedestrian loads, Figures 4.1 show the mean and maximum accelerations produced by a

single pedestrian walking with a mean step frequency µ = 1.8 Hz. The results represent

the passage of this pedestrian across a number of bridges that cover a wide range of

structure-to-pedestrian (fs/fp) frequency ratios. The number of steps taken to cross each

bridge (i.e., π3 = sl/L) and the ratio between pedestrian and bridge weight (i.e., π2 =

ms/mp) are fixed for all analyses (see Section 3.3). The results shown in the figures

124


include a number of different standard deviations for the step frequency to represent the

variability about the mean value of 1.8 Hz. The range of standard deviations span a range

from a deterministic step frequency (σfp = 0 Hz) through to the maximum considered

realistic (σfp = 0.15 Hz).

Mean vertical acceleration [m/s2]

fs / fp

1.0 1.5 2.0 2.5 3.0

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

Maximum vertical acceleration [m/s2]

fs / fp

1.0 1.5 2.0 2.5 3.0

0.005

0.010

0.015

0.020

0.025

0.030

0.035

(a)

fs / fp

1.95 2.0 2.05

0.0050.0100.0150.020

0.025

2.95 3.0 3.05

fs / fp

1.0 1.1 1.2 1.3 1.4

0.05

0.10

0.15

fp

0.045

0.040

0.035

0.030

0.0250.020

0.015

Maximum Vertical Acceleration [m/s2]

Maximum Vertical Acceleration [m/s2]

(b)

(c)

(d)

fp N(1.8,0.0Hz)fp N(1.8,0.025Hz)fp N(1.8,0.050Hz)fp N(1.8,0.075Hz)

fp N(1.8,0.100Hz)fp N(1.8,0.125Hz)fp N(1.8,0.150Hz)

Figure 4.1: Effects of step frequency variability: (a) mean vertical accelerations, (b) maxi-mum vertical accelerations, (20 simulations of the same event with fp = 1.8 Hz), (c) detaileddescription of maximum accelerations around fs = fp, (d) detailed description of maximumaccelerations around fs = 2fp and fs = 3fp.

Figures 4.1 show that when the variability of the step frequency is low results are very

similar to the case of pedestrians walking with a constant step frequency (results for the

model with σfp = 0.0 Hz are equal to those defined by the model with σfp = 0.025 Hz in

most combinations of fs/fp). These results would be similarly predicted with a Fourier

series load model with three harmonics (instead of the load model used here consisting

in the description of the time amplitude of the whole step load). However, for moderate-

to-large variability in the step frequency (σfp = 0.05 - 0.15 Hz), the peaks associated

with resonant conditions (i.e., when the ratio fs/fp is equal to a natural number) are

significantly reduced and the troughs between these peaks have significantly increased

125


accelerations. This smoothing of the peaks and troughs reflects the fact that the points

plotted for a given value of fp really reflect a range of responses at frequencies around

this value.

The results of Figure 4.1(b) have more statistical noise than the average results (Fig-

ure 4.1(a)), but they also reveal some interesting trends. In particular, the results demon-

strate that the maximum response no longer occurs for resonant conditions. This effect

is seen most clearly in Figure 4.1(c), where accelerations for fs/fp = 1.15 are frequently

more than 30% larger than accelerations at fs/fp = 1.0. These larger responses at non-

resonant cases are explained by the different energy transmitted by loads corresponding to

different step frequencies (see Figure 4.2), as the load description is frequency dependent

(see Figure 3.1).

1.4

fp [Hz]

fs/ fp = 1.0 fs/ fp = 2.0

a) b)

Acceleration

[m/s2]

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

1.6 1.8 2.0 2.2 2.4 1.4 1.6 1.8 2.0 2.2 2.4

fp [Hz]

Figure 4.2: Vertical maximum accelerations generated by loads defined with the proposedmodel (with σfp = 0.0 Hz) for: (a) fs/fp = 1; or, (b) fs/fp = 2.

Figures 4.2(a) and (b) compare the maximum vertical accelerations caused by vertical

pedestrian loads generated with mean step frequencies ranging from 1.3 to 2.4 Hz and

σfp = 0.0 Hz when fs/fp = 1 or fs/fp = 2 respectively. The first plot illustrates that the

response when fp = 1.8 is smaller than that when fp > 1.8 and, conversely, the second

displays how the response when fp = 1.8 is similar or larger than those of larger or smaller

step frequencies (except when fp > 2.2).

For multiple-pedestrian scenarios, the consideration of the step-frequency variability

for individuals also generates significant differences in the response (see Figure 4.3). This

figure represents results of crowd flows with densities between 0.2 and 0.6 ped/m2 walking

on structures with fundamental vertical frequencies of fs = 1.8 and 2.0 Hz. To generate

the loads for these multi-pedestrian scenarios each pedestrian is first allocated a mean step

frequency via sampling from a normal distribution centred on the structural frequency,

µ(i)fp

∼ N(fs, 0.175 Hz), with i being an index denoting a particular pedestrian. The intra-

subject step frequency variability is then defined using another normal distribution with

a mean set to this sampled step frequency. That is, the step frequencies of any individual

are represented by f(i)p ∼ N(µ

(i)fp, 0.1 Hz), or simply f

(i)p = µ

(i)fp

in the case that the intra-

subject variability is ignored. This standard deviation σfp = 0.10 Hz is the average of

126


values observed by Maruyama et al. (1992) and Butz et al. (2008). The results obtained

using this approach suggest that both the step frequency variability and the density of

pedestrians on the structure have a significant impact upon the results (see Figure 4.3).

Step frequency variability produces results that differ from their constant counterparts

by an amount that varies with the mean step frequencies of pedestrians (for µ = 1.8

Hz non-constant step defines results 5% larger and for µ = 2.0 Hz 75% larger). The

density of the flow of pedestrians seems correlated to the variability of results (results of

Figure 4.3 for scenarios with flows of 0.6 ped/m2 show a larger spread of results) and to

larger maximums, suggesting that the effects of this intra-subject variability in response

are relevant.

Var. Ct. Var. Ct. Var. Ct. Var. Ct.

fs = 1.8Hz

0.2 ped/m2

fs = 1.8Hz

0.6 ped/m2

fs = 2.0Hz

0.2 ped/m2

fs = 2.0Hz

0.6 ped/m2

Acceleration

[m/s2]

0.14

0.120.10

0.04

0.02

0.06

0.08

Figure 4.3: Effects of step frequency variability on structural vertical response in scenar-ios with multiple pedestrians: maximum vertical accelerations (50 simulations of the sameevent); Ct. corresponds to constant step frequency of each pedestrian and Var. variablestep frequency (µfp = fs and σfp = 0.10 Hz).

4.2.2 Effect of step frequency variability on lateral response

For lateral loads, the effects of the variability of the step frequency of an individual

are studied through the same procedure implemented for vertical loads (applying the

Metropolis-Hastings algorithm for Markov-chain Monte Carlo simulation).

As opposed to vertical loads, results in the lateral direction do not depend on the

magnitude of the mean step lateral frequency fp,l (fp,l = 0.5fp), which is explained by the

fact that the impulse introduced per period of vibration of the structure (any mode) by

the lateral loads is the same, regardless of the lateral frequency fp,l (these results, obtained

with constant step frequency, are depicted in Figure 4.4). In the vertical direction this

statement is not valid due to the different shapes of the vertical loads according to fp, as

depicted in Figure 4.2.

Figure 4.4 depicts as well that lateral loads cause resonant response for fs,l/fp,l = 1

and fs,l/fp,l = 3 (and for any other odd natural number), which is due to the different

signs of the loads associated with consecutive steps. This is opposed to what occurs in

the vertical direction, where there also is resonance at fs,l/fp,l = 2 (and any other even

number).

In relation to intra-subject variability, Figure 4.5 represents the maximum lateral

accelerations caused by a pedestrian crossing a range of bridges under the same conditions

as for the vertical case (Section 3.3). This figure demonstrates how the effect of the step

127


fs,l / fp,l

0.5 1.0 1.5 2.0 2.5 3.0

0.001

0.002

0.003

0.004

0.005

0.006

Acceleration

[m/s2]

Figure 4.4: Maximum midspan lateral accelerations of simply supported structures undersingle pedestrian loads defined by the new load model with σfp,l = 0.0 (valid for any stepfrequency).

frequency variability on response is significant, even when relatively small degrees of

variability are considered. Generally speaking, both the mean and maximum responses

are qualitatively similar and are characterised by their lack of strong peaks that are

described when the step frequency variability is ignored. This is a very important point

as contrary to conventional thinking, the proximity of the lateral step frequency to the

lateral structural frequency does not seem to be particularly important once this intra-

variability is considered.

0.003

0.006

0.009

0.003

0.006

0.009

0.003

0.006

0.009

0.003

0.006

0.009

0.003

0.006

0.009

0.003

0.006

0.009

Lateral Acceleration [m/s2]

1.0 2.0 3.0 1.0 2.0 3.0

1.0 2.0 3.0 1.0 2.0 3.0

1.0 2.0 3.0 1.0 2.0 3.0

fs,l/fp,l

fs,l/fp,l

fs,l/fp,l

fs,l/fp,l

fs,l/fp,l

fs,l/fp,l

fp = 0.025Hz fp = 0.050Hz

fp = 0.075Hz fp = 0.100Hz

fp = 0.125Hz fp = 0.150Hz

Figure 4.5: Effects of step frequency variability on lateral response caused by a singlepedestrian (20 simulations of the same event); the black line correspond to results of constantstep frequency fp, the red line to the maximum accelerations of variable step frequency andthe grey line to the mean accelerations of variable step frequency.

When considering results that involve multiple pedestrians, the effects of intra-subject

variability are even more dramatic than for single pedestrians (Figure 4.6). The results

shown in Figure 4.6 are obtained considering pedestrian flows with different densities

where individual step frequencies of pedestrians are either constant, in which case the fp of

each pedestrian is sampled from a normal distribution f(i)p ∼ N(2.0, 0.175 Hz), or variable,

where the mean step frequencies are obtained according to µ(i)fp

∼ N(2.0, 0.175 Hz) and the

128


actual frequencies are sampled according to f(i)p ∼ N(µ

(i)fp, 0.1 Hz). For these analyses, the

median responses obtained with variable step frequencies are considerably smaller than

for the constant frequency case. Both of the comparisons made here (for individual and

crowd scenarios) suggest that this variability of step frequency is an important parameter

that should be considered when modelling lateral loads and consequently this is included

in the proposed load model.

0.0250.030

0.2 ped/m2 0.6 ped/m2

Ct.Var. Var.

0.020

0.0050.010

0.015

Acceleration

[m/s2]

Ct.

Figure 4.6: Effects of step frequency variability on lateral response caused multiple pedes-trians scenarios: maximum lateral accelerations (50 simulations), where Ct. corresponds toconstant step frequency of each pedestrian and Var. variable step frequency (µfpl = fs andσfp = 0.10 Hz).

4.2.3 Effect of step width variability on lateral response

Despite there being very little precedent for modelling this step width variability,

this effect is considered herein using the same approach adopted for the analysis of step

frequency (implementing the Metropolis-Hasting algorithm). The analysis is conducted

considering loads of a subject generated with step widths that have a marginal normal

distribution with mean half-step width of µws = 5.15 cm and a standard deviation rang-

ing from 0.5 to 2.5 cm. The results obtained under these conditions are presented in

Figure 4.7.

For these analyses, the feedback between structural response and pedestrian gait char-

acteristics (i.e, the change of pedestrian step width as a consequence of the structural

performance) has been ignored. Nonetheless, the results are still valid due to the small

magnitude of the responses generated by the pedestrian lateral loads (see Figure 4.7,

where maximum lateral accelerations are well below 0.01 m/s2). If the resulting lateral

accelerations had been larger in magnitude, this feedback mechanism would have prevailed

and therefore modified the mean step width of the pedestrian and the lateral response of

the structure.

Figure 4.7 suggests that the impact of step width variability is very small (despite the

large variability considered) over the range of variabilities that we have considered and

that, in fact, what prevails in determining the final structural response is the mean step

width or the average step width of all the step loads. Therefore pedestrians taking a few

steps with smaller or larger step width than the average does not disturb the response

generated by the mean step widths. These results justify not considering this component

of the variability in the proposed load model.

129


0.003

0.006

0.009

0.003

0.006

0.009

0.003

0.006

0.009

0.003

0.006

0.009

0.003

0.006

0.009

fs,l/fp,lfs,l/fp,l

fs,l/fp,lfs,l/fp,l

fs,l/fp,l

Lateral Acceleration [m/s2]

1.0 2.0 3.0 1.0 2.0 3.0

1.0 2.0 3.0 1.0 2.0 3.0

1.0 2.0 3.0

fp = 0.5 cm fp = 1.0 cm

fp = 1.5 cm fp = 2.0 cm

fp = 2.5 cm

Figure 4.7: Effects of step width variability on lateral response caused by a single pedes-trian (20 simulations of the same event), where the black line correspond to results of con-stant step width ws,t, the red line to the maximum accelerations with variable step widthand the grey line to the mean accelerations with variable step width.

4.3 Pedestrian inter-subject variability

In the previous section the focus was upon characterising the effect that intra-subject

variability has upon the structural response. However, in that section multi-pedestrian

scenarios were also considered. The purpose of the present section is to focus more

specifically upon these multi-pedestrian scenarios and to characterise the effect that inter-

subject variability (defined already in Section 2.3.5) has upon the acceleration response.

The simulation approach adopted herein emulates that of the previous section, but the

focus is now upon the impact that variations in the subject-specific load characteristics

have from subject-to-subject. The inter-subject variability is considered for the vertical

load amplitudes, pedestrian weights as well as gait characteristics such as step frequency

and step width.

4.3.1 Variability of vertical load amplitudes

The definition of vertical load amplitudes in the proposed model is based upon param-

eters (that are functions of fp) that represent the mean values of those observed by Butz

et al. (2008) (given in Table 3.1). However, there is a significant degree of variability in

these various parameters for any given step frequency fp (see Figure 2.7). This variability

is not considered in the load model and the present section evaluates the validity of this

assumption by demonstrating the modest impact upon resulting movements that exists

when this is accounted for.

The load model proposed for the representation of vertical actions requires the speci-

fication of nine parameters (see Figure 2.7). To appreciate the sensitivity of the response

to variations in each of these parameters, a simple sensitivity analysis is performed. For

these analyses all but one parameter are hold fixed at their expected values and per-

turb the remaining free parameter so as to represent its maximum and minimum value

130


when including 10, 50 and 90% of the measurements of that parameter (equivalent to a

one-factor-at-a-time sensitivity analysis). The corresponding variation in the response is

then observed and compared against the variation associated with changes due to other

parameters. The response variations φi caused by the modification of each parameter i

are compared after a normalization of the result as proposed in Equation 4.3.1, where

∆a is the increment of acceleration caused by the change of the single parameter (on a

simply-supported structure at midspan), ab is the basic acceleration caused by the load

generated with mean parameters on the same structure and measured at midspan, ∆zi is

the difference in magnitude of the parameter i and zi,b is the reference magnitude of the

same parameter.

φi =∆a

ab

(

∆zizi,b

)

−1

(4.3.1)

By using these differing levels of perturbation (including 10, 50 or 90% of the mea-

surements of each parameter) the linear relationship between the change in the parameter

and the associated effect upon the response is qualitatively appraised. The results of this

exercise are presented in Figure 4.8 and indicate that the response is most sensitive to

parameters defining the temporal locations of the peaks as well as the total time of the

load.

slope 2

slope 1

T3

Min

T2

Peak 2

T1

Peak 1

Ttot

Gradient Gradient Gradient

10% 50% 90%fp = 1.9Hz

Factor magnitude variation

Figure 4.8: Sensitivity analysis of vertical load amplitude (φ describes the relative sensi-tivity of response with respect to each parameter, see Equation 4.3.1) (parameters definedin Figure 2.7).

The results presented in Figure 4.8 are based on pedestrian loads with a constant step

frequency fp = 1.9 Hz. Nonetheless, similar results are obtained when loads defined with

frequencies such as fp = 1.6 or fp = 2.4 Hz are evaluated (step frequencies where load

amplitudes are considerably different from those of step frequency 1.9 Hz).

The impact that the load amplitude variability has on structural response is assessed

by comparing responses of the different load shapes that can be defined (according to the

previous sensitivity analyses, only the variability of the time parameters is considered).

Table 4.1 reports the maximum absolute variation of the maximum acceleration (with

131


respect to the acceleration caused by loads defined with mean parameters) that a pedes-

trian causes when his load amplitudes are described using fixed mean values coupled with

values of the time parameters that are independently set to upper and lower bound levels

that collectively encapsulate 10, 50 and 90% of the observations in Butz et al. (2008). For

example, the results for 90% of the data correspond to cases where the temporal parame-

ters are independently raised or lowered to their 5th or 95th percentile values, respectively.

These results show that the response is most sensitive to the total time of the load of a

given footstep, followed by the time of the first peak.

Table 4.1: Effect of the variability in temporal parameters

% of cases %-ile range Ttot T1 T2 T3

10 [45, 55] 10% 5% 1.5% 1.5%

50 [25, 75] 50% 25% 8% 10.%

90 [5, 95] 90% 50% 30% 30%

Further analysis of the impact of these amplitude parameters in the response evalua-

tion is considered by simulating scenarios with multiple pedestrians (Figure 4.9). These

correspond to traffic flows of 0.2 and 0.6 ped/m2 walking at the same step frequency and

including a case where all pedestrians have load amplitudes defined according to mean

values of the parameters (including tT , with a value µtT ,fp) and a case where the loads of

each pedestrian are defined considering different descriptions of tT in terms of fp (only tTis included since it is the parameter with largest impact in response). For this second sce-

nario each pedestrian i has load amplitudes that depend on tiT ∼ N(µtT ,fp , σtT ,fp), where

σtT ,fp is estimated from the dispersion of this parameter as plotted in Butz et al. (2008).

The resulting accelerations of these scenarios (Figure 4.9) show how, despite the fact

that differences for scenarios with a single pedestrian were large, when multiple pedestrians

are considered these differences disappear. This stems from the fact that, on average, the

total load that perturbs the structure for both scenarios, remains fairly similar despite

the variability that is introduced in the second case. Hence, these results justify the fact

that this inter-variability parameter is disregarded in the definition of the proposed load

model summarised in Section 3.2.5.

4.3.2 Variability of weight

Codes and guidelines assume a uniform representative weight for all pedestrians. How-

ever, as seen in Section 3.2.2, a pedestrian’s step frequency is related to his weight (through

speed, age and height of the subject as outlined previously) which in turn influences the

vertical and horizontal load amplitudes. Therefore, when considering a range of pedestrian

step frequencies within a multi-pedestrian simulation these relationships between anthro-

pometric characteristics and gait may lead to consider that the assumption of uniform

mass among pedestrians is unrealistic.

In order to evaluate the impact of representing pedestrian weight as a random variable

132


0.04

0.01

0.02

0.03

Acceleration [m/s2]

Ct. Ttot

0.05

New Model

0.2 ped/m2

New Model

0.6 ped/m2

Var. Ttot Ct. Ttot Var. Ttot

Figure 4.9: Effects of the variability in the definition of vertical load amplitudes, Ct. (con-stant) and Var. (variable), among pedestrians (maximum vertical accelerations at midspanof a simply supported structure).

we compare results obtained for a constant weight of 700 N (see Section 2.3.2) with results

obtained when the weight is coupled to pedestrian characteristics such as age, height and

speed. The distribution of weights for this exercise is taken from the Health and Social

Care Information Centre to be representative of the population of the UK (the average

weight of the UK population is 780 N). Consistent values of weight, age and height are then

used within the equations presented in Section 3.2.2 to define step frequencies, speeds,

etc. The first values (height and weight) are defined according to normal distributions

that represent the population depending on age intervals (not including any potential cor-

relations as these are unknown). Regarding the second set of variables (step frequencies,

speed, etc.), values have been predicted using the expressions proposed by Equations 3.2.5

and 3.2.7 without considering any correlation between the parameters (this assumption

was formally tested for speed and frequency and no statistically significant correlation

was found).

The results of this comparison are presented in Figure 4.10 and correspond to the

acceleration response of a simply-supported bridge with fundamental frequency in the

vertical direction of fs = 2.0 Hz under a flow of commuting pedestrians with a density

0.6 ped/m2. The figure suggests that the accelerations obtained when the inter-subject

weight variability is considered are about 10% greater than for a constant weight in this

particular case (equal to the difference between the constant weight of codes and the mean

weight of UK population).

The results of Figure 4.10 justify the use of a uniform weight of 780 N among repre-

sented pedestrians in the proposed load model (equal to the mean weight of pedestrians

from UK or Western European countries).

4.3.3 Variability of gait characteristics

The definition of the proposed new load model includes an evaluation of the speed

and step frequencies adopted by pedestrians while crossing footbridges (depending on

flow density and the aim of the journey of each pedestrian). The distributions relevant

for pedestrians in Western Europe were shown in Figure 3.4. These distributions suggest

values of step frequencies that can be substantially different from those considered as crit-

ical in guidelines and codes. These usually propose mean values around 1.8-2.0 Hz which

133


Commuter traffic 0.6 ped/m2

0.02

0.04

0.06

Acceleration [m/s2]

Ct. Weight Var. Weight

0.08

Figure 4.10: Effects of variability of weight, Ct. (constant) and Var. (variable), amongpedestrians (maximum vertical accelerations at midspan of a simply supported structure).

0.04

0.05

0.01

0.02

0.03

Acceleration [m/s2]

Setra Loads

0.2 ped/m2

New Model

Leisure

0.2 ped/m2

New Model

Business

0.2 ped/m2

Figure 4.11: Effects of variability of step frequency, according to traffic type, amongpedestrians (maximum vertical accelerations at midspan of a simply supported structure,with fs = 2.0 Hz).

were derived from observations of several researchers such as Pachi et al. (2005) and do

not consider any other parameter to modify these distributions in different circumstances.

Nonetheless, the step frequencies of pedestrians crossing a bridge have a large impact on

its response. Figure 4.11 provides an example of the differences that can exist under

particular circumstances for a structure of fs = 2.0 Hz under flows of pedestrians with a

density of 0.2 ped/m2. The results using both business and leisure conditions in the new

model are compared with results found from the application of the Setra guideline (2006)

that assumes that fp ∼ N(2.0, 0.175 Hz) in this case. The response caused by pedestrians

in business conditions are around half of those predicted by the Setra guideline, whereas

those obtained for leisure conditions are roughly 10% larger.

These results show how models from guidelines or codes are not able to capture re-

sponses caused by pedestrian flows for different contexts. In order to improve this defi-

ciency, the proposed load model includes the relationships between speed, step frequencies,

and use (business, commuting, leisure, etc.) that include these contexts, as described in

Section 3.2.2. Alternatively, as a simplification, it is recommended to consider values of

these characteristics taking into account distributions such as the proposed for Western

Europe population traffic (Figure 3.4).

134


4.3.4 Variability of step width

The proposed load model describes the initial step width adopted by each user in ac-

cordance to a stochastic relationship (Equation 3.2.9) derived from values experimentally

obtained by several researchers. This expression relates gait, pedestrian characteristics

and mean step width used when walking. The basis upon which this magnitude for each

pedestrian is defined by such relationship lies in results described hereunder.

The relevance of the step width magnitudes for multiple pedestrians is evaluated here

considering traffic flows where pedestrian lateral loads are generated with either an initial

uniform or non-uniform step width. For the first scenario pedestrians have a constant step

width (0.0515 m), whereas for the second the initial step width of each pedestrian is defined

considering Equation 3.2.9 and speed and pedestrian characteristics corresponding to a

pedestrian flow of commuters (obtained half-step widths approximately define a normal

distribution ∼ N(0.0515, 0.016 m); the mean and standard deviation of this distribution

are representative of the data gathered from experimental tests referenced in Section

3.2.2).

Figure 4.12 depicts lateral accelerations of a structure with fundamental lateral fre-

quency fs,l = 1.0 Hz and pedestrian crowds with densities of 0.2 or 0.6 ped/m2.

0.025

0.0300.2 ped/m2 0.6 ped/m2

Ct. ws,tVar. ws,t Var. ws,t

0.020

0.005

0.010

0.015

Acceleration [m/s2]

Ct. ws,t

Figure 4.12: Effects of step width variability (among pedestrians in a flow) on response inmultiple pedestrian scenarios, where Ct. ws,t represents pedestrian flows where all pedes-trians have the same initial half-step width whereas Var. ws,t represents the results ofpedestrian flows where each pedestrian has a random half-step width (according to normaldistribution).

The results of Figure 4.12 indicate that the variability of step width magnitudes among

the different pedestrians in a flow has a very large impact on the response. Responses

of constant step width are around three to four times those caused by pedestrians with

non-uniform step width. Accordingly, a very detailed evaluation of this magnitude among

pedestrians appears to be of utmost importance to predict lateral response of structures

under the action of pedestrian flows and must be included in pedestrian load models

attempting to represent real pedestrian lateral loads with accuracy.

4.4 Pedestrian flow interactions

In accordance with results described in previous sections and as highlighted by re-

searchers in Section 2.3.5, the most critical scenarios for the assessment of structural

response involve multiple pedestrians. As reported in that section, several authors have

135


attempted to account for these loading cases in a simplified manner, through Monte

Carlo simulations and relationships between gait characteristics and density of the flow.

Nonetheless, all these simplifications may lead to unrealistic flow characteristics.

A first evaluation of the mean step frequencies used by pedestrians under different flow

conditions is defined by Figure 3.4. However, in order to produce results closer to reality,

for flows with medium and large densities, the step frequencies adopted by each subject

should reflect possible alterations due to changes of trajectory or speed in order to avoid

collisions with other pedestrians.

Including these alterations due to interactions between pedestrians, the proposed load

model considers a simulation of the two-dimensional movement of subjects on a structure

to obtain the gait parameters of each subject crossing structures under different initial

conditions (density and targeted velocities depend on the aim of the journey). This

model predicts the movement of the CoM in plan. This model has been adopted by a

small number of authors in structural assessments of bridges, e.g., Carroll et al. (2012),

and it is of great interest in large assemblies of pedestrians (gatherings in stadia, etc.).

A commonly-used model suitable for such simulations was formulated by Helbing et al.

(2000). The model is based on the representation of pedestrians as particles that move

towards their final destination at their desired speed. The speed and direction of the

movement of each particle (or pedestrian) can be affected by the proximity of obstacles

such as the balustrades of the bridge or other pedestrians at the front and those at the

back at short distances. Each of these effects (aim of the journey, other pedestrians

and obstacles) are characterised through forces, the summation of which describes the

direction and speed of movement of each pedestrian at any moment of the event.

Using the proposed expressions for the evaluation of pedestrian gait characteristics

(Section 3.2), the importance of collective behaviour is evaluated by comparing results

generated with and without these interactions. Results are presented for accelerations of

a structure with a fundamental vertical frequency of fs = 2.0 Hz in Figure 4.13. Two

different crowd densities are considered and for each density three sets of analysis are

presented. The first set of results are obtained considering the model of the Setra guide-

line. The second set of results represent the accelerations obtained considering pedestrian

crowds with the same gait characteristics as those defined in the Setra guideline but cou-

pling these with vertical loads from the newly proposed load model of this study. Finally,

the third set represent the vertical accelerations obtained when pedestrians are simulated

including collective behaviour, with gait characteristics described by the proposed rela-

tionships between speed and step frequency, and including non-constant step frequencies

for individuals.

The main point that can be made from consideration of Figure 4.13 is that the effects

of accounting for crowd interaction are most pronounced for the high pedestrian densities.

This finding is naturally consistent with intuition as pedestrians moving in sparse crowds

have little need to adjust their movement in response to dynamic changes in the flow.

For the traffic flows with 0.2 ped/m2, the three considered methods therefore exhibit very

similar results. However, when the density is increased to 0.6 ped/m2, although there

136


0.2 ped/m2 0.6 ped/m2

Acceleration [m/s2]

N.M.

Ct.

N.M.

Var.

Crowd int.

Setra M.N.M.

Ct.

N.M.

Var.

Crowd int.

Setra M.

0.14

0.02

0.06

0.10

Figure 4.13: Effects of collective behaviour simulation, where Setra corresponds a pedes-trian events characterised according to Setra guideline, N.M. defines pedestrian flows whereloads are described according to the proposed new load model, Ct. or Var. refer to constantor variable step intra-subject frequency and Crowd int. refers to collective behaviour.

remains a reasonable level of consistency in terms of the median predictions, there is a

significant difference in the overall range of responses that are observed. From a proba-

bilistic standpoint, this change in the nature of the distribution of induced accelerations is

important for the assessment of serviceability, and the inclusion of effects associated with

crowd interaction appear to enable more demanding loading scenarios to be acceptable

than in the case that these effects are neglected.

These results justify the inclusion of this microscopic pedestrian simulation in the

proposed load model. Nonetheless, this consideration does increase the complexity of the

analyses. Further work in this area could provide simplified assessment tools, perhaps

related to variance reduction factors, that would allow for collective behaviour to be more

readily implemented in codes and recommended in guidelines.


The evaluation of the impact of each stochastic characteristic introduced in the pro-

posed load model on the structural response (presented in previous sections) justifies the

inclusion of these factors. These assessments lead as well to the identification of the fol-

lowing important considerations to take into account for a precise evaluation of structural

response in general:

• The intra-subject variability of the step frequency has a large impact upon structural

response. Vertical responses under non-resonant conditions may be larger than res-

onant responses, which is explained by the definition of vertical loads considered in

the model (where amplitudes depend on the step frequency). In the lateral direction,

the impact of step frequency variability is considerable as well.

• As a result of modelling the feedback between bridge response and lateral induced

loads, the response of a structure under the action of a stream of pedestrians is caused

by all pedestrians and not only those with step frequencies close to the structural

frequency. Therefore streams of pedestrians should be explicitly modelled.

137


• The predicted accelerations are sensitive to the gait characteristics (speed, step

frequency and step width) attributed to each pedestrian in a flow. These should

be defined according to the situation considered (type of flow and densities most

likely to occur). The proposed method provides a more realistic alternative to the

proposals currently used in practice.

• Finally, the modelling of collective behaviour using sophisticated simulations pro-

vides a very realistic prediction of pedestrian behaviour while crossing a bridge.

However, the use of such approaches leads to a level of complexity significantly

greater than alternative simplified methods.

138

5Girder footbridge design: evaluation

of response in serviceability

conditions

5.1 Introduction

Footbridge design has evolved rapidly in recent decades (as detailed in Section 2.2).

Research has shown that the non-fulfilment of serviceability requirements in some bridges

is not only related to this design progression but also to the unrealistic loading considered

to assess serviceability conditions.

Designers need to be able to predict, from the very early stages of their designs, whether

or not their proposals can satisfy serviceability requirements. However, many design codes

do not define the procedure that should be adopted to perform analyses if needed. In

addition, these have two main disadvantages: on one hand, they are computationally

demanding and, on the other hand, some are based on load models not including the

latest advances in this research area.

With the aim of providing a tool to perform this serviceability assessment and of

presenting a sound basis upon which to compare the performance of girder footbridges

to cable-stayed bridges, this chapter proposes a very simple method to accurately obtain

the maximum vertical and lateral accelerations expected in a girder footbridge due to

pedestrian actions, underpinned by a simplified version of the load model proposed in

Section 3.2.

The method is applicable to footbridges of one-to-three spans, with uniform deck depth

and designed with conventional geometric and material properties as well as materials

introduced more recently in design. The response is derived based on concepts of Section

3.3 and on the structural features of the footbridge (Sections 5.4, 5.5 and 5.6) and traffic-

flow characteristics (Sections 5.7 and 5.8). Section 5.9 includes an assessment of the

performance of the proposed method by comparing the method results to those of detailed-

finite element analyses and to those measured at two real footbridges. Finally, using the

139

5. Girder footbridge design

proposed method, Section 5.10 evaluates the performance in serviceability in the vertical

or lateral directions of a wide range of footbridges with one or two spans.

5.2 Foundations of the method

The overall method presented over the course of the following sections is based upon

a very simple conceptual model. The general idea is that the response of footbridges

under serviceability conditions is essentially a reflection of the resonant characteristics

that arise from the interaction of the structural properties and the pedestrian loading

characteristics. A physically-based equation is therefore developed that identifies the

key frequency ranges that will be most important for the acceleration response of simple

bridges in service conditions. This equation is parameterised using very common and

fundamental geometric and material properties and can therefore be applied to a very wide

range of structural configurations. This parameterisation is founded on the conceptual

basis presented in Section 3.3.

In the sections that follow the approach to determine these basic accelerations is

first defined before the various adjustment factors that can be applied to adapt these

basic values to be appropriate for the structural and loading configuration of interest are

explained and presented.

5.3 Pedestrian loading

The method presented hereunder is based on the load model described in Section 3.2

with simplifications regarding intra-variability and collective behaviour effects (these are

not included in the derivation of the model for the sake of simplicity). The model still

defines individual foot loads according to the step frequency, lateral loads depend on the

lateral movements noticed at previous steps and the step frequency is related to speed,

aim of the journey, pedestrian density and other factors through the consideration of the

distributions described in Figure 3.4.

5.4 Vertical and lateral structural frequencies

For a simply-supported beam of constant geometrical and mechanical characteristics

throughout the length, the vertical, fv,n, and lateral, fl,n, vibration frequencies associated

with the nth vertical and lateral vibrational modes are given by Equation 5.4.1:

fy,n =n2π

2L2

√

EIxρA

, y ∈ v, l (5.4.1)

where L is the span length, E is Young’s modulus, ρ is the material density, A is the

cross-sectional area of the section, and Ix, x ∈ v, l, are the second moments of area in

the vertical and lateral direction of the section, respectively.

The ratio between the second moment of area, in the vertical and lateral directions,

and the cross sectional area, can be defined by Equations 5.4.2 and 5.4.3:

140


IvA

= ηvαv(1− αv)d2h (5.4.2)

IlA

= ηlαl(1− αl)b2 (5.4.3)

where αv is the ratio between the vertical distance from the centroid of the section to

the top extreme fibre and the vertical depth of the section dh; αl is the ratio between the

horizontal distance from the centroid of the section to the closest lateral extreme fibre

and the width of the section b; ηv is the ratio between the depth of the central kern and

the depth of the section dh; and ηl is the ratio between the width of the central kern and

the width of the structural section b.

By substituting Equations 5.4.2 and 5.4.3 into Equation 5.4.1, the following expressions

are obtained:

fv,n =n2π

2L

√

E

ρηvαv(1− αv)

(

dhL

)2

(5.4.4)

fl,n =n2π

2L2

√

E

ρηlαl(1− αl)b2 (5.4.5)

The parameters ηv, ηl, αv and αl take reasonably constant values for each section type.

Figure 5.1 provides values of these parameters for conventional sections that can be used

in footbridge design.

In preliminary design, the vertical and lateral frequencies of the structure can be

directly estimated from Equations 5.4.4 and 5.4.5, whilst in detailed design they can be

estimated from Equation 5.4.1 or directly obtained from finite element (FE) models. For

lateral frequencies, Equation 5.4.1 describes the vibration modes of bridges where bearings

allow the rotation with respect to the line described by the lateral centre of gravity of the

section (for other bearing dispositions FE models will provide a more accurate evaluation

of the modal vibration).

5.5 Resonance parameters

The main parameters that control the vertical and lateral response of a footbridge un-

der pedestrian loading are the ratios between the vertical or lateral structural frequencies

and the corresponding pedestrian frequencies. The ratio between the nth vertical struc-

tural frequency and the pedestrian vertical frequency fp,v (or simply fp, hereafter), rv,n,

is given by Equation 5.5.1, in which φs,n is an adjustment factor to account for cases that

differ from a simply-supported bridge, and ρ∗ differs from ρ as it considers non-structural

mass. Note that the resonance parameters being discussed here are equivalent to the

first nondimensional parameter π1 presented in Section 3.3 in the case that φs,n = 1 and

ρ∗ = ρ.

141


Section geometry Geometric efficiency

Material Generic dimensionsVertical Lateral

Figure 5.1: Summary of geometric properties, usual materials (RC and PC stands forreinforced and prestressed concrete, respectively) and span ranges for different footbridgesections. The slab defining the decking is part of the structural cross section in sectionsS.1-S.5, and a non-structural element for sections S.6-S.9.

rv,n =fv,nfp,v

φs,n =n2π

2Lfp

√

E

ρ∗ηvαv(1− αv)

(

dhL

)2

φs,n (5.5.1)

When walking, consecutive vertical pedestrian loads have the same sign (downwards)

and are characterised by a frequency fp,v = fp (step frequency). For lateral loads, consec-

utive steps have opposite signs, corresponding to loads whose frequency is half the step

frequency (fp,l = fp/2). The ratio between the nth lateral structural frequency and the

lateral pedestrian frequency fp,l, is therefore denoted by rl,n as in Equation 5.5.2.

rl,n =fl,nfp,l

φs,n =n2π

L2fp

√

E

ρ∗ηlαl(1− αl)b2φs,n (5.5.2)

When the parameters rv,n or rl,n are equal to 1, it means that the vertical or lateral

pedestrian loading is inducing resonance in the structure, as the pedestrian frequency is

equal to the nth-mode structural frequency. When these parameters are equal to 2, it

means that the pedestrian loading reinforces the displacements in the nth mode in every

other cycle. In general, when the parameters rv,n or rl,n are equal to a natural number p,

the pedestrian loading will reinforce any existing structural movement at every p cycles

142


of the nth mode.

In the above expressions, ρ∗ is the effective material density when the non-structural

mass per unit length m (accounting for the pavement, parapets, and handrail weight, as

well as the mass of a pedestrian stream) is also considered:

ρ∗ = ρ ·(

1 +m

ρ · A

)

(5.5.3)

The parameter φs,n is related to the number of spans and the geometrical arrangement.

The frequency of the nth mode in a two or three span girder is equal to that in a simple

supported one span girder multiplied by this factor φs,n. For simply-supported spans

φs,n = φ1,n = 1. For two and three span beams, this factor can be directly obtained from

Figure 5.2.

1.50 2.0

0.2 0.4 0.6 0.8 1.0

Lsmall/L

0.00

0.25

0.50

0.75

1.00

1.25

LsmallL

0.2 0.4 0.6 0.8 1.0

LsmallL

s

Mode 1

Mode 2

Mode 3

Mode 4

Lsmall

Figure 5.2: Amplitude of φs,n, according to mode, n, and number of spans.

The pedestrian frequency fp should be defined based on the density of the crowd flow

and the type of use expected (Figure 3.4). For simple calculations, the mean values of

frequencies should be considered (given by the continuous line in Figure 3.4). For more

detailed calculations, a wider range of pedestrian frequencies should be considered (for

this purpose, additional fractiles of the frequency distribution are shown in the same

Figure 3.4).

5.6 Basic vertical and lateral accelerations

The basic accelerations (aby,n y ∈ v, l) linked to the nth mode of vibration are those

associated with the passage of a single pedestrian crossing a simply-supported bridge.

These basic accelerations have been obtained for broad ranges of the resonance param-

eters and fixed values of the remaining dimensionless parameters. The values of the

structure and the pedestrian characteristics that were used to define these fixed nondi-

mensional parameters (π2,...,6 as denominated in Section 3.3) are: (1) a structural mass

of 7440 times the pedestrian mass; (2) a pedestrian step of 2% of the span length; (3) a

damping ratio 0.5%; (4) a mean pedestrian height of 1.70 m; and, (5) a nominal transverse

143


distance between footsteps of 0.10 m. These values considered for the reference cases are

representative of those structures and materials considered in Figure 5.1, as well as the

characteristics of the UK population.

The values of the basic accelerations (as well as accelerations obtained with any other

value of the nondimensional parameters) have been obtained representing the beam struc-

tures as detailed in Section 3.6, analysis that accounts for pedestrian-structure interaction.

Results of multiple cases obtained with the developed numerical model have been vali-

dated with Abaqus (ABAQUS, 2013).

The basic vertical acceleration linked to the nth mode of vibration (abv,n) is obtained

as a function of the vertical resonance parameter (rv,n) and the pedestrian frequency fp(the different load shape, and therefore impulse, for different step frequencies explains

the different accelerations according to fp, as discussed in Section 3.2). The basic lateral

acceleration linked to the nth mode of vibration (abl,n) is obtained as a function of the

lateral resonance parameter (rl,n), but is independent of the step frequency (lateral loads

have the same shape and impulse regardless fp, given particular values of step width and

pedestrian height, as seen in Section 4.2). The values of these functions are listed in Annex

D. Intermediate values not listed in this table can be obtained by linear interpolation.

Basic vertical and lateral accelerations for vertical and lateral resonance parameters not

included in the table can be assumed equal to zero.

5.7 Maximum vertical and lateral accelerations caused by a sin-

gle pedestrian

The maximum vertical (av) and lateral (al) accelerations caused by one pedestrian

are given by the expressions in Equations 5.7.1 and 5.7.2. These maximum accelerations

are the maximum accelerations calculated from the consideration of the first four modes

considered in the analyses, i.e., n = 1, 2, 3, 4. This structural response appraisal has

been considered a reliable evaluation of the total response since results show that in each

case the response is largely dominated by a single vibration mode. Therefore, the results

obtained using this simple approach would not differ much from more elaborate modal

combination rules.

av = maxn

(

abv,nφpmφslφdφsm

)

(5.7.1)

al = maxn

(

abl,nφpmφswφphφslφdφsm

)

(5.7.2)

where φpm and φph are factors related to the mass and height of the pedestrian; φsl and

φsw are factors related the the length and width of the pedestrian step; φd is a factor

related to the damping of the structure; and φsm is a factor related to the mass of the

structure.

The adjustment factors are defined below. The basic vertical and lateral accelerations

given in Annex B have been obtained for certain fixed nondimensional parameters (cited

above), such as the ratio between the pedestrian and the structural mass, the ratio between

144


Table 5.1: Coefficients for obtaining φsl in Equation 5.7.4 for vertical response (y = v),where x is the ratio between the pedestrian step and the span length, i is a natural numbergreater than 2.

rv,n interval B1 B2

[0.975,1]5460x2 − 820x+ 14.34

0.975[1,1.025] 1.025[1.975,2]

1180x2 − 215x+ 3.841.975

[2,2.025] 2.025[i-0.025,i]

101x2 − 58.5x+ 1.12i− 0.025

[i,i+ 0.025] i+ 0.025

Table 5.2: Coefficients for obtaining φsl in Equation 5.7.4 for horizontal response (y = l),where x is the ratio between the pedestrian step and the span length, i is a natural numbergreater than 1, and j = 2i− 1

rl,n interval B1 B2

[0.975,1]5460x2 − 820x+ 14.34

0.975[1,1.025] 1.025[j − 0.025, j]

1750x2 − 201x+ 3.355j − 0.025

[j, j + 0.025] j + 0.025

the step lengths and the main span (i.e, the pedestrian would take 1/0.02 = 50 steps to

cross the main span). Therefore, the φi factors listed above are required to account for

different ratios to those initially considered.

5.7.1 Factor related to the pedestrian mass (φpm)

Codes and guidelines usually consider a standard pedestrian weight of 700 N. The

standard pedestrian mass mp is therefore 71.36 kg. The basic accelerations have been

obtained for a ratio between the masses of the footbridge and the pedestrian of 7440. For

ratios different to this particular value, the maximum acceleration can be obtained from

the basic acceleration by using the factor φpm, which is defined by Equation 5.7.3.

φpm = 7440mp

ρ∗AL(5.7.3)

5.7.2 Factor related to the pedestrian step length (φsl)

The pedestrian step length depends on various parameters, such as speed, gender,

height, etc. Values reported by Pachi et al. (2005) suggest that a value of 0.70 m can

be considered as representative. The basic accelerations have been obtained for a ratio

between the pedestrian step and the span length of 0.02. For different ratios, the maximum

acceleration can be obtained from the basic acceleration by using the factor φsl, which is

defined by Equation 5.7.4 and the coefficients presented in Tables 5.1 and 5.2.

φsl = exp(

B1 × |r2y,n − B22 |)

, y ∈ v, l (5.7.4)

In deriving these factors, results have indicated that the maximum accelerations reg-

145


istered are sensitive to the value of this parameter only near resonance (i.e., when the

resonance parameters take values very close to a natural number) and only for the first

and second vibrational modes. Therefore, this factor φsl needs to be assessed when the

vertical or lateral resonance parameters linked to the first (rv,1, rl,1) or the second mode

(rv,2, rl,2) are in the following intervals: [i ± 0.025], where i is a natural number (i.e.,

i = 1, 2, 3, . . .). For all other cases, φsl = 1.

5.7.3 Factor related to the pedestrian step width (φsw)

The pedestrian step width varies significantly among different pedestrians as well as

for a given pedestrian while they walk. Despite this, a value of ws = 0.10 m is taken as

being representative for the UK population (see Section 3.2.2). This value represents the

total transverse distance between feet, in units of metres. For different pedestrian step

widths, the maximum acceleration can be obtained from the basic acceleration by using

the factor φsw, which is defined by Equation 5.7.5.

φsw =ws

0.10(5.7.5)

5.7.4 Factor related to the pedestrian height (φph)

The basic accelerations have been obtained using a distribution of height suitable for

the UK population (the mean value is approximately 1.70 m). For populations with

similar height distributions to those of the UK, this factor should be considered equal to

1, otherwise the maximum acceleration can be obtained from the basic acceleration by

using the factor φph, which is defined by Equation 5.7.6.

φph =1.70

hpd

(5.7.6)

where here hpd is the mean height of the target population in units of metres.

5.7.5 Factor related to the structural damping (φd)

Due to its different causes, damping is a difficult parameter to appraise. Several

documents listing values of this parameter have been enumerated in Section 2.4. Generally

suggested values for reinforced concrete structures are: 0.8− 1.5%, prestressed concrete:

0.5 − 1.0%, composite sections (steel and concrete): 0.3 − 0.6%, steel: 0.2 − 0.5% and

timber structures: 1.0− 1.5%.

The basic accelerations have been obtained considering a damping ratio of ζ = 0.5%.

For different damping ratios, the maximum acceleration can be obtained from the basic

acceleration by using the factor φd, which is defined by Equation 5.7.7 and Tables 5.3 and

5.4. Results show that the maximum accelerations registered are sensitive to the value of

this parameter only near resonance (i.e., when the resonance parameters take values very

close to a natural number) and for the first and second vibrational modes. Therefore, this

factor φd needs to be assessed when the vertical or lateral resonance parameters linked to

the first (rv,1,rl,1) or the second mode (rv,2, rl,2) are in the following intervals: [i± 0.05],

where i is a natural number (i.e., i = 1, 2, 3, 4, . . .). For the rest of the cases, φd = 1. The

evaluation provided by this factor is valid for values of the damping ratio in the interval

146


0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

LsmallL LsmallLLsmall

Mode 1

Mode 2

Mode 3

Mode 4

Figure 5.3: Values for φsm, according to mode and spans arrangement.

0.002 ≤ ζ ≤ 0.025.

φd = exp(

C1 × |r2y,n − C22 |)

, y ∈ v, l (5.7.7)

Table 5.3: Coefficients of φd for Equation 5.7.7 and vertical response.

r∗v,n interval C1 C2

[0.95,1] −50.9× 104ζ3 + 37.2× 103ζ2 0.95[1,1.05] −1170ζ + 5.05 1.05[1.95,2] −254.5× 103ζ3 + 18.6× 103ζ2 1.95[2,2.05] −585ζ + 2.525 2.05[i− 0.05, i] −169.7× 103ζ3 + 12.4× 103ζ2 i− 0.05[i, i+ 0.05] −390ζ + 1.683 i+ 0.05

Table 5.4: Coefficients of φd for Equation 5.7.7 and lateral response, where j = 2i− 1 andi is a natural number.

r∗l,n interval C1 C2

[0.95,1] −147× 104ζ3 + 77.6× 103ζ2 0.95[1,1.05] −1540ζ + 6.29 1.05[j − 0.05, j] −49× 104ζ3 + 25.87× 103ζ2 j − 0.05[j, j + 0.05] −513.3ζ + 2.097 j + 0.05

5.7.6 Factor related to the structural mass (φsm)

The parameter φsm is related to the number of spans and modal masses. The basic

accelerations have been obtained for simply supported beams, where φsm = 1. For two

and three span beams, this factor can be obtained from Figure 5.3.

147


5.8 Vertical and lateral accelerations caused by groups of pedes-

trians and continuous streams of pedestrians

The issue of estimating the acceleration demands from groups or streams of pedes-

trians is significantly more complex than that of estimating the demands from a single

pedestrian. When considering a single pedestrian, it is possible, and meaningful, to allo-

cate the properties of that pedestrian so that they reflect the population being considered.

However, results of Sections 4.3 and 4.4 have shown that it is important to account for

the variability generated by inter-subject differences and collective behaviour. But the

procedure to include such considerations are far more involved that what is desirable for

preliminary design. For that reason, in what follows in this chapter, a simple approach

for obtaining first-order estimates of the demands from groups and streams is proposed.

5.8.1 Group of pedestrians

Under the assumption that all pedestrians within a group are identical, walk in a

synchronised manner, and apply their load to the same point on the structure, then the

vertical accelerations induced by this group will differ from those of a single pedestrian by

a linear factor equal to the group size. That is, the maximum vertical acceleration would

be defined as in Equation 5.8.1.

av,g = Nav (5.8.1)

It should be noted that when evaluating Equation 5.8.1 the acceleration associated

with the single pedestrian av should be computed using a value of ρ∗ that accounts for the

increased mass of pedestrians associated with the group. That is, the effect of considering

the group is not simply a linear scaling of the accelerations, but also accounts for a small

shift in the resonance parameter.

Despite the fact that the assumptions underpinning the above equation are often

violated, it is proposed to use this very simple expression during preliminary design.

Once one considers lateral accelerations it is important to also consider the influence

that pedestrian-structure interaction can have upon the responses from a group of pedes-

trians. For that reason, the expression proposed for evaluating the accelerations in the

lateral direction is given by Equation 5.8.2 in which an additional term Nnl, which ac-

counts for these nonlinear interaction effects, is introduced. In Equation 5.8.2, the lateral

acceleration al should be computed accounting for the mass of the entire group (that is,

the group mass should influence ρ∗ and φpm).

al,g = (N +Nnl)al = (N +β

φpm

)al (5.8.2)

For a single pedestrian, the lateral force that they impart upon the bridge depends

upon their own relative lateral acceleration with respect to the bridge as well as the global

acceleration of the bridge. For this single pedestrian case, these global accelerations are

the sole result of this same pedestrian. However, when a group is considered, the global

accelerations of the bridge, which affect the lateral forces introduced by each pedestrian,

148


0

2

4

6

8

10

x = 10 20 30 40 50 N pm

1

2

Figure 5.4: Amplification factor β for lateral response, x = Nφpm.

result from the effects of all pedestrians. Therefore the effects of this interaction do not

scale linearly with the group size.

The vertical accelerations induced by a group of N pedestrians are equivalent to those

produced by one pedestrian multiplied by a factor of N, where N is the number of pedes-

trians in the group. However, due to the nonlinear effects explained before, the lateral

accelerations induced by a group of N pedestrians are larger than those produced by

one pedestrian multiplied by the aforementioned factor N. Therefore, in order to account

for this nonlinear interaction within a simple expression appropriate for preliminary de-

sign, an exercise was conducted in which the lateral accelerations were obtained using

a ‘scaled pedestrian’ (the pedestrian mass is set to be N times larger than the nominal

single-pedestrian value, and the other assumptions regarding group behaviour used for

the vertical case are retained). The accelerations obtained from this ‘scaled pedestrian’

are then compared with those found from an individual pedestrian in order to define an

appropriate equivalent number of pedestrians (N + Nnl) that account for the nonlinear

interaction effects.

The term Nnl is represented by the ratio β/φpm in order to enable the nonlinear

interaction effect to be estimated for cases where the value adopted for φpm differs from

that used to derive the values of Nnl. These nonlinear effects have only been found to be

significant in the case that the resonance parameter rl,1 falls within two limited ranges of

[0.95, 1.05] and [2.9, 3.1] or the total linear response is larger than 0.10 m/s2. In any of

these cases, the factor β is found to be a function of the product of the group size and φpm.

For each of the two intervals of the resonance parameter an expression is developed for

β, as shown in Figure 5.4, with β = β1, defined in Equation 5.8.3, corresponding to case

that 0.95 ≤ rl,1 ≤ 1.05 and β = β2 being relevant for 2.9 ≤ rl,1 ≤ 3.1 and being defined in

Equation 5.8.4. When linear accelerations are beyond 0.10 m/s2 β = β2. For any other

value of the resonance parameter, response is assumed to be linearly proportional to the

group size (β = 0).

β1 = 1.5× 10−4x3 − 4.5× 10−3x2 + 6× 10−2x− 0.15 (5.8.3)

β2 = 5.0× 10−4x2 − 3.0× 10−3x+ 7× 10−3 (5.8.4)

149


5.8.2 Continuous streams of pedestrians

In order to enable designers to obtain estimates of the response of girders under streams

of pedestrians the β factors derived as part of this study is coupled with an existing

approach advocated by Setra (2006) and NA to BS EN 1991-2:2003 (BSI, 2008). The

accuracy or validity of this approach was not assessed within the present study and a

more elaborate approach to estimating the response due to streams of pedestrians is

rather provided by the consideration of the model described in Section 3.2.

The maximum vertical (av,s) and lateral (al,s) accelerations of a stream of pedestrians

can be obtained as follows:

av,s = Neq ·av0.6

(5.8.5)

al,s = (Neq +Nnl)al0.6

= (Neq +β

φpm

)al0.6

(5.8.6)

whereNeq is a number of “equivalent” pedestrians and can be adopted from Setra guideline

(2006) or BSI (2008). The guideline defines this equivalence according to the density:

for sparse or dense crowds (ped/m2 ≤ 0.8) Neq = 10.8√ζN and for very dense crowds

(ped/m2 > 0.8) Neq = 1.85√N , where ζ is the damping ratio and N is the total number

of pedestrians on the structure simultaneously. The factor 1/0.6 is introduced to account

for the fact that the flow is continuous as opposed to the event of a single pedestrian, and

is taken from Grundmann et al. (1993).

The magnitude of the nonlinear factor β for lateral loads depends on the the equivalent

number of pedestrians in a stream and is given again by Figure 5.4, with x = Neqφpm.

It is worth highlighting that the load model considered for lateral loads is capable of

reproducing the initial interaction that occurs when pedestrians sense a slight movement

of the platform, but not an actual change of gait to adapt themselves better to the

movement (named synchronisation, which would be a second phase of the interaction

that some researchers point out that takes place between pedestrians and a platform,

although there are still discrepancies between researchers about this phenomenon).

5.9 Verification of the serviceability design appraisal

5.9.1 Comparison of the methodology against FEM models

In this section, accelerations predicted using the detailed numerical analysis and those

predicted by the simplified method just presented are compared. Generally speaking, an

excellent agreement is found, as seen in what follows.

Figure 5.5 compares results obtained numerically and following the methodology to

assess vertical responses of structures with two spans (L+ 0.8L) crossed by a pedestrian

walking at 2.16 Hz and a step length of 0.65 m (the pedestrian weight is 700 N) on

structures with a T-slab (S.2 in Figure 5.1) section with a depth-to-span ratio dh/L = 1/35

and a damping ratio of ζ = 0.005 (i.e, for a set of parameters different to those considered

in the reference case). The proximity of the simplified methodology results to those

150


0.250

0.200

0.150

0.100

0.050

0.00025.0 30.0 35.0 40.0 Lmain span

[m]

Max. Vertical acceleration [ m/s2]

Numerical evaluation

Methodology evaluation

20.0

Figure 5.5: Comparison of vertical response of two-span bridges, L+ 0.8L.

obtained numerically suggests that the accuracy of the methodology is very good given

its computational simplicity (maximum differences near 10% at non-resonant cases).

Figure 5.6 compares the maximum lateral accelerations obtained for simply supported

structures calculated using the detailed numerical procedure and the simplified methodol-

ogy presented in previous sections of this chapter. The structures considered have a com-

posite box girder transverse section (S.4 in Figure 5.1) with dh/L = 1/35 and ζ = 0.003,

whereas pedestrians were assumed to walk at a step frequency fp = 2.16 Hz and with a

step length of 0.65 m. The results shown in Figure 5.6 again suggest that the simplified

methodology proposed herein predicts the response with a very good degree of accuracy

(maximum differences of 25% at non-resonant results, where absolute lateral accelerations

are very small).

0.025

0.020

0.015

0.010

0.005

0.00030.0 35.0 45.0 50.0 55.0 Lspan

[m]

Max. Lateral acceleration [ m/s2]

Numerical evaluation

Methodology evaluation

40.025.0

Figure 5.6: Comparison of lateral response of simply supported bridges.

Figures 5.5 and 5.6 show that the accelerations registered in the deck could be very

sensitive to the span length. A sensitivity analysis should be included at the design stage

in order to consider the uncertainty of this important parameter.

5.9.2 Comparison of the methodology against real responses

In this second section the efficiency of the proposed methodology is assessed by com-

paring movements caused by pedestrians at real footbridges (experimentally recorded)

with those predicted by the method. The similarity of the results highlights the compe-

tence of the method despite its simplicity.

151


5.9.2.1 Footbridge over Hringbraut, (Reykjavik, Iceland)

The first structure whose responses are compared in this section is a footbridge in

Reykjavik consisting of a postensioned girder with 8 spans of longitudes between 15.4 and

27.1 m. The transverse section is a slab with a width of 3.2 m (further details can be

found in Gudmundsson et al., 2008).

Experimental measurements showed that the structure has a vertical frequency of 2.34

Hz and a damping ratio of value 0.006. Tests of the concrete mix used demonstrated that

the concrete Young’s modulus had a value of 45 GPa instead of 33.5 GPa required in the

project.

The proposed method is implemented considering only the three central longest spans

with lengths of 15.5+27.1+15.5 m (in reality these have lengths of 20.6+27.1+23.6 m).

The length of the side spans results from an average value of the individual real lengths

multiplied by 0.7 (20.6+23.62

· 0.7). This factor 0.7 is applied to take into account that these

have side spans that restrict their vibration (see Figure 5.3). With these equivalent side

spans, the three span footbridge to be assessed through the methodology presented before

would be expected to have a dynamic behaviour similar to the real footbridge.

This span disposition, the theoretical Young’s modulus (the one considered in the

project) and an additional mass for the balustrades, yields a vertical mode frequency of

2.06 Hz and if the real Young’s modulus is considered the vertical frequency predicted is

2.38 Hz.

A pedestrian of 800 N of weight walking on the bridge with vertical frequency 2.06

Hz at a step frequency of 2.06 Hz generates a peak vertical response of 0.37 m/s2 and

two pedestrians of the same weight 0.74 m/s2. At the real structure, single pedestrians

walking at a constant step frequency near resonance generated peak responses near 0.38

m/s2 (except one case where peaks of 0.46 m/s2) whereas two pedestrians caused peak

accelerations of 0.71 m/s2. The values measured in real tests and those predicted by this

method are almost the same, with differences below 5%.

5.9.2.2 Aldeas footbridge, (Gouveia, Portugal)

The second footbridge is a steel box girder of variable depth and deck width of 1.5 m

that spans a total distance of 57.8 m through three spans of longitude 17.7+30.0+10.1 m

(further details can be found in Alves et al., 2008).

Designers of the bridge predicted numerically vertical modes at frequencies 3.13 and

4.50 Hz, however dynamic tests at the structure once it was finished described vertical

modes at frequencies 3.68 and 5.16 Hz.

For this structure, the proposed method is implemented considering that both side

spans have the same length (an average of the actual dimensions of these two spans which

are very similar to the actual length). This span disposition (together with the masses

of the deck surfacing, the handrails and protection panels) describes a vertical mode at

frequency 3.64 Hz. If the real span arrangement had been considered, the magnitude of

this frequency would not have changed.

According to the proposed model, a pedestrian of 700 N of weight walking on the

bridge at a step frequency of 1.80 Hz generates peak accelerations of 0.39 m/s2 if the

152


damping ratio considered is 0.006 (as pointed out in Section 2.4) or 0.25 m/s2 if the

magnitude of the damping ratio is 0.012 (introducing the effects of the large protection

panels placed on this bridge that crosses over a highway). Real dynamic tests reflected

that the damping ratio of the first vertical modes is moderately larger than 0.012 and

that movements caused by a single pedestrian were below 0.30 m/s2 (very similar to

those predicted by the model with the second damping ratio).

5.10 Evaluation of the serviceability performance in conven-

tional footbridges

The new methodology presented herein constitutes a tool for evaluating the adequacy

of the different design options usually considered in practice. The purpose of this section

is to compare the different design options available for single span bridges with structural

characteristics and materials listed in Figure 5.1 and damping ratios of ζ = 0.005. The

evaluation is performed considering the characteristics of a pedestrian stream of com-

muters or at leisure (mp = 71.36 kg) with a density of 0.6 ped/m2 (the step frequency for

the stream is the mean value provided in Figure 3.4). For these analyses, it is considered

that all sections have additional deck finishings and balustrades.

Figure 5.7: Evaluation of serviceability of simply supported structures in the vertical andlateral directions under pedestrian streams of density 0.6 ped/m2 with commuting or leisureaim of the journey. Section S.6 has non-structural concrete deck and sections S.7 to S.9 havenon-structural wooden decks.

For a preliminary evaluation of the adequacy of the response in service, it can be

considered that vertical accelerations in the range between 0.5 and 1.0 m/s2 define the

comfort limit, whereas, for lateral accelerations, responses in the range between 0.2 and

0.4 m/s2 represent the limit of serviceable situations (further ranges can be found in codes

and guidelines such as Setra detailed in Section 2.5).

153


The results shown in Figure 5.7 illustrate how the response of certain structures under

the action of these pedestrian streams are large, and do not satisfy serviceability criteria for

certain span lengths. However, it should be appreciated that although the values presented

in Figure 5.7 are often very large, in reality accelerations beyond 2 m/s2, vertically, or 0.4

m/s2, laterally, will probably not be developed due to a change of behaviour of pedestrians

after sensing large vibrations (by either changing the characteristics of their walking, or

even stopping).

It is also clear from Figure 5.7 that there is a very strong dependence upon both the

span length just mentioned but also upon the aim of the journey. Given the strong sen-

sitivity to these parameters, and the simplified nature of the approach proposed herein,

it is clear that any design decisions, even at preliminary stage, should account for uncer-

tainties in these parameters. The uncertainty associated with the assumed damping ratio

should also be considered.

Inspection of Figure 5.7 also suggests that for this level of traffic, sections S.7 and

S.9 have poor performance irrespective of span length. This suggests that for these types

of sections it will be difficult to satisfy serviceability requirements using the slenderness

adopted here and for similar traffics (which have been obtained from footbridges described

in codes and guidelines such as Setra as well as those described in the proceedings of the

Footbridge Conference, e.g., Debell et al., 2014).


The chapter presents a methodology for the serviceability evaluation of beam-type

structures subjected to pedestrian loads. The steps of the procedure together with the

fundamental underlying assumptions are outlined for both vertical and lateral response

caused by these loads. An advantage compared to current proposals is that the method

presented herein does not require the use of any elaborate pieces of software or analy-

sis techniques, yet is still able to provide reliable evaluations that are indispensable for

designers during the early stages of the design. Based on the methodological procedure,

as well as the inherent assumptions and simplifications included for its development, it

should be highlighted that:

a) An adequate evaluation of the response includes a comprehensive description of pedes-

trian loads and structural properties. Despite the fact that the method presented herein

intends to simplify the assessment of pedestrian-induced vibrations, it still includes a

refined evaluation of loads (both load magnitude and step frequencies). Regarding the

structural properties, the method proposes evaluating the dynamic properties through

a small number of parameters that are easy to appraise even during the early stages

of design.

b) The prediction of the lateral response attempts to include the interaction phenomenon

that has been detected for pedestrian loading scenarios through a parameter that

reflects a certain response nonlinearity. This parameter shows that under certain

circumstances the nonlinear response in the lateral direction can be larger than that

154


obtained considering only linear results. However, it should be highlighted that this

model is only able to reproduce what seems to be a first stage of what might happen

in reality (it is not reflecting a change of step frequency according to the response

generated, or synchronisation). This phenomenon could be further investigated for its

inclusion in the assessment of lateral structural response. Although having said that,

it is the opinion of the author that serviceability is unlikely to be satisfied when it

becomes necessary to model these types of phenomena.

c) The comparison of the response for a set of structures calculated both using an ad-

vanced numerical procedure and the proposed simplified approach shows that the

methodology is an adequate tool for the evaluation of vertical and lateral structural

response for preliminary design.

d) The responses predicted by the method of two real structures in service show that the

method is reliable despite its inherent simplifications. A good response evaluation in

service depends more on well founded values of the non-structural mass or damping

ratio of the structure than on a very accurate definition of span lengths.

e) The estimates of the accelerations are extremely sensitive to the structural frequencies

(and therefore to the number of spans) and the type of traffic loading scenario. For

conventional sections commonly considered within design, once values of the span

length and pedestrian loading are defined, the method clearly identifies span ranges

that should be avoided.

f) In summary, it is important to recognise that the simplicity of the proposed approach

allows a designer to obtain estimates of accelerations with very little computational

effort. The proposed method therefore lends itself to undertaking analyses in which

the sensitivity of the results to various design assumptions is quantified. The strong

sensitivity of the results shown in this chapter implies that such sensitivity analyses

are an indispensable component of the design process.

155

Chapter 6Evaluation of the response of a

conventional cable-stayed footbridge

under serviceability conditions

6.1 Introduction

Cable-stayed footbridges are structures that are prone to vibrate under the passage of

pedestrian traffic flows due to their low masses and damping ratios. The detailed study

of the behaviour of cable-stayed footbridges that are representative of this structural type

can be used to gain understanding about their structural performance under pedestrian

loading and to draw conclusions related to their structural behaviour and design criteria.

In addition, such a study would be extremely useful for design purposes. Therefore, the

objective of this chapter is to investigate the structural behaviour of a particular cable-

stayed bridge (with variations that are representative of built bridges with this structural

type) under pedestrian loading. The magnitudes of the selected bridge have been defined

on the basis of a detailed survey conducted for existing cable-stayed footbridges worldwide

(see Chapters 2 and 3).

The detailed analysis of the structural behaviour under pedestrian loading includes

the assessment of vibrations, deflections, and stresses in the deck, the pylons and the stay

cables. In addition to the load model proposed in this thesis, other existing methodologies

are also considered for comparative purposes.

Apart from gaining understanding about the structural behaviour of cable-stayed

bridges under pedestrian loading, the different parameters and scenarios that govern the

design of the different structural members are identified. The study presented in this

chapter is complemented by the following Chapters 7 and 8 where the main parameters

that define the typology are varied within a parametric analyses in order to gain a better

understanding about the structural behaviour under pedestrian loading and to be able to

define a set of design criteria that are applicable to the entire bridge typology.

Thus, Sections 6.2 to 6.4 describe the geometry and dynamic properties of the reference

157

6. Conventional cable-stayed footbridge

cable-stayed footbridges and the main characteristics of pedestrian loads. Sections 6.5 to

6.7 evaluate the vibration performance of these footbridges in service, according to the

proposed or existing load models. Section 6.8 characterises the magnitude of the deck

dynamic deflections during these events and Sections 6.9 to 6.11 describe the magnitudes

of the internal forces of the deck, tower and cables during these serviceability events.

6.2 Geometric characteristics of the footbridge representative

of the cable-stayed bridge typology

The geometric characteristics of conventional cable-stayed footbridges, CSFs, (detailed

in Section 3.5.2) are extracted from the parameters observed in real structures documented

in Section 2.2. According to these, a conventional, or stereotypical, cable-stayed footbridge

has a main span Lm of length 50 m and a side span Ls of length approximately 0.2 Lm

(see Figure 6.1). The main characteristics of each structural element are:

• The cables (parallel strand stays) that support the deck are arranged in two planes

following a fan configuration. The anchorages of these cables at the pylon are ex-

tended over a reduced length to avoid concentration and at the deck are separated

7.0 m (distance that is not related to the main span length).

• The pylon, with a vertical mono-pole configuration, has a transverse section that

consists of a steel circular hollow section.

• Due to the number of planes of cables, the deck transverse section consists of a con-

crete slab depth of 0.20 m supported by two longitudinal steel girders and transverse

steel girders located at the support section on the abutments, on the pylon and also

at sections where the stay cables are anchored on the deck.

The articulation of the deck consists of two Laminated Elastomeric Bearings (LEBs),

which provide vertical support, and a shear key at each of the two abutments. At the

pylon, the deck is simply supported in the vertical direction. Arguments supporting the

use of these support conditions are based on the performance in the lateral direction of the

footbridge (further details are exposed in the Chapter 7). The LEBs, with dimensions of

200 x 200 x 32 mm, are represented numerically by linear springs in both the longitudinal

and transverse directions whereas the shear keys block transverse movements of the deck at

the abutments (see Figure 6.1, where the variables are as those defined in the Chapter 3).

6.3 Fundamental dynamic characteristics of the footbridge rep-

resentative of the cable-stayed bridge typology

Design guidelines emphasise that the sensitivity of footbridges to dynamic pedestrian

loads is characterised by the coincidence of footbridge vibrational frequencies to frequency

ranges of individual pedestrian step loads. Experience (see Section 2.6) has shown that the

existence in the vertical direction of structural modal frequencies between 1.0 and 3.0 Hz is

related to the possible development of notable vertical movements caused by pedestrians.

158


Detail B-B:

0.5

0.50.20

w = 4.0

HEB 200

hgird =

0.3

tflange,bot = 0.025

Sec. A-A:

hto

t = 0

.5

tflange,top = 0.025

tweb = 0.0125

Cable No. Tendons Cable No. TendonsBS

CB#1CB#2

CB#3CB#4CB#5

3422

332

==

Lm= 50.0Ls= 10.0

Dp Dc Dp Dc Dc Dc

HT=

25.5

Hs =

18.0

Hi =

7.5

Ha= 2.0

Detail B-BDext = 0.60

BS#1

#2#3

#4#5

Sec. A-A

LEB

LEB

LEB

LEB

Shear Key

Shear Key

y

xz

(a) Geometry

(b) Deck articulation (deck plan)

(1)

(2)

(3)

(4)

(5)

(6)

(7) Region Restrictions(1)-(4)(5)-(6)

(7)

Kx, Ky, UzUy

Ux, Ux, Uz(*)

(*): Pylon movements

Restricted movements

at supports:

Cable characteristics:

Figure 6.1: (a) Geometric and structural characteristics of the conventional cable-stayedfootbridge; (b) articulation of the footbridge deck (movements restricted by supports).

In the lateral direction, this correlation between frequencies and large responses is not as

straightforward as in the vertical direction. Bridges with lateral frequencies well below

1.0 Hz have registered large lateral movements (Section 2.6) and consequently guidelines

and codes highlight the sensitivity to pedestrian actions of footbridges with lateral modes

below 1.1 Hz or 1.5 Hz (according to NA to BS EN 1991-2:2003), which are the typical

upper bounds of the lateral frequency ranges induced by pedestrians walking. Accordingly,

it is of utmost importance to assess the magnitude and characteristics of the vibration

modes of conventional CSFs.

According to the observations found in the relevant literature (Section 2.3.3), the mass

of pedestrians in a traffic flow is included at the deck of the structure to assess the vibration

characteristics. Table 6.1 describes the first vibration modes of the CSF when it is empty

or being used by pedestrian flows of different densities. The mode shapes associated with

the fundamental frequencies presented in Table 6.1 are shown in Figure 6.2.

The modes described in Table 6.1 highlight the fact that CSFs with composite decks

and moderate span lengths (where moderate span lengths correspond to lengths near 50 m)

present vertical modes with frequencies that are within the range considered critical for

pedestrian loading, whereas in the lateral direction or for torsional movements, frequencies

of these structures are beyond the commonly assumed critical ranges.

The effect of the traffic flow mass on the magnitude of the vibration frequencies is

relatively modest (irrespective of the vibration direction considered), i.e., the vibration of

the CSF when it is empty or loaded with a heavy flow of 1.0 ped/m2 takes place at very

159


Table 6.1: Vibration modes and frequencies (Hz) of the CSF, where ‘VN’, ‘TN’ and ‘LN’denote vertical, torsional and lateral modes with N half-waves in the main correspondingstructural span (i.e., from the pylon to the abutment support section, for vertical and tor-sional modes; and between abutment support sections, for lateral modes), ‘Ld’ longitudinaland ‘P’ pylon modes.

Mode No. Empty 0.2 ped/m2 0.6 ped/m2 1.0 ped/m2 Description

1 1.01 1.00 0.98 0.96 V12 1.14 1.13 1.11 1.09 Ld.3 1.21 1.20 1.20 1.20 P4 2.02 2.00 1.96 1.92 V25 2.23 2.21 2.16 2.12 L16 2.98 2.96 2.93 2.89 T17 3.28 3.25 3.18 3.12 V38 3.82 3.79 3.73 3.67 T29 5.08 5.03 4.92 4.83 V410 5.57 5.52 5.44 5.36 T311 7.02 7.01 6.99 6.96 P12 7.44 7.36 7.20 7.06 V5

similar frequencies (maximum difference between modes with a heavy flow and those of

the empty CSF are 5% for the first 15 modes).

6.4 Characteristics of Pedestrian Traffic

Design guidelines highlight that the sensitivity of a footbridge to the dynamic effects

of pedestrian loads is influenced by the step frequencies adopted by pedestrians when

walking on the structure. Statements to this effect can be found in guidelines and codes

(e.g., NA to BS EN 1991-2:2003) as well as in proposals described in Section 2.3.

The pedestrian traffic crossing the conventional CSF correspond to leisure and com-

muter pedestrian flows with densities of 0.2, 0.6 and 1.0 ped/m2 (as described in Section

3.2.5). These scenarios correspond to service conditions in which pedestrians enter the

main span of the structure and walk along it over the time, with an average number of

pedestrians of 34, 102 and 170 for each density, respectively. The total number of pedestri-

ans in the main span remains fairly constant throughout time (the random arrival process

that forms part of the load model dictates that the exact number at any given time can

vary slightly from these average numbers) except for the first moments of the simulation,

when the first pedestrians start crossing the bridge.

The step frequencies selected by pedestrians of each flow class when crossing the foot-

bridge are described by the distributions shown in Figures 6.3 and 6.4. The step fre-

quencies depicted in these distributions correspond to the mean step frequencies that

each pedestrian aims to adopt while crossing the bridge although they may be modified

due to the influence of other pedestrians (caused by collective behaviour, as highlighted

in Section 4.4). The comparison of these distributions (that represent particular traffic

events) to those proposed in Figure 3.4 of Section 3.2.2 (that describes typical events with

pedestrians belonging to Western Europe countries) pinpoints that these traffic flows are

representative of the events they intend to simulate.

160


Mode 1: V1

Mode 5: L1

Mode 9: V4

Mode 13: L2

Mode 2: Ld

Mode 6: T1

Mode 10: T3

Mode 14: T4

Mode 3: P

Mode 7: V3

Mode 11: P

Mode 15: V5

Mode 4: V2

Mode 8: T2

Mode 12: V5

Mode 16: L2

Figure 6.2: Modal shapes of the first 16 modes of the conventional cable-stayed footbridge.

Contrasting these distributions of step frequencies to those ranges emphasised by Setra

(2006) (step frequencies between 1.7 and 2.1 Hz) or by the British code (BSI, 2008)

(frequencies between 1.8 and 2.0 Hz) one can see that, if these initial step frequencies

determine the magnitude of the accelerations developed in a footbridge, scenarios caused

by light commuting traffic or heavy leisure flows, where the average values are outside

these proposed ranges, may not be adequately predicted by these codes or guidelines.

6.5 Response in service of the CSF

In order to predict the serviceability response of footbridges, it is of utmost importance

to predict the movements (accelerations, as argued in Section 2.5) triggered by pedestrians

while crossing the structure and compare these to the limits that represent the comfort

levels for those pedestrian crossing the footbridge and also for those potential users that

may stand stationary on the footbridge.

The following sections describe the magnitudes of the CSF deck movements predicted

by the proposed load model (Section 3.2) and caused by different classes of pedestrian

traffic (leisure and commuter flows with characteristics detailed in the previous section).

As well, they evaluate the corresponding accelerations felt by the users that are simulated

in the model walking along the bridge (see Section 3.7.1) and compare these to those

recorded at different locations of the deck (that are susceptible of being felt by potential

users of the footbridge who would be staying at those locations rather than crossing the

161


1.5 2 2.50

50

Step frequencies [Hz]

0.2 ped/m2

Num

ber

of

pedestr

ians

1.5 2 2.50

50

100

150


0.6 ped/m2

1.5 2 2.50

100

200

300


1.0 ped/m2

Figure 6.3: Distributions of step frequencies adopted by commuters in flows of 0.2, 0.6or 1.0 ped/m2 (blue bars correspond to simulated events and red lines to the predictionaccording to Figure 3.4).

1 1.5 20

100

200

300


1.0 ped/m2

1 1.5 2 2.50

50

100

150


0.6 ped/m2

1.5 2 2.50

50


0.2 ped/m2

Num

ber

of

pedestr

ians

Figure 6.4: Distributions of step frequencies adopted by leisure pedestrians in flows of 0.2,0.6 or 1.0 ped/m2 (blue bars correspond to simulated events and red lines to the predictionaccording to Figure 3.4).

bridge). Finally, further sections illustrate the movements of the same structure estimated

by other currently available models and compare all these predictions to comfort limits

detailed in Section 3.4.

6.5.1 Structural accelerations predicted by the proposed load model

In the vertical direction, commuter and leisure pedestrian flows simulated with the

proposed load model generate the accelerations described in Figure 6.5 (peak and 1.0

second Root Mean Squared, 1s-RMS, accelerations). From these it can be inferred that:

a) The maximum responses occur at approximately x = 28.0 m and x = 50.0 m, coincid-

ing with the antinodes of the vertical mode V2. The analysis in the frequency domain

of the time-history of accelerations at x = 28.0 m, see Figure 6.6, corroborates the

large participation of this vertical mode V2. This figure represents the Fourier spectra

of the time accelerations recorded at that section and highlights the importance of

modes with frequencies near 2.0 Hz.

b) In the main span, the magnitude of the vertical accelerations increases with the number

of pedestrians in the flow.

c) For commuter traffic flows, the accelerations grow linearly with the pedestrian density,

with an average increment of the peak vertical acceleration of 7.5% (in comparison to

the results of the lightest flow) for each additional 0.1 ped/m2 within the flow. For

leisure traffic flows this increment is nonlinear as there is an increment of the peak

accelerations of 12.5% for every additional 0.1 ped/m2 within the flow for densities

162


0 10 20 30 40 50 600

0.51

1.52

2.50.2 ped/m

2 − Commuter

0 10 20 30 40 50 600

0.51

1.52

2.50.2 ped/m

2 − Leisure

0 10 20 30 40 50 600

0.51

1.52

2.5

Vert

ical accele

ration [

m/s

2]

0.6 ped/m2 − Commuter

0 10 20 30 40 50 600

0.51

1.52

2.50.6 ped/m

2 − Leisure

0 10 20 30 40 50 600

0.51

1.52

2.51.0 ped/m

2 − Commuter

0 10 20 30 40 50 600

0.51

1.52

2.5

Structure length [m]

1.0 ped/m2 − Leisure

0.600.730.34

0.901.06

0.51 0.43

0.851.03

1.20 1.00

1.26

2.07

1.16

0.240.81 0.73

1.62 1.44

0.40

1.96

1.080.37

1.722.15

0.54

0.61

1.70 1.53

1.00

apeak

a1s−RMS

Figure 6.5: Peak and 1s-RMS vertical accelerations recorded at the CSF deck generatedby commuter or leisure flows of 0.2, 0.6 or 1.0 ped/m2. The origin for the abscissa axis islocated at the support section of the side span on the abutment, see Figure 6.1(a).

between 0.2 and 0.6 ped/m2, and an increment of these peak accelerations of 7% for

each additional 0.1 ped/m2 of leisure pedestrians between 0.6 and 1.0 ped/m2.

1 2 3 4 5 6 7 8 9 10

100

Am

plit

ude


1 2 3 4 5 6 7 8 9 10

100

Frequency [Hz]

Am

plit

ude


Figure 6.6: Fourier amplitudes [m/s] of the vertical acceleration response of the CSF at x= 28.0 m under the action of commuter or leisure flows with 0.6 ped/m2.

d) The effects of a pedestrian of the lightest flow are considerably larger than those

of a pedestrian in the medium-density or heavy flow (where effects can be quantified

dividing the acceleration by the number of pedestrians in the flow). These larger effects

are not explained by the different modal masses in each case (effect described by the

factor φsm for girder bridges of Section 5.7.6) but rather by the total loads introduced by

pedestrian flows in each case. The flows with a small number of pedestrians introduce

a total load on the bridge frequency content that resembles that of a single pedestrian

(and approximate amplitude corresponding to that of a pedestrian times the number

163


of pedestrians), whereas flows with a medium or large number of users introduce a

total load with a wider frequency content due to the superposition of the loads of each

user (instead of larger amplitudes at the usual step frequencies of a single pedestrian).

e) In the main span, small density pedestrian flows generate different movement mag-

nitudes according to the aim of the journey (with larger accelerations for commuter

than for leisure). However, pedestrian flows of moderate and large densities cause

peak and 1s-RMS vertical accelerations that are very similar, irrespective of the aim

of the journey. Hence the aim of the journey seems to be important for small traffic

densities exclusively. This is a consequence of the frequency content of the total loads

introduced simultaneously to the footbridge by a large number of pedestrians where, as

stated at the previous point, this larger sample is characterised by a greater dispersion

in frequencies.

The lateral movements caused by commuting or leisure pedestrian flows simulated with

the proposed load model are described in Figure 6.7. Similarly to vertical movements,

these are represented through peak and 1s-RMS accelerations. From these movements it

can be concluded that:

0 10 20 30 40 50 600

0.10.20.30.40.5


0 10 20 30 40 50 600

0.10.20.30.40.5


0 10 20 30 40 50 600

0.10.20.30.40.5

Late

ral accele

ration [

m/s

2]


0 10 20 30 40 50 600

0.10.20.30.40.5


0 10 20 30 40 50 600

0.10.20.30.40.5



0 10 20 30 40 50 600

0.10.20.30.40.5


0.11

0.31

0.20

0.46

0.31

0.40

0.27

0.18

0.13

0.10

0.07

0.16

apeak

a1s−RMS

Figure 6.7: Peak and 1s-RMS lateral accelerations recorded at the CSF deck generatedby commuter or leisure flows of 0.2, 0.6 or 1.0 ped/m2. The origin for the abscissa axis islocated at the support section of the side span on the abutment.

a) In the lateral direction, the maximum responses are located at approximately x =

30.0 m. This deck region corresponds to the antinode of a lateral mode with a single

half-wave (L1). The characterisation of the lateral acceleration at this deck section in

the frequency domain shows a large contribution of the lateral mode L1, with frequency

2.15 Hz approximately (see Figure 6.8).

164


1 2 3 4 5 6 7 8 9 10

100

Am

plit

ude


1 2 3 4 5 6 7 8 9 10

100

Frequency [Hz]

Am

plit

ude


Figure 6.8: Fourier amplitudes [m/s] of the lateral acceleration response of the CSF at x= 33.0m under the action of commuter or leisure flows with 0.6 ped/m2.

b) The amplitudes of the lateral accelerations grow with the flow density for both com-

muter and leisure flows.

c) At commuter scenarios, the accelerations grow linearly with the number of pedestrians

on the footbridge, with an average increment of the peak lateral accelerations of 23.5%

(in comparison to the results of the lightest flow) for every additional 0.1 ped/m2 at

the traffic (between 0.2 and 1.0 ped/m2). At leisure events, peak lateral accelerations

do not grow linearly with the number of walking pedestrians, as these increase 20%

with every additional 0.1 ped/m2 for flows between 0.2 and 0.6 ped/m2 and 55% with

every additional 0.1 ped/m2 for traffic flows between 0.6 and 1.0 ped/m2.

d) The larger lateral movements at the most crowded event with leisure pedestrians is

explained by the higher number of walking pedestrians that adopt a step frequency

fp near 1.5 Hz in comparison to other traffic flows. With this vertical step frequency,

the lateral step frequency fp,l is approximately 1/3 times the first lateral frequency L1

(fL1 = 2.12 Hz). At the other events walking pedestrians have a higher step frequency

thus this coincidence does not occur.

e) The effects caused by commuter pedestrians are greater than those of users at leisure.

This is produced by the slightly larger contribution of mode T1 to the lateral move-

ments in commuter events (see shape of mode T1 in Figure 6.2, fT1 ≈ 2.96 Hz). This

mode is excited by pedestrians walking with lateral step frequencies near 1.0 Hz (equiv-

alent to fp = 2.0 Hz), number of pedestrians that is higher in commuter flows than

leisure events.

6.5.2 Accelerations felt by users predicted by the proposed load model

Alternatively to the movements recorded at multiple sections of the bridge deck (that

could potentially be felt by any stationary user who has stopped on a particular location

for a period of time), the response in service of the CSF is characterised by the magnitudes

of the accelerations felt by pedestrians while crossing the bridge (see Section 3.7.1).

165


Since there are a large number of pedestrians crossing the footbridge and sensing dif-

ferent magnitudes of vertical and lateral accelerations (which depend on accelerations

registered at the particular location where they place their feet at each point in time

moment as they cross the bridge), the representative magnitudes of the movements felt

by users are expressed in statistical terms. Comfort limits usually correspond to bound-

aries distinguishing movements accepted by 50% of the population (µ mean of a normal

distribution representing the movements accepted by users), e.g., as cited in Gierke et al.

(1988). Alternatively, ranges including 95% of the events are usually considered for the

design of structures (µ±2σ, where σ is the standard deviation of the normal distribution).

Hence, the minimum peak movements felt by 50%, 25% and 5% of the users provide an

accurate description of the accelerations induced in order to analyse the comfort levels.

Figure 6.9 illustrates the magnitudes of the peak vertical accelerations felt by users

while crossing the bridge. In Figure 6.9(a) these magnitudes are expressed in relation to

the largest peak acceleration felt by any pedestrian amax,P at the corresponding scenario.

The horizontal red lines show the relative maximum vertical accelerations felt by 50, 25

and 5% of the users in each traffic event.

amax,P i / amax,P

% P

edestr

ians

20%

60%

100%

80%

40%

0.25 1.00 1.25 1.75 2.25

amax,P i [m/s2]

0.50 0.75 1.50 2.00

(a)

(b)

0.2 ped/m2 C (1.15 m/s2)

0.6 ped/m2 C (1.68 m/s2)

1.0 ped/m2 C (2.12 m/s2)0.2 ped/m2 L (0.81 m/s2)

0.6 ped/m2 L (1.54 m/s2)

1.0 ped/m2 L (2.02 m/s2)

50%

25%

5%

% P

edestr

ians

20%

60%

100%

80%

40%

0.2 0.4 0.6 0.8 1.0

Figure 6.9: Relative, compared to amax,P (defined in legend for each scenario), (a) and ab-solute (b) maximum vertical accelerations felt by walking pedestrians vs cumulative numberof users that feel maximum vertical acceleration (according to type and density of flow).

Table 6.2 summarises these vertical accelerations felt by pedestrians and compares

them to the peak vertical accelerations recorded at the deck of the CSF during the same

events.

According to the results illustrated in Figure 6.9(a), for small and medium densities

166


(and irrespective of the flow type) there are similar values of the users exposed to similar

ratios of accelerations in comparison to the maximum accelerations felt by at least one

user. However, for heavy flows it is clear that the proportion of users exposed to the same

ratio of accelerations is smaller. For light and medium density flows, 50% of the users

notice a peak acceleration equal or larger than 0.65 times amax,P, 25% of the users an

acceleration equal or larger than 0.8 times amax,P and 5% of the users 0.95 times amax,P.

For heavy flows these proportions are 0.6, 0.7 and 0.8 amax,P for commuter flows and 0.5,

0.6 and 0.9 for leisure traffic flows. The differences between heavy flows and the rest are

explained by the fact that only very few users (in particular for the leisure event) of these

large flows notice accelerations as high as the peaks recorded at the deck as opposed to

the other flows, which can only be explained by the amount of time where accelerations

have values similar to the peak, which is smaller in scenarios with heavy traffic flows.

The comparison of the absolute peak accelerations felt by different groups of walking

pedestrians (Figure 6.9(b)) highlights that a large portion of the pedestrians of medium

and heavy flows notice fairly similar accelerations (slightly larger at commuting flows),

irrespective of the aim of the journey. In addition, users in light traffic flows feel consid-

erably lower accelerations (even smaller under leisure conditions).

An alternative magnitude measuring the experience of pedestrians while crossing the

structure corresponds to the average movement felt by each user. Peak magnitudes of

Figures 6.9 do not consider whether these magnitudes are felt on several occasions or

describe an instant while crossing the bridge. The basis for representing the experience

of pedestrians according to average movements instead of peaks lies upon the reduced

proportions of event duration and deck area where pedestrians feel large accelerations.

The proportions of the overall crossing time that particular levels of acceleration are

experienced for are shown in Figure 6.10 (for the flow with 0.6 ped/m2 of commuters) with

respect to the fractional length over which these levels are experienced. The maximum

average acceleration felt by users of commuter flows are 0.28, 0.41 and 0.40 m/s2 (for

flows with 0.2, 0.6 or 1.0 ped/m2 respectively) and those felt by leisure pedestrians are

0.16, 0.41 or 0.46 m/s2 (for the same pedestrian densities), while the peak accelerations

recorded at the deck in these cases are 4.5 times larger on average.

0.0 20 40 60 80

20

40

60

80

Main span length [%]

Tim

e e

ve

nt

[%]

0.2

0.40.6

1.4

1.6

0.6 ped/m2 - Commuter

1.21.0

0.8

100

100

Figure 6.10: Percentage of time and of main span surface for which the maximum accel-erations felt by users are larger than the value indicated in the contour curves.

167


Table 6.2: Maximum absolute vertical accelerations [m/s2] at the deck (Max. Deck),maximum average vertical acceleration felt by walking users (Max. Av. Ped.) and minimumpeak vertical acceleration felt by 50% (aP1), 25% (aP2) or 5% (aP3) of the users accordingto the traffic scenario.

Max. Deck

Max. Av. Ped. 0.2 ped/m2 0.6 ped/m2 1.0 ped/m2

aP1 aP2 aP3

1.16 1.70 2.15

Commuter 0.28 0.41 0.40

0.74 0.86 1.04 1.14 1.37 1.60 1.26 1.44 1.70

0.81 1.62 2.07

Leisure 0.16 0.41 0.46

0.51 0.63 0.73 1.06 1.26 1.48 0.97 1.19 1.79

Following a similar analysis for the lateral accelerations perceived by users, Figure 6.11

represents the magnitudes of the maximum lateral accelerations felt by each user while

crossing the bridge where, similarly to the vertical movements, amax,P describes the peak

acceleration felt by at least one user of the bridge.

amax,P i / amax,P

50%

25%

5%

% P

edestr

ians

20%

60%

100%

80%

40%

0.2 0.4 0.6 0.8 1.0

% P

edestr

ians

20%

60%

100%

80%

40%

0.00 0.20 0.50

amax,P i [m/s2]

0.10 0.30 0.40

(b)

0.2 ped/m2 C (0.16 m/s2)

0.6 ped/m2 C (0.31 m/s2)

1.0 ped/m2 C (0.46 m/s2)0.2 ped/m2 L (0.10 m/s2)

0.6 ped/m2 L (0.18 m/s2)

1.0 ped/m2 L (0.40 m/s2)

(a)

Figure 6.11: Relative (a) and absolute (b) maximum lateral accelerations felt by walkingpedestrians vs cumulative number of users that feel the maximum lateral acceleration.

Table 6.3 provides a summary of these lateral accelerations and a comparison to those

recorded at the deck. Figure 6.11(a) shows that the relative magnitudes of the peak

accelerations felt by 50%, 25% or 5% of the users decreases with increasing density of

168


the flow. The analysis of these proportions versus the absolute accelerations felt by

pedestrians, as represented in Figure 6.11(b), highlight the fact that the largest lateral

accelerations are only felt by a few pedestrians: at the worse scenarios less than 10-30% of

pedestrians notice accelerations larger than approximately 0.25 m/s2. This effect is due to

the small time where the footbridge develops accelerations near the peak movements, an

observation that is corroborated by the response shown in Figure 6.12 (percentage of time

and deck surface with largest accelerations than the value indicated in the contour curve).

Similarly to vertical movements, these events can be characterised as well according to the

maximum average acceleration felt by users. These correspond to 0.06, 0.09 and 0.13m/s2

for commuter flows with 0.2, 0.6 or 1.0 ped/m2 respectively, and 0.03, 0.06 and 0.14 m/s2

for leisure flows.

Tim

e e

ve

nt

[%]

0.050.10

0.250.30 0.20

0.15

0.0 20 40 60 80

20

40

60

80

Main span length [%]

0.6 ped/m2 - Commuter

100

100

Figure 6.12: Percentage of the time and of the main span surface for which the maximumlateral accelerations felt by users are larger than the value indicated in the contour curves.

Table 6.3: Maximum absolute lateral accelerations [m/s2] at the deck (Max. Deck), maxi-mum average lateral acceleration felt by walking users (Max. Av. Ped.) and minimum peaklateral acceleration felt by 50% (aP1), 25% (aP2) or 5% (aP3) of the users according to thetraffic scenario.

Max. DeckMax. Av. Ped. 0.2 ped/m2 0.6 ped/m2 1.0 ped/m2

aP1 aP2 aP3

0.16 0.31 0.46Commuter 0.06 0.09 0.13

0.09 0.12 0.15 0.17 0.20 0.29 0.17 0.23 0.43

0.10 0.18 0.40Leisure 0.03 0.06 0.14

0.07 0.08 0.10 0.12 0.15 0.18 0.23 0.31 0.38

6.6 Structural accelerations estimated by alternative proposals

Bridge designers have several alternative procedures to predict the movements caused

by pedestrian traffic flows likely to cross the footbridge. Among the latest proposals, one

can consider the current version of the British Standards (2008) or the Setra guideline

(2006). Older methods such as those proposed by Grundmann et al. (1993) or Rainer

et al. (1988) are used as well hereunder due to the simplicity of the evaluations they

propose.

The movements of the CSF deck predicted by current available methods for the same

traffic events present a large variability. This inconsistency between proposals emphasises

169


that the representation of pedestrian flows is not well captured yet, and also partly related

to the assumptions about the bridge (in those models where the CSF is equivalent to a

two spans bridge).

The British Standard does not consider the different effects of pedestrian flows with

alternative aims of the journey. The accelerations that this code predicts for low and

medium density flows are similar to those of commuter flows (see Table 6.4) whereas results

for heavy flows the predicted accelerations are slightly larger. In the lateral direction the

code does not predict movements.

Similarly to the British code, Setra does not consider the aim of the journey although

it provides a method to assess lateral movements. The vertical accelerations that Setra

approach predicts are substantially larger than those of Figure 6.5 (and similar to results

of models based on dynamic displacements, e.g., Grundmann et al., 1993, or Rainer et al.,

1988), as seen in Table 6.4, whereas in the lateral direction the magnitudes predicted are

considerably smaller than the peak or 1s-RMS accelerations obtained with the sophisti-

cated load model (assuming that this corresponds to the correct behaviour), as seen in

Table 6.5. Therefore this method is less adequate than that of the British Standard.

The method proposed by Georgakis et al. (2008) predicts vertical accelerations (with

the assumption of resonant step frequencies) that are moderately smaller in all the traffic

events. The vertical accelerations provided by extrapolating the method of Chapter 5

would describe very similar values to those of Georgakis et al. (2008) if pedestrians adopt

resonant step frequencies and the equivalent flow is simulated according to the proposal

of Setra.

Table 6.4: Comparison of the cable-stayed footbridge performance in the vertical directionestimated by alternative proposals.

Pedestrian flow [ped/m2]

Vertical response 0.2 0.6 1.0

Proposed method 1.16 (C) / 0.81 (L) 1.70 (C) / 1.62 (L) 2.15 (C) / 2.07 (L)

NA to BS EN 1991-2:2003 1.13 1.99 2.47

Setra (2006) 1.89 3.21 11.89

Georgakis et al. (2008) 0.9 (C) / 0.83 (L) 1.56 (C) / 1.44 (L) 2.01 (C) / 1.86 (L)

Method Chapter 5 0.35 (C) / 1.09 (L) 0.53 (C) / 1.65 (L) 2.38 (C) / 0.79 (L)

Grundmann et al. (1993) 1.56 4.68 7.80

Rainer et al. (1988) 1.10 3.30 5.50

In the lateral direction, Setra underestimates considerably the effects of pedestrians.

The method proposed for girder footbridges describes larger movements although these

are smaller than the proposed method. Nonetheless, if pedestrians are assumed to walk

at a mean step frequency near 2.0 Hz (regardless the journey aim and density), the lateral

accelerations predicted would be more similar to those of the sophisticated method (still

with magnitudes 30% smaller).

170


Table 6.5: Comparison of the cable-stayed footbridge performance in the lateral directionestimated by alternative proposals.

Pedestrian flow [ped/m2]

Lateral response 0.2 0.6 1.0

Proposed method 0.16 (C) / 0.10 (L) 0.31 (C) / 0.18 (L) 0.46 (C) / 0.40 (L)

Setra (2006) 0.01 0.02 0.08

Method Chapter 5 0.05 (C) / 0.03 (L) 0.10 (C) / 0.10 (L) 0.20 (C) / 0.20 (L)

6.7 Comfort appraisal

Despite the large variability of the acceleration predictions described in previous sec-

tions, the appraisal of the comfort level would describe very similar situations since values

would fall within the same comfort ranges. Figures 6.13 and 6.14 assess whether the struc-

ture is acceptable or not (in terms of comfort) according to the comfort range considered

(see figures legend and Section 3.4) and to the type of acceleration used to appraise the

comfort (peak and 1s-RMS accelerations recorded at the deck and maximum accelerations

felt by 75% or 95% of the users).

According to these figures, the footbridge would practically always be comfortable

in the vertical direction if accelerations of maximum magnitude 2.0 m/s2 were deemed

acceptable. If 1s-RMS accelerations recorded at the deck or the maximum accelerations

felt by 75% were considered representative, the comfort would be equivalent to a medium

range (with a limit acceleration of 1.0 m/s2). In the lateral direction, the footbridge

would always be comfortable for walking pedestrians if limits of magnitude 0.80 m/s2

were considered acceptable. Similarly to vertical movements, if 1s-RMS accelerations at

the deck or peak movements felt by 75% of the users were considered, the comfort of this

CSF in the lateral direction would correspond to medium as well (with accelerations just

below 0.3 m/s2).

0.2 C 0.6 C 1.0 C 0.2 L 0.6 L 1.0 L

Traffic event

Vert

ica

l accele

ration [

m/s

2]

2.5

2.0

1.5

3.0

1.0

0.5

0.0

Structure acc. Pedestrian acc.

Peak

1s-RMS

75%

95%

Maximum

Medium

Minimum

Unacceptable (2)

Unacceptable (1)

Figure 6.13: Comfort of walking pedestrians due to vertical accelerations according to traf-fic scenario and representative acceleration magnitude for the event (‘C’ represents commuterflows and ‘L’ leisure flows and there are two limits for ranges of unacceptable accelerations,that of Setra and that of NA to BS EC1).

171


Late

ral accele

ration [

m/s

2]

0.5

0.4

0.3

0.2

0.1

0.00.2 C 0.6 C 1.0 C 0.2 L 0.6 L 1.0 L

Traffic event

Structure acc. Pedestrian acc.

Peak

1s-RMS

75%

95%

Maximum

Medium

Minimum

Figure 6.14: Comfort of walking pedestrians due to lateral accelerations according totraffic scenario and representative acceleration magnitude for the event.

Vertical movements would never be adequate for pedestrians standing or sitting al-

though in the lateral direction they would be acceptable in events with light and medium

densities, as described in Figure 6.15 (movements perceived by these users are charac-

terised by peak and 1s-RMS accelerations recorded at the deck). Therefore, the comfort

would be acceptable or not depending on the use of the structure. If the structure is

expected to be used by people who could spend time staying on the bridge, or looking at

the landscape, then the comfort criteria would not be acceptable.

0.2 C 0.6 C 1.0 C 0.2 L 0.6 L 1.0 L

Late

ral accele

ration [

m/s

2]

0.40

0.30

0.20

0.0

0.10

0.50

Vert

ica

l accele

ration [

m/s

2]

0.2 C 0.6 C 1.0 C 0.2 L 0.6 L 1.0 L

2.5

2.0

1.5

3.0

1.0

0.5

0.0

(a) (b)

Figure 6.15: Comfort of standing and sitting pedestrians in the vertical (a) or lateraldirection (b), according to the traffic scenario.

6.8 Serviceability limit state of deflections

As opposed to rail and road bridges, the dynamic deflections of footbridges have

scarcely been used to predict the magnitudes of the movements in service. Some of

these methods, which relate static deflections with accelerations, are mentioned in Chap-

ter 2. Others such as Bachmann et al. (2001) propose limiting the static deflection to

ensure an adequate response in service. These evaluations are based on an assumed cor-

relation between static and dynamic deflections and between dynamic deflections and

accelerations.

The maximum dynamic vertical deflections of the conventional cable-stayed footbridge

172


under the action of different pedestrian flows are represented in Figures 6.16 - 6.18. These

have peak magnitudes of 13.6, 29.6 and 45.4 mm (Lm/3676, Lm/1689, and Lm/1101

respectively) for flows of low, medium and high densities (the aim of the journey of users

does not influence these magnitudes). These maximum deflections are described near x

= 45.0 m (explained by the important contribution of mode V2, with an antinode at this

point) as opposed to maximum vertical accelerations, which take place at x = 28.0 m

(another antinode of mode V2). The comparison of these dynamic deflections with the

static deformations of the weight of these flows shows that at x = 45.0 m the former are

between 2.3 and 1.5 times larger than the latter.


10.0 20.0 30.0 40.0 50.0 60.0

0

Deflection [

mm

]

10

20

40

30

1213.6

DAFC = 2.61

DAFL = 2.43 DAFC = 2.27

DAFL = 1.93

0.2 ped/m2

Commuter flow

Leisure flow

Static case

Figure 6.16: Dynamic and equivalent static vertical deflections caused by pedestrian flowswith 0.2 ped/m2.

0

Deflection [

mm

]

10

20

40

30


10.0 20.0 30.0 40.0 50.0 60.0

27.4 29.6DAFC = 1.69

DAFL = 1.80DAFC = 1.62

DAFL = 1.63

0.6 ped/m2

Commuter flow

Leisure flow

Static case


0

Deflection [

mm

]

10

20

40

50

30


10.0 20.0 30.0 40.0 50.0 60.0

27.427.4

DAFC = 1.51

DAFL = 1.65

DAFC = 1.48

DAFL = 1.49

1.0 ped/m2

Commuter flow

Leisure flow

Static case


The contrast of the vertical dynamic deflections with the peak accelerations computed

173


for the same scenarios (shown later in Figure 6.22(a)) describes a good correlation between

both magnitudes when the traffic flow has a medium or large number of pedestrians,

whereas for small flows the aim of the journey is a characteristic that influences the

accelerations but not the displacements. Hence, limiting the deflections or predicting the

vertical accelerations according to deflections will not always yield adequate values (in

particular for light flows), thus serviceability should be appraised in terms of accelerations.

A further analysis of these vertical dynamic deflections shows that these would corre-

spond to the static deflections associated with the loads considered for the ULS case (5

kN/m2, equivalent to 6.4 ped/m2), as illustrated in Figure 6.22(b) (this plot represents

the dynamic deflections at x = 45.0 m in comparison to the static deflections of the pedes-

trian traffic weight at the same deck location, i.e., DAFs related to deflection, according

to the number of pedestrians at the flow). In fact, the dynamic effects would be expected

to be negligible for pedestrian flows with densities similar or above 2 ped/m2.

The deflections in the lateral direction (Figures 6.19 - 6.21) have peak values ranging

from 0.5 to 2.6 mm, well below the magnitudes that several authors consider that disturb

pedestrians (10 mm according to P. Fujino et al., 1993). These values emphasize the

importance of density of the traffic as well as the aim of the journey in the evaluation of

lateral deflections.


10.0 20.0 30.0 40.0 50.0 60.0

0

1.0

2.0

DAFC = 13.7

DAFL = 9.0

0.2 ped/m2

Deflection [

mm

]

0.8

Commuter flow

Leisure flow

Static case

Figure 6.19: Dynamic and equivalent static lateral deflections caused by pedestrian flowswith 0.2 ped/m2.


10.0 20.0 30.0 40.0 50.0 60.0

0

1.0

2.0DAFC = 8.5

DAFL = 5.0

0.6 ped/m2

Deflection [

mm

]

1.7

Commuter flow

Leisure flow

Static case


The contrast of the lateral accelerations with the dynamic deflections predicted for the

same scenarios (Figure 6.22(c)) suggests that there is a linear correlation between both

components. The analysis of the average amplitudes of the pedestrian lateral loads in

174



10.0 20.0 30.0 40.0 50.0 60.0

0

1.0

3.0

2.0

DAFC = 6.58

DAFL = 5.6

1.0 ped/m2

Deflection [

mm

]

2.63

Commuter flow

Leisure flow

Static case


each scenario shows that these are proportional to these deflections (and accelerations).

Therefore these deflections (and accelerations and the amplitude of the pedestrian lateral

loads) are a consequence of the traffic event and therefore could be used to characterise

comfort of pedestrian bridges in the lateral direction (for comfort in this direction a limit

in terms of lateral accelerations is equivalent to a limit in terms of deflections).

2.0

1.0

10 20 30

x = 45.0m

Vert

ica

l peak a

cc.

400.0

Vertical peak displ.

0.4

0.2

0.5 1.0 1.5 2.00.0

[m/s2]

[mm]

[m/s2]

[mm]

2.6

2.2

1.8

1.4

1

DAFdeflect,x = 45.0m

0.2 0.6 1.0 2.0 6.4

(5kN/m2)

ped/m2

x = 30.0m

Lateral peak displ.

Late

ral peak a

cc.

(a) (b) (c)

Figure 6.22: (a) Comparison between peak vertical deflections and peak accelerationsdescribed at the same events, (b) DAFs related to vertical deflections at x = 45.0 andpedestrian flow density causing these dynamic deflections, and (c) comparison between peaklateral deflections and peak lateral accelerations at the same events.

6.9 Deck normal stresses

Considering the detailed evaluations of the loads transmitted by walking pedestrians

in service, some guidelines propose implementing these models for additional assessments

of structural performance. This is the case of the Setra guideline (2006), which recom-

mends using these SLS models to appraise stresses and displacements (apart from the

accelerations).

In this and the following sections, appraisals similar to those proposed by Setra are

performed to understand the effects of the pedestrian events on different structural ele-

ments. This section evaluates the magnitudes of the bending moments (BMs) described

at multiple sections of the deck during any of the events and compares these to the BMs

caused by the equivalent static weights of the traffic and to the BMs of the ULS load (due

175


to a uniform distributed load of 5 kN/m2 that could act, or not, at any location of the

deck). These descriptions and comparison are given in Figures 6.23 - 6.25. From these

figures it can be stated that:

a) The principal difference between dynamic and static BMs occurs in the prediction of

hogging moments in the second half of the main span x ≥ 40.0 m. As static loads do

not generate this effect, the corresponding DAFs tend to infinity.

b) The BMs are related to the magnitudes of the vertical displacements and these in

turn to the vertical accelerations, as it has already been shown in Figure 6.22. For

light pedestrian traffics, commuting traffics produce larger BMs than leisure traffics.

Nevertheless, for medium and heavy traffics, both commuting and leisure traffics lead

to similar BMs (as it was observed for the accelerations).

c) The Dynamic Amplification Factors (DAFs) related to BMs correspond to the ratio

between BMs induced by the dynamic events and the BMs induced by the equivalent

static loads. An assessment of these factors for the different traffic scenarios shows

that they attain the largest magnitudes at x = 30.0 m (antinode of the vertical mode

V2) whereas at the section of largest dynamic BMs (x = 50.0 m) these DAFs have

magnitudes that are significantly smaller.

d) The larger the pedestrian intensity the smaller the DAF related to BMs. For small

pedestrian densities, the DAF related to hogging and sagging BMs in critical sections

are 9.81 and 4.16 respectively; whereas for heavy pedestrian frequencies these values

are reduced to 2.67 and 2.18.

e) For a given pedestrian density and aim of the journey, the DAF related to BMs for

critical sections are significantly larger than those related to deflections.

f) For heavy pedestrian densities, the maximum DAF related to BMs in critical sections

are smaller than 2.86. This means that a static analysis with pedestrians densities of

2.86 ped/m2 would lead to identical BMs in critical sections. This static loading is

equivalent to 2.25 kN/m2, which is significantly smaller than the prescribed 5 kN/m2.

Nevertheless, the hogging bending moments in x>40 m would not be captured in these

static analysis, although these BMs are equal or smaller than those critical hogging

bending moments that would be obtained in critical sections between 10 and 40 m.

A comparison of these BMs to those caused by the ULS load of 5.0 kN/m2 at x =

45.0 m (through the associated DAFs) allows one to grasp whether the magnitudes of deck

stresses under ULS conditions are similar or larger than those caused by the dynamic flows

(see Figure 6.26). This figure denotes that the dynamic effects would be expected to be

negligible for 2 ped/m2 and that the effects of 5.0 kN/m2 (equivalent to 6.4 ped/m2)

are well beyond the dynamic normal stresses generated by a moving flow of any realistic

number of pedestrians (except in the second half of the main span where only dynamic

analyses can predict hogging moments).

176



100

50

0

150

50

100

150

200

250

D B

endin

g M

om

ent

[kN

m]

36 35.3

41.573

32.6

19.1

DAFC = 3.68

DAFL = 3.28

DAFC = 9.81

DAFL = 7.03

DAFC = 7.98

DAFL = 6.23DAFC = 4.16

DAFL = 3.24

10.0 20.0 30.0 40.0 50.0 60.0

Commuter flow

Leisure flow

Static case

0.2 ped/m2

Figure 6.23: Dynamic and equivalent static bending moments caused by pedestrian flowswith 0.2 ped/m2 along the length of the deck.

100

50

0

150

50

100

150

200

250

D B

endin

g M

om

ent

[kN

m] 77.3

57.8

70.5

56.9

143

57.3

29.6

DAFC = 2.41

DAFL = 2.61

DAFC = 3.96

DAFL = 3.64

DAFC = 3.79

DAFL = 4.12

DAFC = 2.45

DAFL = 2.49

10.0 20.0 30.0 40.0 50.0 60.0

0.6 ped/m2


Commuter flow

Leisure flow

Static case


100

50

0

150

50

100

150

200

250

112

67

112

208

30

95.5

50

DAFC = 2.06

DAFL = 2.24

DAFC = 2.67

DAFL = 2.62

DAFC = 2.46

DAFL = 2.86

DAFC = 2.18

DAFL = 2.14

D B

endin

g M

om

ent

[kN

m]

10.0 20.0 30.0 40.0 50.0 60.0

1.0 ped/m2


Commuter flow

Leisure flow

Static case


177


11

9

7

5

3

1

DA

FB

M,

x =

50.0

m

0.2 0.6 1.0 2.0 6.4

(5kN/m2)pedestrian density [ped/m2]

5

4

3

2

1

1

Ped.

weig

ht

* D

AF

BM

, x =

50.0

m

0.2 0.6 1.0 2.0

pedestrian density [ped/m2]

[kN/m2]

(a) (b)

Figure 6.26: (a) Comparison between DAFs related to BMs at x = 50.0 m and pedestriantraffic density causing these dynamic stresses, and (b) similar correlation considering theweight of the traffic flow compared to the weight of the live load of ULS.

6.10 Deck shear forces

The comparison of the shear forces (Figures 6.27-6.29) induced in the steel girder webs

during the dynamic events or static weights of pedestrian flows highlights that: a) the

largest dynamic shear forces are induced at the support at x = 60.0 m, b) the flows

with similar densities induce similar shear forces irrespective of the aim of the journey,

c) the DAFs related to these shear forces (where DAF is the ratio between the dynamic

shear forces and static shear forces induced by the weight of the pedestrian traffic flows)

have smaller magnitudes than the DAFs related to bending moments, and that d) DAFs

increase with increasing flow density, as opposed to the DAFs related to BMs.

The comparison between DAFs related to the shear forces caused by the different

pedestrian flows and the DAF of the ULS load of 5.0 kN/m2, Figure 6.30, highlights that

these shear forces are considerably smaller than those of the ULS load.

15

10

0

20

-5

-10

-15

Dynam

ic S

hear

Forc

es [

kN

]

10.0 20.0 30.0 40.0 50.0 60.0

5


Commuter flow

Leisure flow

Static load

0.2 ped/m2

4.4

3.3

7.3

DAFC = 0.76

DAFL = 0.6010.5

Figure 6.27: Dynamic and corresponding static shear forces generated by pedestrian flowswith 0.2 ped/m2.

178


15

10

0

20

-5

-10

-15

Dynam

ic S

hear

Forc

es [

kN

]

10.0 20.0 30.0 40.0 50.0 60.0

5


0.6 ped/m2

4.9

8.8

13.1DAFC = 0.93

DAFL = 1.0

9.4

11.7

Commuter flow

Leisure flow

Static load

Figure 6.28: Dynamic and corresponding static shear forces generated by pedestrian flowswith 0.6 ped/m2.

15

10

0

20

-5

-10

-15

Dynam

ic S

hear

Forc

es [

kN

]

10.0 20.0 30.0 40.0 50.0 60.0

5


1.0 ped/m2

11.3

7.8

18.3

5.3

DAFC = 1.38

DAFL = 1.36

DAFC = 2.80

DAFL = 3.20

Commuter flow

Leisure flow

Static load

Figure 6.29: Dynamic and equivalent static shear forces generated by pedestrian flowswith 1.0 ped/m2.

6

2.0

1.5

1.0

0.5

DA

FS

HE

AR

, x =

57.5

m

0.2 0.6 1.0 2.0 6.4

(5kN/m2)pedestrian density [ped/m2](a)

5

4

3

2

1

Ped.

weig

ht

* D

AF

SH

EA

R,

x =

57.5

m

0.2 0.6 1.0 2.0

pedestrian density [ped/m2]

[kN/m2]

(b)

Figure 6.30: (a) Comparison between DAFs related to shear forces at x = 57.5 m andpedestrian traffic density causing these dynamic stresses, and (b) similar correlation consid-ering the weight of the traffic flow.

179


6.11 Pylon stresses in serviceability

The axial loads, shear forces or bending moments induced in the pylon due to the static

weight of pedestrian flows with 0.2, 0.6 or 1.0 ped/m2 have very moderate magnitudes.

However, in relation to the static stresses, the dynamic scenarios generate significantly

larger stresses (values are represented in Figure 6.31): the DAFs associated to axial loads

range between 1.05 and 1.6, and the DAFs related to bending moments have values

between 3.4 and 7.6 at the base and between 2.6 and 4.1 at sections near the deck. For

bending moments at sections of the pylon near the deck, static and dynamic loads would

cause similar effects for pedestrian flows with densities beyond 8.0 ped/m2, which implies

that the static calculations with a uniform load of 5.0 kN/m2 do not cover the dynamic

effects in the pylon. Therefore these results indicate that the pylon should be designed

on the basis of the stresses generated during these dynamic events.

5.0

10.0

15.0

20.0

25.0

00 20.0 40.0 60.0

Bending Moment [kNm]

Positio

n u

p t

ow

er

[m]

0.2 ped/m2

0.6 ped/m2

1.0 ped/m2

5.0

10.0

15.0

20.0

25.0

00 100 200 300

0.2 ped/m2

0.6 ped/m2

1.0 ped/m2

Axial force [kN]

Commuter flow

Leisure flow

Figure 6.31: Maximum dynamic bending moments and axial loads along the height of thepylon generated by the different traffic scenarios (the intersection of the pylon with the deckis at 7.5 m high).

6.12 Fatigue of cables

The fatigue of stayed cables caused by repetitive loading and unloading during service

is an important consideration for their design given that the failure of individual cables

can generate more severe consequences. This ULS should therefore not be reached. One

approach to guarantee a correct performance of these cables is limiting the maximum

stress and stress variations caused by permanent and variable loads (see Section 3.5.2).

Another approach to guarantee their performance is by considering the definition of dam-

age detailed in Section 3.7.6 (Equations 3.7.2 and 3.7.3, which assess the fatigue of a cable

according to the stress variations and number of cycles with that stress amplitude).

The comparison of the maximum stress variations caused by the pedestrian traffic

flows in each cable (see Figures 6.32- 6.34) illustrates how the cables that endure largest

stress variations are those located at the main span near the pylon (cables with smallest

180


longitudinal inclination). In these cables, light commuter flows cause stress variations

larger than those of leisure pedestrians whereas the aim of the journey in heavy flows has

the contrary effect.

100

150

50

0Backstay CB1 CB2 CB4 CB5CB3

0.2 ped/m2

Commuter flow

Leisure flow

Static load

Ds S

tay C

able

s [

MP

a]

CB1 CB5CB4

CB2CB3

BS

Figure 6.32: Maximum stress variations of the backstay and main span cables generatedby light pedestrian flows.

100

150

50


0.6 ped/m2

Ds S

tay C

able

s [

MP

a]

Commuter flow

Leisure flow

Static load

Figure 6.33: Maximum stress variations of the backstay and main span cables generatedby medium-density pedestrian flows.

100

150

50


1.0 ped/m2

Ds S

tay C

able

s [

MP

a]

Commuter flow

Leisure flow

Static load

Figure 6.34: Maximum stress variations of the backstay and main span cables generatedby heavy pedestrian flows.

If the number of cycles and stress amplitudes of each cable are compared (Table 6.6)

according to the type of flow and density of the serviceability traffic scenario (damage

comparison described by Equation 3.7.3 of Section 3.7.6, it can be stated that:

a) Leisure flows with larger densities cause higher damages at all the cables except at

the backstay, where the largest fatigue effect is generated by the pedestrian flow with

181


Table 6.6: Fatigue performance of the stay cables (C describes commuter events and Lleisure events): effects of the density. Values calculated according to Equation 3.7.3 ofSection 3.7.6.

Comparison Backstay CB1 CB2 CB3 CB4 CB5

0.6 C / 0.2 C 1.51·10−4 11 13 13 13 131.0 C / 0.2 C 3.94·10−4 30 25 25 25 250.6 L / 0.2 L 5.35·106 178 239.5 240 241 2521.0 L / 0.2 L 808 343 460 460 461 477

Table 6.7: Fatigue performance of the stay cables: effects of the aim of the journey (seeEquation 3.7.3 of Section 3.7.6).

Comparison Backstay CB1 CB2 CB3 CB4 CB5

0.2 C / 0.2 L 4.1·105 11.0 11.7 11.8 11.8 12.20.6 C / 0.6 L 1.2·10−5 0.7 0.6 0.6 0.6 0.61.0 C / 1.0 L 0.2 1.0 0.6 0.6 0.6 0.6

smallest density. At the main span, stay cables endure similar stress variations re-

gardless of their anchorage location and longitudinal inclination (in relation to the

deck).

b) The density of leisure flows has effects on the backstay and main span cables that

are opposite to those of commuter flows. Higher density flows generate larger fatigue

damages at the backstay.

Table 6.7 demonstrates that the fatigue damage at any cable caused by light flows is

worse when pedestrians are commuting instead of strolling at leisure (the effects of light

leisure flows are very mild); in medium and heavy flows the effect is the opposite (leisure

traffic classes cause larger damages to cables) and the contribution to the stress variation

of one pedestrian in a leisure or commuter traffic is the same regardless the density.

The effects caused by each type of flow are extrapolated to evaluate the total damage

accumulated in each cable during the lifetime of cable-stayed bridges scarcely, regularly

or heavily used (see Table 3.8 of Section 3.7.6). These evaluations, which exclude effects

caused by temperature variations, wind actions, etc., are listed in Table 6.8.

These results emphasise that the stay cables of bridges with seldom use will not present

fatigue problems due to the passage of pedestrians. At a regularly used bridge the cable

that endures the greatest fatigue effects is CB1, which are generated by the commuter

events principally (see Table 6.6). All the cables anchored in the main span of a CSF with

heavy use need to be replaced at least once (twice for CB1) during the bridge lifetime (50

years) due to the stress variations caused by pedestrians. On the contrary, the performance

of the backstay during the same events and time period is adequate (practically similar

to that of a bridge with seldom usage). This adequate performance is caused by the very

mild effects of commuter flows on the backstay (see Table 6.6).

182


Table 6.8: Fatigue performance of cables of bridges with different usages (described inTable 3.8 of Chapter 3).

Bridge usage Backstay CB1 CB2 CB3 CB4 CB5

Seldom 0.032 0.047 0.033 0.033 0.033 0.033

Regular 0.144 0.777 0.627 0.626 0.626 0.622

Heavy 0.038 2.386 1.668 1.665 1.661 1.635


Conventional cable-stayed footbridges with a main span length of 50 m have vertical

vibration frequencies within ranges that are considered critical for their design in service

(V2 has a frequency near 2.0 Hz). Therefore, designers must appraise their dynamic

response under pedestrian loading in service.

Despite the existence of these vertical critical frequencies, these bridges develop mod-

erate responses under the passage of light traffic flows (the peak vertical accelerations of

commuter and leisure flows with 0.2 ped/m2 are 1.16 and 0.81 m/s2 respectively). For

heavier flows, regardless the type of flow, the vertical accelerations are expected to be

large and will be considered adequate or not depending on the criterion adopted to assess

comfort. For the events simulated, the proposed load model generates peak vertical ac-

celerations of magnitude 1.70 and 1.62 m/s2 under the passage of commuter and leisure

flows with 0.6 ped/m2 and 2.15 and 2.07 m/s2 under the passage of flows with 1.0 ped/m2

respectively.

In the lateral direction these conventional footbridges do not have natural frequencies

below 1.5 Hz. Nonetheless, pedestrians produce noticeable lateral accelerations that ac-

cording to some proposals are equivalent to a minimum comfort (lateral accelerations are

larger than 0.3 m/s2 for the heaviest flows).

The analysis of the dynamic deflections of this footbridge in relation to the acceler-

ations described at the same events shows that the assessment of comfort for vertical

movements can only be based upon acceleration amplitudes. However, in the lateral di-

rection movements are linearly related to the accelerations, therefore comfort limits based

on accelerations or deflections would be equivalent in this direction.

If comfort is appraised in terms of the movements felt by users, it has been seen that

it is likely that at least one pedestrian notices peak movements of similar magnitudes to

those recorded at the deck. However, many other users feel movements well below these

peaks (as detailed in Section 6.5.2). In the vertical direction, most of the users within

medium and heavy flows feel very similar accelerations. In the lateral direction, only

20% of the pedestrians in medium and heavy flows feel accelerations beyond 0.2 m/s2.

Additionally, it has been seen that these peak responses are not felt by users during long

time periods or over many regions of the deck. Hence, values of accelerations weighed

with time durations would provide a more realistic assessment of the comfort of users at

these footbridges.

In relation to currently available methods to predict vertical and lateral accelerations,

the vertical accelerations described by the British code are more similar to those obtained

183


with the load model proposed in this thesis than any other available method. In the

lateral direction, the prediction of Setra is unrealistic and the method described in the

Chapter 5 provides a better evaluation.

Regardless of the method used to predict the accelerations in service, the comfort

associated with vertical movements of these conventional CSFs is considered to be medium

or low (between 0.5 and 1.0 or 1.0 and 2.5 m/s2 respectively, comfort classes described in

Setra, 2006) and that of lateral movements ranges from maximum to minimum (the first

range corresponding to accelerations between 0.0 and 0.15 m/s2 and the second between

0.30 and 0.80 m/s2, as described by the Setra, 2006) according to the traffic scenario.

The evaluation of stresses at the deck and tower emphasise that dynamic assessments

are necessary to predict the worse case scenario at certain sections (in particular hogging

BMs at the deck at 45.0 ≤ x ≤ 60.0 m and, at the pylon, near the deck height) whereas

the rest can be predicted using static loads of amplitudes considerably lower than those

implemented in the ULS evaluations.

Finally, the assessment of the fatigue of the stayed cables shows that the nature of

the usage of the bridge during its lifetime has a large impact on the performance of each

cable. In regularly or heavily used CSFs, many of these cables would have to be replaced

before the end of their design life.

184

Chapter 7Performance of cable-stayed

footbridges with a single pylon:

parameters that govern serviceability

response

7.1 Introduction

Guidelines and codes focused on the design of footbridges emphasise the need for an

accurate prediction of their response in service (under pedestrian loads) when vertical,

torsional and lateral frequencies have magnitudes near or within ranges that are considered

critical. Setra (2006) highlights the range 1.7-2.1 Hz in the vertical direction and 0.5-1.1 Hz

in the lateral direction; the NA to BS EN1991-2:2003 (BSI, 2008) emphasises the range

1.2-2.6 Hz in the vertical direction and below 1.5 Hz in the lateral direction.

As illustrated in Section 6.3, the conventional cable-stayed footbridge with a single

pylon presents vertical vibration modes within the ranges thought to generate large move-

ments and lateral modes that are above those. The resulting movements in service of this

footbridge under the action of several flows of pedestrians correspond to levels of comfort

ranging from medium to low in the vertical direction and maximum or medium in the

lateral direction.

In order to mitigate these responses, damping devices can be installed once the bridge

is built. However, it is far preferable to anticipate these responses and to take steps during

the design stage to prevent their occurrence. There are some guidelines that recommend

the modification of certain structural elements of the bridge (cable diameter, thickness

of steel elements, etc.) in order to reduce these responses (see Section 2.7), however

these recommendations are scarce and in some cases have not been obtained considering

realistic models of pedestrian loads.

This chapter evaluates different alternatives to modify and improve the structural

response in service of CSFs with one tower (such as the one considered in Chapter 6).

185

7. Design of cable-stayed footbridges with a single pylon

Accordingly, the following sections appraise the effect on the accelerations in service of

these CSFs with: (a) alternative boundary conditions (involving laminated elastomeric

bearings, LEBs, or pot bearings, POTs), geometrical characteristics (e.g., section and

dimensions of the tower or the deck, span distribution or cable arrangement) or dimensions

of the structural elements (e.g., cables, steel girders, concrete slab and transverse section

of the tower). These analyses are presented in Section 7.3 for vertical accelerations and in

Section 7.4 for lateral accelerations. Additionally, the impact on the dynamic response of

these characteristics of CSFs with longer spans and alternative depth-to-main span length

ratios are assessed and presented in Section 7.5. Finally, based on the magnitudes of the

serviceability accelerations of these footbridges (which are compared to comfort criteria

in Section 7.6), Section 7.7 summarises the improvements in the dynamic performance

obtained for these CSFs when considering additional dissipation capacity for the structure

(either inherent or externally provided through the use of supplemental damping devices).

The interpretation and prediction of the effects of the different considered measures

on the dynamic movements of the footbridge that are given in these sections are based on

observations and correlations relating these vertical and lateral movements to characteris-

tics of the footbridge and to characteristics of the vertical and lateral loads of pedestrian

flows. These are detailed in Section 7.2. Apart from the effect of these measures on the

dynamic movements of the footbridge and the comfort experienced by users, these analy-

ses also investigate, evaluate and characterise the magnitude of the dynamic deflections,

the normal and shear stresses at the deck (i.e., bending moments and shear forces), axial

and bending moments at the tower and the performance of the stays (in terms of fatigue).

These are detailed in Sections 7.8 to 7.12 respectively.

7.2 Dynamic characteristics of pedestrian loads and the foot-

bridge related to its performance in service

The analysis and comparison of the dynamic movements caused by pedestrian traffic

crossing one pylon cable-stayed footbridges, 1T-CSFs, with alternative geometric charac-

teristics allows one to discern the parameters that are most influential for the magnitude

of these movements. In the vertical direction, the characteristics of these footbridges that

are decisive correspond to the frequency and mass for modes V2 and V3, the partici-

pation from the second torsional mode T2, and the stress (under permanent loads) and

the longitudinal inclination of the stay cables, in particular for the backstay and the stay

anchored at the antinode of mode V2 in the main span closest to the pylon. Therefore the

vertical peak or 1s-RMS acceleration at a location x (av,x) can be generically described

as follows (where fV 2 and fV 3 are the frequencies of V2 and V3, mi denotes the effective

mass associated with these as well as the T2, σi the stress of the backstay i = BS or the

second stay in the main span i = CB2 and φi the longitudinal inclination of these):

av,x = f(fV 2,mV 2, fV 3,mV 3,mT2, σBS, σCB2, φBS, φCB2), x ∈ peak, 1s−RMS(7.2.1)

186


Another conclusion that can be extracted from this analysis is that the influence of

these modes does not decrease if the frequencies associated with these modes are rela-

tively far away from the range considered critical (e.g., V3 has a frequency near 3.2 Hz

and T2 around 4.0 Hz, yet they still provide important contributions to the response).

This observation is supported by the analysis of the frequency content of the total loads

introduced by a flow of pedestrians to the bridge at different deck regions (Figure 7.1

represents the frequency content of the vertical loads introduced at the region located at

25 m ≤ x ≤ 30 m). As illustrated in that figure, the energy introduced by pedestrians

while walking on the bridge is not exclusively introduced at frequencies near 2.0 Hz but

at lower and higher frequencies.

1 1.5 2 2.5 3 3.5 40

5

10x 10

5

Frequency [Hz]

Energ

yA

mplit

ude

Figure 7.1: Energy amplitude [N] of the total vertical loads introduced by the pedestrianflow while stepping at the deck between 25 m ≤ x ≤ 30 m.

The importance of the longitudinal inclination and force introduced by the backstay

and the stay CB2 is substantiated by their impact on the deformation of the main span.

The backstay has an overall control of deformations through the restraint of the movement

of the top of the tower whereas the stay CB2 has a large impact on modes V2 and V3, as

it is anchored near their antinode.

In the lateral direction, the magnitudes of peak (al,p) or 1s-RMS (al,rms) accelerations

are largely correlated to factors such as the characteristics of the first L1 and second

lateral modes L2, the second torsional mode T2, the mass and the second moment of area

in lateral direction of the deck, and the stress and lateral inclination of the stay anchored

at the antinode of the mode L1 x ≈ 28 m. In this direction, analyses show that the

response is sensitive to the frequency associated with mode L1, in particular when this

is near 1.0 Hz. Regarding the torsional mode, its relevance is related to its projection in

the lateral direction, as it increases or decreases the lateral movements accordingly (see

Figure 6.2).

7.3 Strategies to improve the vertical dynamic performance of

1T-CSFs in service

Considering the above-mentioned characteristics of a cable-stayed footbridge, the sub-

sequent sections study the sensitivity of the response of the reference bridge, which was

analysed in the previous chapter, to the bridge articulation (boundary conditions), the

geometrical parameters that define the general configuration of the bridge (such as the

span lengths, the tower height, the spacing between cables, and the cable arrangement)

and the geometrical and mechanical properties that define each of its structural elements

187


(the deck, the stay cables, and the pylon).

7.3.1 Articulation of the deck

The articulation of the deck has a strong influence upon the structural response of

footbridges in service. Hence, the effect in service of different support arrangements is

explored, compared and used to derive conclusions about the best arrangement when

considering the vertical accelerations of the deck for these footbridges.

There are many factors that have an impact on the selection of footbridge articulations.

Depending on the demands, bearings usually consist of laminated elastomeric bearings

(LEBs) and pot bearings (POTs).

The conventional cable-stayed footbridge with a single pylon, 1T-CSF, and main span

length of 50 m, which was considered representative of this typology and was analysed in

Chapter 6, has two vertical LEBs and one lateral shear key (SK) at each abutment and

is simply supported at the pylon restraining the relative vertical, longitudinal and lateral

displacements of the deck and the pylon at this point. This configuration prevents vertical

and transverse horizontal movements of the deck at the abutments, while it allows its

longitudinal displacement. This configuration has been adopted based on performance and

economical criteria as it ensures a minimum user comfort both in the vertical and lateral

directions, it does not considerably restrict movements caused by temperature variations,

and it uses LEBs which are cheaper than POTs. Three alternative support schemes to

the support arrangement in the reference case (Figure 7.2(a)) have been considered: (1)

with the same articulation as the reference case at the abutments and at the pylon, but

without installing the lateral shear keys (Figure 7.2(b)), (2) with a ‘classical’ layout of

POT bearings at the abutments and the same articulation as the reference case at the

pylon (Figure 7.2(c)), and (3) with a statically indeterminate layout of POT bearings at

the abutments and the same articulation as the reference case at the pylon (Figure 7.2(d)).

Classical POT layout Statically indeterminate POT layout

LEBs+SKLEBs

Pylon

LEB support

POT bearing

Free UxFixed UyFree z


Fixed UxFixed Uy

Free UxFixed Uy

Free UxFree Uy

Fixed UxFree Uy

Fixed UxFixed Uy

Fixed UxFixed Uy

Fixed UxFixed Uy

Fixed UxFixed Uy

(a)

(c)

(b)

(d)

y

x

z

KxKy

KxKy

KxKy

KxKy

KxKy

KxKy

KxKy

KxKy

Figure 7.2: Plan view of the support configurations of the CSF with LEB bearing schemesor POT bearing schemes. (a) 2 LEBs and a SK per abutment, (b) 2 LEBs at each abutment,(c) ‘classical’ POT arrangement and (d) statically indeterminate POT arrangement.

Traffic flows of commuters with medium-high densities (0.6 ped/m2) walking on CSFs

188


with support arrangements as those previously detailed generate the peak and 1s-RMS

vertical accelerations illustrated in Figure 7.3. These plots represent the envelope of the

absolute accelerations that occur at any point of each section of the deck along its length.

According to the results of Figure 7.3, bearing schemes such as (b) or (d) produce

vertical accelerations that are larger (1s-RMS are 10% or 20% larger respectively) than

those of the reference scheme (a), whereas the scheme (c) improves the vertical response

of the CSF (1s-RMS accelerations are 10% smaller). The CSF with LEBs (scheme (b))

develops considerably large torsional movements, which can be inferred from Figure 7.3(b)

by observing the differences between the maximum peak accelerations at any location of

the transverse section (with values up to 1.86 m/s2) and those at the centre line of the

section (with values up to 1.5 m/s2). These large torsional movements are due to the large

lateral movements that the traffic generates for this particular structure (this is shown in

more detail in Section 7.4.1).

0 10 20 30 40 50 600

0.5

1

1.5

2

(b)

0 10 20 30 40 50 600

0.5

1

1.5

2

(c)

0 10 20 30 40 50 600

0.5

1

1.5

2

(d)

0 10 20 30 40 50 600

0.5

1

1.5

2

(a)

Vert

ical accele

ration [

m/s

2]

apeak

a1s−RMS

0.78


1.62

0.88

0.96

1.50

1.07

1.86

1.731.45

Figure 7.3: Peak and 1s-RMS vertical accelerations recorded at the deck of CSFs withsupport schemes (a)-(d) according to Figure 7.2. Peak accelerations at the centre line inscheme (c) have been included for comparison purposes.

These modifications of the serviceability response are generated by the changes that

the restrictions of the deck movement produce on modes V2, V3 and T2, i.e., the changes

of their natural frequencies (Table 7.1) and their modal masses. The use of scheme (c)

results in smaller vertical movements due to the lower participation from mode T2 and

scheme (d) produces larger accelerations due to the smaller effective modal masses for

modes V2 and V3 in comparison to those of scheme (a).

Table 7.1: Frequencies [Hz] for the vertical and torsional vibration modes of CSFs fordifferent support arrangements (defined in Figure 7.2), where ‘VN’ and ‘TN’ denote verticaland torsional modes with N half-waves.

Scheme V1 V2 V3 T1 T2

(a) LEBs 0.98 1.96 3.18 2.94 3.75(b) LEBs+SK 0.98 1.96 3.18 2.93 3.73(c) POTs 0.98 1.95 3.18 3.00 3.78(d) POTs 1.00 2.00 3.23 2.88 3.80

The analysis of the accelerations felt by the pedestrians (Figure 7.4) leads to similar

189


conclusions for all schemes, although the differences between schemes (a) and (b) are

smaller. This is explained by the fact that at the scheme (b) the increments of accelera-

tions in most of the areas close to the centre line of the transverse section are negligible,

as the torsional movements (which are enhanced when removing the shear keys) do not

induce significant effects at these locations.

(a)(c)

(b)(d)

25

50

75

100

0

% P

edestr

ians

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0amax,P i / amax,P

50%

25%

5%

amax,P = 1.54

Figure 7.4: Vertical accelerations felt by users amax,Pi compared to amax,P . Curves definedfor CSFs with different support conditions (a)-(d).

Thus, POTs with a classical arrangement (c) improve considerably the performance of

the 1T-CSF (although the decrement of the deck accelerations is only 10%); POTs with

a statically indeterminate arrangement (d) decrease the comfort for users (and increase

the acceleration level in the deck), and the use of two LEBs without a lateral shear key

at each abutment does not reduce the degree of comfort perceived by users in the vertical

direction, despite the fact that the accelerations in the deck at eccentric locations are

amplified by enhanced torsional movements.

7.3.2 Area of backstay cable

0.8

1.0

1.2

1.4

0.5 1.0 1.5 2.0 2.5

um

ax /

um

ax,0

ABS / ABS,0

umax

bsarea

(a)

0.5 1.0 1.5 2.0 2.5

1.0

2.0

3.0

4.0

Fre

qu

ency [

Hz]

V1

V2

V3 T1

T2

5.0

ABS / ABS,0

(b)

Figure 7.5: Static and dynamic behaviour of the CSFs in terms of backstay area ABS

(compared to that of the benchmark CSF ABS,0): (a) main span maximum static deflectionsumax (compared to the deflection at the basic CSF umax,0) and (b) frequencies [Hz] of verticaland torsional modes.

The area of the backstay influences the horizontal displacements of the tower top

(through the backstay elongation in tension) and ultimately the deck vertical deformations

at the main span. Under permanent loads, backstays with smaller area ABS permit larger

vertical deflections whereas backstays with larger areas provide greater restraint to the

tower and limit the deck vertical deflections by an amount that depends upon the stiffness

of the main span stay cables and the deck. Dynamically, only at CSFs with backstay areas

190


smaller than the reference case the modal frequencies would be modified (Figure 7.5(b))

and the modal masses for the first vertical modes (V2 in particular) markedly increased.

0 10 20 30 40 50 600

0.5

1

1.5

2

Peak v

ert

ical

acc.

[m/s

2]

0 10 20 30 40 50 600

0.5

1

1.5

2


1s−

RM

S

acc.

[m/s

2]

0.5 1 1.5 2 2.5

0.6

0.8

1

1.2

ABS

/ ABS,0

acc /

acc

0

0.5 ABS,0

1.5 ABS,0

2.0 ABS,0

2.5 ABS,0

ABS,0

Peak acc.

1sRMS acc.

(a) (b)

Figure 7.6: Vertical service response of the CSF deck according to backstay area ABS : (a)peak and 1s-RMS vertical accelerations and (b) comparison of maximum absolute peak and1s-RMS accelerations to those of the reference case acc0.

The differences in the dynamic characteristics (modal masses for vertical modes) of

the CSF with smaller ABS justify the performance of the CSFs in service according to

the backstay area described in Figure 7.6. The CSF with smaller ABS produces peak

and 1s-RMS vertical accelerations that are 27% lower than the other cases whereas an

increment of this magnitude has negligible impact upon the accelerations (if the area of

the backstay is increased for up to 250%, the vertical accelerations grow 10%).

The accelerations perceived by walking users illustrated in Figure 7.7 (where values

are represented in comparison to the maximum acceleration felt by at least a pedestrian

at the reference CSF, amax,P ) depict a situation similar to that observed for the deck

movements. The accelerations felt by pedestrians are significantly smaller if the area of

the backstay is smaller than the reference case. If the area of the backstay is increased

significantly (doubled) the accelerations felt by the pedestrians are moderately larger than

at the reference case.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

25

50

75

100

amax,Pi

/ amax,P

% P

edestr

ians

25%

5%

50%

amax,P

= 1.54 m/s2

0.5 ABS,0

1.5 ABS,0

2.0 ABS,0

2.5 ABS,0

ABS,0

Figure 7.7: Vertical accelerations felt by users amax,Pi compared to amax,P of the referencecase. Curves defined for CSFs with different backstay areas.

191


By comparing Figures 7.6 and 7.7, it can be concluded that by reducing the vertical

statical deflections of the deck (through the increment of the area of the backstays) it is

not possible to reduce the accelerations in the deck.

7.3.3 Area of main span stays

The dimensions of the main span stays contribute to controlling the vertical displace-

ments of the deck caused by vertical static loads described by umax in Figure 7.8(a), in a

similar manner as it was observed in Figure 7.5(a). Additionally, together with the deck,

they constitute the system that resists loads eccentrically applied on the deck (torques).

For dynamic response, the vertical and torsional frequencies increase notably with

larger areas and vice versa (see Figure 7.8(b), where AS represents the area of the stays)

and modal masses for V2 remain fairly similar regardless of this dimension, those of V3 are

considerably larger with smaller stay areas and those of T1 increase with this magnitude.

0.8

1.0

1.2

1.4

0.5 1.0 1.5 2.0 2.5

um

ax /

um

ax,0

AS / AS,0

AS

umax

(a)

0.5 1.0 1.5 2.0 2.5

1.0

2.0

3.0

4.0

V1

V2

V3

T1

T2

5.0

Fre

qu

ency [

Hz]

AS / AS,0

(b)

Figure 7.8: Static and dynamic behaviour of the CSF in terms of stay cables area AS

(compared to that of the benchmark CSF AS,0): (a) main span maximum static deflectionsumax and (b) frequencies [Hz] of vertical and torsional modes.

This additional stiffness with larger stays (and in relation to the frequency of loads

depicted in Figure 7.1) is the reason why CSFs with larger main span stay areas de-

velop larger vertical response in service under similar traffic scenarios, as illustrated in

Figure 7.9. According to this figure, in CSFs with smaller main stay areas the vertical

response is slightly lower, whereas in CSFs with larger stay areas the vertical movements

increase with increasing main stay area. The contribution of the torsional modes in the

response gets reduced with the increase of the modal masses for these modes associated

to the increment of the area of the stay cables (see Figure 7.8(b)).

The analysis of the footbridge comfort according to the movements felt by pedestrians

yields conclusions similar to those related to the accelerations registered in the deck (see

Figure 7.10). In addition, Figure 7.10 shows very clearly how the reduction of the area of

the main stay cables beyond the values given in the reference case leads to a significant

reduction of the level of accelerations felt by the pedestrians (despite the fact that reduc-

tion of the accelerations in the deck were smaller). This is due to the increment of the

weight of the torsional modes in the response (as a consequence of the reduction of the

area of the stay cables) due to the reduction of the torsional frequencies down to values

that produce a resonant response with the pedestrian load.

Hence, contrary to what some guidelines suggest, increasing the area of cables does not

192


0 10 20 30 40 50 600

0.5

1

1.5

2P

eak v

ert

ical

acc.

[m/s

2]

0 10 20 30 40 50 600

0.5

1

1.5

2


1s−

RM

S

acc [

m/s

2]

0.5 1 1.5 2 2.5

0.8

1

1.2

1.4

1.6

AS / A

S,0

acc /

acc

0

0.5 AS,0

1.5 AS,0

2.0 AS,0

2.5 AS,0

AS,0

Peak acc.

1s−RMS acc.

(a) (b)

Figure 7.9: Vertical service response of the CSF deck according to area of cables AS : (a)peak and 1s-RMS vertical accelerations and (b) comparison of maximum absolute peak and1s-RMS accelerations to those of the benchmark case acc 0.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40

25

50

75

100

amax,Pi

/ amax,P

% P

edestr

ians

amax,P

= 1.54 m/s2

50%

25%

5%

0.5 AS,0

1.5 AS,0

2.0 AS,0

2.5 AS,0

AS,0

Figure 7.10: Vertical accelerations felt by users amax,Pi compared to amax,P . Curvesdefined for CSFs with different areas of stay cables.

favour a better service comfort whereas placing stays with smaller areas is considerably

more beneficial despite the increment of torsional movements. However, this latter alter-

native is not possible unless the current technology of anchorages is improved, allowing

larger stress variations in the cables and thus smaller areas of the cables.

7.3.4 Material of stays: bars vs strands for the stay cables

In short and medium span cable-stayed footbridges, designers commonly propose the

use of strand stay cables (as has been considered for the benchmark footbridge) although

alternatively bars can also be used (with the characteristics described in Section 3.5.2.2).

The use of bars for the cables for the conventional 1T-CSF slightly reduces the static

vertical deflection of the deck when compared to that of the footbridge with strand stays

due to the larger areas of the bars (permitted stress variations of the bars are considerably

lower than those of strands). As a consequence of this stiffness increment, the footbridge

presents vertical and torsional modes at higher frequencies, see Table 7.2 (although modal

masses for these modes remain similar).

The accelerations recorded at the deck and the movements perceived by users for CSF

193


Table 7.2: Frequencies [Hz] for the vertical and torsional vibration modes of CSFs withstrand stay or bar cables.

Cables V1 V2 V3 T1 T2

Strands 0.98 1.96 3.18 2.94 3.75Bars 1.21 2.28 3.47 3.55 3.99

using either strands or bars are very similar. For the deck, 1s-RMS accelerations are 10%

larger for the CSF with bar cables. In relation to the accelerations felt by users, 75% of

the users walking of the footbridge with bars notice peak accelerations below 0.78amax,P

compared to 0.8amax,P at the bridge with stay cables. Hence, despite the modest stiffness

increment of the bars, using bar or stay cables does not improve or worsen the vertical

performance of the footbridge in service.

7.3.5 Section of the steel girders

This geometry modification reduces the static deflection at the CSF main span (see

Figure 7.11(a)). Nonetheless, its efficiency is limited as an increase in the bottom flange

depth by a factor of 2.5 results in a modest reduction of the vertical static deflection of

just 15%. As far as the dynamic characteristics are concerned, the variation of flange

thickness effectively increases the frequencies of the vertical vibration modes of the deck

(modal masses for these modes are fairly similar) but has a relatively weak impact upon

the torsional modes, as shown in Figure 7.11(b).

0.85

0.90

0.95

1.00

1.0 1.5 2.0 2.5

um

ax /

um

ax,0

tbf / tbf,0

0.80

umax

tbf

Trans. deck sec.:

(a)

0.5 1.0 1.5 2.0 2.5

1.0

2.0

3.0

4.0

V1

V2

V3

T1

T2

5.0

Fre

qu

ency [

Hz]

tbf / tbf,0

(b)

Figure 7.11: Static and dynamic behaviour of the CSF in terms of flange girder thick-ness t bf (compared to that of the benchmark CSF t bf,0): (a) main span maximum staticdeflections umax and (b) frequencies [Hz] of vertical and torsional modes.

The increase of the depth of the bottom flange tbf leads to an increase of the frequencies

associated to vertical modes (see Figure 7.11(b)), and in turn to a general increase in the

accelerations in the deck. In addition, when there is a coupling of vertical and torsional

modes (V3 and T2 for tbf/tbf,0=1.8) the accelerations are larger than expected.

The magnitudes of the accelerations noticed by walking pedestrians, illustrated in

Figure 7.13, are similar to the accelerations of the deck: the CSF with a flange depth 1.4

times larger enhances modestly the experience of users whereas the CSF with a flange 1.8

times larger increases considerably the movements noticed by users.

Hence, increasing the depth of the bottom flange increases the level of accelerations in

the deck in general, and in particular if there is a coupling between vertical and torsional

194


0 10 20 30 40 50 600

0.5

1

2

2P

eak v

ert

ical

acc.

[m/s

2]

0 10 20 30 40 50 600

0.5

1

1.5

2


1s−

RM

S

acc.

[m/s

2]

0.5 1 1.5 2 2.5

0.8

1

1.2

1.4

1.6

tbf

/ tbf,0

acc /

acc

0

1.4 tbf,0

1.8 tbf,0

2.2 tbf,0

tbf,0

Peak acc.

1s−RMS acc.

(a) (b)

Figure 7.12: Vertical service response of the CSF deck according to bottom flange depth:(a) peak and 1s-RMS vertical accelerations and (b) comparison of maximum absolute peakand 1s-RMS accelerations to those of the benchmark case acc 0.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30

25

50

75

100

amax,Pi

/ amax,P

% P

edestr

ians

50%

25%

5%

amax,P

= 1.54 m/s2

1.4 tbf,0

1.8 tbf,0

2.2 tbf,0

tbf,0

Figure 7.13: Vertical accelerations felt by users amax,Pi compared to amax,P . Curvesdefined for CSFs with different girder bottom flange.

modes, thus it is not an appropriate measure to control the accelerations as has been

recommended by some guidelines such as Setra (2006).

7.3.6 Concrete slab section

A modification of the depth of the concrete slab forming part of the deck changes both

the mass and the deck second moment of area. In this section, the impact of the slab

depth increment is evaluated whereas a decrement of this magnitude is disregarded due

to construction reasons (smaller depths would complicate placing the bar reinforcement).

In general, the increase of the slab thickness leads to a growth of the vertical and

torsional frequencies (and to a considerable increment of the modal masses for these

modes). The impact on torsional modes is higher than in vertical modes. In addition, the

higher the mode the higher the increment of its frequency due to an increment in the slab

thickness. In fact, there is not an increase of the frequency in the first vertical mode, but

a small decrease of its frequency with the slab thickness, as a consequence of the increase

of the mass (Figure 7.14(b)).

Figure 7.15 describes the peak and 1s-RMS vertical accelerations generated at the deck

195


1.0 1.5 2.0 2.5

1.0

2.0

3.0

4.0

V1

V2

V3

T1

T2

5.0

Fre

qu

ency [

Hz]

tc / tc,0

concrete slab

depth

Transverse deck section:

(a) (b)

Figure 7.14: (a) Transverse section of the deck; (b) dynamic behaviour of the CSF interms of slab depth tc: frequencies [Hz] of vertical and torsional modes.

0 10 20 30 40 50 600

0.5

1

1.5

2

Peak v

ert

ical

acc.

[m/s

2]

0 10 20 30 40 50 600

0.5

1

1.5

2


1s−

RM

S

acc.

[m/s

2]

1.0 1.5 2.0 2.50.2

0.4

0.6

0.8

1

1.2

1.4

tc / t

c,0

acc /

acc

0

1.5 tc,o

2.0 tc,o

2.5 tc,o

tc,o

Peak acc.

1s−RMS acc.

(a) (b)

Figure 7.15: Vertical service response of the CSF deck according to depth of the concreteslab: (a) peak and 1s-RMS vertical accelerations and (b) comparison of maximum absolutepeak and 1s-RMS accelerations to those of the benchmark case acc 0.

of 1T-CSFs with larger slab depths. According to the results shown in this figure, even

moderate increments of the concrete slab depth improve the response considerably (peak

and 1s-RMS accelerations are between 0.25 and 0.50 times smaller), which is explained

by the larger masses for the modes that trigger the movement. However, this decrease in

response is not constant with the mass increment of the deck (the CSF with a slab depth

of 0.5 m generates larger movements than that with 0.3 m).

The evaluation of the dynamic characteristics of each CSF demonstrates that at the

footbridge with a slab depth of 0.3 m modes T2 and V4 have coupled frequencies whereas

the CSF with a slab depth of 0.5 m has an additional torsional mode T2 at the frequency

of mode L2. Both coincidences explain the smaller or larger than expected movements

at each case. From the users’ point of view (Figure 7.16), an increment of the slab depth

considerably improves their comfort. For the footbridges with slabs of depth 0.3, 0.4 or

0.5 m, 75% of the users notice accelerations of magnitudes equal to or smaller than 0.38,

0.46 or 0.48amax,P respectively, compared to 0.8amax,P at the benchmark footbridge.

Thus, enlarging the concrete slab depth is an effective way of improving the vertical

response of the bridge.

196


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

25

50

75

100

amax,Pi

/ amax,P

% P

edestr

ians

50%

25%

5%

amax,P

= 1.54 m/s2

1.5 tc,o

2.0 tc,o

2.5 tc,o

tc,o

Figure 7.16: Vertical accelerations felt by users amax,Pi compared to amax,P . Curvesdefined for CSFs with different slab depths.

7.3.7 Transverse section of the pylon

The pylon of the CSF consists of a single-free standing steel column with a circular

hollow section. The following paragraphs explore the impact of the dimensions of this

column (thickness and diameter of the steel section) on the overall response of the bridge

deck in service.

The alteration of the steel thickness does not result in a noticeable change to the

vertical displacements of the main span under static loads nor to changes in the frequencies

of the first vertical and torsional modes. Under pedestrian loads, this dimension (for

increments of up to 2.5 times the thickness of the reference CSF) does not modify the

response of the CSF in service.

The impact of the pylon diameter on the CSF response in service is larger. Larger di-

ameters moderately reduce the static deflections of the main span and change the response

under eccentric loads. In terms of the dynamic behaviour, these changes are reflected in

the magnitude of vertical and torsional modes, as represented in Figure 7.17.

1.0 1.5 2.5

1.0

2.0

3.0

4.0

V1

V2

V3

T1

T2

5.0

Fre

qu

ency [

Hz]

Dt / Dt,0

V2b

T2b

2.0

Figure 7.17: Dynamic behaviour of the CSF according to diameter of the pylon: frequencies[Hz] of vertical and torsional modes (V2b and T2b are additional modes).

In service, the peak and 1s-RMS accelerations of the deck increase as the pylon di-

ameter increases, as depicted in Figure 7.18. This increment is modest as a diameter 2.5

times greater than the base case only increases the vertical response by 25%. For the

CSF with a diameter 1.7 times greater than that of the benchmark bridge, the response

in service is reduced instead of enlarged, which is caused by the coincidence of the mode

V2 and a new vertical mode with two antinodes at the main span V2b (both have modal

197


0 10 20 30 40 50 600

0.5

1

1.5

2

Peak v

ert

ical

acc.

[m/s

2]

0 10 20 30 40 50 600

0.5

1

1.5

2


1s−

RM

S

acc.

[m/s

2]

1.0 1.50 2.0 2.50.5

0.75

1

1.25

1.5

Dt / D

t,0

acc /

acc

01.3 Dt,0

1.7 Dt,0

2.5 Dt,0

Dt,0

Peak acc.

1s−RMS acc.

(a) (b)

Figure 7.18: Vertical service response of the CSF deck according to pylon diameter Dt:(a) peak and 1s-RMS vertical accelerations and (b) comparison of maximum absolute peakand 1s-RMS accelerations to those of the benchmark case acc 0.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 610

−3

10−1

100

101

Frequency [Hz]

Am

plit

ude [

m/s

]

1.7 Dt,0

2.5 Dt,0

Figure 7.19: Fourier spectrum of the vertical acceleration response at x = 28 m of CSFwith pylon diameter 1.7Dt,0 or 2.5Dt,0.

masses that are considerably larger than those for mode V2 for smaller or larger pylon

diameters) and the coincidence of mode V3 with T1. These coincidences are clear with

the representation of the Fourier spectrum of the vertical accelerations at x = 28 m at

CSFs with diameters 1.7Dt,0 and 2.5Dt,0 (see Figure 7.19), where the first shows 2 peaks

near 2 Hz and above 3 Hz that the second does not have. The assessment of comfort in

terms of the accelerations felt by users shows similar results to those of the deck acceler-

ations except for the largest pylon diameter, where pedestrians feel accelerations similar

to those of the reference CSF.

Hence, increasing the pylon section does not substantially affect the comfort for users

in service unless there is a coincidence of modes that affects the contribution of the

most important modes of the movement (contribution that cannot be relied upon when

designing).

7.3.8 Pylon height

For cable stayed footbridges with a cable fan system and a side span with length Ls =

0.2Lm, the variation of the main span deflection with the magnitude of the relative pylon

198


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

25

50

75

100

amax,Pi

/ amax,P

% P

edestr

ians

1.3 Dt,0

1.7 Dt,0

2.5 Dt,0

Dt,050%

25%

5%

amax,P

= 1.54 m/s2

Figure 7.20: Vertical accelerations felt by users amax,Pi compared to amax,P . Curvesdefined for CSFs with pylon diameters.

height hp/Lm is illustrated in Figure 7.21(a). The minimum deflection occurs for pylon

heights of 0.45Lm, although for heights as small as 0.35Lm this deflection is very similar.

In practice, footbridges with medium span lengths are designed with pylon heights of

0.36Lm (see Section 2.2). Nonetheless, the effect of substantially shorter or higher pylons

on the dynamic response of the bridge are appraised in this section. Pylons with heights

below 0.25Lm or above 0.5Lm are not considered due to reasons involving aesthetics (these

proportions are not used by designers as seen in Chapter 2), the anchorage of cables and

construction costs.

1.0

1.2

1.4

0.2 0.3 0.4

um

ax /

um

ax,0

0.5

umax

hp

Lm

Ls

hp Lm

(a)

0.2 0.3 0.4

1.0

2.0

3.0

4.0

V1

V2

V3 T1

T2

5.0

Fre

qu

ency [

Hz]

0.5

hp Lm

(b)

Figure 7.21: Static and dynamic behaviour of the CSF in terms of pylon height hp: (a)main span maximum static deflections umax and (b) frequencies [Hz] of vertical and torsionalmodes.

Figure 7.21(b) describes the frequencies of the first vertical and torsional modes relative

to the pylon height. These frequencies experience the largest variations for short pylon

heights (in particular torsional mode T1). In terms of modal masses, those of the vertical

modes remain fairly constant with the height of the pylon whereas those of the mode T1

are considerably modified.

The analysis of the accelerations recorded at the deck of CSFs with shorter or higher

pylons shows that the amplitudes of peak and 1s-RMS accelerations tend to increase

with height and this increment is more pronounced for 1s-RMS than peak movements, as

shown in Figure 7.22 (the highest pylon produces 1s-RMS 25% larger and the shortest

38% smaller). These results are related to the inclination of the stays at the main span,

199


which increase the stiffness of the deck with a larger height of the pylon.

0 10 20 30 40 50 600

0.5

1

1.5

2

Peak v

ert

ical

acc.

[m/s

2]

0 10 20 30 40 50 600

0.5

1

1.5

2


1sR

MS

acc.

[m/s

2]

0.2 0.3 0.4 0.50.5

0.75

1

1.25

1.5

hp/L

m

acc /

acc

basic

Peak acc.

1sRMS acc.

hp/L

m = 0.25

hp/L

m = 0.30

hp/L

m = 0.40

hp/L

m = 0.45

hp/L

m = 0.36

Figure 7.22: Vertical service response of the CSF deck according to pylon height hp: (a)peak and 1s-RMS vertical accelerations and (b) comparison of maximum absolute peak and1s-RMS accelerations to those of the benchmark case acc 0.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

25

50

75

100

amax,Pi

/ amax,P

% P

edestr

ians

50%

25%

5%

amax,P

= 1.54 m/s2

hp/L

m = 0.25

hp/L

m = 0.30

hp/L

m = 0.40

hp/L

m = 0.45

hp/L

m = 0.36

Figure 7.23: Vertical accelerations felt by users amax,Pi compared to amax,P . Curvesdefined for CSFs with shorter or higher pylons.

When considering the accelerations noticed by users while walking, depicted in Fig-

ure 7.23, the footbridges with pylon heights most similar to 0.36Lm provide users with a

very similar experience, whereas the footbridges with tallest or shortest pylons consider-

ably increase or reduce these magnitudes. Hence, shorter pylons improve the performance

of CSFs in the vertical direction when compared to that of the reference model.

7.3.9 Inclination of pylon

Cable-stayed footbridges are commonly designed with vertical pylons, however on few

occasions these are inclined in the longitudinal direction (more frequently towards the

side span than the main span). Under static loads, the inclination of the pylon affects

the main span deflections, as represented in Figure 7.24(a). According to this figure, the

maximum deflection takes place with a vertical pylon although an inclination towards the

side span or the main span reduce this magnitude very lightly.

As far as the dynamic characteristics are concerned, the footbridge vibration charac-

teristics are very sensitive to the inclination of the pylon (Figure 7.24(b)). Both vertical

and torsional modes are affected and possess higher frequencies when leaning towards the

200


main span (except for the mode V1 and a new vertical mode with a single half-wave V1b).

In relation to modal masses, those related to the modes V3 and T2 are larger for pylons

inclined towards the side span and smaller for pylons inclined towards the main span.

1.2

0.9

1.0

um

ax /

um

ax,0

-20 -10 10

Tower inclination

200.0

1.1

umax

a

(-)

(+)

a [º]

(a)

-20 -10 10

1.0

2.0

3.0

4.0

V1

V2

V3

T1

T2

5.0

V1b

Fre

qu

ency [

Hz]

Tower inclination

200.0

a [º]

(b)

Figure 7.24: Static and dynamic behaviour of the CSF according to pylon inclination α:(a) main span maximum static deflections umax and (b) frequencies [Hz] of vertical andtorsional modes.

Under the effects of continuous streams of pedestrians, the deck develops peak and

1s-RMS accelerations with amplitudes that are larger as the inclination towards the main

span increases, as described in Figure 7.25 (the variations of the peak accelerations are

more modest). For footbridges where the pylon is inclined towards the main span, which

develop the largest peak accelerations at the deck, pedestrians experience very similar

movements to those of the reference CSF whereas only a very reduced number of users

feel accelerations similar to those observed at the deck. The footbridges with pylons

towards the side span improve the comfort of users notably.

0 10 20 30 40 50 600

0.5

1

1.5

2

Peak v

ert

ical

acc.

[m/s

2]

0 10 20 30 40 50 600

0.5

1

1.5

2


1s−

RM

S

acc.

[m/s

2]

−20 −10 0 10 200.5

0.75

1

1.25

1.5

Tower inclination α [º]

acc/a

cc

0

−20º

−10º

10º

20º

BasicPeak acc.

1s−RMS acc.

Figure 7.25: Vertical service response of the CSF deck according to pylon inclination α:(a) peak and 1s-RMS vertical accelerations and (b) comparison of maximum absolute peakand 1s-RMS accelerations to those of the reference case acc0.

The magnitudes of the accelerations recorded at the deck are justified by the contri-

butions of modes V3 and T2, which increase when the pylon gains inclination towards the

main span (and the backstay becomes longer). However, these large torsional movements

are not noticed by pedestrians (thus comfort for users in these cases is very similar).

Hence, the inclination of pylons away from the main span improves the response in ser-

vice of CSFs in the vertical direction, particularly for the largest angles of inclination that

201


have been considered.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20

25

50

75

100

amax,Pi

/ amax,P

% P

edestr

ians

25%

50%

5%

amax,P

= 1.54 m/s2 α = −20º

α = −10º

α = 10º

α = 20º

α = 0º

Figure 7.26: Vertical accelerations felt by users amax,Pi compared to amax,P . Curvesdefined for CSFs with pylons inclined towards the side span or main span.

7.3.10 Shape of the pylon

The pylon is the structural element that primarily carries the permanent and live loads

of the cable-stayed footbridge to the ground. Its shape influences the arrangement of the

cable system (inclination of the cables in relation to the deck) as well as the deflection

of the deck to vertical and horizontal loads. As highlighted in Chapter 2, cable-stayed

footbridges usually adopt mono-pole pylons. However, according to costs, foundation

characteristics and aesthetic considerations, cables can be supported by pylons with two

free-standing poles (‘H’), a portal shape or an ‘A’ shape, among others (see Figure 7.27).

(a) (b) (c) (d)

Figure 7.27: Shapes of CSF py-lons: (a) mono-pole pylon, (b) twofree-standing poles pylon, (c) portalshape pylon, (d) ‘A’ shape pylon.

Table 7.3: Frequencies [Hz] ofvertical and torsional modes of theCSF according to pylon shape.

Pylon V1 V2 V3 V3b T1 T2

(a) 0.98 1.96 3.18 2.94 3.75

(b) 0.97 1.99 3.20 2.50 4.64

(c) 1.01 1.98 3.04 3.24 2.63 4.71

(d) 1.02 1.99 3.20 2.91 4.73

The footbridge with an ‘H’ shape pylon allows the deck to describe deflections under

eccentric loads that are larger than those observed at other pylon shapes. Furthermore,

these pylon shapes have two supports of the deck at the pylon section, which is related

to the frequencies of the first torsional modes of the footbridge in each case (described in

Table 7.3).

Under pedestrian dynamic loads, the footbridges with ‘H’, portal, and ’A’ pylons

experience maximum vertical accelerations which are 50%, 20%, and 30% larger than

that with a mono-pole pylon (Figure 7.28). The results of pylons (b) and (c) are due

to the additional contribution of torsional modes, whereas in the last case is due to the

higher participation from both vertical and torsional modes.

Despite these large differences in the magnitudes of the accelerations recorded at the

202


0 10 20 30 40 50 600

1

2

Peak v

ert

ical

acc.

[m/s

2]

0 10 20 30 40 50 600

0.5

1

1.5


1s−

RM

S

acc.

[m/s

2]

I H Portal A0.5

1

1.5

2

Pylon shape

acc /

acc

0

Peak acc.

1s−RMS acc.

H

Portal

A

I

Figure 7.28: Vertical service response of the CSF deck according to pylon shape: (a)peak and 1s-RMS vertical accelerations and (b) comparison of maximum absolute peak and1s-RMS accelerations to those of the benchmark case acc0.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

25

50

75

100

amax,Pi

/ amax,P

% P

edestr

ians

’I’ shape (basic)

’H’ shape

Portal shape

’A’ shape

50%

25%

5%

amax,P

= 1.54 m/s2

Figure 7.29: Vertical accelerations felt by users amax,Pi compared to amax,P . Curvesdefined for CSFs with different pylon shapes.

deck, the differences related to the accelerations felt by different groups of users are

significantly smaller and similar to those of the reference footbridge (see Figure 7.29).

This is due to the fact that the differences in accelerations at the deck, which are due to

an enhancement of the torsional response, are not perceived in such a large manner by

the pedestrians, as many of them walk in deck areas which are not very eccentric.

Hence, the performance in service of the CSF is not strongly related to the type of

pylon shape, despite the considerable differences in torsional movements experienced in

each case.

7.3.11 Cable system: anchorage spacing

The spacing between the anchorages of the stay cables at the deck influences the mech-

anisms that transmit the loads that are applied on the deck. In relation to the dynamic

characteristics, the frequencies of the main modes are not significantly modified with the

cable spacing (see Figure 7.30(b)). Nevertheless, the model with the smallest cable spac-

ing has a lower participation from torsional and vertical modes and the footbridge with

largest cable distances has a larger contribution of torsional modes.

203


6.0 7.0 9.0

1.0

2.0

3.0

4.0

V1

V2

V3

T1

T25.0

Fre

qu

ency [

Hz]

Cable distance Dc [m]

10.08.0Dp

Dp

Dc

Dc

(a) (b)

Figure 7.30: (a) Cable-stayed footbridge geometry according to anchorage of stays and(b) frequencies [Hz] of vertical and torsional modes.

In service, the different participations from vertical and torsional modes are related

to the peak accelerations that are observed for each case at the deck (see Figure 7.31(b))

and to the accelerations felt by users (see Figure 7.32). The footbridges with lower

participation from different vertical modes (cable spacing of 6 m or 8 m) improve the

experience of pedestrians whereas the footbridge with differences in torsional modes (cable

spacing of 10 m) produces the less important reduction of discomfort.

6 7 8 9 100.5

0.6

0.7

0.8

0.9

1

1.1


acc /

acc

0

0 10 20 30 40 50 600

0.5

1

1.5

2

Peak v

ert

ical

acc.

[m/s

2]

0 10 20 30 40 50 600

0.5

1

1.5

2


1s−

RM

S

acc.

[m/s

2]

Peak acc.

1s−RMS acc.

6m

8m

10m

7m (basic)

Figure 7.31: Vertical service response of the CSF deck according to cable anchorage dis-tance: (a) peak and 1s-RMS vertical accelerations and (b) comparison of maximum absolutepeak and 1s-RMS accelerations to those of the benchmark case acc0.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

25

50

75

100

amax,Pi

/ amax,P

% P

edestr

ians

Dc = 6m

Dc = 8m

DC = 10m

DC = 7m

50%

25%

5%

amax,P

= 1.54 m/s2

Figure 7.32: Vertical accelerations felt by users amax,Pi compared to amax,P . Curvesdefined for CSFs with alternative stays anchorage spacing.

The level of accelerations at the deck are quite sensitive to the cable spacing. This

204


is due to the change of the contribution of different vertical and torsional modes in the

response, which is induced as a consequence of the change of the modal frequencies. Never-

theless, a specific analysis is required and it is not possible to define a clear tendency with

the change of this parameter (cable spacing) as occurs with other parameters previously

assessed.

7.3.12 Cable system: transverse inclination of cables

Figure 7.33 illustrates the variation of the maximum main span deflection under static

loads and the first vertical and torsional frequencies according to the lateral inclination of

the pylons. Statically, the larger inclination of the pylons allows higher vertical deflections

of the deck (due to the smaller vertical component of the stays). In terms of the dynamic

characteristics, the frequencies of the first vertical and torsional modes are not modified

with modest increments of this inclination. Nevertheless, a larger inclination of the cables

in the transverse direction is related to a larger the projection of the first torsional mode

T1 in vertical direction and a smaller the contribution of the second torsional mode T2.

0.95

1.0

um

ax /

um

ax,0

0.0 10

Lateral inclination [º]

155

1.05

a

a

Figure 7.33: Maximum static de-flections umax at the main span ac-cording to lateral inclination of ‘H’pylon.

Table 7.4: Frequencies [Hz] ofvertical and torsional modes of theCSF according to pylon lateral in-clination.

Inclination V1 V2 V3 T1 T2

‘I’ shape 0.98 1.96 3.18 2.94 3.75

α = 0o 0.97 1.99 3.20 2.50 4.64

α = 5o 0.96 1.99 3.20 2.50 4.65

α = 10o 0.95 1.98 3.19 2.49 4.64

α = 15o 0.93 1.96 3.18 2.50 4.63

0 5 10 15 200.8

0.85

0.9

0.95

1

1.05

1.1

Lateral inclination α [º]

acc /

acc

0

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

Peak v

ert

ical

acc.

[m/s

2]

0 10 20 30 40 50 600

0.51

1.52

2.5


1s−

RM

S

acc.

[m/s

2]

5º

10º

15º

0º Peak acc.

1s−RMS acc.

Figure 7.34: Vertical service response of the CSF deck according to pylon lateral inclinationα: (a) peak and 1s-RMS vertical accelerations and (b) comparison of maximum absolutepeak and 1s-RMS accelerations to those of the benchmark case acc0.

Under the dynamic loads of pedestrians, moderate inclinations (α ≤ 10o) of the two

free standing pylons have a very modest effect on the magnitudes of peak vertical accel-

erations (see Figure 7.34) and only larger inclinations give rise to modest decrements of

205


the accelerations (1s-RMS accelerations are 0.10 times smaller). These light changes in

movements recorded at the deck explain the moderate differences of the magnitudes of

accelerations felt by different proportions of pedestrians during their passage (see Fig-

ure 7.35). Thus, the transverse inclination of the tower has a modest effect on the vertical

response of the CSF (in comparison to that of the base-case CSF with vertical axis pylon).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20

25

50

75

100

amax,Pi

/ amax,P

% P

edestr

ians

α = 0º

α = 5º

α = 10º

α = 15º

50%

25%

5%

amax,P

= 1.54 m/s2

Figure 7.35: Vertical accelerations felt by users amax,Pi compared to amax,P . Curvesdefined for CSFs with stays laterally inclined.

7.3.13 Geometry of deck: deck width

The deck width of footbridges is largely related to the expected usage of the structure.

However, this magnitude determines the flexural stiffness (mainly in transverse direction),

the mass of the deck and the transverse inclination of the stay cables for some pylon types.

These parameters are related to the dynamic behaviour of the bridge, in particular its

torsional modes, as described by the modal frequencies given in Figure 7.36.

4.0 5.0 6.0

1.0

2.0

3.0

4.0

V1

V2

V3 T1

T2

5.0

Fre

qu

ency [

Hz]

wd [m]

Trans. deck sec.: w

Figure 7.36: Dynamic behaviour of the CSF according to deck width: frequencies [Hz] ofvertical and torsional modes.

In service, the CSFs with wider decks experience peak and 1s-RMS accelerations of

considerably smaller magnitude, as depicted in Figure 7.37. The peak accelerations at

CSFs with 5 m and 6 m deck widths are respectively 25% and 40% smaller than that for

a CSF with a 4 m deck width. The accelerations felt by users reflect a similar situation

to that observed for the deck accelerations, as depicted in Figure 7.37 (CSFs with wider

decks improve the comfort for users considerably). Therefore, widening the deck width

enhances the vertical performance in service of the footbridge due to the increment of the

deck mass that this modification introduces.

206


0 10 20 30 40 50 600

0.5

1

1.5

2

Peak v

ert

ical

acc.

[m/s

2]

0 10 20 30 40 50 600

0.5

1

1.5

2


1s−

RM

S

acc.

[m/s

2]

4 5 60.4

0.6

0.8

1

wd [m]

acc /

acc

0

Peak acc.

1sRMS acc.

5.0 m

6.0m

4.0m (basic)

Figure 7.37: Vertical service response of the CSF deck according to deck width dimension:(a) peak and 1s-RMS vertical accelerations and (b) comparison of maximum absolute peakand 1s-RMS accelerations to those of the benchmark case acc 0.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

25

50

75

100

amax,Pi

/ amax,P

% P

edestr

ians

50%

25%

5%

amax,P

= 1.54 m/s2

w = 5.0m

w = 6.0m

w = 4.0m (basic)

Figure 7.38: Vertical accelerations felt by users amax,Pi compared to amax,P . Curvesdefined for CSFs with wider decks.

7.3.14 Side span length

The length of the side span Ls has a significant effect on the magnitude of the maximum

static deflections at the main span, of length Lm, (as illustrated in Figure 7.39(a)). In

terms of the dynamic behaviour, the vertical and torsional modes are significantly affected

by this dimension, as shown in Figure 7.39(b) (for side span lengths larger than 0.3Lm,

the CSF presents an additional vertical mode with three antinodes at the main span,

V3b).

For the CSF with a side span of length 0.3Lm, the dynamic response is smaller due

to the longer side span length and the larger masses for the vertical modes with highest

participation in the vertical movement. At the CSF with longer side span, the additional

mode V3b increases this response at the main span (results represented in Figure 7.40)

and at the side span (these are 5 times larger than those of other CSFs at the same

region). Nevertheless, there are clear ways of controlling the accelerations at the side

span without compromising the efficiency of the cable-stayed footbridge.

If comfort is appraised in terms of the accelerations felt by users while crossing the

main span, the largest accelerations felt by the users in each case are proportional to the

207


0.8

0.9

1.0

1.1

0.2 0.3 0.4

um

ax /

um

ax,0

umax

Ls

Lm

Ls Lm

(a)

0.2 0.3 0.4

1.0

2.0

3.0

4.0

V1

V2

V3

T1

T25.0

V1b

Fre

qu

ency [

Hz]

Ls Lm

(b)

Figure 7.39: Static and dynamic behaviour of the CSF according to side span length Ls:(a) main span maximum static deflections umax and (b) frequencies [Hz] of vertical andtorsional modes.

0 10 20 30 40 50 600

0.5

1

1.5

2

Peak v

ert

ical

acc.

[m/s

2]

0 10 20 30 40 50 600

0.5

1

1.5

2


1s−

RM

S

acc.

[m/s

2]

0.15 0.2 0.25 0.3 0.35 0.4 0.45

0.8

1

1.2

1.4

1.6

Ls/L

m

acc /

acc

0

0.30 Lm

0.40 Lm

0.20 Lm

Peak acc.

1s−RMS acc.

Figure 7.40: Vertical service response of the CSF deck according to side span length Ls:(a) peak and 1s-RMS vertical accelerations and (b) comparison of maximum absolute peakand 1s-RMS accelerations to those of the reference case acc 0.

accelerations recorded at the main span of the bridge: 25% of the users feel accelerations

equal to or larger than 0.87, 0.65 or 1.00amax,P at CSFs with side span lengths of 0.2, 0.3

or 0.4Ls respectively.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

25

50

75

100

amax,Pi

/ amax,P

% P

edestr

ians

Ls/L

m = 0.3

Ls/L

m = 0.4

Ls/L

m = 0.2 (basic)

50%

25%

5%

amax,P

= 1.53 m/s2

Figure 7.41: Vertical accelerations felt by users amax,Pi compared to amax,P . Curvesdefined for CSFs with longer side spans.

Hence, a longer side span length improves the performance in service of the CSF unless

additional modes change this general response. The side spans with a length 30% that

208


of the main span tend to be the optimal configuration in order to reduce the vertical

accelerations felt by the pedestrians.

7.4 Strategies to improve the lateral dynamic performance of

1T-CSFs in service

The magnitude of the lateral forces transmitted by pedestrians while crossing a foot-

bridge have a lower order of magnitude in comparison to the loads that they transmit

vertically. Similarly, the footbridge lateral accelerations that these loads give rise to and

the magnitudes of these that are considered serviceable are smaller than those in the verti-

cal direction. Nonetheless, these lateral loads may increase significantly when pedestrians

become engaged with the structure lateral movements (as described in Section 2.3.4).

As Section 2.3.4 outlines, these loads and their effects on bridges have only been con-

sidered relatively recently and introduced in their analysis. Consequently there is a limited

understanding of the key structural parameters that ensure an adequate performance of

a footbridge in this direction. Hence, the following sections explore the performance of

single pylon cable-stayed footbridges in service and the consequences that different design

characteristics have on that.


In the lateral direction, traffic flows of commuters with medium-high densities (0.6

ped/m2) that cross CSFs with articulation schemes such as those described in Sec-

tion 7.3.1 generate peak and 1s-RMS lateral accelerations with amplitudes depicted in

Figure 7.42(a,b).

0 0.25 0.5 0.75 1 1.25 1.50

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

0 10 20 30 40 50 600

0.2

0.4


Peak late

ral

acc.

[m/s

2]

(a) (b) (c) (d)

1

20

Articulation scheme

acc /

acc

0 Peak acc.

1s−RMS acc.

5%

25%

50%

amax,P

= 0.18 m/s2

POT (b)

POT (c)

LEBs+SK (a)

(a) (b)

Figure 7.42: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the deck of CSFs with support schemes (a), (c) and (d); and (b) lateral ac-celerations felt by users.

Scheme ‘POTs(c)’ worsens the lateral response of the footbridge (lateral accelerations

are 1.5 times higher than those of the CSF with LEBs+SK), whereas a CSF with scheme

209


‘POTs(d)’ improves those movements (peak lateral accelerations are decreased by a factor

of 0.5). The CSF with LEBs produces the most drastic changes, as it causes the bridge

to exhibit peak lateral movements of a higher order of magnitude (2.9 m/s2 at x = 60 m

that would continue to increase with time, see Figure 7.43, as these lateral movements

are unstable due to an engagement of pedestrians with the bridge lateral movement).

0 10 20 30 40 50 600

1

2

3


Peak late

ral

acc.

[m/s

2]

0 20 40 60 80 100

−2

0

2

Time [s]

Late

ral acc.

[m

/s2]

x =

60.0

m(a) (b)

Figure 7.43: (a) Absolute peak lateral accelerations recorded at the deck of CSFs withLEBs; (b) time history acceleration at x = 60 m developed at the CSF with LEBs.

The larger response of the CSF with ‘POTs(c)’ is explained by the horizontal rota-

tion and displacement of the deck at the abutment where POTs have free longitudinal

movements (for this reason mode L1 has a higher contribution to the movement). The

smaller lateral movements of the CSF with scheme ‘POTs(d)’ are due to the restraint of

the transverse rotations of the deck at the abutments (modes L1 and L2 appear at higher

frequencies). Finally, the unstable lateral movement of the CSF with LEBs is caused

by the characteristics of the first lateral modes (mode L1 has an effective mass 4 times

smaller than that of the CSF with LEBs+SK and a frequency below 1.0 Hz).

Table 7.5: First lateral vibration modes [Hz] of CSFs according to support arrangement.

Supports L1 L2

(a) LEBs 0.78 1.10 (L1b)(b) LEBs+SK 2.16 7.34

(c) POTs 2.76 9.38(d) POTs 3.71 10.61

The analysis of the footbridge performance in the lateral direction on the basis of the

accelerations felt by users points towards a slightly different assessment of the support

arrangements (see Figure 7.42(c)).

The accelerations felt by users walking on the footbridge with ‘POTs(d)’ is effectively

better than that of users using the benchmark footbridge. However, the footbridge with

‘POTs(c)’ provides users with a better comfort than that described by the deck move-

ments. This difference is explained by the fact that in this case peak accelerations take

place during very short times at certain locations, hence only very few pedestrians notice

the largest accelerations.

Thus, for cable-stayed footbridges with a pylon, the use of two LEBs at each abutment

as deck articulations should be disregarded as they produce inadequate lateral resonant

accelerations in service. The rest of the support schemes result in lateral movements

210


that are considered acceptable by design guidelines: the ‘POTs(d)’ scheme enhances the

comfort for users and the ‘POTs(c)’ scheme worsens the situation for those sensing the

largest accelerations (25% of the users).

7.4.2 Area of the backstay cable

For dynamic response, a larger elongation of the backstay causes lateral modes to have

smaller lateral frequencies and a larger projection of the torsional mode T1 in the lateral

direction. Larger backstay areas describe lateral modes with very similar characteristics.

These dynamic characteristics explain the magnitude of the lateral response of the

1T-CSF with alternative backstay dimensions. As depicted in Figure 7.44, the footbridge

with the smallest backstay area reproduces the largest lateral response (1.7 times that

of the benchmark bridge) whereas the other cases describe fairly similar lateral accelera-

tions. The differences between the lateral accelerations felt by walking users on a bridge

with a backstay 0.5ABS,0 and the rest are not as large as those of the deck accelerations

(Figure 7.44(b)). In this case, 75% of the users feel lateral accelerations below 1.25amax,P

compared to 0.8amax,P for the other considered cases.

0 10 20 30 40 50 600

0.2

0.4


Peak v

ert

ical

acc.

[m/s

2]

0.5 1 1.5 2 2.50.5

1

1.5

2

ABS

/ ABS,0

acc /

acc

0

0 0.25 0.5 0.75 1 1.25 1.50

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

0.5 ABS,0

1.5 ABS,0

2.0 ABS,0

2.5 ABS,0

ABS,0

Peak acc.

1s−RMS acc.

5%

25%

50%

amax,P

= 0.18 m/s2

(a) (b)

Figure 7.44: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the CSF deck according to backstay area and (b) lateral accelerations felt byusers.

Hence, a CSF with insufficient backstay area experiences larger lateral accelerations

in service. Nevertheless if sufficient area is provided to the backstays, additional areas

would not allow the control of the lateral accelerations. Irrespective of the magnitude

of this parameter, pedestrians do not become engaged with the deck movements and

the magnitudes of these are acceptable in service (peak lateral accelerations are below

0.3 m/s2).

7.4.3 Area of the main span stays

In terms of the dynamic behaviour of the CSF, the area of the stays changes the

characteristics of modes L1 and T1 similarly to the backstay areas.

211


This effect of the main stays on the CSF dynamic behaviour provides an explanation

for the lateral movements in service illustrated in Figure 7.45. The footbridge with the

smallest main stays develops peak responses that are 5 times larger than those of the basic

footbridge, whereas larger main stay areas lightly decrease the lateral accelerations. The

lateral accelerations felt by walking pedestrians are similar to the values recorded at the

deck. At the CSF with the smallest stays, 75% of the users notice lateral accelerations

below 4.0amax,P , as opposed to 0.6-0.8amax,P at the rest of the CSFs.

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1


Peak v

ert

ical

acc.

[m/s

2]

0.5 1 1.5 2 2.50

2

4

6

AS /A

S,0

acc /

acc

0

0 0.5 1 1.5 20

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

0.5 AS,0

1.5 AS,0

2.0 AS,0

2.5 AS,0

AS,0

Peak acc.

1s−RMS acc.

50%

25%

5%

amax,P

= 0.18 m/s2

(a) (b)

Figure 7.45: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the CSF deck according to main span stay area and (b) lateral accelerations feltby users.

Hence, the lateral accelerations do not vary significantly with the area of the main

stay cables. Nevertheless, if an insufficient area of the main stay cables is provided then

the lateral accelerations would increase drastically (although these would not correspond

to an unstable event).

7.4.4 Material of stays: bars vs strands for the stay cables

Despite the differences in the material characteristics of bars and stay cables, CSFs

with stays of one or the other type have first lateral modes with very similar characteristics

(frequencies and modal masses).

This consideration justifies the fact that both footbridges describe lateral accelerations

of similar amplitudes and that pedestrians notice similar lateral accelerations when using

one or the other footbridge. Therefore, similarly to vertical accelerations, the use of

bars instead of stay cables does not modify the response in service of the cable-stayed

footbridge.


An increment of the bottom flange thickness of the steel girders leads to an increment

of the lateral stiffness of the deck. This modification changes the first lateral modes as

well as the first torsional modes, which develop a component in the lateral direction (in

212


particular the mode T1).

In agreement with these dynamic characteristics, the deck of CSFs with deeper flange

thickness trigger lateral accelerations 1.5-1.7 times larger than those of the reference CSF

under the passage of the same traffic of pedestrians (see Figure 7.46(a)). Furthermore,

the CSF with a flange thickness 1.8t bf,0 has an additional lateral contribution of mode

V4 due to the coincidence in frequency magnitude of this mode with that of T3. This

additional effect explains the larger lateral accelerations noticed by users walking on that

bridge (see Figure 7.46(b)).

Therefore, an increment of the bottom flange thickness of the steel girders is not

a beneficial measure for the lateral accelerations of these footbridges since modes that

originally did not contribute to the movement in this direction have larger importance

with this modification.

0 0.5 1 1.50

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

0 10 20 30 40 50 600

0.2

0.4


Peak v

ert

ical

acc.

[m/s

2]

1.0 1.4 1.8 2.2

1

1.5

2

tbf

/ tbf,0

acc /

acc

0

Peak acc.

1s−RMS acc.

1.4 tbf,0

1.8 tbf,0

2.2 tbf,0

tbf,0

5%

25%

50%

amax,P

= 0.18 m/s2

(a) (b)

Figure 7.46: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the CSF deck according to bottom flange thickness and (b) lateral accelera-tions felt by users.


The concrete slab of the CSF is the heaviest element of the structure and both vertical

and lateral vibration modes of the bridge change considerably according to this geomet-

rical characteristic. In terms of the dynamic behaviour, a CSF with larger slab depth

possesses first lateral modes at lower frequencies (from 2.16 Hz at the reference CSF to

2.02 Hz at the CSF with depth 2.5 times larger) and increases considerably the modal

masses for these lateral modes. The magnitude of this modal mass explains the lateral

accelerations that the deck describes under the passage of a pedestrian flow.

Figure 7.47(a) represents the deck accelerations as a function of the slab depth: the

peak and 1s-RMS lateral accelerations are inversely proportional to the slab depth. In

terms of the lateral accelerations felt by users, this positive effect on comfort is clear as

well (as seen in Figure 7.47(b)), as the maximum accelerations noticed by 75% of the

users decrease from 0.8amax,P with a slab depth of 0.2 m to 0.2amax,P for a CSF with a

213


slab depth of 0.5 m. Therefore, an increment of the deck mass of the CSF is an effective

manner of drastically improving the lateral performance of the structure in service.

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

0 10 20 30 40 50 600

0.2

0.4


Peak v

ert

ical

acc.

[m/s

2]

1 1.5 2 2.50

0.5

1

tc / t

c,0

acc /

acc

0

Peak acc.

1s−RMS acc.

1.5 tc,0

2.0 tc,0

2.5 tc,0

tc,0

25%

50%

5%

amax,P

= 0.18 m/s2

(a) (b)

Figure 7.47: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the CSF deck according to concrete slab depth and (b) lateral accelerationsfelt by users.

7.4.7 Transverse section of the pylon

When increasing the cross section of the pylon, the deflections of this structural element

under horizontal loads and its rotation due to torques are reduced. Both these changes

cause the alteration of the first lateral modes of the cable-stayed footbridge as well as

its first torsional modes. The CSFs with larger pylons (1.3Dt,0 or 1.7Dt,0) have modes

L1 and T1 at higher frequencies (the mode T1 adopts a larger component in the lateral

direction). However, for very large diameters (above 2Dt,0 approximately) the bridge

develops a lateral mode L1 at frequencies lower than those of smaller pylon diameters

(2.12 Hz in comparison to 2.59 Hz at 1.7Dt,0). Furthermore the torsional mode T1 of the

CSF with widest pylon has a smaller lateral projection in comparison to those of narrower

pylon diameters. These modal characteristics are related to the magnitudes of the lateral

deck accelerations and to the accelerations noticed by walking pedestrians described in

Figure 7.48(a,b).

If instead of changing the diameter of the pylon, the steel thickness is increased, the

changes of the dynamic behaviour of the CSF are smaller and lateral accelerations are

very similar.

Hence, the alteration of the pylon transverse section does not improve the lateral

response of the CSF and it can even increase the magnitude of lateral accelerations de-

pending on the characteristics of modes L1 and T1.

7.4.8 Pylon height

The pylon height governs the shape and the characteristics of the lateral and torsional

modes. For the CSFs with shorter pylons, the first lateral mode L1 has a lower frequency

214


0 10 20 30 40 50 600

0.2

0.4

Structure length

Peak v

ert

ical

acc.

[m/s

2]

1 1.5 2 2.50

2

4

Dt / D

t,0

acc /

acc

0

0 0.5 1 1.5 2 2.50

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

Peak acc.

1s−RMS acc.

1.3 Dt,0

1.7 Dt,0

2.5 Dt,0

Dt,0

50%

25%

5%

amax,P

= 0.18 m/s2

(a) (b)

Figure 7.48: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the CSF deck according to pylon diameter and (b) lateral accelerations feltby users.

(e.g., L1 has a frequency of 1.89 Hz at the footbridge with pylon height 0.25Lm and

2.15 Hz at the footbridge with 0.4Lm) and smaller modal mass due to the shorter height

of the pylon, shorter length of the stays and their inclination in relation to the deck (the

stays provide a smaller lateral stiffness to the deck). Furthermore, these characteristics

affect as well the first torsional mode T1, which has a larger component in the lateral

direction when the footbridge has a shorter pylon.

0 0.5 1 1.5 20

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

0 10 20 30 40 50 600

0.2

0.4


Peak v

ert

ical

acc.

[m/s

2]

0.2 0.3 0.4 0.50.5

1

1.5

2

hp / L

m

acc /

acc

0

hp/L

m = 0.25

hp/L

m = 0.30

hp/L

m = 0.40

hp/L

m = 0.45

hp/L

m = 0.36

Peak acc.

1s−RMS acc.25%

5%

50%

amax,P

= 0.18 m/s2

(a) (b)

Figure 7.49: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the CSF deck according to pylon height and (b) lateral accelerations felt byusers.

Based on these dynamic characteristics, the footbridges with a lower pylon height

have lateral modes L1 with effective masses that allow larger lateral vibrations than the

215


footbridges with pylon heights similar or larger than 0.36Lm, as illustrated by the results

of Figure 7.49(a) (where the first footbridges develop lateral accelerations 2 times larger

than those of the second group of footbridges). The accelerations noticed by walking users

describe similar conclusions. Therefore, lower pylons increase the lateral accelerations of

CSFs whereas higher pylons do not enhance the comfort for users as this modification

does not have a large impact on the lateral modes of the deck.

7.4.9 Tower longitudinal inclination

With the longitudinal inclination of the pylon, its height remains fairly similar whereas

the length of either the backstay or the main span stay cables are increased or reduced

in a significant manner in each case. The longer cables are related to a smaller stiffness

in the lateral direction and thus to lateral modes L1 with a smaller frequency. The CSFs

with pylons inclined 20 towards the side span (‘PS’) or main span (‘PM’) have a mode

L1 with frequencies 2.07 and 2.04 Hz in comparison to 2.16 Hz of the CSF with vertical

or nearly vertical pylons (lateral modal masses are fairly similar in all the cases). The

CSFs with pylons most inclined (either to the side span or the main span) develop as

well torsional modes T1 with a larger component in the lateral direction (in particular

the CSF with largest inclination towards the side span).

0 10 20 30 40 50 600

0.2

0.4


Peak v

ert

ical

acc.

[m/s

2]

−20 −10 0 10 200.5

1

1.5

2

2.5

Pylon inclination α [º]

acc /

acc

0

0 0.5 1 1.5 20

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

α = −20º

α = −10º

α = 10º

α = 20º

α = 0º

Peak acc.

1s−RMS acc.

50%

amax,P

= 0.18 m/s2

(a) (b)

Figure 7.50: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the CSF deck according to pylon longitudinal inclination and (b) lateral ac-celerations felt by users.

Based on these dynamic characteristics, the CSFs with alternative longitudinal incli-

nations of the pylon describe lateral accelerations as illustrated in Figure 7.50(a,b). The

CSF with PS pylons (high inclination) describes accelerations twice as large as those of

the footbridge with a vertical pylon (75% of the users feel movements below 1.6amax,P ).

The CSF with a pylon moderately inclined towards the main span describes lateral move-

ments similar to the conventional case whereas the CSF with a PM pylon highly inclined

describes accelerations 1.5 times larger (and 75% of the users notice movements below

1.3amax,P instead of 0.8amax,P ).

216


Thus, the inclination of the pylon is not an efficient geometric characteristic that can

be changed in order to reduce the magnitude of the lateral accelerations. Instead, this

alternative can considerably increase the lateral accelerations although these would never

be caused by a full engagement of users with the deck lateral movements.

7.4.10 Pylon shape

Pylons with ‘H’, portal or ‘A’ shapes are considerably more rigid in the transverse

direction than a free standing mono-pole pylon. Related to this stiffness, the first lateral

modes of these cable-stayed footbridges have frequencies that are considerably larger than

the conventional CSF (2.3, 6.7 and 5.9 Hz for CSFs with ‘H’, portal or ‘A’ pylon shapes

respectively, in comparison to 2.16 Hz at the reference CSF). However, similarly to the

reference CSF, both the portal and ‘A’ shapes have torsional modes with an important

projection in the lateral direction. The pylon with an ‘H’ shape does not develop torsional

modes with these shapes.

The contribution in the lateral direction of the torsional modes, rather than the fre-

quency of the lateral mode L1, is related to the peak lateral accelerations of the deck of

the CSF in each case. The CSF with an ‘H’ has lower peak lateral accelerations whereas

the CSF with a portal pylon has the largest peak accelerations (Figure 7.51(a)). However,

the accelerations felt by pedestrians, as seen in Figure 7.51(b), point out towards the fact

that the deck peak lateral accelerations of the CSFs with ‘H’, portal or ‘A’ pylons occur

during a limited time and that, in fact, lateral modes L1 with higher frequencies are re-

lated to lower accelerations felt by the users. Considering both effects, it is clear that the

shape of the pylon has a determinant effect on the lateral response and that any pylon

with larger lateral stiffness improves the CSF response in the lateral direction.

0 10 20 30 40 50 600

0.2

0.4


Peak late

ral

acc.

[m/s

2]

I H Portal A0

1

2

acc /

acc

0

0 0.2 0.4 0.6 0.8 1 1.20

20

40

60

80

100

amax,Pi

/ amax,P

Peak acc.

1s−RMS acc.

‘H’

Portal

‘A’

‘I’

5%

25%

50% amax,P

= 0.18 m/s2

(a) (b)

Figure 7.51: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the CSF deck according to pylon shape and (b) lateral accelerations felt byusers.

217


7.4.11 Transverse inclination

In previous cases, the inclination of the main span stays in the transverse direction

is very modest. However, the models analysed in this section have a pylon with an ‘H’

shape which permits inclining cables in the transverse direction (see Figure 7.4).

The transverse inclination of the stays modifies the lateral and torsional modes. Ir-

respective of this lateral inclination, the lateral mode L1 adopts a very similar frequency

(near 2.3 Hz). However, for very large inclinations (α > 10o) the modal mass for this

mode is drastically increased in comparison to those of lower inclinations. The lateral

inclination of the stays changes the characteristics of the torsional modes in a similar

manner (mode T1 has similar characteristics with low transverse inclinations and consid-

erably larger component in the lateral direction when this inclination is α > 10o).

The dynamic characteristics are related to the magnitudes of the lateral accelerations

of these CSFs. The footbridge with a pylon inclined 15o develops lateral accelerations 4

times larger than those of the footbridge with vertical ‘H’ pylon (Figure 7.52(a)). The

lateral accelerations felt by users exhibit even larger differences: 25% of the users on that

footbridge notice lateral accelerations 6 times larger than those noticed at the vertical ‘H’

pylon footbridge or 1.9 times larger than those noticed at footbridges where the pylon is

moderately inclined.

In comparison to the reference CSF, only the CSF with vertical stay cables improves

the performance in the lateral direction whereas those with a moderate or large transverse

inclination describe very similar or considerably larger lateral accelerations. Hence, the

transverse inclination of the stay cables does not improve the performance of the CSF in

service.

0 0.5 1 1.5 2 2.50

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

0 10 20 30 40 50 600

0.2

0.4


Peak v

ert

ical

acc.

[m/s

2]

0 5 10 150

1

2

3


acc /

acc

0 Peak acc.

1s−RMS acc.

α = 0º

α = 5º

α = 10º

α = 15º

’I’ pylon

25%

5%

50%

amax,P

= 0.18 m/s2

(a) (b)

Figure 7.52: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the CSF deck according to pylon transverse inclination and (b) lateral acceler-ations felt by users.

218


7.4.12 Cable anchorage distance

When modifying the separation between consecutive stay cable anchorages (distances

ranging from 6 m to 10 m), the lateral response developed by the cable-stayed footbridge

is not markedly changed. Only the bridge with a large distance between consecutive

anchorages generates larger movements: the deck accelerations are 1.25 times larger and

25% of the users feel accelerations 20% larger than those felt by pedestrians on the other

three footbridges. These modest differences are explained by the small modifications of

the first lateral and torsional modes introduced by these modifications (exclusively the

CSF with a cable separation of 10 m has a moderately larger lateral projection of the

mode T1). Thus, this characteristic does not have a substantial effect in the performance

of the CSF.

7.4.13 Geometry of deck: deck width

The transverse flexural stiffness of the deck, mass and the inclination of the stay

cables (in relation to the deck) are related to the magnitude of the deck width. With an

increment of this magnitude from 4 m to 5 m, the first lateral mode L1 is combined with

an important torsional component (this becomes a torsional mode for larger deck widths)

and has an effective mass that is smaller than that of mode L1 for CSFs with narrower

or wider deck.

These differences in the effective mass for mode L1 cause the bridge with a deck width

of 5 m to develop lateral accelerations larger than those of the CSFs with a deck width

of 4 m or 6 m (1.8 and 2.2 times larger respectively, see Figure 7.53(a)), since vertical

loads of pedestrians increase the lateral accelerations of the CSF (apart from their lateral

loads). The movements noticed by users have similar trends: 75% of the users walking

on the CSF with a deck width of 5 m notice movements that are 1.5 times larger than

those noticed by 75% of the users walking of the footbridge with a deck width of 4 m and

1.7 times larger than those of the same number of users walking on the bridge with the

widest deck (see Figure 7.53(b)).

Hence, in general, a larger deck width improves the magnitude of the lateral move-

ments. However, if lateral modes coincide with torsional modes (as occurs with the CSF

with a deck width of 5 m), the lateral movements may become larger despite the increment

of the deck stiffness and mass produced by the larger width.


In terms of the dynamic characteristics, a longer side span length decreases the mag-

nitude of the first lateral modes (from 2.16 Hz at the reference case to 1.88 Hz at the

longest side span) due to the longer length of the bridge that vibrates freely in the lateral

direction (i.e., the distance between deck sections at the abutments). This geometrical

characteristic changes as well the torsional modes. Additionally, the lateral mode L1 of

the CSF with a side span of length 0.3Lm has practically the same frequency of mode V2.

The coincidence of modes V2 and L1 in frequency at the CSF with a side span 0.3Lm

is the main cause for the moderately larger peak lateral accelerations of the deck as well as

those felt by walking pedestrians (Figure 7.54). At the CSF with longer side span results

219


0 0.5 1 1.5 20

25

50

75

100

amax,Pi

/ amax,P

% P

edestr

ians

0 10 20 30 40 50 600

0.2

0.4


Peak v

ert

ical

acc.

[m/s

2]

3.5 4 4.5 5 5.5 6 6.5

1

2

3

wd

acc /

acc

0 Peak acc.

1s−RMS acc.

wd = 5.0m

wd = 6.0m

wd = 4.0m

amax,P

= 0.18 m/s2

50%

25%

5%

(a) (b)

Figure 7.53: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the CSF deck according to deck width and (b) lateral accelerations felt byusers.

are practically the same as those at the reference CSF. Therefore, a moderate increment

of the side span length does not change the magnitude of the lateral accelerations of the

1T-CSF.

0 0.5 1 1.50

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

0 10 20 30 40 50 600

0.2

0.4


Peak v

ert

ical

acc.

[m/s

2]

0.15 0.2 0.25 0.3 0.35 0.4 0.450.6

0.8

1

1.2

Ls / L

m

acc /

acc

0 Peak acc.

1s−RMS acc.

0.30 Lm

0.40 Lm

0.20 Lm

5%

25%

50%

amax,P

= 0.18 m/s2

(a) (b)

Figure 7.54: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the CSF deck according to side span length and (b) lateral accelerations felt byusers.

7.5 Cable-stayed footbridges with long main span lengths

The most commonly constructed cable-stayed footbridges have main spans of approx-

imately 50 m length, although some footbridges of this typology have been constructed

with longer lengths (around 100 m), as indicated in Section 3.5.2.

220


The foregoing sections have appraised the effects of multiple design and geometry

modifications of cable-stayed bridges on the serviceability response of these medium span

footbridges. Thus, in order to validate the effect of the afore-mentioned characteristics on

the response of long span cable-stayed footbridges, the following paragraphs describe and

evaluate the effects of multiple parameters (those that develop the largest variations of the

bridge serviceability response) on the performance of long span cable-stayed footbridges

with a single pylon.

Specifically, the following sections describe: (a) the geometrical characteristics and

dynamic behaviour of representative cable-stayed footbridges with long main span lengths,

(b) their performance in service, under the passage of medium-heavy pedestrian flows

and (c) the impact on their serviceability response of some of the most effective design

measures previously evaluated for cable-stayed footbridges of medium span lengths.

7.5.1 Geometry of long span cable-stayed footbridges

Based on the wide range of geometric characteristics that engineers use for the design

of cable-stayed footbridges (described in Section 3.5.2), it is considered that footbridges

of this typology and main span lengths near 100 m are represented by depth-to-main span

length ratios of 1/100 and 1/200, a transverse deck section with two lateral steel girders

and a concrete slab of 0.20 m depth, and a pylon height of 0.36Lm.

Taking into consideration these dimensions, the actions used for their design in ULS

and the characteristics of the bridge materials (both outlined in Section 3.5.2), the cable-

stayed footbridges illustrated in Figure 7.55 correspond to representative footbridges of

this typology and this main span length.

The articulation of the deck of these bridges consists of POT bearings with a stati-

cally indeterminate arrangement (illustrated in Figure 7.2(d)) and, as well, it is simply

supported by the pylon. Arguments supporting this arrangement instead of others are

exposed in the following sections.

7.5.2 Dynamic characteristics of long span cable-stayed footbridges

As has been observed in the previous sections, the characteristics of the vibration

modes of the footbridges are critical for determining the vertical and the lateral responses

of the structure caused by the passage of pedestrian flows.

Table 7.6 describes the first nine modes of these cable-stayed footbridges and Fig-

ures 7.56 and 7.57 illustrate their modal shapes. In the vertical direction, the modes with

frequencies closest to the range of step frequencies used by pedestrians while walking are

V2, V3 and V4 or V5 for the slenderer CSF (for medium span cable-stayed footbridges

these correspond to basically modes V2 and V3). In the lateral direction, the long span

cable-stayed footbridges generate lateral modes with frequencies within the range consid-

ered critical (below 1.5 Hz) regardless of the deck depth, see Table 7.6, as opposed to

cable-stayed footbridges with medium span lengths where these are above 2.0 Hz.

221


HT=

43.5

Hs =

36.0

Hi =

7.5

Lm= 100.0Ls= 20.0

Ha

BS

CB#1CB#2

CB#3 CB#5 CB#7 CB#9 CB#11 CB#13CB#4 CB#6

CB#8CB#10

CB#11

Dp Dc Dp Dc Dc

Detail B-B

Detail B-B:

Dext = 0.60

0.5

0.5

0.20

w = 4.0

HEB 200

hgird

tflange,bot

Sec. A-A

Sec. A-A:

Characteristics CSFB htot/Lm = 1/100

hto

t

htot = 1.0

hgird = 0.8

tflange,top

tweb

tflange,bot = 0.0175

tflange,top = 0.0175

tweb = 0.015

Cable No. Strands Cable No. StrandsBS

CB#1CB#2CB#3CB#4CB#5

CB#7CB#8CB#9

CB#12CB#13

9512333

556632

CB#6

CB#11CB#10

3 1Characteristics CSFB htot/Lm = 1/200

htot = 0.5

hgird = 0.3

tflange,bot = 0.055

tflange,top = 0.025

tweb = 0.015


CB#1CB#2CB#3CB#4CB#5

CB#7CB#8CB#9

CB#12CB#13

10212333

556632

CB#6

CB#11CB#10

3 1

(composite section)

Figure 7.55: Geometric definition of the representative long span CSFs with transversesection depth Lm/100 and Lm/200. Dimensions in meters [m].

Table 7.6: Frequencies [Hz] of the vibration modes of long span CSFs according to theirdepth magnitude, where ‘VN’, ‘LN’ and ‘TN’ denote vertical, lateral and torsional modeswith N half-waves and ‘P’ denotes modes related to the pylon.

Deck depth Lm/100 Deck depth Lm/200

Mode No. Frequency Description Mode No. Frequency Description

1 0.45 P1 1 0.45 P1

2 0.79 V1 2 0.74 V1

3 1.29 L1 3 1.21 V2

4 1.41 V2 4 1.35 L1

5 1.99 T1 5 1.72 V3

6 2.05 V3 6 1.91 T1

7 2.52 T2 7 2.04 V4

8 2.72 V4 8 2.41 T2

9 2.73 T2 9 2.50 V5

222


V1, 0.79Hz

(a)

L1, 1.29Hz

(b)

V2, 1.41Hz

(c)

T1, 1.99Hz

(d)

V3, 2.05Hz

(e)

T2, 2.52Hz

(f)

Figure 7.56: First modal frequencies [Hz] of long span CSF with a deck depth of Lm/100.

V1, 0.74Hz

(a)

V2, 1.21Hz

(b)

L1, 1.35Hz

(c)

V3, 1.72Hz

(d)

T1, 1.91Hz

(e)

V4, 2.04Hz

(f)

Figure 7.57: First modal frequencies [Hz] of long span CSF with a deck depth of Lm/200.

7.5.3 Articulations of the deck

For the cable-stayed footbridges with a main span length of 100 m, deck articulations

such as those enumerated in Section 7.3.1 generate the lateral modal frequencies listed in

Table 7.7. For both deck depths (htot = Lm/100 and htot = Lm/200), only the support

arrangement ‘POTs(d)’, which restricts all the horizontal movements of the deck at the

supports on the abutments, are associated with first lateral modes above 1.1 Hz. The

others produce first lateral modes with frequencies very near 1.0 Hz.

These natural frequencies are related to the fact that only the ‘POTs(d)’ support

configuration allows the bridge to develop stable lateral accelerations under the passage of

pedestrian flows of medium-high density (the other supports generate lateral accelerations

that rapidly increase with time while the traffic is constant or movements that are far

beyond the acceptable range). Figure 7.58 represents the magnitudes of the peak vertical

and lateral movements of these long span CSFs with this support scheme.

The vertical accelerations (represented in Figure 7.58) of these long span CSFs are

triggered by modes V2, V3 and V4 (and V5 at the slenderer CSF, with deck depth

223


htot = Lm/200) and torsional modes have a more noticeable effect than in CSFs of medium

span length, in particular at the deck with depth htot = Lm/200 (differences between the

response at the middle and edges of the deck are 20%, except near the pylon, where these

are larger). This figure illustrates as well that the CSF with larger depth (htot = Lm/100)

describes the smallest vertical accelerations (1.02 m/s2) whereas the CSF with smallest

deck depth (htot = Lm/200) produces the largest vertical accelerations (1.76 m/s2). The

larger vertical accelerations at the slenderer CSF are due to the higher number of vertical

modes that trigger the accelerations as well as the higher contribution of torsional modes.

In the lateral direction, the impact of the deck depth on the response is the opposite,

the CSF with a larger deck depth describes the highest lateral accelerations. This effect

is explained by the considerably larger contribution of the torsional modes in the lateral

direction in comparison to that of the torsional modes at the CSF with slenderer deck.

In terms of the peak accelerations experienced by users, 75% of pedestrians walking on

the bridge with the largest depth feel peak vertical accelerations below 0.7 m/s2 whereas

75% of the users of the footbridge with smallest depth notice movements smaller than

1.15 m/s2. In the lateral direction differences between movements of the deck and those

experienced by users are less considerable: at the former footbridge 75% of the users

notice movements below 1.1 m/s2 and 0.45 m/s2 at the latter. Hence, the CSF with the

greatest depth would correspond to a better solution when considering the comfort for

users in the vertical direction and the CSF with smallest depth would provide the best

solution for the comfort of users in the lateral direction.

Table 7.7: Frequencies [Hz] of lateral modes of long span CSFs according to deck articu-lation and depth magnitude.

Deck depth Lm/100 Deck depth Lm/200Supports L1 L2 L1 L2

(d) POTs 1.29 2.52 1.35 2.73(c) POTs 0.99 2.46 1.04 2.59LEBs+SK 0.99 2.41 1.05 2.57

0 20 40 60 80 100 1200

0.5

1

1.5

2

2.5


Peak V

ert

.

acc.

[m/s

2]

0 20 40 60 80 100 1200

0.5

1

1.5


Peak L

at.

acc.

[m/s

2]

Lm

/200

Lm

/100

(a) (b)

Figure 7.58: Peak vertical (a) and lateral (b) accelerations recorded at the deck of CSFswith main span length 100 m and support scheme (d).

224


7.5.3.1 Dimensions of structural elements

The CSFs with a main span length of 50 m experience the largest modifications of the

response in service (both vertical and lateral) when altering the dimensions of the cables

and the depth of the concrete slab.

In relation to the area of the backstay, this element controls the vertical deflections

of the main span and changes the torsion modes of the deck. When reducing the area of

the backstay to 0.5ABS,0, the CSF with deck depth Lm/100 has peak vertical movements

that are practically the same and the CSF with deck depth Lm/200 has peak vertical

movements that are reduced less than 20%, see Figure 7.59. In the lateral direction, the

movements are the same at the first CSF or moderately lower at the second due to a

smaller contribution of T1 in the lateral direction.

The changes introduced by a different area of the main span stays are modest as well.

Figure 7.60 represents the peak vertical and lateral accelerations described at the CSFs

when increasing the area of the main span stays 2.5 times. In the vertical direction,

responses at the CSF with htot = Lm/100 are 40% larger due to the higher participation

from vertical modes with 4 antinodes at the main span (2 modes) and 10% larger at the

CSF with htot = Lm/200 due to a larger contribution of modes V4 and V5. In the lateral

direction both bridges experience lateral movements that are 15% larger.

The modification of the concrete slab depth (0.3 m instead of 0.2 m) produces the

largest modifications of the serviceability movements of the CSFs: as represented in Fig-

ure 7.61, the vertical movements are 20% or 35% smaller. However, in the lateral direction

the movements remain very similar, in particular at the CSF with largest lateral move-

ments (htot = Lm/100), and this can be explained by the larger contributions of the lateral

and torsional modes despite the increase of the mass of the deck.

0 20 40 60 80 100 1200

0.5

1

1.5

2


Peak v

ert

.

acc.

[m/s

2]

0 20 40 60 80 100 1200

0.5

1

1.5


Peak lat.

acc.

[m/s

2]

Lm

/200 Basic

Lm

/200 Alt.

Lm

/100 Basic

Lm

/100 Alt.

(a) (b)

Figure 7.59: Peak vertical (a) and lateral (b) accelerations recorded at benchmark CSFor CSF with smaller backstay (0.5ABS,0), with depths Lm/100 and Lm/200.

7.5.3.2 Geometric characteristics of the long span cable-stayed footbridge

Regarding the geometric characteristics of the CSFs, the modification of the pylon

height, pylon inclination and deck width produce the most substantial improvements of

the response of CSFs of medium span length in the vertical and lateral directions. These

modifications in long span CSFs trigger the vertical and lateral accelerations depicted in

225


0 20 40 60 80 100 1200

0.5

1

1.5

2


Peak v

ert

.

acc.

[m/s

2]

0 20 40 60 80 100 1200

0.5

1

1.5


Peak lat.

acc.

[m/s

2]

(a) (b)

Figure 7.60: Peak vertical (a) and lateral (b) accelerations recorded at benchmark CSFor CSF with larger stays (2.5AS,0), with depths Lm/100 and Lm/200.

0 20 40 60 80 100 1200

0.5

1

1.5

2


Peak v

ert

.

acc.

[m/s

2]

0 20 40 60 80 100 1200

0.5

1

1.5


Peak lat.

acc.

[m/s

2]

(a) (b)

Figure 7.61: Peak vertical (a) and lateral (b) accelerations recorded at benchmark CSFor CSF with slab depth 2tc,0, with depths Lm/100 and Lm/200.

Figures 7.62-7.64.

Figure 7.62 represents the movements generated in CSFs with a pylon of height 0.25Lm.

For moderate span lengths this measure reduces considerably the vertical response and

enlarges the lateral accelerations. For long span CSFs this reduction of the pylon height

changes the magnitude of the vertical accelerations near the pylon. In the lateral direction,

instead of increasing the accelerations, there is a considerable reduction. This effect is

explained by the dynamic characteristics of the first lateral mode (with smaller modal

mass and higher frequencies, 1.35 Hz instead of 1.29 Hz).

The longitudinal inclination of the pylon (towards the side span) reduces the vertical

accelerations, in particular of the footbridge with smallest depth as it reduces its torsions,

as depicted Figure 7.63, whereas in the lateral direction the response is increased (due to

an increase in the modal masses for the lateral mode). Finally, an increment of the deck

width has a notable effect in reducing the vertical and lateral accelerations of the CSFs,

as seen in Figure 7.64.

In terms of the accelerations experienced by users, in the vertical direction pedestrians

notice accelerations that are between 0.7-0.75 times those recorded at the deck (largest

differences are given in the model with larger deck depth) whereas in the lateral direction,

similarly to the results of the basic models, pedestrians notice accelerations that are 0.8-0.9

226


0 20 40 60 80 100 1200

0.5

1

1.5

2


Peak v

ert

.

acc.

[m/s

2]

0 20 40 60 80 100 1200

0.5

1

1.5


Peak v

ert

.

acc.

[m/s

2]

Lm

/200 Basic

Lm

/200 Alt.

Lm

/100 Basic

Lm

/100 Alt.

(a) (b)

Figure 7.62: Peak vertical (a) and lateral (b) accelerations recorded at benchmark CSFor CSF with shorter pylon (0.25Lm), with depths Lm/100 and Lm/200.

0 20 40 60 80 100 1200

0.5

1

1.5

2


Peak v

ert

.

acc.

[m/s

2]

0 20 40 60 80 100 1200

0.5

1

1.5


Peak lat.

acc.

[m/s

2]

(a) (b)

Figure 7.63: Peak vertical (a) and lateral (b) accelerations recorded at benchmark CSFor CSF with inclined 20o towards the side span, with depths Lm/100 and Lm/200.

times those of the deck.


Some of the structural modifications described in the previous sections change the

serviceability response significantly whereas others do not introduce changes that would

affect the comfort perceived by users.

Figures 7.65, 7.66 and 7.67 summarise the vertical and lateral accelerations recorded

at the deck or experienced by users at each case and compare them to the comfort ranges

described in Section 3.4 (the first two figures correspond to results for medium span CSFs

and the third to results for long span CSFs).

According to the limits outlined in the Chapter 3, the vertical movements of medium

span length cable-stayed footbridges generally correspond to medium and low levels of

comfort and, in the lateral direction, the magnitudes of the accelerations correspond

to a maximum or medium comfort for users. In the vertical direction, neither of the

modifications introduced at the CSF reduce the accelerations to the range of maximum

comfort, i.e., below 0.5 m/s2.

In relation to the movements recorded at long span CSFs, regardless of the depth

227


0 20 40 60 80 100 1200

0.5

1

1.5

2


Peak v

ert

.

acc.

[m/s

2]

0 20 40 60 80 100 1200

0.5

1

1.5


Peak lat.

acc.

[m/s

2]

(a) (b)

Figure 7.64: Peak vertical (a) and lateral (b) accelerations recorded at benchmark CSFor CSF with deck width of 5 m, with depths Lm/100 and Lm/200.

Basic BC BS S t_f t_c H_T Inc. Pylon Anch. L. Inc. W_d L_s0

0.5

1

1.5

2

2.5

Vert

ical accele

ration [

m/s

2]

Medium

Minimum

Min. / Unacc.

Maximum

aV,DECK

aV, PED 75%

Figure 7.65: Comfort assessment of CSF according to the measures implemented to modifyvertical response (where Basic refers to the reference CSF, BC to deck articulation, BS tobackstay, S to main span stays, tf to thickness of the bottom flange of the steel girder, tcto the thickness of the concrete slab, hp to the height of the pylon, Inc. to the inclinationof the pylon, ‘Pylon’ to its shape, Anch. to the distance between stay anchorage, L. Inc. tolateral inclination of stays, wd to deck width and Ls to side span length).

of the deck and the design characteristics, vertical accelerations are in most occasions

equivalent to a medium or low comfort. In the lateral direction, the depth of the deck

has an enormous impact on the accelerations: the cable-stayed footbridges with a depth

of Lm/200 are associated with accelerations that users would consider to be minimally

comfortable, whereas in most occasions the footbridge with a larger depth, despite the

fact that lateral movements are not unstable, would not be considered suitable for users.

Both in medium and long span cable-stayed footbridges, the assessment performed

through the accelerations noticed by users is less restrictive than that based on the accel-

erations of the deck. Nonetheless, designers usually obtain the comfort evaluation from

the accelerations recorded at the deck, since the appraisal of those experienced by users

makes the evaluation procedure more complex. However, the comparison of the results

obtained in both medium and long cable-stayed footbridges suggests that those noticed

by users are approximately 0.70-0.75 times the peak response of the deck, as depicted in

228


Basic BC BS S t_f t_c h_p Inc. Pylon Anch. L. Inc. w_d L_s0

0.1

0.2

0.3

0.4

0.5

Late

ral accele

ration [

m/s

2]

Minimum

Medium

Maximum

aL,DECK

aL, PED 75%

Figure 7.66: Comfort assessment of CSF according to the measures implemented to modifylateral response.

Basic BS S t_c h_p Incl W_d0

0.5

1

1.5

2

2.5

Vert

ical accele

ration [

m/s

2]

0

0.5

1

1.5

Late

ral accele

ration [

m/s

2]

Medium

Maximum

Min. / Unacc.

Minimum

Maximum

MediumMinimum

UnacceptableaV,DECK

aV, PED 75%

Figure 7.67: Comfort assessment of the long span CSF according to the measures imple-mented to modify vertical and lateral response.

Figure 7.68. Thus this relationship could be used by designers to assess more realistically

the accelerations that pedestrians notice while walking.

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

accL,DECK

acc

L,P

ED

75%

0.5 1 1.5 2 2.50.5

1

1.5

2

2.5

accV,DECK

acc

V,P

ED

75%

Figure 7.68: Comparison of maximum vertical and lateral movements recorded at the deckand maximum accelerations felt by 75% of the walking pedestrians.

229


7.7 Additional dissipation of the serviceability movements: in-

herent or external movement control

As highlighted in the previous section, most of the design alternatives for cable-stayed

footbridges of medium and long span lengths do not reduce the amplitudes of vertical or

lateral (in the lateral direction, exclusively for long span bridges) accelerations to levels

within the range considered of maximum comfort. Therefore, if designers aim to de-

velop cable-stayed footbridges with these low level of accelerations, additional dissipation

elements should be considered.

Section 2.4 summarises the characteristics of the most frequent damping devices used

in these structures (Tuned Mass Dampers in particular). Based on those damping devices,

Figure 7.71 describes the vertical and lateral accelerations computed for the medium span

length CSF with an inherent damping ratio of 0.6% (instead of 0.4%), which corresponds

to the mean damping ratio of composite structures (detailed in Section 2.4) as well as the

movements described by the same CSF with a TMD located at x = 28 m or x = 49 m

(antinodes of mode V2). The TMDs have a mass that is 5% of the modal mass for mode

V2 and its stiffness is described by expressions of Section 2.4.

0 10 20 30 40 50 600

0.5

1

1.5

2

Peak v

ert

ical

acc.

[m/s

2]

0 10 20 30 40 50 600

0.1

0.2

0.3


Peak late

ral

acc.

[m/s

2]

0 0.2 0.4 0.6 0.8 10

25

50

75

100

% P

edestr

ians

0 0.2 0.4 0.6 0.8 10

25

50

75

100

amax,Pi

/ amax,P

% P

edestr

ians

ζ = 0.6%

D1

D2

Basic25%

25%

amax,P

= 0.18 m/s2

amax,P

=1.54 m/s2

(a) (b)

Figure 7.69: (a) Absolute peak vertical and lateral accelerations recorded at medium spanlength reference CSF with higher inherent damping ζ = 0.6%, with TMD located at x =28 m (D1) or at x = 49 m (D2); (b) accelerations noticed by users.

The results represented in that figure highlight the fact that a CSF with a slightly

larger inherent damping is associated with substantially lower vertical accelerations (peak

acceleration is 1.37 m/s2) and 75% of the users notice movements corresponding to a

medium comfort (below 1.0 m/s2). The placement of a supplemental damping device

produces the same effects (peak accelerations are 1.14 m/s2 and 75% of the users notice

movements smaller than 0.92 m/s2).

At CSFs with long main spans, a larger inherent damping has a similar effect in

the vertical direction (it reduces movements by 20-30%, as peak accelerations are 1.2 or

230


0.8 m/s2 for large or small deck depths) and it drastically reduces the movements in the

lateral direction (peaks are 45% smaller, being reduced from 0.56 m/s2 to 0.3 m/s2 and

from 1.3 m/s2 to 0.68 m/s2, for small and large deck depths respectively). In relation

to external devices, vertical TMDs located at x = 30 m and x = 90 m (total mass 5%

of mode V3) reduce the vertical peak response by 20% (for both deck depths) as well,

whereas a lateral TMD located near x = 70 m decreases the peak lateral response of both

deck depths by 35%.

Hence, the use of external damping devices improves considerably the serviceability

response in both vertical and lateral directions. Nonetheless, the accelerations described

by CSFs with a slightly larger inherent damping ratio also enhances the response in

service. Therefore, designers should make provisions to ensure, to the greatest possible

extent, the largest inherent dissipation of the footbridge through measures indicated in

Section 2.4.


In the vertical direction, the dynamic deflections are related to the vertical accelera-

tions as well as characteristics such as the inclination of the stays, the mass of the deck, its

transverse second moment of area, etc. For this reason, the maximum static deflections,

the maximum dynamic deflections and the DAFs related to vertical deflections do not

occur at the same structures.

The maximum dynamic vertical deflections occur at regions of the deck near x =

45 m (as depicted in Figure 7.70(a)). For the conventional CSF, these have a value of

27.5 mm whereas other CSFs may have deflections that differ from this value by ±50%.

The footbridges with the smallest deflections are those with the largest main stay areas,

larger slab depth or with an ‘H’ pylon inclined laterally, whereas the cases with largest

dynamic deflections are the bridges with the smallest main stay areas or a pylon most

inclined longitudinally (towards the side span or the main span).

Despite these large variations in the deflection magnitudes, the DAFs related to vertical

deflections at x = 46.5 m have a mean value of 1.45 (similar to that of the reference

footbridge), a maximum value of 1.65 and a minimum of 0.82. The maximum DAFs

occur at bridges with the largest stay areas, with a ‘POTs(b)’ support scheme, larger

thickness of the bottom flange girder and the highest pylon. The minimum DAFs occur

at footbridges with the largest deck width.

If these deflections and those generated by 0.2 or 1.0 ped/m2 (Chapter 6) are compared

to the static deflections produced with 5.0 kN/m2 (equivalent to 6.4 ped/m2), results

show that a flow with 7.0 ped/m2 would describe peak vertical deflections larger than any

dynamic scenario, with traffic flows of pedestrians (as depicted in Figure 7.72).

In the lateral direction, the dynamic deflections have peak magnitudes below 4.0 mm

approximately, two times smaller than the values that some authors point out as the cause

for the engagement of pedestrians with the structure. The lateral deflections are linearly

related to the lateral accelerations (as represented in Figure 7.70(b)). An analysis of the

amplitudes of the pedestrian lateral loads at each scenario shows that the average load

231


Deflection [

mm

]


(a)

0 0.2 0.4 0.6 0.8 10

2

4

6

accpeak,y

[m/s2]

x =

31.5

m

0.5 1 1.5 2 2.50

20

40

dis

ppeak,y

[m

m]

x =

46.5

m

(b)

Figure 7.70: (a) Static and maximum dynamic vertical deflections generated by medium-high density pedestrian flows on medium span length CSFs; (b) relationship between peakvertical (top, y = V) or lateral (bottom, y = L) dynamic deflections generated by pedestriansand corresponding peak accelerations.

amplitude is linearly related to the peak lateral accelerations and the dynamic deflections

(which would be equivalent to DAFs related to lateral deflections with similar magnitude

irrespective of the scenario).

These lateral accelerations highlight as well the fact that the amplitude of the lateral

loads of the pedestrians are a consequence of the movement and not vice versa, hence

the use of lateral pedestrian load models where these load amplitudes depend upon the

movements noticed by the users is of utmost importance for a realistic prediction of the

structure performance in this direction.


The footbridge models of the CSFs described in the previous sections are used hereun-

der to assess: (a) whether the effect of a static weight of 5 kN/m2 includes the maximum

dynamic normal stresses, and (b) the structural characteristics that are related to the

largest magnitude of these stresses.

Figure 7.71 represents the static and dynamic bending moments along the deck gen-

erated by the same traffic flow crossing cable-stayed footbridges of medium main span

length (50 m) defined in the previous sections. From this figure, it can be highlighted

that:

• The peak dynamic bending moments (BMs) take place at the antinodes of mode V2

and the hogging BMs at the region 40 m≤ x ≤ 60 m are not included by the effects

of a static traffic load.

• When introducing variations at the design of the CSF, at regions near x = 7.5, 25-

32 m or 48-52 m, the magnitudes of the dynamic BMs can be two times larger or

smaller than those of the reference CSF.

When contrasting these dynamic BMs with the characteristics that are changed in the

CSF, it can be highlighted that:

232


Figure 7.71: Static bending moments (BM) of the deck produced by the weight of a flowwith 0.6 ped/m2 and dynamic bending moments (and DAFs related to these) generated bythe dynamic actions of this flow at CSFs with alternative dimensions or geometry.

• The largest hogging BMs at both the main and side span are described at the deck

of CSFs with stiffer pylons. The CSFs with smaller backstay or main span stays,

shorter towers or towers inclined towards the side span produce considerably smaller

hogging moments at the main span (x = 31.5 m).

• The CSF models with largest sagging moments at x = 31.5 or 52.5 m correspond to

those where the pylon section is modified (thickness and diameter).

The comparison of dynamic BM magnitudes with the accelerations described at the

deck at the same time highlights that the largest normal stresses are not concomitant

with the largest accelerations.

In relation to the magnitude of the DAFs related to BMs, these can be as large as

8.9 and small as 0.9 at sections near x = 31.5 and 52.5 m (at the reference footbridge,

DAFs at x = 31.5 or 52.5 m have magnitudes near 3.8 and 2.3 respectively). The largest

DAFs related to hogging BMs at these sections are described by CSFs with modified

pylon transverse section or by CSFs with a pylon longitudinally inclined towards the side

span. The largest DAFs related to sagging BMs are given in CSFs with different pylon

sections, footbridges with a pylon inclined towards the main span and footbridges with

larger main span stays.

A comparison of the DAFs related to sagging bending moments near x = 52.5 m

(section with largest dynamic sagging BMs), given in Figure 7.72, with traffics of 0.2,

1.0 ped/m2 and those of the previous sections with 0.6 ped/m2 highlights that the dynamic

bending moments would be predicted by the static bending moments generated with the

load of 7.0 ped/m2 (5.5 kN/m2), slightly larger than 5.0 kN/m2.

233


0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

ped/m2

DA

Fdeflection,

x =

60.0

m

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

ped/m2

DA

FB

M,

x =

52.5

m

(a) (b) (c)

Figure 7.72: (a) Maximum DAFs related to deflections and (b) DAFs related bendingmoments according to pedestrian flow density.

7.10 Deck shear stresses

Figure 7.73 illustrates the magnitudes of the static and dynamic shear forces at the

conventional medium span length footbridge and the dynamic shear forces at CSFs with

alternative characteristics. The largest shear forces occur near the pylon and near the

supports at x = 60 m.

Depending on the characteristics of the bridge and its dynamic response, the dynamic

shear forces can be 10 times larger or 3.3 times smaller than those given in the conven-

tional footbridge. The largest shear forces are described in footbridges with a ‘POT(d)’

scheme, with larger backstay or main span stay cables, thicker steel girder and larger py-

lon diameter and height. The smallest shear forces are produced at footbridges with wider

deck or thicker slab. Nonetheless, the magnitudes of these shear forces are considerably

small. In this sense, the DAFs related to shear forces at x = 60 m have a value near 1.0

(1.12 at the reference CSF).

Figure 7.73: Static shear forces (SF) at the steel girders of the deck generated by the weightof a flow with 0.6 ped/m2 and dynamic shear forces produced by the dynamic actions ofthis flow at CSFs with alternative dimensions or geometry.

234


7.11 Normal stresses at the pylon

The magnitude of the normal stresses endured by the pylon during serviceability are

substantially influenced by the geometry of this pylon, the characteristics of the stays and

the deck mass, apart from the dynamic response of the structure under the actions of

pedestrians.

At the top of the pylon of the reference 1T-CSF (see Figure 7.74), the serviceability

scenario produces bending moments with a magnitude of 38.5 kNm. This moment is two

times larger with the smallest backstay, the smallest or largest main span cables and when

the pylon is inclined towards the main span, and two times smaller when the deck has a

larger concrete slab or deck width.

At the pylon section near the deck (S2 in Figure 7.74), the dynamic bending moment

can be 5 times larger than that depicted in this figure when using support schemes such

as ‘POTs(c)’ or ‘POTs(d)’ and larger main span cables. When using a larger deck width,

a large longitudinal inclination or a damping device this moment is increased 10 times.

The longitudinal inclination and the damping device have a large impact on the bending

moment at section S3 of the pylon, increasing the bending moment 5 times in relation to

that of the conventional footbridge.

In terms of axial forces, the largest increments are produced when introducing back-

stays or main span stays with larger areas (axial forces 9 times larger).

Thus the performance in service of the pylon is basically related to the characteris-

tics of the stays and its inclination in relation to the vertical axis. The introduction of

damping devices produces large increments of the bending moments for pylon sections

near and below the deck, therefore its introduction once the footbridge is built should be

contemplated during design stages.

5.0

10.0

15.0

20.0

25.0

00 100 200 300

Axial force [kN]

5.0

10.0

15.0

20.0

25.0

00 20.0 40.0 60.0


Tow

er

heig

ht

[m]

24.0

15.0

38.5

S1

S2

S3

167

186

Figure 7.74: Dynamic bending moments and axial forces at critical sections of the pylonof CSFs with alternative dimensions or geometry.

235


7.12 Performance of the stay cables

The performance of the cables in cable-stayed footbridges is related to the amplitude

and number of stress cycles endured by the cables under a particular loading event. For

these cable-stayed footbridges, this evaluation is made according to the Equation 3.7.2,

which evaluates the damage caused by each stress variation during a loading event, and

to the comparison of this accumulated damage to that of the conventional bridge (Equa-

tion 3.7.3).

Based on these evaluations, the backstay endures the smallest stress variations when

the cable-stayed footbridge has smaller main span stays, larger slab depths, wider decks

or shorter pylon heights. The same stay resists the largest effects of this stress variation

when the footbridge has a support scheme with ‘POTs(d)’, larger main span stays, steel

girders with larger thickness, larger pylon diameters, pylons inclined towards the side span

and with a TMD at the main span. In fact, this device causes an accumulated damage at

the backstay that is 42 times larger than that for the conventional footbridge. Therefore

the implementation of TMDs to dissipate vertical movements is disadvantageous for the

backstay.

When considering the behaviour of the main span stays, except for the most vertical

stay, these endure stress variations of similar proportion regardless of their anchorage

position and length. For all these stay cables, their accumulated damage is smaller when

their area is larger, the area of the backstay is smaller, the deck has a larger slab or

width and the tower has a lower height. Their stress variations are larger with supports

‘POTs(d)’, steel girders with larger thickness, longer side spans or higher pylons.

Figure 7.75 provides an overall comparison of the performance of each stay (damage is

compared to the damage of the same stays for the conventional footbridge). This figure

highlights that: (1) on average, the stress variations at each cable and each cable-stayed

footbridge model are very similar to those of the conventional footbridge (damage near

1.0), (2) there are multiple cases that reduce the stress variations endured by the stay

and that (3) the cases that increase the stress cycles at each cable are few but produce

very large modifications.

BS CB1 CB2 CB3 CB4 CB50

2

4

6

Dam

age /

Dam

age

bas

Average Damage

Figure 7.75: Comparison of accumulated damage at each stay of the CSF produced atCSFs with geometric and structural characteristics detailed in previous sections (comparedto accumulated damage of stay cables of the benchmark CSF).

236



The analyses of the performance of cable-stayed footbridges with a single pylon in

service detailed in the previous sections describe the effect that parameters related to

deck articulation, dimensions of structural elements and geometric characteristics of these

bridges have on that serviceability response.

In the vertical direction, results demonstrate that increasing the vertical stiffness of

the deck is unfavourable for vertical movements and vice versa. In the lateral direction,

results of the previous sections show that a larger stiffness of the deck is beneficial for the

lateral accelerations.

Both the vertical and lateral movements are governed by several characteristics of the

bridge. The modification of the geometric characteristics considered in the previous sec-

tions may substantially increase the natural frequencies of some of the important modes,

however their participation is still substantial, pointing towards the fact that modes with

frequencies above the range considered critical (1.7-2.1 Hz) do not ensure low magnitude

responses in service. Nonetheless this assertion is valid for lateral modes with frequencies

below 1.3 Hz as has been seen for medium and long span cable-stayed footbridges.

The serviceability vertical accelerations of the medium span length cable-stayed foot-

bridges ranges from medium to low comfort, although occasionally it can be higher. For

long span cable-stayed footbridges these vertical movements are slightly more moderate

although they correspond to medium or low comfort, in particular for the deck with the

largest depth (Lm/100).

Individually, the characteristics considered in the previous sections describe the fol-

lowing effects on vertical serviceability response:

1. Support conditions: The deck articulations that restrain the longitudinal deck move-

ments (one end relative to the other) generate larger movements.

2. Structural elements:

(a) The main span stays and the backstay areas have a very large impact on the

magnitude of the vertical response. Smaller areas develop lower vertical accel-

erations.

(b) Bar stays and cable stays produce fairly similar vertical accelerations.

(c) The vertical response increases with the thickness of the girder bottom flange,

although this effect is very moderate.

(d) Larger slab depths decrease the vertical movements drastically.

(e) The alteration of the geometry of the pylon transverse section has practically

no effects on vertical response (similarly to the girder thickness).

3. Structure geometry:

(a) The vertical movements increase with the height of the pylon. However, for

users this is only noticeable for very short or very high pylons.

237


(b) The longitudinal inclination of the pylon towards the side span or the main

span improve or do not substantially modify the comfort for users.

(c) The pylon shape does not change the movements perceived by users.

(d) The distance between consecutive cables does modify vertical response, modi-

fication that depends on the characteristics of vertical modes.

(e) A moderate transverse inclination of the pylon with ‘H’ shape does not affect

the vertical movements perceived by users.

(f) The vertical accelerations decrease with larger deck widths, basically due to the

larger mass of the deck.

(g) The effect of the side span length depends on the impact of this geometric

characteristics on modes V2 and V3.

In the lateral direction, the accelerations recorded at the deck and experienced by users

while walking on medium span length cable-stayed footbridges have magnitudes within

the ranges considered to provide maximum or medium comfort. Very few characteristics

produce larger lateral accelerations, although none of these correspond to an unstable

lateral response (except for the 1T-CSF with LEBs as supports). The long span cable-

stayed footbridges display considerably larger movements in this direction, with values

between minimum comfort and uncomfortable (the latter are described in bridges with

larger deck depth, Lm/100). In this lateral direction, the effects of each characteristic of

the cable-stayed footbridge considered on the serviceability response are as follows:

1. Support conditions: The smallest movements are described by the bridge with

‘POTs(d)’ support scheme.


(a) Smaller main span stays and backstay areas increase the lateral movements. In

the case of the backstay this increment gives rise to acceptable movements for

pedestrians, as opposed to those of the main span stays.

(b) Bar stays and cable stays produce similar accelerations in the lateral direction.

(c) The increment of the girder bottom flange thickness fairly increases lateral

response, despite the contribution of this dimension to the lateral stiffness of

the deck.

(d) Similarly to vertical movements, a larger depth of the concrete slab decreases

the lateral movements considerably.

(e) The alteration of the geometry of the pylon transverse section may increase the

lateral response.


(a) The lateral accelerations increase moderately when heights of the pylon are very

low and remain fairly similar otherwise.

238


(b) The longitudinal inclination of the pylon towards the side span or the main

span generate larger accelerations, in particular for large angles of inclination.

(c) Pylon shapes with two legs reduce the magnitude of the lateral movements

perceived by users.

(d) A large transverse inclination of the pylon with ‘H’ shape increases the lateral

accelerations.

(e) The lateral accelerations are not largely affected by the location of the anchor-

ages of the stays or the magnitude of the side span length.

For longer span cable-stayed footbridges, the results highlight the advantage of a thin-

ner deck to reduce lateral movements, the better performance of a stiffer deck for vertical

movements and the similar effects of other measures studied on medium span cable-stayed

footbridges.

Despite the multiple characteristics considered, only few alternatives allow medium

and long span cable-stayed footbridges to describe vertical or lateral accelerations (for

the longest footbridges) corresponding to a maximum comfort. These accelerations are

considerably reduced (to a maximum comfort level) if there is a higher inherent damping

(an average value of ζ = 0.6% is sufficient to drastically reduce the magnitude of the

vertical and lateral accelerations) or external damping devices (e.g., TMDs).

The comparison of the levels of comfort extracted from accelerations recorded at the

deck and from accelerations experienced by users indicate considerable differences, in

particular for those structures with large torsional movements. A general comparison of

these results has shown that, in vertical direction, the second are 0.7-0.75 times the first

magnitude (when considering 75% of the users). Thus it is proposed that this proportion

should be used to assess the comfort for users.

The analysis of other characteristics of the performance of these footbridges in service

highlights that:

• The magnitude of the dynamic deflections are very related to the areas and lengths

of the stays (backstay and those at the main span). The magnitude of DAFs related

to these dynamic deflections are related to the stiffness of the deck in the vertical

direction.

• The lateral dynamic deflections are linearly related to the lateral peak accelerations

and to the average amplitude of the lateral loads introduced by pedestrians in each

case. These lateral movements are a consequence of the magnitude of the lateral

loads therefore only pedestrian load models where lateral load amplitudes are related

to the structure movements will provide realistic assessments.

• The dynamic bending moments at the deck and the corresponding DAFs are related

to the stiffness of the pylon and the length and dimensions of the cables.

• The largest dynamic shear forces and related DAFs are described near x = 60 m at

footbridges with ‘POTs(d)’ support scheme, larger stays, thicker steel girder, larger

pylon section or higher pylons.

239


• Deflections, and normal and shear stresses at the deck are included by ULS if the

design loads are larger than 5.5 kN/m2.

Thus, under the action of pedestrian flows, cable-stayed footbridges with medium and

long span lengths describe serviceability accelerations that considerably depend on the

structural characteristics of the footbridge. Multiple geometric and structural elements

can be used to modify and reduce this serviceability response: the characteristics of the

stays, the deck stiffness and deck mass are the principal.

However, it has been seen that, for medium density flows, the magnitudes of these

accelerations generally correspond to a medium comfort (maximum in the lateral direction

for medium span length bridges). In these cases only the use of external damping devices

or additional inherent damping provides this maximum comfort for pedestrians crossing

these footbridges.

240

Chapter 8Performance of cable-stayed

footbridges with two pylons:

parameters that govern serviceability

response

8.1 Introduction

The most common cable arrangement for cable-stayed footbridges is a solution con-

sisting of a single fan system and a backstay cable supported by one pylon. Alternatively,

a solution involving two single fans with a backstay cable each, supported by two pylons,

may be adopted according to the footbridge site conditions (e.g, foundations), aesthetic

and economical limitations. Analogously to the previous chapter, the following sections

describe and substantiate the serviceability dynamic response of these cable-stayed foot-

bridges with two pylons produced by the action of pedestrian flows (of densities near

0.6 ped/m2) and evaluate the impact on this response of multiple design characteristics in

order to appraise which of these enhance the performance of these footbridges in service.

In addition to the characterisation of the serviceability response of these footbridges in

terms of the accelerations, the dynamic stresses recorded at the deck, pylons and cables

during these events are described and related to concomitant accelerations.

Thus, Sections 8.2 and 8.3 report the geometric characteristics of these cable-stayed

footbridges with two pylons, their dynamic behaviour properties and the amplitudes of

their dynamic serviceability accelerations produced by the passage of medium-high density

flows of pedestrians. Section 8.4 emphasises the parameters of these footbridges as well

as those of the traffic that have the largest influence on their serviceability performance.

According to these main factors, Sections 8.5 and 8.6 are focused on the description

and substantiation of the impact on the accelerations of these footbridges of different

characteristics involving structural elements and geometric characteristics. Geometry

involving longer span bridges are considered in Section 8.7.

241

8. Performance of cable-stayed footbridges with two pylons

Based on the comfort assessment detailed in Section 8.8, Section 8.9 presents the

serviceability accelerations that these footbridges would produce when considering alter-

native measures to dissipate the accelerations in service. Finally, Sections 8.10 to 8.14

appraise and describe the dynamic deflections and the amplitudes of the dynamic stresses

at the deck, pylon and cables.

8.2 Geometry of conventional cable-stayed footbridges with two

pylons

Based on the geometric characteristics of the distinct cable-stayed footbridges that can

be found, it is considered that a representative cable-stayed footbridge with two pylons,

2T-CSF, (see Figure 8.1) has a main span length of 50 m (Lm) and two side spans of

length 0.2Lm (i.e., 10 m each). The deck transverse section consists of a concrete slab

of depth 0.2 m and width 4 m supported by two steel girders located at the edges of the

slab and a cable system with cables disposed in two semi-fan arrangements supported by

two pylons of height 0.20Lm above the deck. The deck has a depth that corresponds to a

depth-to-main span length ratio of 1/100 whereas the thickness and sections of the steel

girders and cables have been obtained considering the corresponding ULS and material

characteristics given in Section 3.5.2 (these dimensions are represented in Figure 8.1).

Detail B-B:

0.50.5

0.2

w = 4

HEB 200

hgird

tflange,bot

Sec. A-A:

Characteristics 2TCSFB htot/Lm = 1/100

hto

t

htot = 0.5

hgird = 0.3

tflange,top

tweb

tflange,bot = 0.008

tflange,top = 0.008

tweb = 0.005


CB#1CB#2CB#3

152

24

HT=

17.5

Hs =

10

Hi =

7.5

Lm= 50.0Ls= 10.0 Ls= 10.0

BS

Detail B-B

Dext = 0.375

Dp Dc Dc Dc Dc Dc Dp

CB#1CB#2

CB#3

Sec. A-A

Figure 8.1: Geometry and structural characteristics of CSF with two pylons and transversesection depth Lm/100. Dimensions in meters [m].

The deck is articulated with POTs in a classical arrangement (see Figure 8.4(a)), with

longitudinal deck movements permitted at x = 70 m and transverse deck movements

allowed in one of the two POTs located at each abutment. At the sections of the pylons,

the deck has the relative movements between the deck and the pylon restricted but not

the relative rotations. The reasons for implementing this arrangement instead of that

including LEBs or LEBs and a shear key limiting the deck transverse movements at the

embankment sections as in CSFs with a single pylon are based on the performance of the

footbridge in the lateral direction and described in sections below.

242


8.3 Dynamic characteristics and response in service of conven-

tional cable-stayed footbridges with two pylons

Cable-stayed footbridges with main span lengths near 50 m and two pylons are struc-

tures with masses and stiffness that give place to the vertical, transversal and torsional

modes of vibration described in Table 8.1 (mode shapes are represented in Figure 8.2). As

illustrated in this table, these footbridges have vertical and torsional modes with frequen-

cies near 2.0 Hz (principally modes V2 and T1) whereas lateral modes have frequencies

larger than 1.2 Hz.

Table 8.1: Frequencies [Hz] for the vertical, lateral and torsional modes of conventionalCSF with two pylons, where ‘VN’, ‘LN’ and ‘TN’ denote vertical, lateral and torsional modeswith N half-waves and ‘P’ denotes modes involving the pylons.

Mode No. Frequency Description Mode No. Frequency Description

1 1.22 V1 8 4.59 T1+P

2 1.82 L1 9 4.82 T2+P

3 1.84 V2 10 5.10 T3+P

4 2.23 T1 11 5.34 V5

5 2.91 T2 12 6.13 V6

6 3.03 V3 13 6.19 L2+T4

7 4.17 V4 14 6.65 L2+T4

V1, 1.22Hz

(a)

L1, 1.82Hz

(b)

V2, 1.84Hz

(c)

T1, 2.23Hz

(d)

T2, 2.91Hz

(e)

V3, 3.03Hz

(f)

V4, 4.17Hz

(g)

T1+P, 4.59Hz

(h)

T2+P, 4.82Hz

(i)

T3+P, 5.10Hz

(j)

V5, 5.34Hz

(k)

V6, 6.13Hz

(l)

Figure 8.2: Modal frequencies of CSFs with two pylons.

The characteristics of these footbridges that have the largest influence on the magni-

tudes of the vertical modes (in addition to the length of the main span) correspond to

the deck second moment of area in the vertical direction and its mass per unit length,

the section and length of the main span stay cable with smallest horizontal inclination,

243


and the length, section and longitudinal inclination of the backstay cable. For torsional

modes, the parameters that change the magnitude of these modes correspond to the mass,

the vertical and the lateral second moments of area of the deck, the second moment of

area of the pylon, the section of the backstay and the transverse inclination of the main

span stays. The magnitudes of the lateral modes are largely affected by the boundary

conditions of the deck at the support sections over the abutments as well as the mass

and the lateral second moment of area of the deck, the tensions of the backstay under

permanent loads as well as the second moment of area of the pylon.

Under the passage of medium-high density flows of pedestrians (densities near 0.6

ped/m2), this cable-stayed footbridge has the largest vertical accelerations at one third

of the main span and the largest lateral accelerations at mid-span (see Figure 8.3). The

highest vertical response (peak vertical acceleration of 1.63 m/s2) is produced at deck

sections near x = 27 m and 43 m, which are located between the antinode of the vertical

mode V1 (located at midspan) and the antinodes of the vertical mode V2 (located at

x = 14.5 m and 45.5 m), and this conveys the importance of these two modes in the

total vertical response. The lateral response (with a peak magnitude of 0.365 m/s2) is

at the main span, at sections between x = 30 m and 50 m. This slightly non-symmetric

distribution of the peak lateral accelerations is due to the support arrangement (with

largest amplitudes nearer to the abutment where POTs do not restrain the longitudinal

movements).

0 10 20 30 40 50 60 700

0.5

1

1.5

2


Peak v

ert

ical

acc.

[m/s

2]

0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5


Peak late

ral

acc.

[m/s

2]

(a) (b)

Figure 8.3: Peak vertical (a) and lateral (b) accelerations described at the deck of theconventional CSF with two pylons.

8.4 Principal dynamic characteristics of the pedestrian loads

and the footbridge related to its performance in service

The evaluation of the dynamic accelerations of cable-stayed footbridges with two py-

lons caused by pedestrian flows in serviceability scenarios highlights some of the dynamic

and structural characteristics that have the greatest impact on this dynamic response. In

the vertical direction, the accelerations developed during these serviceability events are

principally related to the participation from modes V1 to V4, T1 and T2 (with frequencies

up to 5 Hz), and to the stress under permanent loads and the length of the most vertical

stay cables anchored at the main span (those located nearest to the pylons). Among these

characteristics some factors are more critical, which correspond to:

244


av,x = f(fV 1,mV 1, fV 3,mV 3, fT2, σCB1,LCB1), x ∈ p, 1s−RMS (8.4.1)

where fV 1 and fV 3 are the frequencies of the vertical modes V1 and V3, mi denotes the

modal mass of these or T2 torsional modes, σCB1 the stress under permanent loads of the

most vertical stay cable anchored at the main span (see Figure 8.1) and LCB1 represents

the length of this stay cable.

Similarly to what occurs for 1T-CSFs, the contribution to the vertical accelerations

of modes V1 to V4 does not decrease when they adopt frequencies larger or smaller than

those used individually by pedestrians (with mean value near 1.8-2.0 Hz). Some of the

results presented in the following sections show that vertical modes with frequencies near

1.2 Hz or 4.5 Hz have a contribution in the overall vertical accelerations in service. These

observations are substantiated by the frequency amplitudes of the total loads introduced

by the whole traffic of pedestrians crossing the footbridge represented in Figure 7.1, which

exhibit large load amplitudes not only near 2.0 Hz but well above and below this frequency.

In comparison to 1T-CSFs, the small contribution of the backstay cable and other

main span stays to the total vertical response in this case is due to their shorter length

and to the importance in response of the supports arrangement.

In the lateral direction there are fewer parameters that affect the amplitude of the

serviceability accelerations developed by a traffic flow crossing these 2T-CSFs. The mag-

nitude of peak al,p or 1s-RMS al,rms accelerations is correlated to the frequency of the first

lateral mode L1, the first torsional mode T1 and the mass of the deck (per unit of length).

The importance of the torsional mode T1 on the amplitude of the lateral responses is re-

lated to its modal shape, with a component in the lateral direction. As opposed to the

lateral response of 1T-CSFs, stay cable properties do not noticeably affect the amplitude

of the lateral accelerations in 2T-CSFs.

8.5 Strategies to improve the vertical dynamic performance of

1T-CSFs in service


As described in Section 8.3, the deck of the reference cable-stayed footbridge with

two pylons (2T-CSF) is articulated by two POT bearings at each abutment (movements

restricted correspond to a ‘classical’ layout) and simply supported at the sections of the

pylons.

Alternatively to this arrangement, the following paragraphs describe the serviceability

accelerations recorded at the same cable-stayed footbridge with support schemes com-

prised by exclusively 2 LEBs or two LEBs and a shear key at each abutment, POTs

with a statically indeterminate layout or POTs with unrestricted longitudinal movements

(illustrated in plots (b) to (e) of Figure 8.4).

The same flow of pedestrians crossing 2T-CSFs with each bearing scheme generates

peak and 1s-RMS vertical accelerations that are very similar except for the footbridge

245


Longitudinally unrestricted POT layout

LEB support

POT bearing

(d)

y

x

z

Statically indeterminate POT layout

Fixed UxFixed Uy

Fixed UxFree Uy

Free UxFixed Uy

Free UxFree Uy

Fixed UxFixed Uy

Fixed UxFixed Uy

Fixed UxFixed Uy

Fixed UxFixed Uy

Pylon Pylon

KxKy

KxKy

KxKy

KxKy

Free UxFixed Uy

Free UxFree Uy

Free UxFixed Uy

Free UxFree Uy

LEBs


KxKy

KxKy


KxKy

KxKy

LEBs+SK

Classical POT layout

(Basic support arrangement)

(b)

(a)

(e)

(c)

Figure 8.4: Plan view of the support configurations of the CSF with LEB bearing schemesor POT bearing schemes. (a) ‘classical’ POT arrangement (arrangement of the benchmark2T-CSF), (b) 2 LEBs at each abutment, (c) 2 LEBs and a SK, (d) statically indeterminatePOT arrangement and (e) POT support scheme with unrestricted longitudinal movements.

with a scheme ‘POT(d)’ (longitudinal movements of the deck are fully restricted), where

the peak accelerations are 45% larger than the rest (see Figure 8.5). The accelerations

experienced by different groups of walking users describe results analogous to those of

the deck: users notice the same peak vertical accelerations except at the CSF with a

support scheme ‘POT(d)’, where these are considerably larger (25% of the users notice

accelerations above 1.4amax,P instead of 0.78amax,P ).

Similarly to the results observed in a 1T-CSF, the larger accelerations generated by

the scheme ‘POT(d)’ are explained by the larger contribution of the vertical modes, in

particular V2 (the longitudinal restriction of the deck movements in this case notably

modifies the modal mass of this mode). The other four bearing schemes have very similar

vertical modes (see Table 8.2), which indicates that the LEB bearings allow longitudinal

deformations of the deck that are similar to those of the deck with unrestricted longi-

tudinal movements and hence the pylons, rather than the bearings, control the relative

longitudinal displacement of the deck.

Thus, a bearing arrangement such as ‘POT(d)’ worsens the vertical response in ser-

vice of cable-stayed footbridges with two pylons (as it was also observed for 1T-CSFs)

whereas alternative bearing arrangements do not modify the amplitude of these vertical

accelerations.

8.5.2 Area of backstay cable

Under static loads, the area of the backstay partly controls the deflections at the main

span through the displacements of the pylon top that it permits. For a cable-stayed foot-

bridge with two pylons, a reduction of the area of this cable increases rapidly the vertical

246


0 10 20 30 40 50 60 700

1

2


Peak v

ert

ical

acc.

[m/s

2]

Basic (b) (c) (d) (e)0.5

1

1.5

2

acc /

acc

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

20

40

60

80

100

% P

edestr

ians

amax,Pi

/ amax,P

LEBs (b)

LEBs+SK (c)

POTs (d)

POTs (e)

Basic

Peak acc.

1s−RMS acc.

5%

25%

50%

amax,P

= 1.296

(a) (b)

Figure 8.5: (a) Absolute peak vertical, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to support schemes (a)-(e); and (b) vertical ac-celerations felt by users.

Table 8.2: Frequencies [Hz] of vertical and torsional vibration modes of CSFs accordingto support arrangement, described in Figure 8.4, where ‘VN’ and ‘TN’ denote vertical andtorsional modes with N half-waves.

Supports V1 V2 V3 V4 T1 T2

(a) Basic 1.22 1.84 3.03 4.17 2.23 2.91(b) LEBs 1.22 1.85 3.03 4.17 2.15 3.10(c) LEBs+SK 1.22 1.85 3.03 4.17 2.15 2.91(d) POTs 1.24 1.88 3.08 4.23 2.04 2.92(e) POTs 1.22 1.84 3.03 4.17 2.15 2.91

deflections of the main span and a slight increment modestly reduces these deflections

(values depicted in Figure 8.6(a)). Backstays with areas larger than 1.5ABS,0 produce

similar vertical deflections at the main span since the relative effect of the backstay is

more modest and structural elements such as the deck or the main span cables become

more important in the control of these deformations. In terms of the dynamic behaviour,

this alteration of the vertical stiffness of the deck according to the backstay area is practi-

cally unnoticed in terms of the modal frequencies of the first vertical and torsional modes,

see Figure 8.6(b), but it is reflected on their modal masses (footbridges with smaller back-

stays cable have larger effective modal masses for V1 and smaller for modes V2 to V4 and

remain practically constant for backstay areas larger than 1.5ABS,0), similarly to what

has been observed in CSFs with a single pylon.

These static and dynamic characteristics explain the amplitude of the vertical accelera-

tions recorded at the deck of the CSF with alternative backstays, depicted in Figure 8.7(a).

Backstays with area 0.5 times that of the reference CSF exhibit peak and 1s-RMS accel-

erations that are 25% or 40% smaller respectively, backstay cables with areas 0.5 times

larger produce accelerations that are 15% or 30% larger respectively, whereas backstays

with areas equal or larger than 1.5ABS,0 have responses similar to those bridges with

247


0.8

1.0

1.4

1.6

um

ax /

um

ax,0

1.2

0.61.0 2.0 3.0 4.0

ABS / ABS,0

ABS

umax

(a)

V4

V3

V2

V1

T1

T2

0.8

1.0

1.4

1.6

Fre

qu

ency [

Hz]

1.2

0.61.0 2.0 3.0 4.0

ABS / ABS,0

(b)

Figure 8.6: (a) Static and dynamic behaviour of the 2T-CSF in terms of the backstayarea ABS (compared to that of the benchmark CSF ABS,0): (a) main span maximum staticdeflections umax and (b) frequencies [Hz] of vertical and torsional modes.

0 10 20 30 40 50 60 700

0.5

1

1.5

2


Peak v

ert

ical

acc.

[m/s

2]

0.5 1 1.5 2 2.50.5

1

1.5

2

ABS

/ ABS,0

acc /

acc

0

0 0.2 0.4 0.6 0.8 1 1.2 1.41.40

20

40

60

80

100%

Pedestr

ians

amax,Pi

/ amax,P

Peak acc.

1s−RMS acc.

0.5 ABS,0

1.5 ABS,0

2.0 ABS,0

2.5 ABS,0

ABS,0

5%

25%

50%

amax,P

= 1.296

(a) (b)

Figure 8.7: (a) Absolute peak vertical, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to backstay area and (b) vertical accelerationsfelt by users.

backstay areas 1.5ABS,0. The accelerations noticed by walking pedestrians, Figure 8.7(b),

illustrate similar trends as 25% of the users feel similar peak accelerations if the backstay

has a section equal or larger than 1.5ABS,0 (1.07amax,P ) or peak vertical accelerations

above 0.6amax,P if the backstay has an area of 0.5ABS,0 in comparison to 0.78amax,P at

the benchmark footbridge.


As argued for cable-stayed footbridges with a single pylon, the area of stays at the main

span affect the vertical deflections that deck of the CSF can exhibit as well as its rotations

generated by loads eccentrically located at the deck (larger stays produce smaller vertical

deflections, as illustrated in Figure 8.8(a), and smaller longitudinal rotations or torsional

twist of the deck). Regarding the dynamic behaviour, the footbridge with smaller stays

has modes with lower frequencies (and effective modal masses that are smaller than those

for CSFs with larger stays) and the footbridges with larger stay areas (and higher stiffness)

exhibit vertical modes at considerably higher frequencies (torsional modes adopt similar

248


frequencies), as shown in Figure 8.8(b).

0.8

1.0

1.4

1.6u

max /

um

ax,0

1.2

0.61.0 2.0 3.0 4.0

AS / AS,0

AS

umax

(a)

V4

V3

V2

V1

T1

T2

1.0

2.0

4.0

5.0

Fre

qu

ency [

Hz]

3.0

0.01.0 2.0 3.0 4.0

AS / AS,0

(b)

Figure 8.8: (a) Static and dynamic behaviour of the 2T-CSF according to stays area AS

(compared to that of the benchmark CSF AS,0): (a) main span maximum static deflectionsumax and (b) frequencies [Hz] of vertical and torsional modes.

These different vertical and torsional stiffness are related to the amplitude of the

peak vertical accelerations of the deck of these footbridges in service, represented in Fig-

ure 8.9(a). CSFs with two pylons have larger vertical accelerations with larger stays (CSFs

with stay areas 1.5AS,0 exhibit peak accelerations that are larger by 50%), similarly to the

response of CSFs with a single pylon, although they give place to moderately larger verti-

cal accelerations with smaller cables as well (CSFs with main stay areas 0.5AS,0 generate

peak accelerations higher by 35%), as opposed to 1T-CSFs.

0 10 20 30 40 50 60 700

0.5

1

1.5

2

2.5


Peak v

ert

ical

acc.

[m/s

2]

0.5 1 1.5 2 2.5

1

1.5

2

AS / A

S,0

acc /

acc

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

20

40

60

80

100

% P

edestr

ians

amax,Pi

/ amax,P

0.5 AS,0

1.5 AS,0

2.0 AS,0

2.5 AS,0

AS,0

Peak acc.

1s−RMS acc.

50%

25%

5%

amax,P

= 1.296

(a) (b)

Figure 8.9: (a) Absolute peak vertical, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to stays area and (b) vertical accelerations feltby users.

The higher accelerations with larger stay cables are explained by the larger vertical

stiffness of the deck and the contribution of the vertical modes (in particular modes V2).

The larger accelerations at the footbridge with smaller cables occur despite the decrement

of the deck stiffness, due to the contribution of torsional modes T1 and T3 and that of

mode V3 (with a frequency considerably lower than the rest of the footbridges).

In terms of the accelerations experienced by walking pedestrians, the footbridges with

249


larger stays give place to larger magnitudes of the accelerations felt by pedestrians and

the footbridge with smaller stays transmits larger accelerations due to the contribution

of mode V3 (the larger contribution of torsional modes is not noticed by walking pedes-

trians).

Hence, increasing the area of the stays does not improve the vertical performance of

the footbridge, and a large reduction the area does not enhance the vertical response as

well due to the considerable contribution of higher vertical and torsional modes.


The alteration of the thickness of the steel girder bottom flange does not drastically

modify the stiffness of the deck under vertical loads, as represented in Figure 8.10(a),

whereas in terms of the dynamic behaviour of the footbridge, it increases the frequency

of the vertical modes, in particular for modes above V2.

1.0 2.0 3.0 4.0

tbf / tbf,0

umax

0.90

1.00

um

ax /

um

ax,0

1.10

Trans. deck sec.:

tbf

(a)

V4

V3

V2

V1

T1

T2

1.0

2.0

4.0

5.0F

requ

ency [

Hz]

3.0

0.01.0 1.5 2.0 2.5

tbf / tbf,0

3.00.5

(b)

Figure 8.10: (a) Static and dynamic behaviour of the 2T-CSF according to thickness ofbottom flange tbf (compared to that of the benchmark CSF tbf,0): (a) main span maximumstatic deflections umax and (b) frequencies [Hz] of vertical and torsional modes.

The vertical serviceability response is moderately affected by this dimension. A flange

thickness 2.2 times deeper than that for the benchmark footbridge tbf,0 produces peak

vertical and 1s-RMS accelerations that are larger by 25% and 60% respectively. Flanges

smaller than that have intermediate vertical accelerations except for the footbridge with

flange thickness 1.4tbf,0. The unexpected increment of the vertical response in this last

case is due to a larger contribution of the mode V4 and a smaller contribution of the

mode T2b (the first has a frequency exactly two times larger than that of the second),

contribution that does not take place in the other footbridges. The larger vertical response

(in comparison to the reference 2T-CSF) at other footbridges is explained by a modest

increment of the contribution of main vertical modes (due to the additional stiffness of

the girder). Walking pedestrians notice peak vertical accelerations that increase when

walking on footbridges with deeper steel bottom flanges, see Figure 8.11(b).

Thus, the increment of the bottom flange thickness of the steel girders is not advanta-

geous for the performance of these footbridges in the vertical direction, similarly to what

occurs for cable-stayed footbridges with a single pylon.

8.5.5 Concrete slab thickness

As observed in cable-stayed footbridges with a single pylon, the modification of the

concrete slab depth has drastic effects on the total mass of the footbridge and more mod-

250


0 10 20 30 40 50 60 700

0.5

1

1.5

2

2.5


Peak v

ert

ical

acc.

[m/s

2]

0.2 0.3 0.40.75

1

1.25

1.5

tbf

/ tbf,0

acc /

acc

0

0 0.2 0.4 0.6 0.8 1 1.2 1.40

20

40

60

80

100

% P

edestr

ians

amax,Pi

/ amax,P

Peak acc.

1s−RMS acc.

1.4 tbf,0

1.8 tbf,0

2.2 tbf,0

tbf,0

5%

25%

50%

amax,P

= 1.296

(a) (b)

Figure 8.11: (a) Absolute peak vertical, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to bottom flange of steel girder and (b) verticalaccelerations felt by users.

erate effects on the vertical second moment of area of the deck. Due to these alterations,

the magnitudes of the vertical and torsional modes change rapidly with this magnitude,

as seen in Figure 8.12(b) (the increment of the vertical or torsional stiffness of the deck

is larger than that of the masses contributing to these modes).

concrete slab

depth

Transverse deck section:

1.0 1.5 2.0 2.5

1.0

2.0

3.0

4.0

V1

V2

V3

T1T2

5.0

Fre

qu

ency [

Hz]

tc / tc,0

6.0

V4

(a) (b)

Figure 8.12: (a) Transverse section of the deck; (b) frequencies of vertical and torsionalmodes (for a slab depth of 1.75tc,0, the sudden change of the frequency of mode T2 concurswith the coincidence in frequencies of modes V2 and L2).

A moderate increment of the slab depth produces peak and 1s-RMS accelerations at

the footbridge deck that are smaller than those for the benchmark footbridge whereas

those of the footbridge with a large slab depth are unexpectedly higher (peak accelera-

tions are merely smaller by 18% and 1s-RMS accelerations are higher by 5%), as it is

illustrated in Figure 8.13(a). Likewise, the accelerations noticed by walking pedestrians

describe similar situations (values are represented in Figure 8.13(b)). Both footbridges

with moderately larger slab improve the experience of walking users as 25% of them no-

tice peak accelerations above 0.5amax,P instead of 0.78amax,P , whereas that with largest

slab slightly worsens the accelerations for some pedestrians (25% of the users feel peak

accelerations above 0.82amax,P ).

251


Both the accelerations recorded at the deck and those experienced by users are related

to the characteristics of the first vertical and torsional frequencies developed in each case.

Footbridges with slab depths of 0.3 m and 0.4 m have fairly similar modes (with similar

shapes), however that with a depth of 0.5 m has modes V3 and T2 with the same frequency,

coincidence that affects the modal characteristics of these modes and their contribution

in the serviceability response.

0 10 20 30 40 50 60 700

0.5

1

1.5

2

2.5


Peak v

ert

ical

acc.

[m/s

2]

0.2 0.3 0.4 0.50.5

0.75

1

tc / t

c,0

acc /

acc

0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

% P

edestr

ians

amax,Pi

/ amax,P

Peak acc.

1s−RMS acc.

1.5 tc,0

2.0 tc,0

2.5 tc,0

tc,0

50%

25%

5%

amax,P

= 1.296

(a) (b)

Figure 8.13: (a) Absolute peak vertical, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to concrete slab thickness and (b) vertical acceler-ations felt by users.

Hence, the modification of the slab depth generally reduces the amplitude of the

vertical accelerations (due to an increment of the deck mass) except for those cases when

modes with fundamental contribution are coupled (as it occurs with the 2T-CSF with

depth 0.5 m).

8.5.6 Transverse section of the pylons

The two pylons of the 2T-CSF consist of a single free-standing column with a steel

circular hollow section of constant diameter and thickness throughout their height. Under

static loads, the modification of the diameter of the pylons affects the amplitude of the

vertical deflections at the main span. Smaller pylon diameters enlarge the vertical static

deflections and larger pylon diameters reduce moderately these vertical deflections (see

Figure 8.14(a)). The effect of its thickness is significantly less noticeable.

Similarly, in terms of the dynamic behaviour, the footbridges with smaller or larger

thickness of the pylon section have very similar vertical and torsional modes, as opposed

to those with different diameter of the pylons (illustrated in Figure 8.14(b)), in particular

for smaller pylon diameters.

Under pedestrian loads, the dimension of the thickness of the pylons does not modify

the amplitudes of the vertical accelerations recorded at the deck or noticed by users.

At footbridges with larger pylons, the serviceability traffic loads produce slightly larger

vertical accelerations (see Figure 8.15(a)). This increment is moderate (as a footbridge

252


0.75

1.00

1.25

1.50

um

ax /

um

ax,0

0.501.0 2.0 3.0

Dt / Dt,0

umax

(a)

V4

V3

V2

V1

T1

T2

1.0

2.0

4.0

5.0

Fre

qu

ency [

Hz]

3.0

0.01.0 2.0 3.0

Dt / Dt,0

(b)

Figure 8.14: (a) Static and dynamic behaviour of the 2T-CSF according to pylon diameterDt (compared to that of the benchmark CSF Dt,0): (a) main span maximum static deflectionsumax and (b) frequencies [Hz] of vertical and torsional modes.

with pylons 2.5 times larger exhibit peak and 1s-RMS accelerations 5% and 20% larger

respectively) except for the footbridge with pylons 1.7 times larger (where peak and 1s-

RMS accelerations are higher by 25% and 60%). At this particular 2T-CSF, the modes T1

and V3 are coupled and are the cause for these unexpectedly large vertical accelerations.

Pedestrians notice vertical accelerations that are proportional to these peak accelerations

recorded at the deck (as described in Figure 8.15(b)).

0 10 20 30 40 50 60 700

0.5

1

1.5

2

2.5


Peak v

ert

ical

acc.

[m/s

2]

1.0 1.5 2.0 2.50.75

1

1.25

1.5

1.75

Dt / D

t,0

acc /

acc

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

20

40

60

80

100

% P

edestr

ians

amax,Pi

/ amax,P

1.3 Dt,0

1.7 Dt,0

2.5 Dt,0

Dt,0

Peak acc.

1s−RMS acc.

5%

25%

50%

amax,P

= 1.296

(a) (b)

Figure 8.15: (a) Absolute peak vertical, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to pylon diameter and (b) vertical accelerationsfelt by users.

Thus, in general, larger pylon diameters produce slight increases of the vertical stiffness

of the deck and the vertical response in service of these footbridges unless the character-

istics of any vertical mode contributing to the movement are locally modified.

8.5.7 Height of pylons

The smallest static deflections of the main span of 2T-CSFs (Figure 8.16(a)) are ob-

tained for pylons of height equal or larger than 0.25Lm, although the difference between

the performance of the 2T-CSFs with these pylons or slightly smaller (hp ≥ 0.17Lm) is

253


practically unnoticeable. In practice, shorter pylons (height near 0.2Lm) are used due to

economical reasons.

1.0

1.4

1.8

0.1 0.15 0.2

um

ax /

um

ax,0

0.25

hp Lm

2.2

0.3

umax

hp

Ls

Lm

Ls

(a)

0.1 0.15 0.2

1.0

2.0

3.0

4.0

V1

V2

V3

T1

5.0

Fre

qu

ency [

Hz]

0.30.25

hp

Lm

V4 T2

(b)

Figure 8.16: (a) Static and dynamic behaviour of the 2T-CSF according to pylon heighthp (compared to that of the benchmark CSF): (a) main span maximum static deflectionsumax and (b) frequencies [Hz] of vertical and torsional modes.

At 1T-CSFs, a larger height of the pylon is related to increasing vertical accelerations.

For 2T-CSFs, as illustrated in Figure 8.17(a), the vertical accelerations of the deck remain

very similar if towers are shorter and increase with higher pylons (although maximum ac-

celerations are given for pylon heights of 0.22Lm). The similar accelerations recorded at

CSFs with shorter pylons are explained by the large contribution of the first torsional

modes (stays with such a small vertical component do not control these rotational move-

ments of the deck). The larger vertical accelerations at CSFs with higher pylons are due

to the larger vertical projection of the stays, which increase the vertical stiffness of the

deck. The largest vertical response of the bridge with pylon heights of 0.22Lm is explained

by an unexpected contribution of mode V2 due to the coincidence of this modal frequency

with that of mode T1. The accelerations noticed by users depict results similar to those

of the accelerations of the deck (Figure 8.17(b)).

0.15 0.2 0.25

1

1.5

2

hp / L

m

acc /

acc

0

0 10 20 30 40 50 60 700

0.5

1

1.5

2

2.5


Peak v

ert

ical

acc.

[m/s

2]

Peak acc.

1s−RMS acc.

0 0.4 0.8 1.2 1.6 20

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

hp/L

m = 0.13

hp/L

m = 0.17

hp/L

m = 0.22

hp/L

m = 0.25

Basic

50%

25%

5%

amax,P

= 1.296

(a) (b)

Figure 8.17: (a) Absolute peak vertical, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to pylon height and (b) vertical accelerations feltby users.

254


8.5.8 Longitudinal inclination of the pylon

Under static loads, the variation of the maximum deflection at the main span of CSFs

with longitudinally inclined pylons is moderate, as illustrated in Figure 8.19(a). In terms

of their dynamic behaviour, this longitudinal inclination does not significantly change

the frequencies of the first vertical and torsional modes (represented in Figure 8.19(b)).

This modest modification is explained by the height of the pylon and lengths of both

the backstay and the main span stays, which remain almost identical regardless of the

inclination of the pylons.

1.8

1.0um

ax /

um

ax,0

-20 -10 10

Tower inclination

200.0

1.4

a [º]

umax

(-)

(+)

a

(a)

1.0

2.0

3.0

4.0

V1

V2

V3 T1

5.0

Fre

qu

ency [

Hz]

-20 -10 10

Tower inclination

200.0

a [º]

V4

T2

(b)

Figure 8.18: (a) Static and dynamic behaviour of the 2T-CSF according to pylon longi-tudinal inclination α (compared to that of the benchmark CSF): (a) main span maximumstatic deflections umax and (b) frequencies [Hz] of vertical and torsional modes.

Under the passage of pedestrians, the 2T-CSFs with inclined pylons exhibit very similar

vertical accelerations at the deck and pedestrians notice practically the same vertical

accelerations (as seen in Figure 8.19). Hence, this geometric characteristics does not

sensibly modify the serviceability performance of the 2T-CSFs.

0 10 20 30 40 50 60 700

0.5

1

1.5

2


Peak v

ert

ical

acc.

[m/s

2]

−10 −5 0 5 10 150.5

0.75

1

1.25

1.5


acc /

acc

0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

50%

25%

5%

amax,P

= 1.296

α = −10α = −5α = 5α = 10α = 15Basic

Peak acc.

1s−RMS acc.

(a) (b)

Figure 8.19: (a) Absolute peak vertical, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to pylon longitudinal inclination and (b) verticalaccelerations felt by users.

255



Analogously to the performance of 1T-CSFs, the following paragraphs evaluate the

modifications that pylon shapes such as ‘H’, portal shape or ‘A’ (represented in Fig-

ure 7.27) introduce to the response in service of cable-stayed footbridges with two pylons

in comparison to that of the footbridge with two single mono-pole shaped pylons (pylon

shapes of the reference 2T-CSF).

The shape of the pylons considerably affects the lateral and rotational (if the deck is

supported at the pylon section at two points) rigidity of the deck. Nonetheless, in the

vertical direction, the shape of the pylons has a moderately low impact (as described by the

first vertical and torsional modes of the CSF with these pylon shapes given in Table 8.3)

since both their static and dynamic responses depend more on the span arrangement and

the longitudinal inclination of the stays in relation to the deck (which are very similar in

each case regardless of the pylon shape).

Table 8.3: Frequencies [Hz] of the vertical and torsional modes of CSF according to theshape of pylons.

Pylon V1 V2 V3 V4 T1 T2

(a) ‘I’ 1.22 1.84 3.03 4.17 2.23 2.91(b) ‘H’ 1.20 1.78 3.09 4.21 2.92 4.93

(c) Portal 1.24 1.86 3.06 4.20 2.78 4.95(d) ‘A’ 1.23 1.86 3.10 4.24 3.43 5.23

In terms of the vertical accelerations recorded at the deck or those noticed by pedes-

trians, none of the pylon shapes generate accelerations larger or smaller than 10% in

comparison to those of the reference 2T-CSF (as depicted in Figure 8.20(a,b)). Thus, the

vertical response in service of these cable-stayed footbridges is not considerably affected

by the shape of the pylons.

0 10 20 30 40 50 60 700

0.5

1

1.5

2

2.5


Peak v

ert

ical

acc.

[m/s

2]

I H Portal A

0.8

1

1.2

acc /

acc

0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

Peak acc.

1s−RMS acc.

’H’ pylon

Portal pylon

’A’ pylon

Basic (’I’ pylon)

25%

5%

amax,P

= 1.296

(a) (b)

Figure 8.20: (a) Absolute peak vertical, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to pylon shape and (b) vertical accelerations feltby users.

256



The inclination of the stay cables in transverse direction (represented in Figure 7.33)

decreases lightly the frequencies of the first vertical and torsional modes of the footbridge,

due to the smaller vertical component of the stays (values represented in Table 8.4) and

changes the modal masses of mode V2 and the torsional modes (in particular at the

footbridge with largest transverse inclination). These changes in the dynamic behaviour

are related to the amplitudes of the accelerations recorded at the deck and noticed by

users (represented in Figure 8.21).

When the pylons are vertical or slightly inclined, the vertical modes (in particular

V2) have characteristics that are close to those of the benchmark footbridge with two

mono-pole pylons. The participation from this mode V2 decreases with increasing lateral

inclination (which is the case for CSFs with transverse inclination α = 10o). However,

further increments of this inclination trigger larger responses of the torsional modes (due

to the reduced vertical projection of the stays) and generate vertical accelerations that

are slightly larger than those described for the CSFs where stays are less inclined.

Hence, a moderate transverse inclination of the pylons (5o ≤ α ≤ 10o) exhibit a

lightly improved service performance in the vertical direction (both at the deck and the

amplitude of the vertical accelerations felt by pedestrians) whereas further transverse

inclination increases the accelerations recorded at the deck of the footbridge.

Table 8.4: Frequencies [Hz] of vertical and torsional modes of CSF according to the lateralinclination of pylons.

Inclination V1 V2 V3 V4 T1 T2

‘I’ shape 1.22 1.84 3.03 4.17 2.23 2.91α = 0o 1.20 1.78 3.09 4.21 2.92 4.93α = 5o 1.18 1.76 3.09 4.21 2.94 4.94α = 10o 1.17 1.74 3.08 4.20 2.95 4.92α = 15o 1.14 1.70 3.06 4.18 2.94 4.89


The distance between consecutive anchorages of the stay cables at the main span

(ranging from 6 m to 9 m) has a limited effect on the torsional modes of vibration of

the footbridge and even smaller on the vertical modes (illustrated in Figure 8.22), as the

vertical inclination of the stays is relatively similar in each of these scenarios and modal

frequencies are more related to the length and arrangement of the spans.

The vertical accelerations recorded at the deck and the accelerations felt by walk-

ing pedestrians in these scenarios with modified anchorage spacing, both represented in

Figure 8.23(a,b), show that the distance between consecutive stay cables has a limited

impact on the amplitude of the vertical accelerations. Only the model with largest dis-

tance between stays, which has a smaller participation in the response of torsional modes,

introduces a moderate improvement of the serviceability performance (the peak vertical

accelerations are smaller by 15%, and 25% of the users notice peak accelerations that are

25% smaller than for the other three footbridges).

257


0 10 20 30 40 50 60 700

0.5

1

1.5

2


Peak v

ert

ical

acc.

[m/s

2]

0 5 10 150.5

0.7

0.9

1.1


acc /

acc

0

0 0.2 0.4 0.6 0.8 1 1.20

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

50%

25%

5%

amax,P

= 1.296

α = 0º

α = 5º

α = 10º

α = 15º

Peak acc.

1s−RMS acc.

(a) (b)

Figure 8.21: (a) Absolute peak vertical, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to lateral inclination of stays and (b) verticalaccelerations felt by users.

1.0

2.0

3.0

4.0

V1

V2

V3 T1

5.0

Fre

qu

ency [

Hz]

6.0 7.0 9.0

Cable distance Dc

8.0

[m]

V4

T2

Figure 8.22: Frequencies [Hz] of vertical and torsional modes of CSFs with anchorages ofstays spaced different distances.

0 10 20 30 40 50 60 700

0.5

1

1.5

2


Peak v

ert

ical

acc.

[m/s

2]

6 7 8 9

0.8

1

1.2


acc /

acc

0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

Peak acc.

1s−RMS acc.

6.0m

8.0m

9.0m

Basic

50%

25%

5%

amax,P

= 1.296

(a) (b)

Figure 8.23: (a) Absolute peak vertical, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to distance of cable anchorages and (b) verticalaccelerations felt by users.

Thus, the anchorage spacing reduces the vertical accelerations if it changes the partic-

258


ipation from the torsional and vertical modes. However, due to the limited modifications

that these different anchorage arrangements introduce in the characteristics of the stays

(both length and inclination), in general this characteristic does not alter the serviceability

response of 2T-CSFs in a noticeable manner.

8.5.12 Geometry of the deck: deck width

A larger dimension of the deck width notably increases the structural mass that pedes-

trian dynamic loads excite while walking on it. This increment of the mass causes the

bridge to develop vertical and torsional modes of vibration at lower frequencies, as it can

be seen in Figure 8.24.

4.0 5.0 6.0

1.0

2.0

3.0

4.0

V1

V2 V3 T1

T2

5.0

Fre

qu

ency [

Hz]

Deck width [m]

Trans. deck sec.: w

V4

Figure 8.24: Dynamic behaviour of the 2T-CSF according to deck width: frequencies [Hz]of vertical and torsional modes.

In service, under the passage of walking pedestrians, this additional mass generally re-

duces the peak and the average vertical accelerations of the deck of these cable-stayed foot-

bridges as well as the accelerations noticed by walking users (illustrated in Figure 8.25).

This is what occurs at the 2T-CSF with a deck width of 5 m (where the peak and 1s-RMS

vertical accelerations are lower by 50%, and 25% of the users feel accelerations larger than

0.4amax,P instead of 0.78amax,P ). The 2T-CSF with a deck of 6 m width has modes V2

and T1 coupled, with very similar frequencies, which explains that this footbridge ex-

hibits vertical accelerations that are larger than those for the 2T-CSFs with smaller deck

masses.

Hence, as it has been observed with 1T-CSFs, the increment of the deck width is

a beneficial measure to reduce the vertical accelerations unless some of the modes with

largest contribution (e.g., V2) are distorted by other vibration modes.


The side span Ls regulates the static and dynamic behaviour of the cable-stayed foot-

bridge, as depicted in Figure 8.26. Regarding the latter, both the torsional and vertical

modes are considerably changed with this magnitude, due to the magnitude of the total

length (which is free to rotate longitudinally between embankments) and the total mass

of the deck in each case. In fact, with side spans longer than 0.3Lm, the footbridge has

additional vertical modes with two, three or even four antinodes at the main span (V3b

or V4b in Figure 8.26(b)).

259


0 10 20 30 40 50 60 700

0.5

1

1.5

2

2.5


Peak v

ert

ical

acc.

[m/s

2]

1.0 1.125 1.25 1.375 1.50.25

0.5

0.75

1

1.25

wd / w

d,0

acc /

acc

0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

% P

edestr

ians

amax,Pi

/ amax,P

wd = 5.0m

wd = 6.0m

Basic

Peak acc.

1sRMS acc.

50%

25%

5%

amax,P

= 1.296

(a) (b)

Figure 8.25: (a) Absolute peak vertical, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to deck width and (b) vertical accelerations felt byusers.

0.3 0.40.2 0.5

1.10

1.0

um

ax /

um

ax,0

1.05

Ls Lm

Ls

Lm

Ls

umax

(a)

1.0

2.0

3.0

4.0

5.0

Fre

qu

ency [

Hz]

(b)

Figure 8.26: (a) Static and dynamic behaviour of the 2T-CSF according to side span lengthLs (compared to that of the benchmark CSF): (a) main span maximum static deflectionsumax and (b) frequencies [Hz] of vertical and torsional modes.

Despite the fact that a longer span would have been expected to decrease the ampli-

tudes of the vertical accelerations (as argued in Chapter 5 with girder footbridges), both

footbridges with medium and long side spans have moderately or even considerably larger

vertical accelerations. For the 2T-CSF with side span length of 0.3Lm this larger response

is due to the participation from the new modes (V3b and V4b, with three and four antin-

odes at the main span at frequencies 3.3 and 4.5 Hz) that compensate the slightly lower

movements triggered by mode V2 and V3. For the footbridge with a side span length of

0.4Lm, part of these vertical accelerations are generated by the additional modes V2b and

V3b (1.9 and 3.1 Hz) and part are caused by the concomitant large lateral accelerations

generated by the same pedestrians (the frequency content of the vertical accelerations at

locations such as x = 26.4 m show a large peak at frequency 1.15 Hz, coinciding with the

first lateral mode of the footbridge).

Hence, the effect of the length of the side span is related to the type of vertical modal

frequencies that the bridge exhibits within the range 1.5-5.0 Hz.

260


0 10 20 30 40 50 60 700

0.5

1

1.5

2

2.5


Peak v

ert

ical

acc.

[m/s

2]

0.15 0.2 0.25 0.3 0.35 0.4 0.450.8

1

1.2

1.4

1.6

Ls / L

m

acc /

acc

0

0 0.2 0.4 0.6 0.8 1 1.2 1.40

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

Ls = 0.3 L

m

Ls = 0.4 L

m

Basic

Peak acc.

1s−RMS acc.

50%

25%

5%

amax,P

= 1.296 m/s2

(a) (b)

Figure 8.27: (a) Absolute peak vertical, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to side span length and (b) vertical accelerationsfelt by users.

8.6 Strategies to improve the lateral dynamic performance of

cable-stayed footbridges with two pylons in service

The sections below, describe and discuss the magnitudes of the lateral serviceability

accelerations generated by pedestrian flows of medium-high density at cable-stayed foot-

bridges with two pylons and structural and geometric characteristics as those considered

in the previous sections.


The lateral accelerations generated by a continuous pedestrian flow with medium-high

density at the deck of cable-stayed footbridges with two pylons and different articulations

of the deck are illustrated in Figure 8.28(a). The footbridges with ‘POTs(e)’ articula-

tions (with unrestricted longitudinal movements) give rise to the largest lateral acceler-

ations, with peak magnitudes near 0.8 m/s2, the footbridges with the deck articulated

by LEBs or LEBs with a shear key generate modestly lower lateral accelerations (peaks

near 0.75 m/s2), whereas the footbridges with a classical POT scheme (a) support (placed

at the benchmark cable-stayed footbridge) or a statically indeterminate POT scheme (d)

produce more moderate accelerations (peaks below 0.37 and 0.2 m/s2 respectively). The

differences among the lateral accelerations felt by users in each case are large as well: the

best scenario causes pedestrians to notice peak lateral accelerations above 0.5amax,P and

the worst 2.15amax,P .

None of these support arrangements give place to unstable (or almost unacceptable)

lateral accelerations (as occurred at 1T-CSFs with LEBs), although the accelerations

developed at the footbridges with ‘POTs(e)’, LEBs and LEBs with a shear key as supports

are unacceptable in serviceability conditions (as classified by comfort classes such as those

261


outlined by Setra, 2006).

The magnitude of these lateral accelerations are founded on the lateral rigidity that

each support arrangement introduces to the deck and are related to the magnitude of the

first lateral modal frequencies developed in each case (given in Table 8.5). The ‘POTs(e)’

support arrangement and those involving LEBs do not restrict the horizontal rotation

of the deck at the support sections over the abutments (the first arrangement provides

the smallest restriction), whereas the support scheme of the benchmark footbridge allows

these rotations at only one of the support sections over the abutments and the support

scheme ‘POTs(d)’ restrains rotations at both abutments.

Table 8.5: Frequencies [Hz] of lateral vibration modes of 2T-CSFs according to supportarrangement.

Supports L1 L2

(a) Basic 1.82 6.19(b) LEBs 0.93 1.76

(c) LEBs+SK 1.34 5.76(d) POTs 2.67 7.42(e) POTs 1.34 5.76

Hence, similarly to the serviceability response of 1T-CSFs, the performance in the

lateral direction of 2T-CSFs is related to the lateral rigidity that the deck support scheme

generates and it is improved with those support configurations that increase the deck

lateral rigidity.

0 10 20 30 40 50 60 700

0.25

0.5

0.75

1


Peak late

ral

acc.

[m/s

2]

(a) (b) (c) (d) (e)0

0.5

1

1.5

2

2.5

acc /

acc

0

0 0.4 0.8 1.2 1.6 20

20

40

60

80

100

% P

edestr

ians

amax,Pi

/ amax,P

Peak acc.

1s−RMS acc.

(b) LEBs(c) LEBs+SK(d) POT 1(e) POT 2Basic

50%

25%

5%

amax,P

= 0.365 m/s2

(a) (b)

Figure 8.28: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck with alternative support schemes and (b) lateral accelera-tions felt by users.

8.6.2 Area of backstay cables

In the lateral direction, the cross section of the backstay cable does not notably change

how the deck of the CSF deflects under static loads (these depend more on the total length

262


of the deck between sections at the abutments).

From the point of view of the dynamic behaviour of the footbridge, this moderate effect

leads to very similar lateral frequencies irrespective of the characteristics of this cable (in

particular the first two lateral modes, with frequencies near 1.8 and 6.20 Hz respectively).

The first torsional mode, T1, adopts very similar frequencies as well although it has a

smaller lateral component and larger longitudinal rotation component when the backstay

cable has smaller areas. At 1T-CSFs smaller areas of the backstay produce the contrary

effect due to the larger height of the pylon.

These changes at the modal shape of T1 are related to the smaller lateral accelerations

of the deck of the CSF with smaller backstay cables and the similar lateral accelerations in

the other cases, both illustrated in Figure 8.29(a) (peak and 1s-RMS lateral accelerations

are smaller by 30% at the first scenario).

Despite the differences in the peak and 1s-RMS lateral accelerations recorded at the

deck of CSFs with smaller backstays, the magnitudes of the accelerations felt by walking

pedestrians are more similar. For the footbridge with smaller backstay cables, 25% of

the users feel peak accelerations equal or larger than 0.68amax,P in comparison to 0.78-

0.85amax,P for the other footbridges.

0 10 20 30 40 50 60 700

0.2

0.4


Peak late

ral

acc.

[m/s

2]

0.5 1 1.5 2 2.50.6

1

1.4

ABS

/ ABS,0

acc /

acc

0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

% P

edestr

ians

amax,Pi

/ amax,P

0.5 ABS,0

1.5 ABS,0

2.0 ABS,0

2.5 ABS,0

ABS,0

Peak acc.

1s−RMS acc.

50%

25%

5%

amax,P

= 0.365 m/s2

(a) (b)

Figure 8.29: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to backstay area and (b) lateral accelerations feltby users.

Thus, smaller backstay cable areas allow the deck to reproduce smaller lateral acceler-

ations, although the impact of this reduction, from the users’ point of view, is moderate.

If the height of the pylons of the footbridge was considerably larger, as that of 1T-CSFs,

the effect would be the opposite. It should be highlighted that this measure is not feasible

unless the technology of cables allows larger stresses and stress cycles.

263



The area of the main span stay cables does not vary the lateral static deformation

described by the deck of 2T-CSFs under the same static loads (similarly to the backstay

cable). In relation to the dynamic behaviour of the footbridge, the first lateral frequencies

remain fairly constant despite these stay areas (mode L1 has frequencies near 1.8 Hz and

L2 near 6.2 Hz). Torsional modes, which are related as well to the performance of the CSF

in the lateral direction, adopt fairly similar frequencies regardless of the section of these

cables. The projection of these torsional modes in the lateral direction does not change

according to the area of the main span stays, as opposed to what occurs at 1T-CSFs

(which is due to the short height of the pylons, in comparison to that of 1T-CSFs).

These dynamic characteristics are related to the similar magnitudes of the peak and

1s-RMS lateral accelerations exhibited at the 2T-CSFs with alternative main span stays,

illustrated in Figure 8.30(a), and to the accelerations felt by walking users illustrated in

Figure 8.30(b) (25% of the users feel peak accelerations above 0.78-0.85amax,P regardless

of the scenario).

0 10 20 30 40 50 60 700

0.2

0.4


Peak v

ert

ical

acc.

[m/s

2]

0.5 1 1.5 2 2.5

0.8

1

1.2

AS / A

S,0

acc /

acc

0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

% P

edestr

ians

amax,Pi

/ amax,P

0.5 AS,0

1.5 AS,0

2.0 AS,0

2.5 AS,0

AS,0

Peak acc.

1s−RMS acc.

50%

25%

5%

amax,P

= 0.365 m/s2

(a) (b)

Figure 8.30: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to main stays area and (b) lateral accelerationsfelt by users.

Hence, in cable-stayed footbridges with two pylons, the area of the main span stays

neither improves nor worsens the dynamic performance of these footbridges under traffics

of pedestrians.


Despite the fact that the thickness of the steel girder bottom flanges change notably

the deck lateral second moment of area, the dynamic lateral modes of the footbridge

practically remain constant (L1 near 1.8 Hz and L2 at 6.2 Hz). Torsional modes appear

at similar frequencies as well although its projection in the lateral direction increases with

this flange thickness.

264


These dynamic characteristics are not reflected on the magnitude of the peak and

1s-RMS lateral accelerations, as these are the same regardless of the magnitude of the

bottom flange girder (both have magnitudes near 0.37 m/s2, as seen in Figure 8.31(a)).

However, the lateral accelerations felt by walking users depend on the major contribution

of the torsional mode T1 in the lateral direction, although the impact is relatively small

(25% of the users notice peak lateral accelerations above 0.92amax at the footbridge with

deepest flange thickness, in comparison to 0.78amax,P at the benchmark footbridge).

In comparison to the 1T-CSFs, in these footbridges with two pylons this magnitude

has smaller effects on the lateral accelerations due to the smaller dimension of this thick-

ness. Nonetheless, it can be stated that, similarly to 1T-CSFs, steel girders with deeper

thickness increase slightly the amplitude of the lateral accelerations, therefore they do not

represent an adequate measure to improve the lateral response in service.

0 10 20 30 40 50 60 700

0.2

0.4


Peak late

ral

acc.

[m/s

2]

0.2 0.3 0.4

0.75

1

1.25

tbf

/ tbf,0

acc /

acc

0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

% P

edestr

ians

amax,Pi

/ amax,P

1.4 tbf,0

1.8 tbf,0

2.2 tbf,0

tbf,0

Peak acc.

1s−RMS acc.

50%

25%

5%

amax,P

= 0.365 m/s2

(a) (b)

Figure 8.31: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to bottom flange steel girder thickness and (b)lateral accelerations felt by users.


When increasing the deck concrete slab depth, the higher lateral second moment of

area of the deck section increases the magnitude of the first lateral modes despite the

concomitant increment of the deck mass per unit length (L1 increases from 1.8 to 1.96 Hz

and L2 from 6.2 to 6.4 Hz). Additionally, a larger depth of the concrete slab reduces the

lateral projection of the first torsional modes.

The larger mass of the deck, the higher frequencies of the lateral mode L1 and the

smaller projection of mode T1 in the lateral direction constitute the main reasons for a

considerably smaller lateral acceleration of CSFs with larger concrete slab depths (de-

picted in Figure 8.32(a)). The peak and 1s-RMS lateral accelerations decrease from 0.37

and 0.24 m/s2 at the benchmark CSF to accelerations below 0.03 m/s2 in both cases at

CSFs with slab depths of 0.4 or 0.5 m.

This large impact in the lateral response is visible as well in the amplitude of the peak

265


lateral accelerations felt by users (illustrated in Figure 8.32(b)), as 25% of the users notice

peak accelerations above 0.4amax,P or 0.0amax,p at footbridges with slabs of depth 0.3 m

or 0.4 m respectively.

Hence, the increment of the slab depth is an substantially effective measure to reduce

the lateral accelerations of CSFs with two pylons.

0 10 20 30 40 50 60 700

0.2

0.4


Peak v

ert

ical

acc.

[m/s

2]

1 1.5 2 2.50

0.25

0.5

0.75

1

tc / t

c,0

acc /

acc

0

0 0.2 0.4 0.6 0.8 1

20

40

60

80

100

% P

edestr

ians

amax,Pi

/ amax,P

Peak acc.

1s−RMS acc.

1.5 tc,0

2.0 tc,0

2.5 tc,0

tc,0

50%

25%

5%

amax,P

= 0.365 m/s2

(a) (b)

Figure 8.32: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to concrete slab thickness and (b) lateral accel-erations felt by users.

8.6.6 Pylon section

A larger diameter of the pylons restrains the lateral accelerations of the deck of the

cable-stayed footbridge. In terms of dynamic behaviour, this characteristic leads to first

lateral modes at higher frequencies (from 1.82 to 2.17 Hz when increasing the diameter

from Dt,0 to 2.5Dt,0) and to a reduced lateral projection of torsional modes, in particular

of mode T1 (with a drastic reduction when decreasing the diameter from Dt,0 to 1.3Dt,0).

These dynamic characteristics are the reasons that explain the smaller amplitudes

of the peak and 1s-RMS accelerations at the cable-stayed footbridges with larger pylon

diameters (represented in Figure 8.33(a), with decrements of 55% in both cases). In terms

of the lateral peak accelerations noticed by walking pedestrians, these have amplitudes

with trends similar to those observed for the lateral accelerations recorded at the deck

(Figure 8.33(b)), with peak accelerations for 25% of the users that are reduced from 0.78

to 0.3amax,P .

Hence, the increment of the diameter of the pylons increases the first lateral mode

frequencies and reduces the contribution of the torsional modes in the lateral direction.

Therefore it represents and effective measure to reduce the lateral accelerations of these

cable-stayed footbridges with two mono-pole pylons.

If instead of the diameter, the thickness of the cross section of the pylons is increased,

both the lateral and torsional modes adopt very similar characteristics, and this can be

related to the lack of modifications introduced by this measure in the amplitude of the

266


0 10 20 30 40 50 60 700

0.2

0.4


Peak late

ral

acc.

[m/s

2]

1.0 1.5 2.0 2.50

0.5

1

1.5

Dt / D

t,0

acc /

acc

0

0 0.2 0.4 0.6 0.8 1

20

40

60

80

100

% P

edestr

ians

amax,Pi

/ amax,P

Peak acc.

1s−RMS acc.

1.3 Dt,0

1.7 Dt,0

2.5 Dt,0

Dt,0

50%

25%

5%

amax,P

= 0.365 m/s2

(a) (b)

Figure 8.33: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to pylon diameter and (b) lateral accelerationsfelt by users.

peak and 1s-RMS lateral accelerations recorded at the deck as well as the peak lateral

accelerations felt by walking users in each of these cable-stayed footbridges. Therefore the

modification of the thickness of the pylons’ section does not affect the lateral response in

service of CSFs with two pylons.

8.6.7 Pylon height

The alteration of the height of the pylons changes the length of the backstay and main

span stays as well as the transverse inclination of these main span stays (in relation to

the deck). In terms of dynamic behaviour, as the pylon height is increased, the smaller

transverse inclination of the stays is related to smaller lateral and torsional frequencies,

and a progressive larger contribution of mode T1 to the lateral vibration. With a pylon

height of 0.25Lm approximately, both L1 and T1 have very similar frequencies (there is a

coupling of these vibration modes).

The projection in the lateral direction of modes L1 and T1 explain the amplitudes

of the accelerations recorded at the deck and given in Figure 8.34(a). The lateral accel-

erations increase with the height of the pylons except for CSFs with pylon height near

0.25Lm, where the modification of the principal modes that generate the lateral accel-

erations produce lower lateral accelerations. The lowest accelerations at the CSF with

shortest pylons are 0.5 times smaller than those for the reference CSF. The effect on

the amplitude of the lateral accelerations felt by walking users is similar, as the shortest

cable-stayed footbridge generates peak accelerations above 0.43amax,P for 25% of the users

in comparison to 0.78amax,P at the reference footbridge.

Thus, higher pylons produce larger lateral accelerations due to higher lateral projec-

tions of modes T1 and L1 unless frequencies of these modes are suddenly modified as it

occurs for the CSF with pylons’ height of 0.25Lm.

267


0.15 0.2 0.250.2

0.4

0.6

0.8

1

hp / L

m

acc /

acc

0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

0 10 20 30 40 50 60 700

0.2

0.4


Peak v

ert

ical

acc.

[m/s

2]

Peak acc.

1s−RMS acc.

hp/L

m = 0.13

hp/L

m = 0.17

hp/L

m = 0.22

hp/L

m = 0.25

Basic

50%

25%

5%

amax,P

= 0.365 m/s2

(a) (b)

Figure 8.34: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to pylon height and (b) lateral accelerations felt byusers.

8.6.8 Longitudinal inclination of the pylon

The longitudinal inclination of the pylons of cable-stayed footbridges with two pylons

does not change in a noticeable manner the magnitudes of the lateral and torsional modes

(L1 and L2 have magnitudes near 1.8 and 6.2 Hz respectively and T1 and T2 near 2.2

and 2.87 Hz). This effect is caused by the very similar length of the backstay and main

span stays regardless of this longitudinal inclination of the pylons.

The modest modifications of the lateral and torsional vibration modes originate lateral

serviceability accelerations at these CSFs that are very similar to those of the reference

CSF (as illustrated in Figure 8.35(a)). Only the footbridge with largest inclination towards

the abutments produces smaller lateral accelerations, which is due to the coincidence of

modes V2 and L1 in frequency magnitude.

The lateral peak accelerations felt by users walking on these footbridges are fairly

similar as well, as represented in Figure 8.35(b) (25% of the users notice lateral peak

accelerations larger than 0.78-0.9amax,P ). Therefore the longitudinal inclination of the

pylons neither improves nor deteriorates the comfort in the lateral direction of pedestrians

crossing these footbridges.


The shape (number of poles and their transverse inclination in relation to the deck) is

a characteristic of the cable-stayed footbridge that is critical to the lateral and torsional

stiffness of the deck of the footbridge. Pylons with ‘H’, portal or ‘A’ shapes allow the

deck to exhibit smaller transverse accelerations (each of these pylon shapes has two poles

instead of one as it is the case of the pylon at the reference footbridge, with an ‘I’ shape).

Furthermore, with these three pylon shapes the deck has a larger torsional stiffness as there

are two supports at the pylon sections. These two characteristics are related to higher

lateral frequencies of vibration (these have magnitudes of 1.93, 1.97 and 2.28 Hz for pylons

268


0 10 20 30 40 50 60 700

0.2

0.4


Peak late

ral

acc.

[m/s

2]

−10 0 10 200.5

0.75

1

1.25

1.5


acc /

acc

0

0 0.2 0.4 0.6 0.8 1 1.20

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

Peak acc.

1s−RMS acc.

α = −10ºα = −5ºα = 5ºα = 10ºα = 15ºBasic

25%

5%

50%

amax,P

= 0.365 m/s2

(a) (b)

Figure 8.35: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to pylon longitudinal inclination and (b) lateralaccelerations felt by users.

with ‘H’, portal or ‘A’ shape respectively, in comparison to 1.82 Hz at the reference CSF)

and to larger first torsional modes (with frequencies 2.92, 2.78 and 3.43 Hz for pylons

with ‘H’, portal or ‘A’ shape respectively, as opposed to 2.23 Hz at the reference case).

0 10 20 30 40 50 60 700

0.2

0.4


Peak late

ral

acc.

[m/s

2]

I H Portal A0

0.5

1

acc /

acc

0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

50%

25%

5%

amax,P

= 0.365 m/s2

Peak acc.

1s−RMS acc.

’H’ pylon

Portal pylon

’A’ pylon

’I’ pylon

(a) (b)

Figure 8.36: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to pylon shape and (b) lateral accelerations felt byusers.

These dynamic characteristics give place to considerably smaller serviceability accel-

erations in the lateral direction, as represented in Figure 8.36(a). The footbridges with

portal and ‘A’ pylons correspond to the most adequate response as peak and 1s-RMS ac-

celerations are 70% smaller than those for the footbridge with ‘I’ pylons and pedestrians

feel considerably smaller peak lateral accelerations (none of them feel accelerations above

269


0.075 m/s2).

Thus, cable-stayed footbridges with pylon shapes that are more rigid in transverse

direction and restrict the lateral and torsional rotations of the deck substantially reduce

the amplitude of the lateral accelerations developed in service.


A larger transverse inclination of the backstay and main span stays in a cable-stayed

footbridge increases the transverse stiffness of the deck, since the stays have larger hori-

zontal components that react to the transverse deflections of the deck. From the dynamic

point of view, the lateral inclination of the stays does not drastically change the first

lateral modes of vibration of 2T-CSFs (regardless of the magnitude of this inclination,

the first lateral mode L1 has a frequency near 1.9 Hz) or the first torsional mode (T1

adopts a frequency of 2.95 Hz). However, this transverse inclination affects the vertical

and transverse projection of mode T1, in particular for large inclinations.

In accordance to these dynamic characteristics, the cable-stayed footbridges with ver-

tical or lightly inclined ‘H’ pylons describe lateral accelerations that are very similar, as

depicted in Figure 8.37(a), whereas those with large angle of inclination do not control

effectively the transverse and torsional accelerations (peak and 1s-RMS accelerations are

15% larger than those for the CSF with vertical ‘H’ pylons and 65% larger than those

for the CSF with pylons that are inclined α = 10o). The amplitudes of the accelerations

felt by pedestrians convey similar results, as 25% of those walking on the CSF with py-

lons that are most inclined feel peak accelerations larger than 0.5amax,P in comparison to

0.43amax,P at the footbridge with vertical ‘H’ pylons (see Figure 8.37(a)), although both

are better than the lateral accelerations felt when walking on the footbridge with two

mono-pole pylons.

0 10 20 30 40 50 60 700

0.2

0.4


Peak late

ral

acc.

[m/s

2]

0 5 10 15

0.5

1

1.5


acc /

acc

0

0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

50%

25%

5%

amax,P

= 0.365 m/s2

α = 0º

α = 5º

α = 10º

α = 15º

Peak acc.

1s−RMS acc.

(a) (b)

Figure 8.37: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to lateral pylon inclination and (b) lateral ac-celerations felt by users.

270



The alteration of the spacing between consecutive anchorages of the main span stay

cables (distance ranging from 6 m to 9 m) leaves the first lateral modes of the footbridge

practically unchanged (the frequency of the first lateral mode remains constant at 1.82-

1.83 Hz). The lack of effect of this characteristic on the footbridge dynamic behaviour is

explained by the low restrain of the deck lateral displacement that these stays produce

(due to their small transverse inclination towards the deck).

Figures 8.38(a) and (b) illustrate the amplitudes of the lateral accelerations recorded at

the deck and felt by users while crossing CSFs with alternative anchorage spacing. These

highlight the low impact of this magnitude on the footbridge serviceability performance

in the lateral direction.

0 10 20 30 40 50 60 700

0.2

0.4


Peak late

ral

acc.

[m/s

2]

6 7 8 9

0.8

1

1.2


acc /

acc

0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

Peak acc.

1s−RMS acc.

6 m

8 m

9 m

Basic

50%

25%

5%

amax,P

= 0.365 m/s2

(a) (b)

Figure 8.38: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to cable anchorage spacing and (b) lateral ac-celerations felt by users.

8.6.12 Geometry of the deck: deck width

Similarly to the magnitude of the slab depth, the width of the deck modifies the lateral

second moment of area of the deck as well as its mass. With larger deck widths, the

increment of the deck second moment of area in the lateral direction rises the frequency

of the first lateral mode rapidly despite the concomitant increment of mass (L1 has a

frequency of 1.82 Hz with a deck width of 4 m and 2.77 Hz with a width of 6 m).

At CSFs with wider decks, the substantially larger deck masses as well as the higher

frequencies of the lateral mode L1 give rise to considerably smaller accelerations, as de-

scribed in Figure 8.39(a). A CSF with a deck width of 5 m defines peak and 1s-RMS

lateral accelerations that are 75% smaller than those for the reference footbridge (with a

deck width of 4 m) and that with a deck width of 6 m, lateral accelerations 83% smaller

(the reduction of the lateral accelerations in this case is not as large as that of a narrower

deck width due to the coincidence in frequency magnitude of modes V3 and L1). The

amplitudes of the lateral accelerations felt by walking users are in line with those accel-

271


erations of the deck. At cable-stayed footbridges with deck widths of 5 m and 6 m, 75%

of the users notice accelerations smaller than 0.3amax,P or 0.2amax,P .

Hence, larger deck widths improve the performance of these cable-stayed footbridges

in the lateral direction basically due to the larger mass of the deck in each case.

0 10 20 30 40 50 60 700

0.2

0.4


Peak late

ral

acc.

[m/s

2]

1 1.125 1.25 1.375 1.5

0.250.5

0.751

1.25

wd / w

d,0

acc /

acc

0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

% P

edestr

ians

amax,Pi

/ amax,P

Peak acc.

1s−RMS acc.

1.25 wd,0

1.5 wd,0

Basic

50%

25%

5%

amax,P

= 0.365 m/s2

(a) (b)

Figure 8.39: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to deck width and (b) lateral accelerations feltby users.


As has been noticed in Section 8.4, one of the main parameters that is related to

the amplitude of the footbridge lateral acceleration in service is the magnitude of the

first lateral frequency L1. The frequency of this mode is more similar to that for a

simply supported girder with a single span of length equal to the whole footbridge length

(1.4Lm) than to that for a simply supported girder with three spans of length 0.2Lm +

Lm + 0.2Lm due to the small restriction of the pylons to the lateral accelerations of the

deck. Accordingly, the first lateral mode of CSFs with two pylons decreases from 1.82 Hz

(when the side span has a length of 0.2Lm) to 1.43 Hz (with 0.3Lm) or 1.15 Hz (with

0.4Lm).

The lateral accelerations of the deck of a 2T-CSF with side spans of length 0.3Lm are

fairly similar to those of the benchmark footbridge (with side spans length of 0.2Lm), as

illustrated in Figure 8.40(a). However, those of the footbridge with side spans of length

0.4Lm are substantially larger and correspond to an unstable response as the lateral ac-

celerations would continuously increase unless pedestrians stopped walking or the density

of users was drastically reduced (with the simulated duration of the event, peak and 1s-

RMS accelerations are more than 6 times larger). The results of the first case are justified

by the very similar modal characteristics (effective modal masses in the lateral direction

of modes L1 and T1 are fairly similar), whereas those of the second are justified by the

magnitude of the first lateral frequency (near 1.1 Hz). The accelerations felt by users,

represented in Figure 8.40(b), are as those recorded at the deck.

272


0 10 20 30 40 50 60 700

0.25

0.5

0.75

1


Peak late

ral

acc.

[m/s

2]

0.2 0.25 0.3 0.35 0.4

2

4

6

Ls / L

m

acc /

acc

0

0 0.2 0.4 0.6 0.8 1 1.2 1.40

20

40

60

80

100

amax,Pi

/ amax,P

% P

edestr

ians

Peak acc.

1s−RMS acc.

0.3 Lm

0.4 Lm

Basic

50%

25%

5%

amax,P

= 0.365 m/s2

(a) (b)

Figure 8.40: (a) Absolute peak lateral, and relative peak and 1s-RMS accelerationsrecorded at the 2T-CSFs deck according to side span length and (b) lateral accelerationsfelt by users.

Hence, the magnitude of the length of the side spans has an important effect on the

footbridge performance in the lateral direction as it is directly related to the characteristics

of the first lateral modes of the footbridge.

8.7 Cable-stayed footbridges with long main span lengths

As reported in the previous chapter in relation to cable-stayed footbridges with a

single pylon, the most frequent main span length of cable-stayed footbridges is around to

50 m. However, footbridges with moderately longer spans also exist. Thus, the following

sections evaluate the magnitude of the serviceability response of cable-stayed footbridges

with two spans and a main span length of 100 m. Additionally, some of the geometric

and structural characteristics with greatest impact on the vertical or lateral responses of

medium span length cable-stayed footbridges are implemented in the footbridge in order

to assess their validity in longer span bridges.

The following sections describe: (a) the geometrical characteristics and dynamic be-

haviour of representative cable-stayed footbridges with long main span lengths, (b) their

performance under the passage of medium-heavy pedestrian traffic flows and (c) the im-

pact in their serviceability response of design measures previously evaluated at cable-

stayed footbridges of medium span lengths.

8.7.1 Geometry of long span cable-stayed footbridges with two pylons

According to the geometric proportions and structural characteristics of cable-stayed

footbridges that are more common (described in Section 3.5.2), long cable-stayed foot-

bridges with two pylons have main span lengths of 100 m and depth-to-main span length

ratios between 1/100 and 1/200. Their deck consists of a concrete slab of depth 0.2 m

supported by two steel girders located at the edges of this slab of total length 140 m

273


arranged in three spans of length 0.2Lm+Lm+0.2Lm supported by two pylons of height

0.2Lm. Dimensions of the stays, steel girders and cross section of the pylons are obtained

based on this geometry, the ULS enumerated in Chapter 3 and the material characteristics

outlined as well in Chapter 3.

An elevation of the cable-stayed footbridge and detailed definition of the main struc-

tural characteristics of these long span cable-stayed footbridges with two pylons are il-

lustrated in Figure 8.41. The support configuration of these footbridges consists of POT

bearings with a statically indeterminate arrangement (all the horizontal movements are

restricted, as depicted in Figure 8.4(d)). At the pylon sections, the deck is simply sup-

ported by the deck. Results supporting the use of this articulations are described in

Section 8.7.3.


htot = 1.0

hgird = 0.8

tflange,bot = 0.015

tflange,top = 0.015

tweb = 0.010


CB#1CB#2CB#3

CB#4CB#5CB#6

40135

434

HT=

27.5

Hs =

20.0

Hi =

7.5

Lm= 100.0Ls= 20.0 Ls= 20.0

Dp Dc Dc = = = = = = = = Dc Dp

BS

CB#1CB#2

CB#3

CB#4

CB#5

CB#6Sec. A-A

0.20

w = 4.0

HEB 200

h

gird

tflange,bot

Sec. A-A:

hto

t

tflange,top

tweb

Dext = 0.60(steel hollow

section)


htot = 0.5

hgird = 0.3

tflange,bot = 0.020

tflange,top = 0.045

tweb = 0.010


CB#1CB#2CB#3

CB#4CB#5CB#6

43135

434

Figure 8.41: Geometric definition of the representative long span CSFs with transversesection depth Lm/100 and Lm/200. Dimensions in meters [m].

8.7.2 Dynamic characteristics of long span cable-stayed footbridges

The cable-stayed footbridges with main span length of 100 m and depth-to-main span

length ratios of 1/100 or 1/200 have a dynamic behaviour characterised by modal fre-

quencies listed in Table 8.6 and mode shapes depicted in Figures 8.42 and 8.43.

In the vertical direction, both footbridges have multiple modes with frequencies be-

tween 1.0 and 3.0 Hz. The CSF with a deck depth-to-main span length ratio of 1/100

has modes V2, V3 and V4 within that range whereas that with a ratio 1/200 has modes

V2, V3, V4 and V5 at similar frequencies. In the lateral direction both footbridges have

lateral modes with frequencies well below 1.0 Hz. The first lateral mode appears at the

same frequency, pointing towards the reduced impact that the additional mass or stiffness

of a deck section with larger depth has in this case.

274


Table 8.6: Frequencies [Hz] of the vibration modes of long span CSFs with two pylonsaccording to their depth magnitude, where ‘VN’, ‘LN’ and ‘TN’ denote vertical, lateral andtorsional modes with N half-waves and ‘P’ denotes modes related to the pylon.

Deck depth Lm/100 Deck depth Lm/200Mode No. Frequency Description Mode No. Frequency Description

1 0.67 L1 1 0.67 L12 0.95 P+T2 2 0.92 V13 0.96 V1 3 0.94 P+T24 1.03 L1+T1 4 1.04 L1+T15 1.50 V2 5 1.34 V26 2.13 V3 6 1.78 V37 2.16 T2+L2 7 2.06 V48 2.24 T1 8 2.16 T19 2.58 T2 9 2.22 L2+T210 2.61 T3 10 2.46 V511 2.83 V4 11 2.47 T2

L1, 0.67Hz

(a)

P+T2, 0.95Hz

(b)

V1, 0.96Hz

(c)

L1+T1, 1.03Hz

(d)

V2, 1.50Hz

(e)

V3, 2.13Hz

(f)

Figure 8.42: First modal frequencies [Hz] of the long span CSF with a deck depth ofLm/100.

L1, 0.67Hz

(a)

V1, 0.92Hz

(b)

T2+P, 0.94Hz

(c)

L1+T1, 1.04Hz

(d)

V2, 1.34Hz

(e)

V3, 1.78Hz

(f)

Figure 8.43: First modal frequencies [Hz] of the long span CSF with a deck depth ofLm/200.

275


8.7.3 Articulations of the deck

As has been observed for medium span 2T-CSFs (Section 8.5.1) and for medium and

long span 1T-CSFs (Chapter 7), deck articulations that restrict transverse rotations of the

deck at both abutments provide the best performance in service in the lateral direction

(since this larger transversal stiffness of the deck provided by supports is related to first

lateral modes at higher frequencies).

At long span 2T-CSFs with supports that consist of POTs restricting all the horizontal

movements at both abutments, the lateral modes have frequencies well below 1.0 Hz

regardless of the depth of the deck. If instead of these supports, an arrangement with

POTs with a classical layout or LEBs with a shear key at each abutment were considered,

the first lateral frequencies would appear at even lower values, as enumerated in Table 8.7.

Table 8.7: First lateral vibration modes [Hz] of long span CSFs according to deck articu-lation and depth magnitude (see Figure 8.4).

Deck depth Lm/100 Deck depth Lm/200Supports L1 L1b L1 L1b

(a) POTs 0.67 1.03 0.57 1.04(d) POTs 0.56 1.01 0.57 1.02

(b)LEBs+SK

0.46 0.93 0.47 0.95

Despite the support arrangement (POTs (d)), the lateral response of both long span

2T-CSFs (with deck depth Lm/100 or Lm/200) is unstable. At the end of an event

of time duration equivalent to three times the time taken by an average pedestrian to

cross the main span, the lateral response of the 2T-CSF with depth Lm/100 or Lm/200

is larger than 3.0 m/s2. In both cases the lateral accelerations would continue to grow

unless pedestrians stopped walking or the traffic flow had considerably lower density

of pedestrians. Alternative support arrangements would not avoid these large lateral

accelerations (due to the magnitude of their first lateral modes, which are very close

regardless of their arrangement).

In the vertical direction, the response is altered by the unstable lateral accelerations

with unusually large torsional movements that would not occur with stable lateral response

(e.g., the footbridge with largest depth has a peak vertical acceleration recorded at the

middle of the deck of 1.2 m/s2 and a peak vertical acceleration at the deck edge of

2.76 m/s2).

8.7.4 Dimensions of structural elements

Similarly to the analyses performed for 2T-CSFs with main span length of 50 m,

the effect of alternative structural modifications on the serviceability response of these

footbridges has been explored.

As detailed in Section 8.6.2, smaller backstays produce a reduction of the projection

of torsional modes in the transverse direction and hence a reduction of the lateral ac-

celerations of medium span 2T-CSFs. For these longer span footbridges, backstays with

area 0.5ABS,0 exhibit peak lateral accelerations of smaller amplitude, below 2.35 m/s2

276


instead of 3.0 m/s2, although these lateral accelerations are still unstable. Hence, smaller

backstay cables (which would not satisfy ULS requirements) do not improve the lateral

serviceability performance of these footbridges.

Similarly to the area of the backstay, the reduction of the area of the main span

stays (0.5AS,0, which would not satisfy the ULS requirements) reduces the transverse

component of the lateral mode with frequency 1.03-1.04 Hz (mode 4, L1+T). Based

on this effect, both long span cable-stayed footbridges experience a reduction of the peak

lateral acceleration recorded during the same period of time, from 3.0 m/s2 to a maximum

peak lateral acceleration 2.33 m/s2, although these lateral accelerations are still unstable

(only increasing at a lower rate, similarly to the effect of the smaller backstay cable).

One of the parameters that most effectively modifies the serviceability accelerations of

CSFs corresponds to the modification of the slab depth. An increment of this geometric

characteristic from 0.2 m to 0.3 m improves both the vertical and the lateral responses of

these long span 2T-CSFs (regardless of the deck depth) as the lateral accelerations become

stable. As depicted in Figure 8.44, the peak vertical accelerations have amplitudes below

1.0 m/s2 and the peak lateral accelerations below 0.35 m/s2. As can be observed, the

lateral accelerations are reduced dramatically (one order of magnitude) when increasing

the slab depth from 0.2 to 0.3 m.

0 20 40 60 80 100 120 1400

0.25

0.5

0.75

1


Peak v

ert

ical

acc.

[m/s

2]

0 20 40 60 80 100 120 1400

0.1

0.2

0.3

0.4

0.5


Peak late

ral

acc.

[m/s

2]

Lm

/100

Lm

/200

(a) (b)

Figure 8.44: Peak vertical (a) and lateral (b) accelerations recorded at CSF with concreteslab of 0.3 m and deck depths Lm/100 and Lm/200 (continuous lines).

8.7.5 Geometric characteristics of the cable-stayed footbridge

Among the characteristics considered in the previous sections related to geometric

characteristics of medium span length 2T-CSFs, the alteration of the deck width generates

large reductions of the lateral accelerations. For these long span 2T-CSFs, an increment

of the deck width from 4 m to 5 m does not reduce the amplitude of the peak lateral

accelerations but triggers a faster enlargement of these lateral responses (instead of peak

lateral accelerations near 3.0 m/s2 these bridges develop peak movements below 5.0 m/s2

as well as large torsional movements and concomitant high vertical accelerations). Despite

the increment of the lateral second moment of area and deck mass, these responses are

higher due to the shape of the lateral modes (similar to those depicted in Figure 8.2), with

larger contribution in the lateral direction in this case. These results confirm (as it has

277


been observed in other cases) that variations such as these are not always equivalent to

a better performance but depend on what modifications they introduce to the vibration

modes with greatest impact on the serviceability response.


Based on the results described in the previous sections, there are several measures that

notably improve the vertical or the lateral serviceability response (and comfort of users)

of 2T-CSFs. However, others do not modify, or even enlarge, these accelerations.

Figures 8.45 and 8.46 represent the peak vertical and lateral accelerations recorded

at the deck of medium span length 2T-CSFs as well as the maximum accelerations felt

by 75% of the users at the same scenarios together with the analogous comfort ranges

detailed in Section 3.4.

From the comparison of the vertical accelerations to the comfort limits, illustrated in

Figure 8.45, it can be seen how the accelerations recorded at the deck generally correspond

to a minimum level of comfort. If the reference values to assess the comfort of the

footbridge are those felt by 75% of the users, the footbridge has a minimum or medium

level of comfort on few occasions.

Basic BC BS S t_bf t_c D_t h_p Inc. Pylon D_c Lat. Inc. w_d L_s0

0.5

1

1.5

2

2.5

Vert

ical accele

ration [

m/s

2] a

V,DECKa

V, PED 75%

Maximum

Medium

Minimum

Figure 8.45: Comfort assessment of CSF according to the measures implemented to modifyvertical accelerations.

In the lateral direction, regardless of what values are used to represent the serviceability

movements, the level of comfort of the structure corresponds to medium and on few

occasions to either maximum or even minimum (among these there are the scenarios

where the lateral responses become unstable), as seen in Figure 8.46.

In both directions, the evaluation based on the accelerations felt by users is less re-

strictive than that for the accelerations recorded at the deck, in particular in the vertical

direction. This effect can be observed as well from the comparison between accelerations

recorded at the deck and those felt by users at the same events (depicted in Figures 8.47).

From these figures it is seen that the vertical accelerations felt by users are approxi-

mately 0.7 times those of the deck, similar to what occurs in 1T-CSFs, whereas in the

lateral direction both coincide in practically all the occasions (the first is 0.9 times the

second), contrary to what is observed in 1T-CSFs. Thus, in the lateral direction comfort

of pedestrians relative to the accelerations of the footbridge is worse in these cable-stayed

278


Basic BC BS S t_bf t_c D_t h_p Inc. Pylon D_c Lat. Inc. w_d L_s0

0.1

0.2

0.3

0.4

0.5

Late

ral accele

ration [

m/s

2]

aL,DECK

aL, PED 75%

Minimum

Medium

Maximum

Figure 8.46: Comfort assessment of CSF according to the measures implemented to modifylateral accelerations.

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

accL,DECK

acc

L,P

ED

75%

0.5 1 1.5 2 2.50.5

1

1.5

2

2.5

accV,DECK

acc

V,P

ED

75%

Figure 8.47: Comparison of maximum vertical and lateral movements recorded at the deckand maximum accelerations felt by 75% of the walking pedestrians.

footbridges than that for cable-stayed footbridges with one pylon.

At long span cable-stayed footbridges with two pylons, most of the evaluated scenar-

ios generate unstable lateral accelerations (and in consequence enlarged vertical accelera-

tions). Only the footbridges with a slab depth of 0.3 m exhibit acceptable accelerations in

the lateral and vertical directions. The comparisons of these movements against comfort

limits show that in both cases these would correspond to a medium level of comfort.

8.9 Additional dissipation of the serviceability movements: in-

herent or external movement control

The serviceability accelerations illustrated in the previous section depict serviceability

events at medium span 2T-CSFs where the vertical accelerations generally correspond

to a minimum comfort for pedestrians and where the lateral accelerations are equivalent

to medium and maximum comfort (minimum comfort on few occasions). At long span

2T-CSFs the lateral accelerations are excessive and unstable for most of the occasions

(equivalent to unserviceable events) whereas the vertical accelerations would correspond

to practically medium comfort if the concomitant lateral accelerations were stable (based

on the accelerations recorded at the middle of the deck at footbridges with large lateral

accelerations). Therefore, if these footbridges are designed to develop events in service

279


with maximum serviceability response, additional dissipation of these movements should

be introduced (in the vertical direction for medium span footbridges and in the lateral

direction for long span footbridges).

This additional dissipation can be gained through additional inherent damping of the

structure or external dissipation or damping devices (Tuned Mass Dampers described

in Section 2.4). Regarding the first measure, the magnitude of the inherent damping

considered at these footbridges corresponds to the minimum value that guidelines propose

for cable-stayed footbridges. If a mean damping ratio of composite footbridges is adopted

for their design (ζ = 0.6%), the peak vertical and lateral accelerations developed at

medium span 2T-CSFs are 1.41 m/s2 and 0.32 m/s2 respectively (only further increments

of the inherent damping would substantially decrease the response, e.g., ζ = 1.0% would

generate peak vertical and lateral accelerations of 1.15 and 0.26 m/s2 respectively). The

same inherent damping at long span 2T-CSFs does not avoid unstable lateral responses

and exhibits peak vertical and lateral accelerations below 1.1 m/s2 and 2.1 m/s2 regardless

of the depth of the deck (the vertical accelerations have large torsional movements due to

the large lateral accelerations).

If instead of inherent damping (which is a characteristic that depends on a large

number of characteristics of the footbridge and highly unpredictable) the accelerations

are dissipated through external damping devices (TMD), the peak vertical accelerations

at the medium span 2T-CSF are reduced by 40% (TMD with 5% of the modal mass of

mode V2 located at x = 25 and 45 m), from 1.63 m/s2 to 0.95 m/s2. At long span 2T-

CSFs, a TMD with 5% of the modal mass of L1 located at midspan (50% of the mass at

x = 70 m and 25% at x = 45 and 95 m, with a total mass of 10700 kg) does not avoid the

unstable lateral response, thus only substantial increments of the structure mass (such as

that produced by a larger concrete slab depth) allow these type of footbridges to develop

adequate serviceability response.


Analogously to 1T-CSFs, the amplitude of the vertical and lateral dynamic deflections

produced in 2T-CSFs during the serviceability events have a different order of magnitude

and a dissimilar relationship with the accelerations recorded during the same events.

In the vertical direction, the largest dynamic deflections take place at x = 29 m and

43 m (as seen in Figure 8.48(a)), positions that coincide with the sections that produce

the maximum vertical accelerations and the antinodes of mode V2 at the main span.

In this direction, the amplitude of the dynamic vertical deflections is not only related to

the vertical accelerations that take place at the same time (as the values in Figure 8.48(b)

describe). Therefore, an evaluation of the vertical comfort based on the vertical dynamic

deflections would not be adequate as relatively similar values of the dynamic deflections

generate considerably different peak vertical accelerations.

The analysis of the amplitudes of these vertical deflections shows that the largest

deflections are generated at 2T-CSFs with ‘POTs(d)’ deck articulations, those with shorter

pylons and those with largest side spans (deflections are 75% larger than those for the

280


(a)

0 0.5 1 1.5 2 2.50

10

20

30

40

accpeak,vert

[m/s2]

dis

ppeak,v

ert [

mm

]

x =

29.5

m

disppeak,vert

= e(0.3 acc

peak,vert + 2.5)

(b)

Figure 8.48: (a) Static and maximum dynamic vertical deflections generated by medium-high density pedestrian flows on medium span length CSFs; (b) relationship between peakvertical dynamic deflections at x = 29.5 m and concomitant peak vertical accelerations.

reference case). The footbridges with smallest deflections are those with larger concrete

slab or portal pylons (the latter footbridge generates peak vertical accelerations similar

to those of the reference 2T-CSF).

When comparing these dynamic deflections to the corresponding static deflections, the

benchmark 2T-CSF has DAFs related to deflections at x = 26.5 and 43.5 m with values

of 1.36 and 1.70 respectively. The largest DAFs related to deflections are 30 or 40% larger

than those for the benchmark footbridge (maximum DAF at x = 26.5 m is 1.90 and that

at x = 43.5 m is 2.56) and are described by CSFs with larger backstay or main span stay

areas and higher pylons. On the other hand, footbridges with smaller backstay stay or

shorter pylons have DAFs that are smaller by 15-25% (1.07 at x = 26.5 m and 1.35 at x

= 43.5 m).

In the lateral direction, the peak lateral dynamic deflection takes place near the centre

of the main span and has amplitudes that are below 2.0 mm if the lateral accelerations

are stable (in this case the dynamic deflections and the lateral accelerations are linearly

related, as seen in Figure 8.49). At footbridges with unstable lateral response, these

dynamic deflections are larger (this occurs at footbridges with smaller backstay or con-

siderably longer side span) and are not linearly related to the lateral accelerations (as

depicted in Figure 8.49(b)).

The evaluation of the average amplitude of the pedestrian lateral loads in each case

shows that these amplitudes are linearly related to the magnitude of the lateral dynamic

deflections and accelerations. Consequently, as it has already been observed in 1T-CSFs,

the lateral loads introduced by pedestrians while crossing a footbridge depend on the

movement of the bridge, therefore the load models representing pedestrians that do not

consider an interaction between those and the lateral movements are not adequate to

predict the serviceability response of structures such as footbridges.

281


0 0.5 1 1.5 2 2.50

20

40

accpeak,lat

[m/s2]

dis

ppeak,lat [

mm

]

x =

36.5

m

0 0.1 0.2 0.3 0.4 0.50

2

4

accpeak,lat

[m/s2]

dis

ppeak,lat [

mm

]

x =

36.5

m

(a) (b)

Figure 8.49: (a) Comparison of peak lateral dynamic deflections at x = 36.5 m andconcomitant peak lateral accelerations at scenarios with stable lateral response; (b) similarcomparison including scenarios with unstable lateral response.


The normal stresses developed at the deck during the serviceability events, with pedes-

trian flows of density 0.6 ped/m2, are considerably larger than the normal stresses gener-

ated by the static weight of these pedestrian flows, as it can be observed in Figure 8.50(a).

From the values represented in this figure, it can be highlighted that:

• The dynamic events generate sagging bending moments (BMs) at the side spans, as

opposed to the static BMs produced by the weight of the pedestrian flow.

• The largest absolute dynamic BMs correspond to the sagging BMs near the pylons,

which are described at CSFs with side spans longer than 0.2Lm or with a slab depth

thicker than 0.3 m).

• Excluding the cases outlined in the previous point, the largest BMs are produced

at locations near the antinodes of mode V2 (both sagging and hogging BMs) and

near the pylons, where hogging BMs have magnitudes that are similar to the largest

dynamic hogging BMs near the centre of the main span.

• The dynamic sagging BMs can be 2.76 times larger than those for the reference

2T-CSF (described at footbridges with thicker concrete slab or wider deck) and the

dynamic hogging moments can be 18.0 times larger than those for the reference 2T-

CSF (described at footbridges with thicker concrete slabs, thicker steel girders or

largest stays).

The dynamic BMs are considerably larger than the static BMs generated by the weight

of the pedestrian flows at the same footbridge (as seen in Figure 8.50):

• In relation to sagging BMs, the footbridges with higher pylons, larger stays, or

larger deck mass, exhibit DAFs between 24.6 and 31.1 at x = 26.5 and 43.5 m (in

comparison to 9.6 and 13.9 at the reference CSF).

• In relation to hogging BMs, the footbridges with POTs (d) supports, larger main

span stays, thicker steel girders or taller pylons, have DAFs at x = 26.5 m that range

282


from 31.0 to 49.0 (at the reference CSF the DAF related to hogging moments at this

location has a value of 16.0).

The comparison of the DAFs related to hogging and sagging BMs at the antinodes of

mode V2 with the peak vertical accelerations recorded at the same CSFs shows that there

is a slight correlation between both magnitudes although the large variability points out

to other factors influencing the magnitudes of the normal stress at the deck during these

serviceability events.

10.0 20.0 30.0 40.0 50.0 60.0


200

0.0 70.0

150

100

50

0.0

50

100

150

D B

endin

g M

om

ent

[kN

m]

57.4

104.2

148.5

72.2 79.1

53.3 55.5

3.2

(a)

0.0 0.5 1 1.5 2 2.50

10

20

30

40

accpeak,V

DA

FB

M

SBM x = 26.5m

SBM x = 43.5m

HBM x = 26.5m

HBM x = 43.5m

(b)

Figure 8.50: (a) Static and dynamic moments described at the deck of 2T-CSFs; and (b)comparison of the peak vertical accelerations recorded at the deck and DAFs related to hog-ging and sagging bending moments (HBMs, SBMs) at x = 26.5 and 43.5 m at correspondingscenarios.

8.12 Deck shear stresses

The dynamic shear stresses recorded at the steel girders of the deck of 2T-CSFs have

very modest magnitudes at any section of the CSF length in comparison to those produced

by the static weight of the pedestrian flow. The static and dynamic shear forces at the

benchmark 2T-CSF and the maximum and minimum shear stresses exhibited at 2T-CSFs

with characteristics previously enumerated are illustrated in Figure 8.51(a).

According to the results depicted in that figure, the dynamic shear stresses at sections

near the abutments, pylon sections and anchorages are considerably lower than those for

the static loads at the same locations (the former with peaks below 7 kN and the latter

below 20 kN). At other sections of the deck, these dynamic stresses may locally be higher

than the static shear stresses although their reduced magnitudes would not be critical in

design.

Figure 8.51(b) compares the DAFs related to shear stresses at x = 0 and 70 m with

the absolute peak vertical accelerations recorded at the same events and it is obtained

that the first do not depend on the magnitude of the second and that DAFs related to

shear stresses at the critical deck sections are well below 1.0.

283


10.0 20.0 30.0 40.0 50.0 60.0


0.0 70.030

20

10

0.0

10

20

30

Shear

forc

e [

kN

] 15.7

19.7

19.5

(a)

0 0.5 1.0 1.5 2.0 2.50

0.5

1

1.5

accpeak,V

DA

FS

F

SF x = 0.0m

SF x = 70.0m

(b)

Figure 8.51: (a) Static and dynamic shear forces (SFs) described at the deck of the 2T-CSFs; and (b) comparison of the peak vertical accelerations recorded at the deck and DAFsrelated to shear forces at x = 0 and 70 m at corresponding scenarios.

8.13 Normal stresses at the pylon

Under the equivalent static weight of the pedestrian flow, both pylons of the 2T-

CSF endure practically the same normal stresses in spite of the unsymmetrical support

arrangement (values described in Figure 8.52, where pylon heights of other 2T-CSF have

been scaled to match that of the reference 2T-CSF). The maximum static stresses are

equivalent to an axial load of 87 kN and BMs below 10 kNm at any point along their

height.

5.0

10.0

15.0

00

Tow

er

heig

ht

[m]

50.0 100.0 150.0

Axial force [kN]

200.00 20.0 40.0 60.0


60.0

5.0

10.0

15.0

0

Tow

er

heig

ht

[m]

190.0

(a) (b)

Figure 8.52: Static and dynamic axial forces (N) or bending moments (BMs) at the pylonsof the benchmark 2T-CSFs and dynamic N/BMs at alternative 2T-CSFs. (a) Static anddynamic BMs, where (*) represents dynamic BMs at footbridges with larger pylon diameteror side span length; and (b) static and dynamic N.

Under the dynamic loads of the pedestrian flows, the reference cable-stayed footbridge

endures axial loads that are approximately 1.5 times larger than those for the static event

and BMs that are between 3 and 4 times larger except for the anchorage area where these

are 5.7 times larger (values depicted in Figure 8.52).

If the dynamic stresses of other 2T-CSFs (with alternative geometry or characteristics

284


of the structural elements) are considered, the dynamic axial forces are ±10% larger or

smaller than those for the benchmark 2T-CSF (N0) except for CSFs with ‘POTs(d)’

supports, larger backstay or main span stays, and higher pylons, where these are larger

by 30-50%.

In relation to the dynamic BMs at the pylons, the differences between the results for

the reference 2T-CSF and the rest are more substantial. On average, the BMs at the

foundation of any 2T-CSF are 4.1 times those at the basic 2T-CSF (BM0,f ) and near the

deck these are 2.3 times larger (BM0,d). The largest BMs (depicted in Figure 8.52) are

described at 2T-CSFs with a pylon that is inclined towards the main span (BMs at the

foundation are 38 times those of the reference 2T-CSF and BMs at the deck section are

17 times those at the reference 2T-CSF). The comparison of these dynamic BMs to the

static BMs at the reference 2T-CSF describes DAFs related to BMs that are 3.7 and 2.9

at the foundation and deck height pylon sections respectively. The DAFs related to BMs

at the foundation of any 2T-CSF have an average value of 12.8 (the largest DAF related

to BMs at the foundation has a value of 107 and it occurs at the CSF with pylons that

are inclined towards the main span). These magnitudes emphasise the importance of an

assessment of the stresses at the pylons under these serviceability events for their design.

8.14 Performance of stay cables

According to the evaluation of Damage of Equation 3.7.3, (a comparison of the accumu-

lated damage at each scenario to that of the benchmark footbridge, DC = Dacc,i/Dacc,0),

most of the alternative cable-stayed footbridges evaluated in previous sections generate

larger accumulated damage to the stays (either due to larger stress variations or to higher

number of cycles), as seen in Figure 8.53. As depicted in that figure, the damage com-

parison DC ranges from very low values to very high. The average DC is considerably

higher for the backstay cable than for main span stays (the average DC is 340 for the

first and 10 for the others).

For the backstay, the larger average accumulated damage is due to the considerably

high stress variations that it endures in some particular structures. For this stay cable,

the largest stress variations are given when placing a ‘POT(d)’ support arrangement, for

larger areas of the main span stays, larger diameter or height of the pylons and longer side

spans. The use of TMDs dissipating vertical accelerations at the main span corresponds

as well to a measure that causes a large accumulated damage to the backstay cable. On

the contrary, measures such as smaller backstays, increased slab depths or larger deck

widths in particular reduce the stress variations of this stay.

In relation to the stays of the main span, the four stays closest to the main span centre

have very similar stress variations and accumulated damage regardless of the scenario.

The stay cables closest to each pylon endure stress variations that are slightly larger

or smaller than those for the other main span stays although the accumulated damage

is of a similar order of magnitude. For these cables, the characteristics that give rise

to considerably worse performance in service correspond to ‘POTs(d)’ supports, smaller

main span cables, thicker steel girders, larger pylon diameter and higher pylon height. On

285


the contrary, laterally inclined pylons and larger slab depths or wider decks considerably

improve the performance of these stays in service. The modification of the pylon shape

does not change their performance.

Thus, considering both the backstay and the main span stays, the characteristics of the

footbridge with greatest impact on the stays’ stress variations correspond to dimensions of

the stays, diameter and height of the pylon, length of side spans and support arrangement

(in particular the arrangement restraining horizontal movements of the deck).

BS CB1 CB2 CB310

−2

100

102

104

CD

= D

acc,i /

Dacc,b

as

Average CD

Figure 8.53: Comparison of accumulated damage at each stay of the CSF produced CSFswith geometric and structural characteristics detailed in previous sections (compared toaccumulated damage of stay cables of the benchmark CSF).


The analyses of the serviceability response of cable-stayed footbridges with two pylons

produced by the passage of pedestrians that have been presented in the previous sections

illustrate what modifications of these structures (related to the deck articulation, dimen-

sions of the structural elements and geometric characteristics) improve or worsen their

response in service.

In general, similarly to the cable-stayed footbridges with a single pylon, results show

that larger vertical stiffness of the deck increase the vertical accelerations of the deck and

improve the amplitude of the lateral accelerations.

Further analyses of these results allow one to discern the characteristics of these foot-

bridges with greatest impact on the vertical and the lateral movements. In the first case

these correspond to the characteristics of modes V1 to V4 (similar for long span 2T-CSFs),

the first torsional modes and the length and stress under permanent loads of the most

vertical stays. The contribution of these various vertical modes, which adopt frequencies

that are considerably larger or smaller than 1.7-2.1 Hz, highlight the fact that footbridges

with frequencies out of this range may still develop large vertical accelerations in service.

In the second case (in the lateral direction), the mass of the deck and the characteristics

of the first lateral and torsional modes have the greatest influence on the response in

this direction. When compared to the results observed for 1T-CSFs, the 2T-CSFs with

first lateral frequencies below 1.30 Hz do not always develop unstable lateral response.

A cable-stayed footbridge with two pylons and first lateral modes at 0.93 Hz has large

but stable lateral movements (as seen in Section 8.6.1 for footbridges with LEB supports)

286


whereas that with a first lateral mode at 1.15 Hz, corresponding to a footbridge with

a side span of length 0.4Lm, develops lateral accelerations that constantly increase with

the passage of pedestrians. These results point out to an additional factor such as the

total deck length free to laterally vibrate as a parameter to consider when evaluating the

amplitude of the lateral accelerations, as opposed to what codes such as the NA to BS

EN 1991-2:2003 (BSI, 2008) propose.

The vertical serviceability responses of medium span length cable-stayed footbridges

described in the previous sections are equivalent to a low or medium comfort level. At

long span footbridges with stable lateral response these correspond to medium comfort

accelerations (regardless of the deck depth). When the comfort is evaluated in the basis

of the accelerations experienced by walking users, this is improved, particularly in the

vertical direction. The comparison of the accelerations recorded at the deck and those

experienced by 75% of the users (as depicted in Section 8.8) describes that, similarly to 1T-

CSFs, the vertical accelerations felt by users are 0.7 times smaller than the accelerations

recorded at the deck whereas in the lateral direction these differences are not produced

(opposite to what occurs at 1T-CSFs).

Considering individually each characteristic enumerated in the previous sections, these

levels of comfort are improved or worsened as follows:

1. Support arrangements: All the deck articulations considered, except for that re-

straining all the longitudinal deck movements (where these are higher), generate the

same vertical accelerations. These results imply that the restriction of longitudi-

nal movements of the deck deteriorates the amplitude of these vertical accelerations

(similarly to what has been observed for 1T-CSFs). The equal vertical accelerations

at footbridges with supports (a)-(c) and (e) emphasises the fact that longitudinal

movements of the deck are similar in those cases and are more restricted by the

pylons than by the type of supports.


(a) Both backstays and main span stays are critical for the amplitude of the vertical

accelerations. Smaller backstays reduce the vertical movements, as opposed to

the area of the main span stays (in this last case this is due to large torsional

movements). Larger stays increase the vertical accelerations whereas larger

backstays do not considerably affect these movements.

(b) A larger thickness of the steel girder bottom flange enlarges slightly the vertical

accelerations.

(c) Increments of the concrete slab depth reduce the amplitude of the vertical ac-

celerations moderately.

(d) The modification of the section of the pylons (either their diameter or thickness)

does not have substantial effects on the vertical response.


287


(a) Only higher pylons increase the vertical accelerations of these CSFs.

(b) The longitudinal inclination of the pylons does not noticeably change the am-

plitude of the vertical accelerations. The effect of this characteristic as well

as the previous one are explained by the very light modification introduced by

these measures on the longitude and permanent stresses of backstays and main

span stays and the pylon height (for heights equal or smaller than 0.2Lm).

(c) In the vertical direction the effect of the pylon shape is not noticeable since

the vertical vibration modes are not affected by vertical stiffness introduced by

each of these pylons at the deck.

(d) Similarly to the vertical accelerations of 1T-CSFs, the transverse inclination of

the stays does not modify the amplitude of the vertical accelerations unless this

lateral inclination is large enough to considerably change the torsional modes.

(e) The arrangement of the main span stays (distance between consecutive main

span stays) does not change the vertical and torsional modes and thus nor the

amplitude of the vertical accelerations.

(f) A larger deck width generally decreases the peak vertical accelerations, due to

the additional mass introduced by these.

(g) The length of the side spans is closely related to the number of vertical mode

shapes with frequencies between 0 and 5.0 Hz. According to the characteristics

of these vertical modes response will be enlarged or decreased. For the foot-

bridge with side span length 0.4Lm, the response is always increased due to

additional vertical modes with one to three antinodes at the main span. A side

span length of 0.3Lm reduces the vertical accelerations.

The effects on the vertical response of each measure that has been enumerated in the

previous points correspond to a general effect that may not always occur if some of the

modes with highest contribution to this vertical accelerations are disturbed or modified

by other modes. This effect is very important since it may generate unusually large

accelerations at structures where these would have been expected to decrease (e.g., at

footbridges with larger slab depth or deck width).

In the lateral direction, the medium span length 2T-CSFs usually describe lateral

accelerations that correspond to a medium or even large comfort for walking users. At

long span 2T-CSFs these are unstable unless the total deck mass is considerably increased

(as it happens when modifying the slab depth from 0.2 m to 0.3 m). In this lateral

direction, each parameter of these footbridges modifies the amplitude of the serviceability

accelerations as follows:

1. Support conditions: The smallest lateral accelerations are generated at the foot-

bridge with ‘POTs(d)’, which restrict the transverse rotations of the deck at the

embankments. The 2T-CSF with lowest lateral modal frequency (that with LEBs)

exhibits lateral accelerations that are smaller than those for the footbridge with

‘POTs(e)’ (with unrestricted longitudinal movements).

288



(a) Smaller backstay cables reduce the amplitude of the lateral accelerations. Main

span stays with larger section reduce moderately the lateral accelerations.

(b) The thickness of the steel girder bottom flange changes the amplitude of the

lateral accelerations experienced by walking pedestrians (these increase slightly

with the thickness due to the changes this introduces at torsional modes).

(c) A larger concrete slab depth reduces enormously the lateral accelerations of the

deck (these become practically null for slab depths equal or larger than 0.4 m).

(d) Pylons with larger transverse sections reduce the lateral accelerations as they

increase the transverse stiffness of the deck.


(a) Pylons of shorter height reduce the lateral accelerations and vice versa (due to

the changes introduced to mode T1 by these shorter pylons).

(b) Similarly to the accelerations in the vertical direction, the pylon longitudinal

inclination does not largely affect the movements in the lateral direction (only

large inclinations towards the embankment decrease the lateral movements).

(c) The shape of the pylon substantially modifies the amplitude of the lateral ac-

celerations. Pylons with two legs decrease these lateral accelerations due to the

larger stiffness introduced to the deck by these.

(d) The transverse inclination of the stays improve the lateral response unless they

notably modify the characteristics of the torsional vibration modes.

(e) As in the vertical direction, the distribution of the anchorages of the stays has

a small effect on the lateral accelerations.

(f) At these footbridges, a larger deck width decreases the amplitude of the lateral

accelerations. Despite the higher contribution of torsional modes in the lateral

direction, this effect is caused by the larger mass at the deck.

(g) The side span length has a large impact on the lateral accelerations as it in-

creases the free vibration length of the deck in that direction and decreases the

first lateral modal frequencies. Both magnitudes are related to whether the

lateral movements are stable or not under the passage of pedestrian flows.

At 2T-CSFs with longer main span, the lateral accelerations are unstable regardless of

the depth of the deck. In consequence, the concomitant vertical accelerations have unusu-

ally larger amplitudes due to the torsional movements. These unstable lateral responses

are not improved when modifying the characteristics of the stays, the deck width or even

with a larger inherent damping or TMDs acting in the lateral direction. Only a larger

concrete slab depth (which introduces a considerably large mass to the deck) avoids this

large lateral accelerations. Thus, the design of long span cable-stayed footbridges with a

289


main span length near 100 m is feasible if the mass of the deck is considerably larger than

that disposed at the benchmark footbridge.

The evaluation of other characteristics of these footbridges during the serviceability

events shows that:

1. Maximum vertical dynamic deflections are between 1.9 and 2.56 times larger than

the static vertical deflections generated by the static loads at the same cable-stayed

footbridges. These dynamic vertical deflections are related to the vertical acceler-

ations recorded at the deck although the variability indicates that there are other

characteristics involved in the magnitude of the dynamic deflections. In the lateral

direction, the dynamic deflections are clearly related to the lateral accelerations:

the relationship is linear if the lateral accelerations are stable and exponential when

these are unstable.

2. The dynamic BMs at the deck of these footbridges have magnitudes that are con-

siderably larger than those produced by the static loads. Excluding the CSFs with

larger deck depth, which produce unusually large sagging moments at the pylon sec-

tion, the largest deck normal stresses are described near the antinodes of mode V2 at

the main span (DAFs of hogging and sagging moments at these locations can be as

large as 40). A comparison of these BMs to the accelerations recorded at the deck at

the same events highlight that these stresses are related to the vertical accelerations

although the large variability of values suggests the influence of other parameters

(such as mass and stiffness of the deck).

3. As opposed to normal stresses at the deck, the shear stresses at the girders are

considerably smaller than those produced by the static loads (and thus covered by

any ULS analysis). When comparing these stresses to the accelerations produced

during the same events it is visible that these are not related.

4. Despite the support arrangement, both pylons endure fairly similar normal stresses

during the dynamic events. The largest BMs occur at the foundation of the pylon

and are more related to the geometry and stiffness of its section rather than the

dynamic accelerations that the bridge develops at the same time. These explain

how the largest BMs occur at footbridges with largest pylon diameters.

5. In relation to the performance of the stay cables, similarly to what occurred in 1T-

CSFs, the backstay endures the largest stress variations. Those characteristics that

produce its worse performance are the supports ‘POTs(d)’ and the TMDs.

290

Chapter 9Conclusions and recommendations

for future work

9.1 Summary of the developed research work

The research work developed and presented in this thesis is focused on the analysis of

the performance of footbridges in serviceability conditions under the normal passage of

pedestrians, with a special emphasis on footbridges with a cable-stayed typology.

The attention to this particular footbridge typology is founded on the specific charac-

teristics of these structures, i.e., long spans and light decks, parameters that experience

has shown are related to large accelerations. The cable-stayed footbridges considered in

this research work consist of decks with a concrete slab and steel girders, and span lengths

of 50 m and 100 m (values that correspond to the most conventional and frequent span

length, and to a long span length, for this bridge type respectively).

This research work is based on the development of the following topics:

1. The proposal of a new methodology with a non-deterministic approach founded

on a realistic and accurate description of the pedestrian loading extracted from

existing but disconnected multidisciplinary research outcomes. The definition of

this methodology includes:

• The identification and description of the essential variables involved in pedes-

trian actions. The realistic description of the magnitudes of these characteristics

for the population crossing the footbridge (using probabilistic and deterministic

descriptions of these).

• The description of pedestrian movement identifying key parameters and devel-

oping a model that reproduces such description (using both deterministic and

probabilistic relationships as well).

• A vast and accurate knowledge of pedestrian actions described while walking

including models proposed by researchers and those considered in codes and

291

9. Conclusions

design guidelines. Most appropriate models for vertical and horizontal loading

are used to develop the proposed new load model.

• A literature review of the different criteria to be considered to judge if the

serviceability limit state of footbridge vibration is fulfilled under pedestrian

actions.

2. The comparison of the performance predicted by this methodology with existing

methods for the assessment of footbridges, and comparison with experimental data.

3. The development of nonlinear finite element models used to implement this stochas-

tic approach in the structural schemes studied in this thesis.

4. The analysis of the structural response in service of girder footbridges with alter-

native structural sections and materials through a simplified version of the complex

methodology and the proposal of a simplified assessment method to obtain accurate

appraisals of the response of these footbridges through very simple calculations.

5. The characterisation of the average and range of variation for the different structural

parameters of footbridges with a cable-stayed typology through the development of

a data base of existing footbridges of this typology.

6. The implementation of this complex methodology to a set of cable-stayed footbridges

that are representative of this structural typology by means of a set of parametric

analyses with the aim of providing understanding about the structural response of

these bridges under pedestrian loading. The parameters evaluated in these analyses

correspond to the support arrangement, the dimensions of the structural elements,

the technology of the cables, the geometry of the pylon (shape, height, longitudinal

and transverse inclination), the span arrangement, the deck depth for long span

footbridges and movement dissipation capacity (inherent and external).

7. The development of a comprehensive set of design criteria for cable-stayed foot-

bridges.

9.2 Conclusions

The conclusions that can be drawn from this research work are enumerated below

grouping them in the following categories: (1) related to the state-of-the-art (Section 9.2.1),

(2) related to the methodology for the analysis of the response in footbridges (Sec-

tion 9.2.2), (3) related to the response of girder footbridges (Section 9.2.3), and (4) related

to the response of cable-stayed footbridges (Section 9.2.4). The conclusions have been

numbered (CX) to allow cross referencing between them.

9.2.1 Conclusions related to the state-of-the-art

C.1 During the last decades, bridge engineers have more frequently proposed and con-

structed footbridges with lighter decks and longer spans, with innovative structural

arrangements. An example of these trends can be seen in footbridges such as the

292

9. Conclusions

London Millennium Bridge in the UK or the Nesciobrug Bridge in Germany. The

use of cables as structural elements has become more common and thus there are

multiple footbridges with this structural element.

C.2 Together with this evolution in footbridge design, multiple footbridges have expe-

rienced unexpected and substantially large accelerations in service, fact that has

led engineers to usually implement additional devices to dissipate and control the

magnitude of this serviceability response (e.g., Tuned Mass Dampers, Viscous Fluid

Dampers...).

C.3 Over the last fifteen years there has been a significant amount of research focused

on the loads introduced by pedestrians, highlighting the lack of understanding in

relation to both the nature and the magnitude of these loads. This research interest

was generated by the inadequate response in service of multiple footbridges, but also

by the development of the biomechanics field.

C.4 Despite the advances in this field produced by previous research works, there has

not been proposed yet a model for the analysis of footbridges in serviceability con-

ditions under pedestrian loading that includes these recent, but scattered, research

outcomes. Existing models have serious limitations due to the following reasons:

• Loads described through Fourier series do not include a rigorous description of

the energy introduced by individual footsteps.

• Lateral loads described through Fourier series do not capture their nonlinear

nature.

• Intra-subject variability is not considered when deriving the currently used load

models.

• Current load models are based on deterministic assumptions that have not been

validated.

C.5 There are few and limited design criteria (Setra, 2006; fib Bulletin 32, 2006) focused

on the improvement of the serviceability response of footbridges, in particular for

cable-stayed footbridges.

9.2.2 Conclusions related to the methodology for the analysis of the response

C.6 In order to assess the serviceability response of a footbridge under the passage of

pedestrians, engineers should use a pedestrian load model that includes, or has been

derived considering, the following points:

• Pedestrian load models where both vertical and lateral loads are represented

(longitudinal loads are irrelevant as they act over modes of vibrations that

mobilises the entire mass of the deck).

293

9. Conclusions

• The amplitude of the vertical loads depends on the walking velocity and step

frequency of the pedestrian.

• The vertical loads consider all the energy introduced by each step.

• The model of the lateral loads must include their nonlinear nature (their am-

plitude is related to the magnitude of the transverse movements of the bridge).

• The intra-subject variability or inability of pedestrians to walk with consecutive

steps of equal temporary and spatial characteristics for both vertical and lateral

loads.

• The inter-subject variability or different characteristics of users related to:

Step frequency.

The step width amplitude.

• The aim of the journey of users.

• The mass of pedestrians of the flow, considered at least as a uniform and de-

terministic value equivalent to the average mass of the population crossing the

footbridge.

• The collective behaviour, describing the pedestrian-pedestrian interaction as

particles interaction, in particular for pedestrian traffic flows of medium or high

density.

C.7 In order to assess the serviceability response through a simpler methodology, among

the points listed in conclusion C.6, the following are considered absolutely essential

for having a minimal approximation to the behaviour:

• For vertical load models:

The range of vertical structural frequencies that can be excited by pedes-

trians corresponds to a wide range between 1.5 and 3.2 Hz. A pedestrian

flow introduces energy at that frequency range due to their intra-subject

variability and the total amplitude of their step loads (despite the fact that

pedestrians in the flow walk with step frequencies below 2.4 Hz).

• For lateral load models:

Nonlinear nature of these loads.

• For both vertical and lateral load models, the pedestrian weight should be

represented by at least the average weight of the population (780 N for current

populations in the UK and in Western Europe).

C.8 Despite the large value of the current design load for footbridges (equal to 5 kN/m2

in many design codes), the design internal forces obtained from static analyses using

this design load are smaller than those internal forces obtained from a dynamic anal-

ysis using a model with the characteristics described in conclusion C.6. Therefore,

it is crucial to perform an evaluation of the response of the different structural ele-

ments during these dynamic events, as the evaluation with static loads of 5 kN/m2

can be unsafe at some sections of the deck:

294

9. Conclusions

• Depending on the span arrangement of the footbridge, static loads (perma-

nent, or permanent and live load) can only describe either hogging or bending

moments.

• Due to the dynamic nature of the pedestrian loads, these trigger bending mo-

ments of positive and negative signs at every section.

• Nevertheless, these dynamic bending moments do not correspond to critical

values for design (i.e, the absolute maximum along the structural member).

This fact may not be critical in cases where every single section of the structural

member is design to resist the maximum internal forces in critical sections (an

approach that leads to over-design), but it would be critical in cases where

every section is designed to resist the maximum internal forces acting in that

particular section.

9.2.3 Conclusions related to the response of girder footbridges

C.9 The magnitude of the vertical and lateral accelerations of footbridges generated by

the passage of pedestrians can be described on the basis of a series of non-dimensional

parameters:

• ry,n, y ∈ v, l that relates the vertical or lateral response of the footbridges

to the ratio between their fundamental structural frequencies (vertical fv,n or

lateral fl,n) and the pedestrian step frequencies (vertical fp,v or lateral fp,l)

described in Equations 9.2.1 and 9.2.2:

rv,n =fv,nfp,v

φs,n =n2π

2Lfp

√

E

ρ∗ηvαv(1− αv)

(

dhL

)2

φs,n (9.2.1)

rl,n =fl,nfp,l

φs,n =n2π

L2fp

√

E

ρ∗ηlαl(1− αl)b2φs,n (9.2.2)

where φs,n is an adjustment factor to account for cases that differ from a simply-

supported bridge of main span length L, n refers to the nth mode of vibration,

fp = fp,v, fp = 2fp,l, E is the Young’s modulus, ρ∗ is an equivalent material

density introducing the non-structural mass of the footbridge, αv is the ratio

between the vertical distance from the centroid of the section to the top ex-

treme fibre and the vertical depth of the section dh, αl is the ratio between

the horizontal distance from the centroid of the section to the closest lateral

extreme fibre and the width of the section b, and ηv or ηl are the ratio between

the depth of the central kern and the depth of the section dh, or the width of

the structural section b, respectively.

The footbridge response in service is larger when rv,n = 1, 2, 3... (vertical) and

rl,n = 1, 3, 5... (lateral), due to resonant effects.

• The ratio between the footbridge and the pedestrian (or pedestrians in a traffic

flow) masses.

295

9. Conclusions

• The ratio between the main span length of the footbridge and the magnitude

of the average step length of the walking pedestrians.

• The damping ratio.

C.10 In the vertical direction, a pedestrian walking with the same step frequency fp,v on

two footbridges where the magnitudes of these nondimensional parameters (listed in

C.9) are equal triggers the same vertical accelerations.

C.11 If the step frequency adopted by the walking pedestrian is higher at one footbridge

compared to another, and the magnitudes of these nondimensional factors (listed

in C.9) are equal, the vertical accelerations triggered by the pedestrian walking

at higher step frequency are larger (except for fp,v < 1.4 Hz and fp,v > 2.2 Hz,

where the responses for the same set of nondimensional parameters are not frequency

dependent).

C.12 At events where these nondimensional parameters (listed in C.9) adopt the same

magnitudes, the lateral response is the same irrespective of the step frequency

adopted by the walking pedestrian.

C.13 When considering a probabilistic description of the actions of walking pedestrians,

it is observed that:

• The vertical response is very sensitive to the consideration of the intra-subject

variability. The vertical response at resonant ratios rv,n = 1, 2, 3... when the

intra-subject variability of the pedestrian loads is considered is 0.5 times that

generated by pedestrians walking at a constant step frequency.

• The consideration of the intra-subject variability in the pedestrian vertical loads

triggers peak accelerations for non-resonant values of the rv,n nondimensional

parameter that can be larger than those at resonant values (integer numbers)

of this parameter. Values in the rages rv,n = (i± 15%), i = 1, 2 and at rv,n > 2

are larger than those at rv,n = 1, 2 and rv,n = 3, 4... respectively.

• The larger the pedestrian density, the larger the relevance of the intra-subject

variability in the structural response.

• The consideration of the intra-subject variability cancels the resonant effects

in the lateral direction (i.e., the accelerations at rl,n = 1, 3, 5... are not larger

than those when rl,n adopts other values) unless there is an interaction between

the pedestrian and the horizontal movement of the footbridge, i.e., the lateral

movement of the deck substantially increases the amplitude of the step width

of the user.

• The horizontal response is very sensitive to the consideration of the intra-subject

variability. Its relevance is not related to the pedestrian density.

• The amplitude of the lateral load of a pedestrian is linearly related to the

pedestrian step width of this user.

296

9. Conclusions

• The events with multiple pedestrians where each pedestrian adopts his (inter-

subject variability) lateral step width (which depends on height and age) trigger

lateral accelerations that are considerably lower (0.65-0.75 times smaller) than

those produced by the same pedestrian flows where all pedestrians walk with

the same initial step width.

• The interaction among pedestrians (modifying their direction, speed, and aver-

age step frequency as a consequence of the presence of other pedestrians close to

them) while walking depends on the density of the flow. For pedestrian events

with medium and large numbers of users (similar or larger than 0.6 ped/m2)

in the flow, the collective behaviour changes noticeably (medium density flows

have peak vertical accelerations that are 0.15 times smaller) the serviceabil-

ity response in comparison to those events where this interaction effect is not

introduced.

9.2.4 Conclusions related to the response of cable-stayed footbridges

9.2.4.1 Relevant parameters

C.14 The vertical response in service of medium (around 50 m) and long (around 100 m)

span length cable-stayed footbridges substantially depends on:

• The characteristics (modal frequencies, modal shapes, and the components of

the different modal shapes on the vertical direction, as well as the modal masses)

of the first vertical and torsional modes of vibration. For medium span cable-

stayed footbridges with one tower (1T-CSFs), the largest effect is produced by

modes V2, V3 and T2. For medium span cable-stayed footbridges with two

pylons (2T-CSFs), the highest influence is produced by modes V1 to V3 and

T2.

• For 1T-CSFs of medium span length, the characteristics of two key stay cables:

the backstays, and the stay cables anchored in the deck at the main span at

the mode-V2 antinode which is closest to the pylon. The most relevant param-

eters for these two stay cables are the stress under permanent loads and their

orientation (measured by the angle of inclination of the stay cable in relation

to the deck). Both these two stay cables have the largest influence in the mod-

ification of the vertical stiffness of the deck (stiffness that is related in turn to

the magnitude of the vertical accelerations).

• For 2T-CSFs of medium span length, the stresses under permanent loads and

the length of the stay cables anchored near the pylons at the main span. This is

the stay with largest effect on the stiffness of the deck in the vertical direction.

C.15 The lateral response in service of medium (around 50 m) and long (around 100 m)

span length cable-stayed footbridges depends on:

• The characteristics (modal frequencies, modal shapes, and the components of

the different modal shapes on the lateral direction, as well as the modal masses)

297

9. Conclusions

of the first two lateral and torsional modes of vibration.

• The mass of the deck per unit of length.

• For 1T-CSFs of medium span length, the lateral second moment of area of the

deck and the transverse inclination of the main span stay anchored at the deck

section corresponding to half the total deck length (length between support

sections on the abutments), which acts at the mid-span section in the lateral

direction.

C.16 For both medium and long span cable-stayed footbridges, the coincidence (cou-

pling) of frequencies of vertical and torsional modes or lateral and tor-

sional modes (in particular of those with largest contribution to the serviceability

response) produces a notorious alteration of the response in service. Thus, it is highly

recommended to avoid designing cable-stayed footbridges where vertical, lateral or

torsional modes coincide in frequency magnitude.

9.2.4.2 Magnitude of the response

C.17 The serviceability response of footbridges is usually assessed in terms of the ab-

solute peak (vertical or lateral) acceleration recorded at the deck of the footbridge

during an event. However, the magnitude of the acceleration weighted with duration

has been argued to better assess the comfort of users. At cable-stayed footbridges,

the 1s-RMS (1 second Root Mean Squared) vertical accelerations are 0.55 times the

peak vertical accelerations. The 1s-RMS lateral accelerations are 0.65 times the peak

lateral accelerations.

C.18 As an alternative, the evaluations detailed in the body of the thesis have shown

that the maximum peak acceleration felt by 75% of the users corresponds to magni-

tudes that are 0.75-0.8 times those maximum values recorded at the deck for vertical

and lateral accelerations except for 2T-CSFs in the lateral direction, where both have

similar values.

C.19 The levels of vertical and lateral accelerations (peak accelerations recorded at the

deck and minimum peak responses felt by 25% of the users) used to assess the

comfort of 1T-CSFs, (detailed description for medium span length footbridges is

given in Table 9.1), are as follows:

• The serviceability response of these footbridges is usually equivalent to medium

(between 0.5-1.0 m/s2) and minimum (between 1.0-2.0 m/s2) comfort, with

predominantly minimum comfort when span lengths are close to 100 m (for

traffic flows that have densities similar to 0.6 ped/m2).

• The lateral accelerations usually correspond to a maximum (between 0.0 and

0.15 m/s2) or medium (between 0.15 and 0.30 m/s2) comfort range for medium

span length 1T-CSFs. For long span lengths these lateral accelerations increase

to levels that are beyond acceptability (above 0.8 m/s2) - see conclusion C.30.

298

9. Conclusions

C.20 For 2T-CSFs, the serviceability accelerations triggered by a medium-large density

flow of pedestrians are as follows:

• In most occasions, the vertical serviceability response of these footbridges is

equivalent to minimum (between 1.5-2.5 m/s2) comfort and occasionally to

medium (below 1.0 m/s2) comfort for medium span footbridges, and to medium

comfort for long span cable-stayed footbridges.

• The lateral accelerations are generally between 0.0 and 0.40 m/s2, (which ranges

from maximum to minimum comfort), typically near 0.40 m/s2 for long span

2T-CSFs.

C.21 At these footbridges (events detailed in C.19 and C.20), the magnitude of the

peak vertical and lateral accelerations are unacceptable for standing and seating

users, except on few occasions at 2T-CSFs of medium span length in horizontal di-

rection. Hence, at cable-stayed footbridges located in urban areas, where users of

these structures are more likely to stand or seat, the SLS of vibrations of these foot-

bridges should be assessed considering these type of users and their more restrictive

comfort ranges.

Table 9.1: Magnitude of maximum accelerations [m/s2] at medium span length 1T-CSFsin serviceability events.

[m/s2] Pedestrian flow density dp [ped/m2]

Direction dp ∼ 0.2 dp ∼ 0.6 dp ∼ 1.0

Vertical 0.5 - 1.01.0 - 2.5

Occasionally: 0.5 - 1.0

15% larger

than dp ≈ 0.6

Lateral 0.0 - 0.150.0 - 0.30

Occasionally: > 0.30

Slightly above

0.40

9.2.4.3 Enhancement of the serviceability response of CSFs

C.22 The measures in CSFs that notably decrease their vertical response in service

correspond to:

• A larger deck mass: the mass of the deck can be increased by increasing the

depth of the slab or the width of the deck. In both cases these measures increase

the mass and also, although is smaller proportion, the vertical stiffness of the

deck. The beneficial effect of this measure is related to the nondimensional

parameter in C.9 which links the structural and the pedestrian mass. The

larger the structural mass, the smaller the accelerations

• The boundary conditions: The support schemes that allow the longitudi-

nal movements of the deck reduce the vertical accelerations, as this measure

increases the modal mass and reduces the participation of the vertical modes

with largest contribution to the response.

299

9. Conclusions

• A higher dissipation capacity.

C.23 Additional measures in CSFs of medium span length (around 50 m) that notably

decrease their vertical response in service correspond to:

• Smaller backstay cable cross-sectional area: this measure would be valid

if the technology of the stays allowed increasing the maximum stress beyond

the current limits. The decrement of the vertical response described with this

measure is related to the smaller participation of vertical modes, which is related

to the smaller vertical stiffness of the deck cable-staying system in the vertical

direction.

• Shorter pylon: at 1T-CSFs shorter pylons reduce the magnitude of the verti-

cal accelerations due to the smaller participation of torsional modes (which is

related to the inclination of the main span stays in relation to the deck). At

2T-CSFs a shorter pylon does not reduce the response since this modification

does not notably change the inclination of the stays in relation to that of the

reference 2T-CSFs.

• Longer side span: A longer side span generally increases the mass of the

vertical modes with relevant participation to the service response and reduces

considerably the stiffness of the deck. If these changes do not introduce ad-

ditional vertical modes within the frequency range (1.0-4.0 Hz) this measure

reduces the vertical response as it increases the modal masses. If this measure

introduces further modes within that frequency range the responses increase

due to this additional contribution.

C.24 The measures in CSFs of medium span length that do not affect their vertical

response in service correspond to:

• Technology of the cables (bars): the modal masses of the main modes

participating in the response are similar.

• Pylon shape: the shape of the pylon does not notably change the participation

of the vertical modes (although, depending on the shape, torsional modes are

affected due to the number of supports at the pylon section). Hence, vertical

accelerations are similar irrespective of the pylon shape.

• Pylon section: similarly to the shape of the pylon, a modification of the

characteristics of the section of the mono-pole pylon does not change the par-

ticipation of vertical modes or the vertical response.

• Pylon longitudinal inclination: the longitudinal inclination of the pylon

affects modestly the characteristics of the pylon, stays, vertical and torsional

stiffness and thus modes of vibration of the footbridge. Hence, vertical acceler-

ations are not affected by this measure.

• Pylon transverse inclination: for 1T-CSFs this measure does not improve

the vertical response due to the additional contribution of torsional modes. For

300

9. Conclusions

2T-CSFs this measure notably increases the stiffness of the deck to torsional

rotations, decreases the components of the torsional modes in the vertical di-

rection and accordingly reduces the magnitude of the vertical accelerations.

• Main span stays spacing: The spacing between anchorages of the stay cables

in the deck at the main span modifies or not the vertical response according to

the changes introduced by this magnitude to vertical and torsional modes.

C.25 The measures in CSFs of medium span length that increase their vertical response

in service correspond to:

• Larger area of the stays: a larger area of the main span stays increases the

stiffness of the deck (and reduces its longitudinal movements), the participation

of the vertical modes and thus moderately enlarges the vertical accelerations.

• Increment of the bottom flange thickness of the steel girder: similarly

to the area of the stays, a larger thickness moderately increases the deck vertical

and stiffness and the response.

C.26 At long span cable stayed footbridges the measures enumerated in C.23, C.24

and C.25 do not notably affect the vertical response. However, the depth of the

deck has a large impact on the magnitude of the vertical response. A smaller deck

depth describes larger vertical accelerations. This is due to the additional vertical

modes that the flexible deck describes at the range 1.0-4.0 Hz in comparison to a

deck with larger depth.

C.27 The measures in CSFs that substantially decrease their lateral response in ser-

vice correspond to:

• A larger deck mass: the effect of this measure is related to the nondimensional

parameter in C.9 which links the structural and the pedestrian mass.

• The boundary conditions: the support schemes that restrain the rotation of

the deck at the abutment sections, and when possible at the support section

over the pylon, describe smaller lateral accelerations.

• The pylon shape: the pylons with shapes that have two legs considerably

improve the lateral response of footbridges due to the restriction of the deck

rotations and displacements introduced by the pylon.

• A higher dissipation capacity.

C.28 The measures in CSFs of medium span length that considerably increase their

lateral response in service correspond to:

• Smaller backstay cable: this measure causes footbridges to describe torsional

modes with larger lateral components, which is related to the larger accelera-

tions in service produced by this measure. At 2T-CSFs, due to the different

height of the pylon, the effect is the opposite (although its impact is moderate).

301

9. Conclusions

• Smaller main span stays cross-sectional areas: similarly to the area of the

backstay cable, the area of these stays is related to the modal shape of the

torsional modes which has larger lateral component if their area is smaller. For

2T-CSFs this modification is modest, therefore it does not practically modify

the lateral accelerations.

• Thicker bottom flange of the steel girder: an increment of this thickness

increases the projection in the lateral direction of the torsional modes, due to

the increment of the lateral second moment of area of the deck. For 2T-CSFs

this effect is not noticeable since the bottom flange is thinner (in comparison to

that of the reference 1T-CSF) and this modification does not notably increase

the transverse stiffness of the deck.

• Shorter pylon height: for 1T-CSFs, the reduction of the height of the pylons

increases the lateral component of the torsional modes (due to the modification

of the modal shape of the pylon at these modes). For 2T-CSFs, since the pylon

is considerably shorter than those of 2T-CSFs, this modification does not occur.

• Large transverse pylon inclination: A large transverse inclination of the

pylon and the stay cables does not reduce the lateral accelerations but increase

them since this large inclination does not control the torsional modes of the

deck.

• Longer side span: this measure affects notably the first lateral modes of vi-

bration as the footbridge vibrates in the lateral direction as if it has a single

span of length equivalent to the distance between the support sections at the

abutments.

C.29 The measures in CSFs of medium span length that do not affect their vertical

response in service correspond to:

• The cable technology

• The pylon section: a pylon with larger stiffness does not modify the modal

shapes of the CSF with one pylon but it does change those of the 2T-CSF.

These changes are related to the lateral responses described in each case.

• The longitudinal inclination of the pylon: this measure does not consider-

ably affect the dimensions of the pylon, backstay and cables. Consequently the

changes that it produces at lateral and torsional modes are moderate, which

explain the similar service responses of CSFs with this measure considered.

• Cable anchorage spacing: similarly to vertical response, the effect of this

measure is related to how it modifies the lateral and torsional modes.

C.30 At long span cable-stayed footbridges the measures enumerated in C.28 and C.29

do not considerably change the lateral response. Similarly to the vertical response,

the depth of the deck has a large impact on the response as it changes the lateral

components of the first torsional modes (similarly to what occurs at medium span

length 1T-CSFs when increasing the thickness of the steel girder bottom flange).

302

9. Conclusions

C.31 The trend effect of each structural modification on the response at medium span

length 1T-CSFs is detailed in the following Tables 9.2 and 9.3:

Table 9.2: Effect on serviceability response of 1T-CSFs of medium span length of alter-native measures (Part 1), where the subindex ‘0’ refers to the reference case in Chapter 7,ABS,0 describes the area of the backstay of the reference CSF, AS,0 the area of the main spanstays, tbf,0 the thickness of the steel girder bottom flange, tc,0 the thickness of the concreteslab, Dt,0 and tt,0 the diameter or thickness of the pylon, Lm the mains span length, α thepylon longitudinal or lateral (‘Lat.’) inclination (‘incl.’), wd,0 the deck width and Ls theside span length.

Deck Accelerations Trend effect

Parameter adeck,V adeck,L Vertical Lateral

Supports

-Bearings

LEBs - Unstable -Unstable lateral

response

LEBs+SK 1.62 0.18

Moderate

restriction of deck

long. movements

Supports allow

transverse deck

rotation

POTs(c) 1.45 0.28

Partly restriction of

deck long.

movements

Supports partly

restrict deck

transverse rotations

POTs(d) 1.73 0.07Restricted deck

long. movements

Supports restrict

transverse deck

rotations

Backstay 0.5 ABS,0 1.20 0.30

-10% ABS,0 ⇒-5% adeck,V

-10% ABS,0 ⇒+10% adeck,L

2.5 ABS,0 1.64 0.22+10% ABS,0 ⇒

∼ adeck,V

+10% ABS,0 ⇒∼ adeck,L

Staycable 0.5 AS,0 1.58 0.85

-10% AS,0 ⇒∼ adeck,V

-10% AS,0 ⇒+85% adeck,L

2.5 AS,0 2.21 0.13+10% AS,0 ⇒+9% adeck,V

+10% AS,0 ⇒-2% adeck,L

Bars - 1.61 0.17 ∼ adeck,V ∼ adeck,L

Girder 2.2 tbf,0 1.86 0.27

+120% tbf,0 ⇒

+15% adeck,V

+120% tbf,0 ⇒

+150% adeck,L

Slab

2 tc,0 0.98 0.07+100% tc,0 ⇒-40% adeck,V

+100% tc,0 ⇒-60% adeck,L

Pylonsect.

2.5 Dt,0 2.01 0.16+150% tc,0 ⇒+24% adeck,V

+150% tc,0 ⇒∼ adeck,L

2 tt,0 1.59 0.22+100% tc,0 ⇒

∼ adeck,V

+100% tc,0 ⇒∼ adeck,L

303

9. Conclusions

Table 9.3: Effect on serviceability response of 1T-CSFs of medium span length of alterna-tive measures (Part 2).



Pylonheigh

t

0.25 Lm 1.13 0.33-30% hp,0 ⇒-30% adeck,V

-30% hp,0 ⇒+85% adeck,L

0.45 Lm 1.84 0.14+25% hp,0 ⇒+14% adeck,V

+25% hp,0 ⇒-24% adeck,L

Pylonincl. α ≤ 20

Side1.48 0.28

+5(to side span)

⇒-2.5% adeck,V

+5(to side span)

⇒+14% adeck,L

α ≤ 20

Main2.08 0.25

+5(to main span)

⇒+7% adeck,V

+5(to main span)

⇒+10% adeck,L

Pylonshap

e

‘H’ 2.33 0.09

Low torsion stiffness

⇒+45% adeck,V

(torsions)

-50% adeck,L

‘A’ 1.86 0.16 +15% adeck,V∼ adeck,L

Spacing

Var.

1.09

(Dc =

8.0m)

0.18

Effect depends on

modification of V

and

Similar to vertical

response

T modes

Lat.incl. α ≤ 10 2.27 0.20

+40% adeck,V (‘I’)

∼ adeck,V (‘H’)∼ adeck,L

α > 10 2.37 0.39+45% adeck,V (‘I’)

∼ adeck,V (‘H’)

+100% adeck,L (‘I’)

+300% adeck,L (‘H’)

Width wd ≥

wd,0

0.96

(1.5

wd,0)

0.14

+25% wd,0 ⇒

-20% adeck,V

+25% wd,0 ⇒

-10% adeck,L

Sidesp.

Ls ≥ Ls,0

1.25

(1.5

Ls,0)

0.22

+50% Ls,0 ⇒

-25% adeck,V

+50% Ls,0 ⇒

+20% adeck,L

C.32 The trends described in Tables 9.2 and 9.3, are valid unless there is a coincidence

(coupling) in modal frequencies of vertical and torsional modes or lateral

and torsional modes. This is the case of:

• At medium span length 1T-CSFs, in the vertical direction there is a coincidence

of vertical and torsional modes that disturb the vertical response with: a girder

bottom flange 1.8fbf,0, a slab of depth 0.3 or 0.5 m, a pylon diameter 1.7Dt,0, a

deck width of 5 m and a side span of length Ls = 0.3Lm.

• At medium span length 1T-CSFs, in the lateral direction, there is a coincidence

of lateral, vertical and torsional modes that affect the magnitude of the peak

304

9. Conclusions

lateral accelerations with: a girder bottom flange 1.8fbf,0, a pylon diameter 1.3

or 1.7Dt,0, a large longitudinal inclination of the pylon towards the main span

and a deck width of 5 m.

• At long span 1T-CSFs, the peak lateral accelerations are modified due to the

coincidence of lateral and torsional modes when increasing the depth of the slab

to 0.3 m, with a width of 5 m and with a longitudinal inclination of the pylon

towards the main span (for a deck depth of Lm/100).

C.33 The trend effect of each structural modification on the response at medium span

length 2T-CSFs is detailed in the following Tables 9.4 and 9.5:

Table 9.4: Effect on the serviceability response of medium span length 2T-CSFs of alter-native measures (Part 1), where the parameters considered are represented with the namesintroduced in Chapter 8.



Supports

-Bearings

POTs(a) 1.63 0.37

Partly restricted

deck long.

movements

Supports partly

restrict deck

transverse rotations

LEBs 1.71 0.62

Moderate

restriction of deck

long. movements

Supports allow

transverse deck

rotation and

displacement

LEBs+SK 1.61 0.72

Moderate

restriction of deck

long. movements

Supports allow

transverse deck

rotation

POTs(d) 2.21 0.21Restricted deck

long. movements

Supports restrict

deck transverse

rotations

POTs(e) 1.73 0.81Unrestricted deck

long. movements

Supports allow

transverse deck

rotations

Backstay 0.5 ABS,0 1.24 0.27

-10% ABS,0 ⇒-5% adeck,V

-10% ABS,0 ⇒-5% adeck,L

2.5 ABS,0 1.89 0.39+10% ABS,0 ⇒

+1% adeck,V

+10% ABS,0 ⇒∼ adeck,L

Staycable 0.5 AS,0 2.22 0.33

-10% AS,0 ⇒+7% adeck,V

-10% AS,0 ⇒-2% adeck,L

2.5 AS,0 2.49 0.30+10% AS,0 ⇒+4% adeck,V

+10% AS,0 ⇒-1% adeck,L

Girder 2.2 tbf,0 2.14 0.35

+120% tbf,0 ⇒

+30% adeck,V

+120% tbf,0 ⇒∼ adeck,L

305

9. Conclusions

Table 9.5: Effect on serviceability response of 1T-CSFs of medium span length of alter-native measures (Part 2), where the parameters considered are represented with the namesintroduced in Chapter 8.



Slab

2 tc,0 1.15 0.03+100% tc,0 ⇒-30% adeck,V

+100% tc,0 ⇒-90% adeck,L

Pylonsect.

2.5 Dt,0 1.69 0.12+150% tc,0 ⇒

∼ adeck,V

+150% tc,0 ⇒-70% adeck,L

2 tt,0 1.94 0.31+100% tc,0 ⇒+20% adeck,V

+100% tc,0 ⇒-15% adeck,L

Pylonheigh

t

0.13 Lm 1.69 0.18-35% hp,0 ⇒

∼ adeck,V

-30% hp,0 ⇒-50% adeck,L

0.25 Lm 2.12 0.27+25% hp,0 ⇒+30% adeck,V

+25% hp,0 ⇒-25% adeck,L

Pylonincl. α ≤ 10

Side1.58 0.28

+5 (side) ⇒∼ adeck,V

+5 (side) ⇒-13% adeck,L

α ≤ 20

Main1.60 0.36

+5 (main) ⇒∼ adeck,V

+5 (main) ⇒∼ adeck,L

P.shap

e

‘H’ 1.50 0.21 ∼ adeck,V -45% adeck,L

‘A’ 1.45 0.08 -10% adeck,V -80% adeck,L

Spacing

Var.1.38 (Ds

= 10.0m)0.31

Effect depends on

modification of V

and T modes

Similar to vertical

response

Lat.incl. α ≤ 10 1.17 0.15

-30% adeck,V (‘I’)

-20% adeck,V (‘H’)

-60% adeck,V (‘I’)

-30% adeck,V (‘H’)

α > 10 1.22 0.24-25% adeck,V (‘I’)

-20% adeck,V (‘H’)

-35% adeck,L (‘I’)

+15% adeck,L (‘H’)

Width wd ≥

wd,0

0.84 (1.25

wd,0)0.12

+25% wd,0 ⇒-50% adeck,V

+25% wd,0 ⇒-70% adeck,L

Sidesp.

Ls ≥ Ls,01.78 (1.5

Ls,0)0.25

+50% Ls,0 ⇒

-25% adeck,V

+50% Ls,0 ⇒-30% adeck,L

• Similarly to the statement of conclusion C.27, the trends described in previous

tables are valid unless there is a coincidence in modal frequency of vertical and

torsional modes or lateral and torsional modes. At medium span length 2T-

CSFs, in the vertical direction there is a coincidence of vertical and torsional

modes that disturb the vertical response with: a slab of depth 0.5 m, a pylon

diameter 1.7Dt,0, a deck width of 6 m and a tower height hp = 0.20Lm.

C.34 Similarly to the statement of conclusion C.8, it is recommended to perform an

evaluation of the response of structural elements such as the deck and the pylons

306

9. Conclusions

with the dynamic stresses described during these dynamic events since, as pointed

out in Chapters 7 and 8, a ULS load of 5 kN/m2 does not always produce the largest

stresses at these structural elements.

9.2.5 Review of the current available design guidelines

Both the Setra guideline (2006) and the fib Bulletin 32 (2006) recommend an increase

of the stiffness of the structure in order to move the structural frequencies beyond the range

that is considered critical (however it has been proved in this thesis that this action triggers

larger serviceability accelerations). These guidelines suggest that the following measures

should be adopted to reduce the serviceability response of cable-stayed footbridges (mainly

focused to vertical accelerations):

• Increment of the area of the stays.

• An increment of the thickness of the bottom flange of the steel girders.

• An increase of the strength of the concrete modulus.

• The replacement of normal concrete for lightweight concrete.

• The implementation of taller pylons.

C.35 The research work developed in this thesis has shown that these measures rec-

ommended by the guidelines produce the opposite effect: the accelerations are not

reduced but increased. This is because the actions that lead to a reduction of the

static deflections do not lead to a reduction of the accelerations, as a consequence of

the increase of the deck stiffness and modal frequencies. Therefore, assuming that

the accelerations are reduced when the deflections are decreased is clearly incorrect.

In addition, the pedestrian loading has very large components at a very wide spec-

trum, and exclusively considering the effects of these pedestrian loads at particular

frequency ranges (near the mode of the pedestrians’ step frequency, 1.8-2.0 Hz) is

quite a simplistic approach (see C.7). Therefore a revision of these proposals is

recommended.

9.3 Future work

Following the research work carried out in this thesis, there are multiple areas that

would require further investigation in this field. Some of them are as follows:

• The analysis of footbridges with other structural typologies where long spans and

light decks are fundamental characteristics of their designs, e.g., suspension foot-

bridges. It would be interesting to evaluate, using the pedestrian load model devel-

oped in this doctoral research, the magnitude of the response in service of footbridges

with this structural type as well as whether, similarly to cable-stayed footbridges,

the control of their response is fundamentally related to the mass of their decks, and

what dimensions of these footbridges would describe the most adequate serviceability

response.

307

9. Conclusions

• The evaluation of the performance of cable-stayed footbridges with other structural

decks, e.g., those with a steel deck section and those with precast segmental

concrete decks. In relation to the first alternative, the response in service of these

footbridges will probably follow principles similar to those of cable-stayed footbridges

with a composite deck. However, the smaller dissipation capacity of steel footbridges

and the smaller magnitude of the vibration modal masses of these footbridges in

comparison to those with a composite deck and similar span arrangement could

correspond to a critical design factor. In relation to the second, due to the large

influence recognised in this thesis of the mass of the deck, it would be interesting to

accurately appraise the response of footbridges with this deck type and its design

constraints.

• Due to the results obtained in the analysis of cable-stayed footbridges, which em-

phasise that the most effective measure to control the serviceability accelerations of

the deck corresponds to the increment of the mass of the deck, it would be extremely

interesting to develop an exhaustive investigation of the type of damping devices

and their disposition in the footbridge to evaluate the most efficient arrangements,

in particular for long span footbridges.

• An experimental study that focused on this bridge typology in order to confirm

experimentally the conclusions related to behaviour and design criteria proposed in

this thesis. This thesis is underpinned by experimental tests that have confirmed

the predictions related to the vertical response in girders, and the appropriateness of

the lateral model (Carroll et al., 2012). Nevertheless, a comprehensive experimental

study would consolidate the set forward in the knowledge area given here, and might

also allow to obtain further conclusions about the behaviours that are not captured

by current models.

308

Characteristics of cable-stayed

footbridges

The following tables summarise the characteristics of cable-stayed footbridges used to

extract the geometry of the reference cable-stayed footbridge (units in [m]):

Name Year Lm L

No. py-

lons

Pylon

shape hp

Name Year

Main

span

length

Total

length

Pylons

No

Pylon

shape

Pylon to-

tal height

Malta Park 2011 67 134.1 1 H 40

Rosenwood Golf 1993 37.8 75.6 1 I 39.5

I-5 Ped. Bridge 2009 31.8 63.6 1 A 21.3

Uhersky Brod 2010 49 98 1 A 25

Ruda Slaska 2000 54.58 61.6 1 A 23.66

Krzywykij 2000 60.4 NA 1 A 21.67

Bridge across Elbe 2014 156 242 2 A 39.2

Delta Pond 2010 52 104 1 H 25.5

Golden Jubilee 2002 50 320 6 I 28

Bridge across D1 2011 58.29 113 1 I 25.5

Bridge across D47 - 59.2 110.5 1 I 25.1

Neckar (Mannheim) 1975 139.5 252.5 2 I 36.35

Hessenring 2002 46 76 1 Y 13.75

FB over Ticino 2011 60 120 1 A 37

Kaisermuhlenbrucke 1993 87 204 2 H -

Glorias Catalanas 1974 54 97.5 1 I 34.1

Media City 2011 65 83 1 I 31

FB over Ibar river 2011 120 264.1 1 I 42.8

Voluntariado 2008 145 239 1 I 75

309

Appendix A

Name hp Hi Inclination

No. cable

plans

Cable ar-

rang.

Malta Park 25.6 14.28 No 2 Fan mixt

Rosenwood Golf 22.5 17 No 1 Fan mixt

I-5 Ped. Bridge 16 5.3 No 2 Fan mixt

Uhersky Brod 18.5 6.5 No 2 Fan mixt

Ruda Slaska 16.85 6.81 Yes, out 2 Fan mixt

Krzywykij 18 3.67 Yes, in 2 Fan mixt

Bridge across Elbe 32 7.2 No 2 Fan mixt

Delta Pond 18 7.5 No 2 Fan mixt

Golden Jubilee 20 8 No 2 Fan

Bridge across D1 17.45 8.03 No 1 Fan mixt

Bridge across D47 17.8 7.3 No 1 Fan mixt

Neckar (Mannheim) 29.1 7.25 No 1 Fan mixt

Hessenring 9 4.75 No 2 Fan

FB over Ticino 28 9 No 2 Fan

Kaisermuhlenbrucke - - No 2 Fan mixt

Glorias Catalanas 24.6 9.5 No 1 Harp

Media City - - Yes, out 1 Fan mixt

FB over Ibar river 36 6.8 No 1 Harp

Voluntariado - - No 1 Fan mixt

310

Name Year Lm L

No. py-

lons

Pylon

shape hp

Manzanares 2008 147 NA 1 I 42

Euro 2012 Stadium 2012 30 45.7 1 H 14.6

Saint Irinej 1993 192.5 297.5 2 I -

Malecon 1996 60 60 1 I 15

Milwaukee Art 2001 73 73 1 I 50

Zrodlowa Street 2000 29.06 42 1 H 13.75

Zlotnicka 1999 34 68 1 H 19.3

Literatuurwijk 2001 12 24 1 I 7.7

Delftlanden 2005 20 84 1 I 15.5

Viana footbridge 2013 36.3 44.7 1 I 19.9

Passerelle sur la Tet 2013 73.5 107 1 I 30

Nessebrucke 2006 82 NA 2 I 15.5

Katehaki 2004 80 80 1 I 50

Feijenoord 1993 91 NA 1 A -

South quay 1997 48 80 1 I -

Passerelle des deux Rives 2004 177 177 2 I 37

Ohio and Erie Canalway 2011 65.9 175.5 2 H -

7th Av Ped. Bridge 2010 52 NA 1 I 28.65

Turtle Bay Sundial Bridge 2004 150 230 1 I 66

Name hp Hi Inclination

No. cable

plans

Cable ar-

rang.

Manzanares - - No 1 Fan mixt

Euro 2012 Stadium 10.6 4 Yes, out 2 Fan

Saint Irinej 40 - No 1 Fan mixt

Malecon - - No 1 Fan mixt

Milwaukee Art - - Yes, out 1 Fan mixt

Zrodlowa Street 9.3 4.45 Yes, out 2 Harp

Zlotnicka 12.8 6.5 No 2 Fan

Literatuurwijk 5 2.7 No 1 NA

Delftlanden 9.5 6 No 1 Fan mixt

Viana footbridge 16.9 3 Yes, in 2 Fan

Passerelle sur la Tet 22.5 7.5 Yes, out 2 Fan mixt

Nessebrucke 10.5 5 Yes, in 2 Fan

Katehaki - - Yes, out 1 Harp

Feijenoord - - Yes, in 2 Fan mixt

South quay 17.5 - No 1 Harp

Passerelle des deux Rives 27 10 Yes, out 2 Fan

Ohio and Erie Canalway 14.5 - No 2 Fan mixt

7th Av Ped. Bridge 20.5 8.15 Yes, out 1 Fan mixt

Turtle Bay Sundial Bridge - - Yes, out 1 Harp

311

Numerical modelling and response

prediction of girder footbridges

The impact of pedestrian actions on the dynamic response of multiple girder footbridges has been studied

according to the different magnitudes of the main parameters that affect this dynamic response:

1. Related to the structure, the following can be highlighted: span arrangement and dimensions; struc-

tural transverse section (dimensions such width and depth and materials); mass of non-structural

elements; and damping dissipation of the structure.

2. Related to the loads, factors such the following should be considered for dynamic analysis: activity

of pedestrian while walking (commuting or while in leisure); magnitudes of vertical and lateral

loads (related to step frequency, weight of the user and step width); step length; and number and

distribution of pedestrians on the deck.

As highlighted in Chapter 3, such assessment and evaluation of the impact of each of the above

parameters on the overall dynamic response of footbridges has been developed through the numerical

representation of the girder footbridge with a finite element model developed in Matlab. Following there

is a summarised overview of how this numerical model has been generated and how the factors considered

to have an impact on dynamic response have been considered. The code presented hereunder corresponds

to the resolution for a girder bridge of two spans although the procedure for other cases is practically the

same.

Structure characteristics

The structure is represented numerically by multiple elements connecting nodes with a single degree

of freedom:

1 %%% SPAN GEOMETRY AND NUMERICAL REPRESENTATION

2 % Span1

3 L1 = 50 ; % Span length [m]

4 n1 = round (L1/(L1/30) ) ; % Number o f e lements

5 nodes1 = n1−1; % Nodes

6 l 1 = L1/n1 ; % Length o f element

7

8 % Span2

9 a l f a = 0 . 2 ; % Late ra l span length

10 L2 = a l f a ∗L1 ;11 n2 = round (L2/(L1/30) ) ; % Number o f e lements

12 nodes2 = n2−1; % Nodes

13 l 2 = L2/n2 ; % Length o f element

14

313

Appendix B

15 %%% GIRDER STRUCTURAL CHARACTERISTICS ( t r an sv e r s e s e c t i o n )

16 b = 4 ; % Deck width

17 th = L1/100 ; % Deck depth

18 E = 1.00∗35000∗10ˆ6 ; % Young ’ s mod .

19 A = b∗0.2+(L1/35−0.2) ∗2 ; % Deck area [m2]

20 CoG = (4∗0 . 2∗ ( th−0.1)+2∗(th−0.2) ˆ2/2) /A;% Sect i on g rav i ty cen t r e

21 I = 1/12∗b∗0.2ˆ3+4∗0.2∗( th−0.1−CoG) ˆ 2+ . . . ;% Second mom. o f area [m4]

22 d = 25000/9 .81 ; % Mater ia l dens i ty

23 da = 0 . 0 0 4 ; % Damping r a t i o

Pedestrian loads

The magnitudes of loads (document below corresponds to vertical load simulation) are defined ac-

cording to the proposed methodology defined in Chapter 3. These depend on the step frequency

(which is chosen at the beginning of the simulation, according to the characteristics of the

traffic flow expected to cross the bridge) and additionally the step length and weight of

the pedestrian can be modified. Initially, the position of the first step could be introduced,

although the null effect on response led to disregarding such factor:

1

2 %%% PEDESTRIAN CHARACTERISTICS

3 fp = 1 . 6 1 ; % Step f requency ( constant )

4 s l = 0.05∗L1 ; % Step l ength s l /L constant

0 .05∗L5 ip = 0 . 0 ; % I n i t i a l s t ep po s i t i o n

6 Force = −800; % Pedest r ian weight

7

8 %%% PRELIMINARY CHARACTERISTICS FOR CALCULATIONS

9 nsteps = round ( ( L1+L2−ip ) / s l ) ; % Number o f s t ep s

10 i f ( ( nsteps −1)∗ s l+ip ) >= (L1+L2) ;

11 nsteps = nsteps − 1 ;

12 e l s e

13 nsteps = nsteps ;

14 end

15

16 f o r j = 1 : ns teps % Frequency o f each step

17 ct ( j , 1 ) = fp ;

18 end

19

20 pos s t eps = ze ro s ( nsteps , nodes1+nodes2 ) ;% Pos i t i on o f each step

in r e l a t i o n to nodes

21 f o r i = 1 : ns teps

22 pos = ( ( ( i −1)∗ s l )+ip ) ;

23 . . .

314

Appendix B

24 end

25

26 %%% PEDESTRIAN LOADS

27 f o r j = 1 : ns teps

28 fp1 = ct ( j , 1 ) ;

29 td = round ((1.554−0.459∗ fp1 ) /0 .001) ;% Contact time

accord ing to s tep f requency

30 ampt = ze ro s (1 , td+1) ; % Load amplitude

31 ampa = ze ro s (1 , td+1) ;

32 ampb = ze ro s (1 , td+1) ;

33 ampc = ze ro s (1 , td+1) ;

34 . . .

35 ampt = ampa + ampb + ampc ;

36 end

Modal characterisation of the structure

Based on the previous definitions, the Matlab code assembles a stiffness matrix and,

based on this, the vibration frequencies are obtained:

1 %%% STIFFNESS MATRIX ASSEMBLY

2

3 % Local s t i f f n e s s matrix f o r span1

4 k1 = E∗ I / l 1 ˆ3∗ [ 12 6∗ l 1 −12 6∗ l 15 6∗ l 1 4∗ l 1 ˆ2 −6∗ l 1 2∗ l 1 ˆ26 −12 −6∗ l 1 12 −6∗ l 17 6∗ l 1 2∗ l 1 ˆ2 −6∗ l 1 4∗ l 1 ˆ 2 ] ;

8

9 % Local s t i f f n e s s matrix f o r span2

10 k2 = E∗ I / l 2 ˆ3∗ [ 12 6∗ l 2 −12 6∗ l 211 6∗ l 2 4∗ l 2 ˆ2 −6∗ l 2 2∗ l 2 ˆ212 −12 −6∗ l 2 12 −6∗ l 213 6∗ l 2 2∗ l 2 ˆ2 −6∗ l 2 4∗ l 2 ˆ 2 ] ;

14

15 % Global s t i f f n e s s matrix

16 totnodes = nodes1 + nodes2 + 3 ; % Three cor responds to the nodes

with supor t s

17 K = ze ro s ( totnodes ∗2 , totnodes ∗2) ;18 f o r i = 1 : ( totnodes −1)

19 . . .

20 end

21

22 % Remove va lue s cor re spond ing to supor t s

315

Appendix B

23 e l = [ 1 ( nodes1+2) ( nodes1+nodes2+3) ] ;

24 f o r i = 1 : l ength ( e l )

25 . . .

26 %K2 new matrix

27 end

28

29 % Sta t i c condensat ion

30 K0t = K2( ( nodes1+nodes2+1) :m1 , 1 : ( nodes1+nodes2 ) ) ;

31 K00 = K2( ( nodes1+nodes2+1) :m1, ( nodes1+nodes2+1) :m2) ;

32 Kf ina l = K2( 1 : ( nodes1+nodes2 ) , 1 : ( nodes1+nodes2 ) ) − K0t ’∗ inv (K00)

∗K0t ;

33

34 % Mass matrix

35 M = zero s ( ( nodes1+nodes2 ) , ( nodes1+nodes2 ) ) ;

36 f o r i = 1 : ( nodes1 )

37 M( i , i ) = d∗A∗ l 1 ;

38 end

39 f o r i = ( nodes1+1) : ( nodes1+nodes2 )

40 M( i , i ) = d∗A∗ l 2 ;

41 end

42

43 % Modes

44 [V,D] = e i g ( Kf inal ,M) ; % , ’ chol ’

45

46 % Vector with f i r s t 5 natura l f r e qu en c i e s

47 Freq = ze ro s (5 , 1 ) ;

48 f o r i = 1 :5

49 Freq ( i , 1 ) = D( (m1+1− i ) , (m1+1− i ) ) ˆ0 .5/2/ p i ;

50 end

Dynamic response by modal superposition

Considering the obtained modes and the pedestrian loads at each node generated by

each step of the user, the code predicts the footbridge response at multiple span locations:

1 %%% CALCULATION OF RESPONSE

2 t i = 0 . 0 0 1 ; % Ca l cu l a t i on i n t e r v a l

3 mv = 7 ; % Only f i v e modes cons ide r ed

4 mmod = ze ro s (mv, 1 ) ;

5 kmod = ze ro s (mv, 1 ) ;

6 cmod = ze ro s (mv, 1 ) ;

7 f o r i = 1 :mv

316

Appendix B

8 mmod ( i , 1 ) = V( : , ( nodes1+nodes2+1− i ) ) ’ ∗ M ∗ V( : , ( nodes1+

nodes2+1− i ) ) ;

9 kmod ( i , 1 ) = V( : , ( nodes1+nodes2+1− i ) ) ’ ∗ Kf ina l ∗ V( : , (

nodes1+nodes2+1− i ) ) ;

10 cmod ( i , 1 ) = 2 ∗ D(( nodes1+nodes2+1− i ) , ( nodes1+nodes2+1− i ) )

ˆ0 .5 ∗ mmod( i , 1 ) ∗ da ;

11 end

12

13 % Load that each DOF r e c e i v e s

14 s = length (amp) ;

15 ampsteps = ze ro s ( nsteps , r ) ;

16 ampn = ze ro s (mv, r ) ;

17 tn = ( 0 : t i : t i ∗( r−1) ) ;

18

19 f o r y = 1 :mv;

20 % Load o f 1 s tep

21

22 tacum = 1/ ct (1 , 1 ) ;

23 f o r j = 2 : ns teps ; %%% Number o f s tep

24 % Load at each node caused by a l l s t ep s

25 . . .

26 end

27 % Modal load

28 ampn(y , : ) = V( : , ( nodes1 + nodes2 + 1 − y ) ) ’ ∗ possteps ’ ∗ampsteps ( : , : ) ;

29 % Calcu l a t i on o f the re sponse

30 u(y , 1 ) = 0 ;

31 v (y , 1 ) = 0 ;

32 a (y , 1 ) = (ampn(y , 1 )−cmod(y , 1 ) ∗v (y , 1 )−kmod(y , 1 ) ∗u(y , 1 ) ) /mmod(

y , 1 ) ;

33 kb (y , 1 ) = kmod(y , 1 )+3∗cmod(y , 1 ) / t i +6∗mmod(y , 1 ) / t i ˆ2 ; %%

modi f i ed s t i f f n e s s

34 f o r j = 1 : r−1;

35 pb = ampn(y , j +1)+mmod(y , 1 ) ∗(6∗u(y , j ) / t i ˆ2+6∗v (y , j ) / t i +2∗a (y , j ) )+cmod(y , 1 ) ∗(3∗u(y , j ) / t i +2∗v (y , j )+t i /2∗a (y , j ) ) ;

36 u(y , j +1) = pb/kb (y , 1 ) ;

37 v (y , j +1) = 3/ t i ∗(u(y , j +1)−u(y , j ) )−2∗v (y , j )−t i /2∗a (y , j ) ;38 a (y , j +1) = 6/ t i ˆ2∗(u(y , j +1)−u(y , j ) )−6/ t i ∗v (y , j )−a (y , j )

∗2 ;39 end

40 end

317

Appendix B

41

42 % Total r e sponse

43 % Based on prev ious r e s u l t s , c a l c u l a t i o n o f r e sponse s at any

node o f i n t e r e s t

318

UAMP Abaqus subroutine

The load model reproducing the actions of pedestrians while crossing a footbridge is

defined in Chapter 2. According to this proposed model, vertical loads can be numerically

simulated by placing nodal loads with amplitude that is known prior to the beginning of

the analysis (hence these can be defined in Abaqus as a list of values that depend on time).

For lateral loads, these amplitudes depend on previous steps and movements, hence they

need to be defined through an Abaqus subroutine, UAMP. This subroutine consists of

a Fortran script called by Abaqus when the input file of the structure model has this

amplitude defined as:

1 ∗Amplitude , name=P001−H002−T1 , DEFINITION=USER, VARIABLES=15

This input line describes how an amplitude with name P001-H002-T1 is defined by the

user and, from calculations defined within the subroutine, its amplitude at a particular

time of the simulation is extracted from the subroutine. Apart from this amplitude, there

are other 15 variables defined by the subroutine author that are given as results and kept

for following steps of the same pedestrian.

The first lines of the subroutine are as follow:

1 c user amplitude subrout ine

2 Subrout ine UAMP(

3 C passed in f o r in fo rmat ion and s t a t e v a r i a b l e s

4 ∗ ampName, time , ampValueOld , dt , nProps , props , nSvars

,

5 ∗ svars , lF l ag s In f o ,

6 ∗ nSensor , sensorValues , sensorNames ,

7 ∗ jSensorLookUpTable ,

8 C to be de f i n ed

9 ∗ ampValueNew ,

10 ∗ lF lag sDe f ine ,

11 ∗ AmpDerivative , AmpSecDerivative , AmpIncIntegral ,

12 ∗ AmpDoubleIntegral )

13 i n c l ude ’ aba param . inc ’

14 C svar s − add i t i o na l s t a t e va r i ab l e s , s im i l a r to (V)UEL

15 dimension sensorValues ( nSensor ) , sva r s ( nSvars ) ,

16 ∗ props ( nProps ) , ut0 (8 ) , wp(8) , fp (8 ) , bmin (8 ) ,

319

Appendix C

17 ∗ aux i l 1 (4 ) , aux i l 3 (5 ) , ampl it (8 )

18 I n t e g e r ∗4 k4 , l , l e v e r e r r

19 Real∗4 tim , i , ib , tent , tent2 , f r eq , td , t f , ta , tb , wpg ,

20 ∗ bming , auxa , auxb , auxc , auxd , tim3 , ut , x , v , ac , j , h ,

21 ∗ k1x , k1v , k2x , k2v , k3x , k3v , k4x , k4v , tent b , f r eq b ,

i b ,

22 ∗ ib b , td b , t f b , ta b , tb b , tim4 , aux i l2 , num ped

23 cha rac t e r ∗80 sensorNames ( nSensor )

24 cha rac t e r ∗80 ampName

25 C time i n d i c e s

26 parameter ( iStepTime = 1 ,

27 ∗ iTotalTime = 2 ,

28 ∗ nTime = 2)

29 C f l a g s passed in f o r in fo rmat ion

30 parameter ( i I n i t i a l i z a t i o n = 1 ,

31 ∗ iRegu la r Inc = 2 ,

32 ∗ iCuts = 3 ,

33 ∗ i kStep = 4 ,

34 ∗ nFlags In fo = 4)

35 C opt i ona l f l a g s to be de f i ned

36 parameter ( iComputeDeriv = 1 ,

37 ∗ iComputeSecDeriv = 2 ,

38 ∗ iComputeInteg = 3 ,

39 ∗ iComputeDoubleInteg = 4 ,

40 ∗ i S topAna ly s i s = 5 ,

41 ∗ iConcludeStep = 6 ,

42 ∗ nFlagsDef ine = 6)

43

44 parameter ( ze ro =0.0d0 , one=1.0d0 , two=2.0d0 ,

45 ∗ f ou r =4.0d0 , t i n =0.02d0 )

46

47 dimension time (nTime) , l F l a g s I n f o ( nF lags In fo ) ,

48 ∗ l F l ag sDe f i n e ( nFlagsDef ine )

49 dimension jSensorLookUpTable (∗ )50 l F l ag sDe f i n e ( iComputeDeriv ) = 1

51 l F l ag sDe f i n e ( iComputeSecDeriv ) = 1

52 l F l ag sDe f i n e ( iComputeInteg ) = 1

53 l F l ag sDe f i n e ( iComputeDoubleInteg ) = 1

These lines correspond to several definitions established by Abaqus and the definition

of other variables that are used within the calculations of the subroutine (parameters that

describe each pedestrian). Immediately after the definition of the variable names, the code

320

Appendix C

of the subroutine has several lines with the numerical values of these variables. Since the

same subroutine is called for any pedestrian and step, the subroutine distinguishes what

values of each variable should be considered according to the name of the load amplitude

(e.g. P001-H002-T1).

1 ut0 ( 1 : n ) = (/ −0 . 0 753 , . . . ,

2 ∗ −0.0391/)

3

4 wp( 1 : n) = ( / 3 . 0 5 6 6 , . . . ,

5 ∗ 3 .1196/)

6

7 fp ( 1 : n ) = ( / 2 . 4 1 0 0 , . . . ,

8 ∗ 2 .2100/)

9

10 bmin ( 1 : n ) = ( / 0 . 0 3 3 0 , . . . ,

11 ∗ 0 .0153/)

12

13 ampl it ( 1 : n ) = ( / 0 . 0 0 0 0 , . . . ,

14 ∗ 0 .0000/)

15

16 i f (ampName( 1 : 1 ) . eq . ’P ’ ) then

17 tim = time ( iTotalTime )

18 i f (ampName( 1 : 9 ) . eq . ’P001−H001 ’ ) then

19 aux i l 1 ( 1 : 4 ) = (/1 . 00 , 2 . 4 1 , 1 . 0 0 , 1 . 0 0 / )

20 e l s e i f (ampName( 1 : 9 ) . eq . ’P001−H002 ’ ) then

21 . . .

22 . . .

23 end i f

24 . . .

Based on the values of each variable and according to the step and pedestrian, the

subroutine recalculates the amplitude of the lateral load and retains the values of the

variables (15, as previously highlighted) in order to define the amplitude and movements

that will define the load in further steps:

1 c f i r s t s tep o f each pedes t r ian , which does not need p r i o r

r e s u l t s o f movement

2 i f ( tim . eq . 1 . 0 ) then

3 c d e f i n i t i o n o f v a r i a b l e s f o r next s tep

4 . . .

5 sva r s (1 )

6 . . .

7 sva r s (15)

8 c Load amplitude

321

Appendix C

9 ampValueNew = ampl it ( aux i l 1 (3 ) )

10 e l s e

11 c I f i t i s not the f i r s t step , f o r each pede s t r i an and f o r each

step o f each pede s t r i an :

12 c Obtain the s t r u c t u r a l movements o f i n t e r e s t

13 c and de f i n e time i n t e r v a l s where the subrout ine c a l c u l a t e s the

ampl itudes

14 i f (ampName( 1 : 4 ) . eq . ’ P001 ’ ) then

15 i f ( ( tim . ge . 1 . 0 0 ) .AND. ( tim . l t . 1 . 4 2 ) ) then

16 aux i l 2 = GetSensorValue ( ’N−194A ’ ,

jSensorLookUpTable ,

17 ∗ sensorValues )

18 aux i l 3 = ( / 1 . 0 0 , 1 . 4 2 , 2 . 4 1 , 1 . 0 0 , 1 . 0 0 / )

19 e l s e i f ( ( tim . ge . 1 . 4 2 ) .AND. ( tim . l t . 1 . 8 3 ) )

then

20 aux i l 2 = GetSensorValue ( ’N−203A ’ ,

jSensorLookUpTable ,

21 ∗ sensorValues )

22 aux i l 3 = ( / 1 . 4 2 , 1 . 8 3 , 2 . 4 1 , 1 . 0 0 , 2 . 0 0 / )

23 . . .

24 . . .

25 . . .

26 end i f

27 end i f

28 c Aux i l i a r data : time when a c c e l e r a t i o n s o f the deck are

recorded and kept f o r f o l l ow i n g s t ep s

29 t ent

30 tent2

31 f r e q

32 auxa = tent2 −0.1

33 auxb = tent2 −0.1+0.04

34 auxc = tent + 0 .04

35 auxd = tent + 0 .08

36

37 c Acc e l e r a t i on s r e co rd ing and assignment to v a r i a b l e s that are

ex t rac t ed from subrout ine

38 i f ( ( tim . ge . ( t ent ) ) .AND. ( tim . l t . auxa ) ) then

39 i f (ABS( aux i l 2 ) . ge . ABS( sva r s (13) ) ) then

40 sva r s (13) = aux i l 2

41 e l s e

42 sva r s (13) = svar s (13)

322

Appendix C

43 end i f

44 sva r s (14) = svar s (14)

45 sva r s (15) = svar s (15)

46 e l s e i f ( ( tim . ge . ( auxa ) ) .AND. ( tim . l t . auxb ) )

then

47 sva r s (13) = svar s (13)

48 i f (mod( ib , 2 . 0 ) /= zero ) then

49 sva r s (14) = svar s (13)

50 sva r s (15) = svar s (15)

51 e l s e

52 sva r s (15) = svar s (13)

53 sva r s (14) = svar s (14)

54 end i f

55 e l s e i f ( ( tim . ge . ( auxb ) ) .AND. ( tim . l t . ( tent2 ) ) )

then

56 sva r s (13) = zero

57 sva r s (14) = svar s (14)

58 sva r s (15) = svar s (15)

59 end i f

60

61 c Step width , CoM pos and speed f o r f o l l ow i n g step d e f i n i t i o n

62 i f ( ( tim . ge . t ent ) .AND. ( tim . l t . auxc ) ) then

63 sva r s (1 )

64 sva r s (2 )

65 sva r s (3 )

66 c F i r s t o f a l l , v a r i a b l e s r e l a t e d to a c c e l e r a t i o n

67 tim3 = tb−(ta +0.001)

68 ut = svar s (1 )

69 x = svar s (2 )

70 v = svar s (3 )

71 sva r s (4 ) = (x+v/wpg) + bming∗(−1) ∗∗( ib+1)

72 sva r s (5 ) = x

73 sva r s (6 ) = v

74 . . .

75 c i f amplitude i s l a r g e r than permitted l im i t , p r ev ious

v a r i a b l e s are r e c a l c u l a t e d

76 end i f

77

78 cc Amplitude o f l oads

79 t ent b

80 f r e q b

323

Appendix C

81 i b

82 i b b

83 td b

84 t f b

85 ta b

86 tb b

87 wpg

88 bming

89 i f ( tim . l e . ta b ) then

90 ampValueNew = zero

91 e l s e i f ( ( tim . gt . ta b ) .AND. ( tim . l t . tb b ) ) then

92 i f (mod( ib b , 2 . 0 ) /= zero ) then

93 . . .

94 ampValueNew = −wpg∗∗2∗( ut−x )95 e l s e

96 . . .

97 ampValueNew = −wpg∗∗2∗( ut−x )98 end i f

99 e l s e i f ( tim . ge . tb b ) then

100 ampValueNew = zero

101 end i f

Finally, some additional values are printed in text files in order to retain several internal

variables.

1

2 open ( f i l e =”c :\ Users . . . \TIME ALL. txt ” ,

3 ∗ Unit=15, i o s t a t=l e v e r e r r , e r r =100 , STATUS = ’UNKNOWN’ )

4 wr i t e (15 ,FMT=”(( f10 . 2 ) , ( f10 . 2 ) , ( f10 . 2 ) , ( f10 . 6 ) ) ” , e r r

=200) tim , REAL( i b ) ,

5 ∗ REAL( ib b ) , ampValueNew

6 100 i f ( l e v e r e r r /= 0 ) stop ”Be Care fu l − Open”

7 200 i f ( l e v e r e r r /= 0 ) stop ”Be Care fu l − W”

8

9 c Recording o f other va lue s o f i n t e r e s t in t ex t f i l e s

10 . . .

11 . . .

12 . . .

13

14 end i f

15 end i f

16 r e turn

17 end

324

Subroutines such that presented above are developed for each subinterval in which the

whole simulation time is divided. The length of this subinterval is given by the number

of times that the input file calls the subroutine, i.e. roughly related to the number of

pedestrians that are on the structure (a higher number of subroutine calls is equivalent

to a much higher computational cost of each time step).

325

Evaluation of response in

serviceability conditions (girder

footbridges)

The basic or reference vertical and lateral accelerations used to describe the serviceabil-

ity response of girder footbridges of one, two or three spans proposed in chapter 4 are

described below.

327

Appendix D

abv,n [m/s2]

fp,v [Hz]

rv,n, rl,n 1.3 1.4 1.5 1.6 1.7 1.8 1.9

0.5 9.39E-04 8.53E-04 1.01E-03 1.07E-03 1.46E-03 1.74E-03 2.62E-03

0.548 2.09E-03 1.90E-03 1.58E-03 1.26E-03 2.95E-03 4.31E-03 6.04E-03

0.592 7.71E-04 7.40E-04 9.86E-04 1.20E-03 1.57E-03 2.29E-03 2.66E-03

0.632 6.30E-04 6.69E-04 9.59E-04 1.41E-03 1.76E-03 2.71E-03 3.13E-03

0.671 8.63E-04 1.89E-03 2.98E-03 3.39E-03 5.52E-03 8.95E-03 1.05E-02

0.707 6.49E-04 7.55E-04 1.15E-03 1.54E-03 1.70E-03 2.46E-03 3.06E-03

0.742 7.68E-04 6.91E-04 1.19E-03 1.76E-03 1.86E-03 2.22E-03 3.11E-03

0.775 2.44E-03 2.53E-03 4.06E-03 4.99E-03 4.37E-03 2.85E-03 4.69E-03

0.806 1.03E-03 8.48E-04 1.64E-03 2.51E-03 2.59E-03 2.75E-03 3.49E-03

0.837 8.85E-04 7.87E-04 1.61E-03 2.32E-03 2.91E-03 3.01E-03 3.48E-03

0.866 8.67E-04 1.00E-03 1.62E-03 2.63E-03 3.35E-03 3.52E-03 4.45E-03

0.894 2.01E-03 1.44E-03 2.11E-03 4.35E-03 5.39E-03 4.07E-03 6.91E-03

0.922 1.27E-03 1.03E-03 1.91E-03 3.06E-03 4.10E-03 5.07E-03 7.30E-03

0.949 1.54E-03 1.13E-03 2.48E-03 3.98E-03 5.31E-03 6.93E-03 9.68E-03

0.975 3.49E-03 1.72E-03 4.70E-03 8.71E-03 1.26E-02 1.64E-02 2.36E-02

0.987 5.70E-03 2.68E-03 7.67E-03 1.42E-02 2.08E-02 2.76E-02 3.96E-02

1 6.49E-03 3.93E-03 1.12E-02 2.01E-02 2.89E-02 3.65E-02 5.15E-02

1.025 1.85E-03 2.47E-03 6.21E-03 1.03E-02 1.44E-02 1.83E-02 2.51E-02

1.049 1.28E-03 1.33E-03 2.78E-03 4.23E-03 5.64E-03 7.46E-03 1.05E-02

1.072 7.93E-04 1.03E-03 2.13E-03 3.14E-03 4.22E-03 5.33E-03 7.23E-03

1.095 8.05E-04 1.19E-03 1.94E-03 2.36E-03 3.42E-03 4.16E-03 5.06E-03

1.118 1.08E-03 1.74E-03 2.44E-03 2.49E-03 3.49E-03 4.33E-03 4.82E-03

1.225 1.53E-03 1.45E-03 2.03E-03 2.06E-03 1.99E-03 2.67E-03 4.14E-03

1.323 4.95E-04 7.27E-04 1.02E-03 1.32E-03 1.49E-03 1.84E-03 2.85E-03

1.414 5.50E-04 6.23E-04 9.86E-04 1.29E-03 1.63E-03 1.63E-03 1.49E-03

1.5 5.06E-04 6.15E-04 9.94E-04 1.04E-03 1.11E-03 1.15E-03 1.55E-03

1.581 4.68E-04 9.44E-04 8.61E-04 1.10E-03 1.34E-03 1.25E-03 1.68E-03

1.658 7.17E-04 7.95E-04 1.18E-03 1.48E-03 1.23E-03 1.71E-03 1.32E-03

1.732 4.72E-04 7.52E-04 1.11E-03 1.26E-03 1.42E-03 1.27E-03 1.31E-03

1.803 5.47E-04 9.45E-04 1.42E-03 1.86E-03 1.71E-03 1.98E-03 1.21E-03

1.871 6.73E-04 1.35E-03 2.15E-03 2.34E-03 2.55E-03 2.53E-03 1.80E-03

1.936 7.27E-04 2.21E-03 3.64E-03 4.30E-03 4.59E-03 4.25E-03 2.27E-03

1.949 8.23E-04 2.68E-03 4.30E-03 5.12E-03 5.45E-03 5.15E-03 2.70E-03

1.962 9.16E-04 3.62E-03 5.89E-03 7.18E-03 7.61E-03 7.16E-03 3.61E-03

1.975 1.07E-03 5.33E-03 8.82E-03 1.07E-02 1.15E-02 1.10E-02 5.37E-03

1.987 1.25E-03 7.75E-03 1.31E-02 1.61E-02 1.71E-02 1.62E-02 8.17E-03

2 1.41E-03 9.80E-03 1.59E-02 1.93E-02 2.06E-02 1.96E-02 1.04E-02

2.012 1.33E-03 8.54E-03 1.37E-02 1.67E-02 1.77E-02 1.67E-02 9.26E-03

2.025 1.10E-03 6.05E-03 9.51E-03 1.15E-02 1.24E-02 1.17E-02 6.37E-03

2.037 9.59E-04 4.14E-03 6.37E-03 7.62E-03 8.14E-03 7.72E-03 4.39E-03

2.049 8.08E-04 2.90E-03 4.41E-03 5.26E-03 5.51E-03 5.34E-03 2.99E-03

2.062 7.04E-04 2.57E-03 3.80E-03 4.53E-03 4.75E-03 4.58E-03 2.69E-03

2.121 7.22E-04 1.37E-03 1.97E-03 2.59E-03 2.87E-03 3.02E-03 1.90E-03

2.179 5.58E-04 1.05E-03 1.45E-03 1.71E-03 1.77E-03 1.80E-03 1.14E-03

2.236 7.06E-04 8.14E-04 1.30E-03 1.59E-03 1.52E-03 1.88E-03 1.45E-03

2.291 5.06E-04 8.21E-04 1.13E-03 1.14E-03 1.19E-03 1.35E-03 9.36E-04

2.345 6.00E-04 8.75E-04 1.20E-03 1.26E-03 1.25E-03 1.64E-03 9.76E-04

2.398 4.91E-04 6.89E-04 8.01E-04 9.26E-04 9.97E-04 1.10E-03 8.39E-04

2.449 5.25E-04 9.58E-04 9.45E-04 8.70E-04 1.29E-03 1.32E-03 8.81E-04

2.5 6.34E-04 7.01E-04 8.27E-04 7.94E-04 9.93E-04 1.00E-03 9.32E-04

2.55 6.77E-04 7.92E-04 9.27E-04 9.72E-04 1.12E-03 1.02E-03 1.16E-03

2.598 6.90E-04 6.61E-04 6.80E-04 6.83E-04 7.91E-04 8.60E-04 8.57E-04

2.646 8.39E-04 7.29E-04 8.04E-04 6.82E-04 7.35E-04 7.85E-04 9.94E-04

2.693 8.31E-04 7.65E-04 8.18E-04 6.73E-04 7.27E-04 8.44E-04 1.10E-03

2.739 8.13E-04 8.00E-04 7.61E-04 6.58E-04 7.23E-04 9.12E-04 1.24E-03

2.784 9.53E-04 9.64E-04 7.73E-04 7.47E-04 1.01E-03 1.16E-03 1.73E-03

2.828 1.06E-03 9.81E-04 8.45E-04 7.57E-04 9.03E-04 1.08E-03 1.47E-03

2.872 1.37E-03 1.20E-03 9.62E-04 9.05E-04 1.15E-03 1.21E-03 1.80E-03

2.915 1.84E-03 1.62E-03 1.39E-03 1.17E-03 1.45E-03 1.78E-03 2.45E-03

2.958 3.35E-03 2.73E-03 2.16E-03 1.98E-03 2.61E-03 2.98E-03 4.26E-03

2.966 4.06E-03 3.28E-03 2.62E-03 2.45E-03 3.19E-03 3.67E-03 5.26E-03

2.975 4.89E-03 3.95E-03 3.11E-03 3.11E-03 4.13E-03 4.51E-03 6.54E-03

2.983 5.95E-03 4.75E-03 3.81E-03 3.72E-03 4.95E-03 5.28E-03 7.96E-03

2.992 6.89E-03 5.40E-03 4.29E-03 4.24E-03 5.76E-03 6.34E-03 9.01E-03

3 7.40E-03 5.75E-03 4.40E-03 4.59E-03 6.19E-03 6.80E-03 9.72E-03

3.008 7.10E-03 5.52E-03 4.16E-03 4.34E-03 5.89E-03 6.74E-03 9.30E-03

3.017 6.01E-03 4.84E-03 3.52E-03 3.57E-03 5.17E-03 5.88E-03 8.16E-03

3.025 5.06E-03 4.12E-03 3.10E-03 3.07E-03 4.30E-03 4.72E-03 6.50E-03

3.033 4.06E-03 3.40E-03 2.57E-03 2.54E-03 3.45E-03 3.73E-03 5.16E-03

3.041 3.37E-03 2.78E-03 2.11E-03 1.99E-03 2.79E-03 3.01E-03 4.25E-03

328

Appendix D

abv,n [m/s2]

fp,v [Hz]

rv,n, rl,n 2.0 2.1 2.2 2.3 2.4 [0.65-1.2]

0.5 2.76E-03 3.13E-03 2.72E-03 3.25E-03 3.10E-03 4.27E-04

0.548 5.85E-03 4.65E-03 3.46E-03 6.20E-03 1.05E-02 1.64E-03

0.592 2.90E-03 2.39E-03 3.45E-03 3.43E-03 3.96E-03 5.92E-04

0.632 3.05E-03 2.67E-03 3.52E-03 3.68E-03 3.49E-03 3.97E-04

0.671 8.82E-03 3.63E-03 7.48E-03 1.10E-02 9.52E-03 6.08E-04

0.707 3.27E-03 3.05E-03 3.62E-03 4.10E-03 3.85E-03 6.86E-04

0.742 3.02E-03 3.15E-03 3.71E-03 4.25E-03 3.49E-03 3.11E-03

0.775 4.87E-03 4.68E-03 5.13E-03 7.54E-03 7.12E-03 1.75E-03

0.806 3.63E-03 4.16E-03 4.27E-03 4.61E-03 5.01E-03 8.64E-04

0.837 4.22E-03 4.90E-03 4.89E-03 5.27E-03 5.31E-03 7.75E-04

0.866 5.26E-03 5.72E-03 5.85E-03 5.66E-03 5.68E-03 7.24E-04

0.894 9.02E-03 8.83E-03 7.93E-03 8.19E-03 6.24E-03 7.47E-04

0.922 8.99E-03 9.26E-03 9.28E-03 9.66E-03 8.79E-03 1.23E-03

0.949 1.15E-02 1.28E-02 1.29E-02 1.29E-02 1.16E-02 1.96E-03

0.975 2.83E-02 3.24E-02 3.30E-02 3.15E-02 2.82E-02 3.70E-03

0.987 4.70E-02 5.33E-02 5.55E-02 5.34E-02 4.76E-02 4.83E-03

1 6.28E-02 7.18E-02 7.34E-02 6.82E-02 6.13E-02 5.88E-03

1.025 3.05E-02 3.48E-02 3.54E-02 3.42E-02 3.08E-02 4.09E-03

1.049 1.21E-02 1.34E-02 1.39E-02 1.39E-02 1.32E-02 2.23E-03

1.072 8.60E-03 9.82E-03 1.03E-02 1.07E-02 1.05E-02 1.48E-03

1.095 6.56E-03 7.35E-03 7.95E-03 7.70E-03 8.67E-03 9.30E-04

1.118 6.18E-03 6.60E-03 7.24E-03 6.53E-03 8.80E-03 7.55E-04

1.225 3.14E-03 5.04E-03 6.02E-03 4.92E-03 6.25E-03 1.44E-03

1.323 1.90E-03 3.18E-03 3.12E-03 3.16E-03 5.84E-03 7.09E-04

1.414 1.53E-03 1.91E-03 2.57E-03 3.32E-03 3.97E-03 4.74E-04

1.5 1.28E-03 1.78E-03 2.27E-03 3.00E-03 4.01E-03 3.60E-04

1.581 1.48E-03 1.78E-03 2.57E-03 3.22E-03 4.40E-03 3.78E-04

1.658 1.62E-03 1.50E-03 3.28E-03 4.10E-03 5.60E-03 5.86E-04

1.732 1.16E-03 1.71E-03 2.63E-03 3.81E-03 5.37E-03 1.11E-03

1.803 1.33E-03 2.14E-03 3.26E-03 4.99E-03 6.58E-03 6.21E-04

1.871 1.54E-03 2.52E-03 4.19E-03 6.24E-03 9.52E-03 4.32E-04

1.936 2.03E-03 4.15E-03 7.61E-03 1.19E-02 1.76E-02 4.78E-04

1.949 2.20E-03 4.60E-03 8.66E-03 1.38E-02 2.04E-02 4.72E-04

1.962 2.84E-03 6.66E-03 1.25E-02 2.00E-02 2.94E-02 4.15E-04

1.975 4.03E-03 1.01E-02 1.87E-02 3.11E-02 4.45E-02 3.98E-04

1.987 5.09E-03 1.43E-02 2.70E-02 4.40E-02 6.46E-02 3.13E-04

2 5.72E-03 1.58E-02 3.11E-02 5.12E-02 7.55E-02 3.04E-04

2.012 5.11E-03 1.32E-02 2.57E-02 4.31E-02 6.31E-02 3.14E-04

2.025 3.81E-03 8.85E-03 1.82E-02 2.97E-02 4.42E-02 3.38E-04

2.037 2.72E-03 6.08E-03 1.26E-02 2.05E-02 2.99E-02 3.90E-04

2.049 2.10E-03 4.55E-03 9.13E-03 1.46E-02 2.12E-02 4.95E-04

2.062 1.95E-03 3.83E-03 7.69E-03 1.24E-02 1.80E-02 5.27E-04

2.121 1.55E-03 2.35E-03 4.99E-03 7.27E-03 1.02E-02 5.96E-04

2.179 1.08E-03 1.72E-03 3.32E-03 4.95E-03 7.24E-03 4.96E-04

2.236 1.07E-03 1.61E-03 2.70E-03 4.17E-03 6.06E-03 1.03E-03

2.291 1.01E-03 1.31E-03 2.22E-03 3.76E-03 5.33E-03 6.73E-04

2.345 1.07E-03 1.68E-03 2.53E-03 4.17E-03 5.51E-03 5.34E-04

2.398 9.77E-04 1.13E-03 2.12E-03 3.07E-03 4.65E-03 3.27E-04

2.449 1.26E-03 1.39E-03 2.12E-03 2.96E-03 4.74E-03 4.69E-04

2.5 1.08E-03 1.10E-03 1.78E-03 2.85E-03 4.14E-03 3.51E-04

2.55 1.31E-03 1.21E-03 1.90E-03 3.04E-03 4.54E-03 6.28E-04

2.598 1.18E-03 1.12E-03 1.71E-03 2.96E-03 4.53E-03 4.08E-04

2.646 1.15E-03 1.03E-03 1.71E-03 3.12E-03 4.87E-03 4.03E-04

2.693 1.16E-03 1.08E-03 1.70E-03 3.06E-03 5.23E-03 5.87E-04

2.739 1.13E-03 1.18E-03 1.79E-03 3.28E-03 4.88E-03 9.55E-04

2.784 1.47E-03 1.49E-03 2.07E-03 3.86E-03 5.40E-03 8.56E-04

2.828 1.50E-03 1.30E-03 1.85E-03 3.97E-03 6.25E-03 6.85E-04

2.872 1.83E-03 1.36E-03 1.97E-03 4.63E-03 7.64E-03 8.46E-04

2.915 2.46E-03 1.60E-03 2.38E-03 5.98E-03 1.04E-02 1.02E-03

2.958 4.30E-03 2.49E-03 3.59E-03 1.00E-02 1.86E-02 2.24E-03

2.966 5.18E-03 2.97E-03 4.17E-03 1.22E-02 2.32E-02 2.51E-03

2.975 6.55E-03 3.52E-03 4.75E-03 1.51E-02 2.88E-02 2.84E-03

2.983 7.98E-03 4.24E-03 5.35E-03 1.81E-02 3.33E-02 3.27E-03

2.992 9.00E-03 4.81E-03 5.82E-03 2.05E-02 3.91E-02 3.68E-03

3 9.49E-03 4.85E-03 6.11E-03 2.10E-02 4.08E-02 3.72E-03

3.008 9.17E-03 4.71E-03 5.85E-03 1.98E-02 3.85E-02 3.86E-03

3.017 8.21E-03 4.37E-03 5.23E-03 1.70E-02 3.41E-02 3.62E-03

3.025 6.41E-03 3.74E-03 4.58E-03 1.44E-02 2.72E-02 3.39E-03

3.033 5.18E-03 3.05E-03 3.83E-03 1.19E-02 2.18E-02 3.07E-03

3.041 4.30E-03 2.49E-03 3.22E-03 9.64E-03 1.83E-02 2.61E-03

329

Appendix D

abv,n [m/s2]

fp,v [Hz]

rv,n, rl,n 1.3 1.4 1.5 1.6 1.7 1.8 1.9

3.082 1.88E-03 1.71E-03 1.35E-03 1.08E-03 1.53E-03 1.67E-03 2.28E-03

3.122 1.55E-03 1.48E-03 1.10E-03 1.05E-03 1.17E-03 1.26E-03 1.72E-03

3.162 1.18E-03 1.12E-03 9.07E-04 7.77E-04 9.17E-04 1.01E-03 1.40E-03

3.202 1.06E-03 9.70E-04 8.53E-04 7.42E-04 8.45E-04 9.75E-04 1.20E-03

3.24 1.01E-03 9.59E-04 8.21E-04 8.48E-04 8.81E-04 1.08E-03 1.16E-03

3.279 9.37E-04 9.25E-04 7.53E-04 7.49E-04 6.85E-04 8.40E-04 1.08E-03

3.317 8.72E-04 8.95E-04 7.78E-04 7.57E-04 6.43E-04 7.77E-04 1.06E-03

3.354 9.23E-04 8.71E-04 8.30E-04 8.23E-04 7.49E-04 7.11E-04 1.03E-03

3.391 8.85E-04 8.83E-04 7.81E-04 6.67E-04 6.71E-04 6.86E-04 9.17E-04

3.428 9.18E-04 9.14E-04 7.78E-04 6.77E-04 6.53E-04 6.51E-04 1.02E-03

3.464 9.72E-04 9.63E-04 7.50E-04 6.44E-04 6.85E-04 7.45E-04 1.17E-03

3.5 9.42E-04 9.20E-04 7.56E-04 6.44E-04 7.08E-04 7.20E-04 8.49E-04

3.536 9.64E-04 9.61E-04 7.87E-04 6.96E-04 7.58E-04 7.47E-04 7.88E-04

3.571 9.97E-04 1.13E-03 8.51E-04 7.29E-04 8.75E-04 7.17E-04 8.71E-04

3.606 9.63E-04 1.03E-03 8.62E-04 6.93E-04 7.20E-04 7.43E-04 8.04E-04

3.64 1.07E-03 1.03E-03 9.52E-04 7.06E-04 6.76E-04 7.91E-04 7.71E-04

3.674 1.16E-03 1.13E-03 1.08E-03 8.21E-04 7.24E-04 8.77E-04 8.53E-04

3.708 1.21E-03 1.27E-03 1.09E-03 8.45E-04 7.79E-04 7.90E-04 8.30E-04

3.742 1.31E-03 1.35E-03 1.16E-03 8.87E-04 8.21E-04 7.90E-04 8.34E-04

3.775 1.53E-03 1.44E-03 1.38E-03 9.56E-04 9.07E-04 8.79E-04 9.19E-04

3.808 1.75E-03 1.59E-03 1.55E-03 1.04E-03 8.99E-04 9.52E-04 1.07E-03

3.841 1.98E-03 1.88E-03 1.67E-03 1.19E-03 1.01E-03 1.01E-03 1.16E-03

3.873 2.38E-03 2.29E-03 1.96E-03 1.48E-03 1.20E-03 1.20E-03 1.34E-03

3.905 3.09E-03 2.92E-03 2.50E-03 1.94E-03 1.53E-03 1.59E-03 1.68E-03

3.937 4.37E-03 4.13E-03 3.53E-03 2.72E-03 1.93E-03 2.08E-03 2.26E-03

3.969 7.57E-03 7.33E-03 6.19E-03 4.57E-03 3.15E-03 3.12E-03 3.79E-03

3.975 8.64E-03 8.55E-03 6.89E-03 5.19E-03 3.51E-03 3.63E-03 4.37E-03

3.981 9.85E-03 9.74E-03 7.98E-03 6.09E-03 4.12E-03 4.01E-03 4.98E-03

3.987 1.09E-02 1.08E-02 8.96E-03 6.59E-03 4.48E-03 4.38E-03 5.44E-03

3.994 1.16E-02 1.13E-02 9.43E-03 7.09E-03 4.68E-03 4.57E-03 5.76E-03

4 1.19E-02 1.17E-02 9.64E-03 7.19E-03 4.67E-03 4.53E-03 5.85E-03

4.006 1.16E-02 1.13E-02 9.47E-03 7.00E-03 4.52E-03 4.49E-03 5.86E-03

4.012 1.08E-02 1.08E-02 8.68E-03 6.32E-03 4.21E-03 4.23E-03 5.48E-03

4.019 1.01E-02 9.75E-03 7.90E-03 5.83E-03 3.79E-03 3.84E-03 4.97E-03

4.025 8.85E-03 8.48E-03 6.85E-03 5.32E-03 3.45E-03 3.39E-03 4.53E-03

4.031 7.66E-03 7.64E-03 6.17E-03 4.78E-03 3.11E-03 3.00E-03 3.98E-03

4.062 4.40E-03 4.19E-03 3.51E-03 2.62E-03 1.81E-03 1.84E-03 2.37E-03

4.093 3.06E-03 2.90E-03 2.41E-03 1.86E-03 1.32E-03 1.38E-03 1.93E-03

4.123 2.46E-03 2.29E-03 1.89E-03 1.52E-03 1.06E-03 1.18E-03 1.79E-03

4.153 2.00E-03 1.82E-03 1.54E-03 1.29E-03 9.06E-04 1.09E-03 1.58E-03

4.183 1.72E-03 1.56E-03 1.33E-03 1.09E-03 8.40E-04 1.00E-03 1.29E-03

4.213 1.57E-03 1.38E-03 1.20E-03 1.01E-03 7.83E-04 9.46E-04 1.21E-03

4.243 1.44E-03 1.25E-03 1.14E-03 9.80E-04 7.52E-04 9.30E-04 1.20E-03

4.272 1.26E-03 1.13E-03 1.01E-03 8.51E-04 6.66E-04 9.18E-04 1.13E-03

4.301 1.15E-03 1.05E-03 9.26E-04 7.59E-04 6.26E-04 8.67E-04 1.10E-03

4.33 1.03E-03 9.93E-04 8.71E-04 7.16E-04 6.58E-04 8.69E-04 1.17E-03

4.359 1.04E-03 9.60E-04 8.25E-04 6.96E-04 6.78E-04 9.53E-04 1.18E-03

4.387 1.00E-03 8.75E-04 7.63E-04 6.39E-04 6.12E-04 1.02E-03 1.10E-03

4.416 9.52E-04 8.22E-04 7.41E-04 6.07E-04 5.77E-04 9.76E-04 1.11E-03

4.444 9.25E-04 7.96E-04 7.30E-04 6.10E-04 6.02E-04 9.84E-04 1.14E-03

4.472 8.93E-04 7.98E-04 7.28E-04 5.98E-04 6.59E-04 1.05E-03 1.20E-03

4.5 8.45E-04 7.68E-04 6.84E-04 5.69E-04 6.24E-04 1.01E-03 1.19E-03

4.528 8.30E-04 7.43E-04 6.80E-04 5.41E-04 6.11E-04 9.91E-04 1.17E-03

4.555 8.44E-04 7.52E-04 6.89E-04 5.39E-04 6.17E-04 1.06E-03 1.20E-03

4.583 8.27E-04 7.63E-04 6.87E-04 5.72E-04 6.36E-04 1.07E-03 1.23E-03

4.61 7.85E-04 7.24E-04 6.45E-04 5.48E-04 6.72E-04 1.04E-03 1.39E-03

4.637 7.83E-04 6.99E-04 6.24E-04 5.12E-04 6.79E-04 1.01E-03 1.49E-03

4.664 8.04E-04 6.84E-04 6.19E-04 5.06E-04 7.46E-04 1.02E-03 1.58E-03

4.69 8.38E-04 7.04E-04 6.34E-04 5.34E-04 8.51E-04 1.10E-03 1.63E-03

4.717 8.60E-04 7.48E-04 6.16E-04 5.16E-04 8.65E-04 1.19E-03 1.70E-03

4.743 8.77E-04 7.39E-04 6.16E-04 5.02E-04 8.48E-04 1.24E-03 1.69E-03

4.77 9.14E-04 7.49E-04 6.25E-04 4.97E-04 9.08E-04 1.33E-03 1.83E-03

4.796 9.24E-04 7.97E-04 6.73E-04 4.91E-04 9.42E-04 1.42E-03 2.01E-03

4.822 9.57E-04 8.51E-04 7.05E-04 4.82E-04 1.03E-03 1.57E-03 2.20E-03

4.848 1.02E-03 9.05E-04 7.07E-04 5.27E-04 1.10E-03 1.76E-03 2.36E-03

4.873 1.13E-03 9.77E-04 7.50E-04 5.47E-04 1.21E-03 2.03E-03 2.57E-03

4.899 1.30E-03 1.09E-03 8.73E-04 5.91E-04 1.38E-03 2.33E-03 3.05E-03

4.924 1.60E-03 1.31E-03 1.05E-03 6.55E-04 1.70E-03 2.99E-03 3.73E-03

4.95 2.08E-03 1.72E-03 1.32E-03 7.67E-04 2.30E-03 4.23E-03 4.85E-03

4.975 3.16E-03 2.57E-03 2.07E-03 1.01E-03 3.52E-03 6.21E-03 7.84E-03

330

Appendix D

abv,n [m/s2]

fp,v [Hz]

rv,n, rl,n 2.0 2.1 2.2 2.3 2.4 [0.65-1.2]

3.082 2.33E-03 1.65E-03 2.10E-03 5.39E-03 9.55E-03 1.33E-03

3.122 1.84E-03 1.55E-03 1.73E-03 4.02E-03 6.97E-03 8.92E-04

3.162 1.44E-03 1.35E-03 1.41E-03 3.36E-03 5.18E-03 6.57E-04

3.202 1.34E-03 1.16E-03 1.41E-03 2.84E-03 4.36E-03 7.03E-04

3.24 1.45E-03 1.17E-03 1.42E-03 2.71E-03 4.03E-03 9.10E-04

3.279 1.23E-03 1.12E-03 1.31E-03 2.43E-03 3.59E-03 7.40E-04

3.317 1.17E-03 1.09E-03 1.32E-03 2.20E-03 3.25E-03 4.73E-04

3.354 1.26E-03 1.26E-03 1.45E-03 2.28E-03 3.16E-03 4.50E-04

3.391 9.46E-04 1.20E-03 1.32E-03 2.19E-03 2.70E-03 3.76E-04

3.428 9.81E-04 1.19E-03 1.33E-03 2.07E-03 2.45E-03 4.24E-04

3.464 1.07E-03 1.20E-03 1.62E-03 2.02E-03 2.36E-03 5.74E-04

3.5 9.43E-04 1.13E-03 1.61E-03 1.96E-03 2.29E-03 3.84E-04

3.536 9.41E-04 1.16E-03 1.54E-03 2.02E-03 2.18E-03 3.74E-04

3.571 1.00E-03 1.34E-03 1.58E-03 2.16E-03 2.09E-03 3.32E-04

3.606 1.04E-03 1.38E-03 1.62E-03 2.15E-03 2.18E-03 3.52E-04

3.64 1.09E-03 1.44E-03 1.77E-03 2.16E-03 2.21E-03 3.42E-04

3.674 1.24E-03 1.52E-03 1.91E-03 2.26E-03 2.32E-03 5.29E-04

3.708 1.24E-03 1.58E-03 2.15E-03 2.29E-03 2.47E-03 4.70E-04

3.742 1.18E-03 1.81E-03 2.30E-03 2.54E-03 2.30E-03 5.96E-04

3.775 1.26E-03 2.08E-03 2.52E-03 2.88E-03 2.33E-03 6.05E-04

3.808 1.46E-03 2.43E-03 2.90E-03 3.43E-03 2.57E-03 3.51E-04

3.841 1.68E-03 2.55E-03 3.29E-03 3.62E-03 2.73E-03 3.75E-04

3.873 1.93E-03 3.08E-03 3.91E-03 4.13E-03 3.10E-03 4.13E-04

3.905 2.51E-03 4.09E-03 5.28E-03 5.18E-03 4.01E-03 4.76E-04

3.937 3.64E-03 5.94E-03 7.71E-03 7.18E-03 5.59E-03 3.53E-04

3.969 6.25E-03 1.03E-02 1.26E-02 1.19E-02 9.16E-03 3.39E-04

3.975 7.30E-03 1.19E-02 1.47E-02 1.40E-02 1.06E-02 3.20E-04

3.981 8.41E-03 1.36E-02 1.69E-02 1.63E-02 1.23E-02 2.96E-04

3.987 9.31E-03 1.44E-02 1.89E-02 1.73E-02 1.32E-02 2.85E-04

3.994 9.97E-03 1.57E-02 1.97E-02 1.89E-02 1.45E-02 2.91E-04

4 1.02E-02 1.62E-02 2.05E-02 1.95E-02 1.47E-02 3.00E-04

4.006 9.86E-03 1.58E-02 1.99E-02 1.94E-02 1.45E-02 3.15E-04

4.012 9.54E-03 1.47E-02 1.92E-02 1.82E-02 1.40E-02 3.28E-04

4.019 8.71E-03 1.39E-02 1.73E-02 1.60E-02 1.24E-02 3.41E-04

4.025 7.79E-03 1.22E-02 1.50E-02 1.48E-02 1.14E-02 3.49E-04

4.031 6.82E-03 1.04E-02 1.26E-02 1.28E-02 9.58E-03 3.49E-04

4.062 3.95E-03 6.16E-03 7.21E-03 7.39E-03 5.45E-03 3.19E-04

4.093 2.78E-03 4.20E-03 5.06E-03 5.24E-03 4.04E-03 4.08E-04

4.123 2.25E-03 3.28E-03 3.97E-03 4.03E-03 3.31E-03 5.08E-04

4.153 1.93E-03 2.71E-03 3.31E-03 3.40E-03 2.91E-03 3.30E-04

4.183 1.70E-03 2.42E-03 2.82E-03 3.07E-03 2.69E-03 3.64E-04

4.213 1.53E-03 2.17E-03 2.51E-03 2.87E-03 2.55E-03 4.43E-04

4.243 1.45E-03 1.96E-03 2.25E-03 2.59E-03 2.48E-03 5.53E-04

4.272 1.41E-03 1.79E-03 2.08E-03 2.36E-03 2.50E-03 5.67E-04

4.301 1.30E-03 1.65E-03 1.88E-03 2.23E-03 2.37E-03 3.85E-04

4.33 1.26E-03 1.58E-03 1.81E-03 2.14E-03 2.32E-03 4.34E-04

4.359 1.26E-03 1.58E-03 1.75E-03 2.05E-03 2.20E-03 4.83E-04

4.387 1.24E-03 1.49E-03 1.65E-03 2.03E-03 2.13E-03 3.53E-04

4.416 1.18E-03 1.28E-03 1.53E-03 1.92E-03 2.17E-03 3.09E-04

4.444 1.17E-03 1.25E-03 1.46E-03 1.82E-03 2.13E-03 3.50E-04

4.472 1.20E-03 1.24E-03 1.39E-03 1.84E-03 2.10E-03 3.56E-04

4.5 1.26E-03 1.28E-03 1.36E-03 1.84E-03 2.12E-03 3.43E-04

4.528 1.22E-03 1.20E-03 1.32E-03 1.79E-03 2.02E-03 4.09E-04

4.555 1.22E-03 1.15E-03 1.29E-03 1.76E-03 2.02E-03 4.83E-04

4.583 1.20E-03 1.18E-03 1.23E-03 1.71E-03 2.11E-03 4.60E-04

4.61 1.26E-03 1.16E-03 1.24E-03 1.71E-03 2.23E-03 4.26E-04

4.637 1.30E-03 1.18E-03 1.18E-03 1.69E-03 2.34E-03 3.97E-04

4.664 1.34E-03 1.20E-03 1.10E-03 1.71E-03 2.56E-03 3.82E-04

4.69 1.41E-03 1.27E-03 1.11E-03 1.79E-03 2.56E-03 4.29E-04

4.717 1.51E-03 1.32E-03 1.14E-03 1.87E-03 2.64E-03 5.25E-04

4.743 1.52E-03 1.34E-03 1.17E-03 1.94E-03 2.77E-03 6.21E-04

4.77 1.60E-03 1.39E-03 1.15E-03 1.99E-03 2.98E-03 6.60E-04

4.796 1.73E-03 1.50E-03 1.15E-03 2.06E-03 3.24E-03 7.02E-04

4.822 1.92E-03 1.63E-03 1.20E-03 2.22E-03 3.51E-03 6.24E-04

4.848 2.15E-03 1.75E-03 1.23E-03 2.51E-03 4.02E-03 6.49E-04

4.873 2.38E-03 1.87E-03 1.27E-03 2.75E-03 4.60E-03 7.76E-04

4.899 2.76E-03 2.10E-03 1.36E-03 3.27E-03 5.50E-03 8.75E-04

4.924 3.56E-03 2.59E-03 1.50E-03 3.91E-03 6.89E-03 1.11E-03

4.95 4.93E-03 3.48E-03 1.90E-03 5.02E-03 1.03E-02 1.56E-03

4.975 7.55E-03 5.13E-03 2.51E-03 7.75E-03 1.57E-02 2.08E-03

331

Appendix D

abv,n [m/s2]

fp,v [Hz]

rv,n, rl,n 1.3 1.4 1.5 1.6 1.7 1.8 1.9

4.98 3.29E-03 2.75E-03 2.20E-03 1.06E-03 3.69E-03 6.63E-03 8.38E-03

4.985 3.51E-03 3.07E-03 2.43E-03 1.11E-03 3.89E-03 7.08E-03 8.83E-03

4.99 3.60E-03 3.20E-03 2.51E-03 1.13E-03 4.09E-03 7.45E-03 9.25E-03

4.995 3.75E-03 3.30E-03 2.55E-03 1.12E-03 4.25E-03 7.72E-03 9.49E-03

5 3.77E-03 3.34E-03 2.60E-03 1.17E-03 4.28E-03 7.80E-03 9.49E-03

5.005 3.74E-03 3.33E-03 2.62E-03 1.17E-03 4.15E-03 7.75E-03 9.26E-03

5.01 3.62E-03 3.23E-03 2.57E-03 1.15E-03 4.05E-03 7.36E-03 8.93E-03

5.015 3.53E-03 3.14E-03 2.47E-03 1.14E-03 3.92E-03 6.94E-03 8.52E-03

5.02 3.27E-03 3.01E-03 2.21E-03 1.13E-03 3.70E-03 6.44E-03 7.97E-03

5.025 3.12E-03 2.70E-03 2.07E-03 1.07E-03 3.45E-03 6.04E-03 7.36E-03

5.05 1.94E-03 1.83E-03 1.43E-03 8.45E-04 2.39E-03 4.26E-03 5.05E-03

5.074 1.41E-03 1.33E-03 1.07E-03 7.32E-04 1.88E-03 3.21E-03 3.79E-03

5.099 1.13E-03 1.10E-03 8.62E-04 6.55E-04 1.56E-03 2.57E-03 3.16E-03

5.123 9.62E-04 9.38E-04 7.50E-04 6.24E-04 1.33E-03 2.16E-03 2.66E-03

5.148 8.39E-04 8.64E-04 6.81E-04 6.40E-04 1.21E-03 1.91E-03 2.34E-03

5.172 7.62E-04 7.56E-04 6.16E-04 6.33E-04 1.13E-03 1.76E-03 2.15E-03

5.196 6.79E-04 6.79E-04 5.86E-04 5.83E-04 1.05E-03 1.63E-03 1.96E-03

5.22 6.36E-04 6.32E-04 5.70E-04 5.53E-04 1.00E-03 1.55E-03 1.86E-03

5.244 6.14E-04 6.13E-04 5.77E-04 5.30E-04 9.61E-04 1.48E-03 1.76E-03

5.268 5.89E-04 6.04E-04 5.64E-04 5.12E-04 9.39E-04 1.44E-03 1.69E-03

5.292 5.37E-04 5.65E-04 5.26E-04 5.00E-04 8.89E-04 1.41E-03 1.62E-03

5.315 5.01E-04 5.39E-04 5.14E-04 5.04E-04 8.27E-04 1.34E-03 1.55E-03

5.339 4.86E-04 5.24E-04 5.12E-04 5.14E-04 8.14E-04 1.29E-03 1.52E-03

5.362 4.78E-04 5.25E-04 5.01E-04 5.31E-04 8.24E-04 1.27E-03 1.50E-03

5.385 4.66E-04 5.25E-04 5.14E-04 5.45E-04 8.34E-04 1.27E-03 1.50E-03

5.408 4.56E-04 5.05E-04 5.06E-04 5.40E-04 8.17E-04 1.27E-03 1.51E-03

5.431 4.51E-04 4.82E-04 5.04E-04 5.25E-04 8.13E-04 1.25E-03 1.45E-03

5.454 4.58E-04 4.70E-04 5.03E-04 5.15E-04 8.09E-04 1.24E-03 1.41E-03

5.477 4.53E-04 4.76E-04 5.06E-04 5.15E-04 8.21E-04 1.23E-03 1.40E-03

5.5 4.09E-04 4.89E-04 4.92E-04 5.04E-04 8.40E-04 1.23E-03 1.41E-03

5.523 3.96E-04 4.86E-04 4.79E-04 5.03E-04 8.67E-04 1.25E-03 1.46E-03

5.545 3.77E-04 4.91E-04 4.71E-04 5.03E-04 8.74E-04 1.28E-03 1.47E-03

5.568 3.74E-04 5.06E-04 4.71E-04 5.09E-04 8.83E-04 1.30E-03 1.49E-03

5.59 3.74E-04 5.08E-04 4.75E-04 5.25E-04 9.11E-04 1.30E-03 1.53E-03

5.612 3.76E-04 4.91E-04 5.03E-04 5.40E-04 9.49E-04 1.34E-03 1.58E-03

5.635 3.81E-04 4.79E-04 5.18E-04 5.56E-04 9.72E-04 1.41E-03 1.67E-03

5.657 3.83E-04 4.74E-04 5.46E-04 5.74E-04 9.84E-04 1.47E-03 1.70E-03

5.679 3.90E-04 4.87E-04 5.64E-04 6.02E-04 1.00E-03 1.50E-03 1.77E-03

5.701 4.11E-04 5.07E-04 6.06E-04 6.34E-04 1.05E-03 1.57E-03 1.83E-03

5.723 4.15E-04 5.11E-04 6.40E-04 6.69E-04 1.11E-03 1.64E-03 1.92E-03

5.745 4.07E-04 5.04E-04 6.48E-04 7.01E-04 1.14E-03 1.74E-03 1.99E-03

5.766 4.10E-04 5.12E-04 6.55E-04 7.24E-04 1.20E-03 1.85E-03 2.12E-03

5.788 4.07E-04 5.34E-04 6.83E-04 7.50E-04 1.26E-03 1.95E-03 2.26E-03

5.809 4.16E-04 5.67E-04 7.41E-04 7.96E-04 1.34E-03 2.10E-03 2.41E-03

5.831 4.38E-04 6.08E-04 8.14E-04 8.62E-04 1.49E-03 2.36E-03 2.61E-03

5.852 4.53E-04 6.37E-04 8.88E-04 9.59E-04 1.63E-03 2.63E-03 2.87E-03

5.874 4.66E-04 6.85E-04 9.88E-04 1.07E-03 1.85E-03 2.93E-03 3.22E-03

5.895 4.93E-04 7.74E-04 1.13E-03 1.19E-03 2.10E-03 3.46E-03 3.93E-03

5.916 5.28E-04 8.81E-04 1.34E-03 1.41E-03 2.57E-03 4.08E-03 4.72E-03

5.937 6.06E-04 1.04E-03 1.68E-03 1.82E-03 3.12E-03 5.06E-03 6.00E-03

5.958 6.89E-04 1.29E-03 2.21E-03 2.32E-03 4.30E-03 6.53E-03 8.05E-03

5.979 8.31E-04 1.74E-03 2.90E-03 3.25E-03 5.50E-03 9.59E-03 1.09E-02

5.983 8.31E-04 1.85E-03 2.90E-03 3.25E-03 5.93E-03 9.59E-03 1.09E-02

5.987 8.45E-04 1.92E-03 3.09E-03 3.43E-03 5.93E-03 1.03E-02 1.17E-02

5.992 8.50E-04 1.92E-03 3.22E-03 3.65E-03 6.25E-03 1.03E-02 1.17E-02

5.996 8.42E-04 1.98E-03 3.25E-03 3.76E-03 6.41E-03 1.07E-02 1.23E-02

6 8.21E-04 2.03E-03 3.25E-03 3.76E-03 6.41E-03 1.08E-02 1.25E-02

6.004 7.97E-04 2.02E-03 3.26E-03 3.73E-03 6.48E-03 1.08E-02 1.25E-02

6.008 8.01E-04 1.97E-03 3.25E-03 3.59E-03 6.32E-03 1.06E-02 1.22E-02

6.012 8.01E-04 1.97E-03 3.15E-03 3.59E-03 6.32E-03 9.98E-03 1.22E-02

332

Appendix D

abv,n [m/s2]

fp,v [Hz]

rv,n, rl,n 2.0 2.1 2.2 2.3 2.4 [0.65-1.2]

4.98 7.55E-03 5.56E-03 2.63E-03 8.43E-03 1.74E-02 2.27E-03

4.985 8.09E-03 5.56E-03 2.81E-03 8.43E-03 1.74E-02 2.41E-03

4.99 8.53E-03 5.95E-03 3.04E-03 9.02E-03 1.89E-02 2.50E-03

4.995 8.91E-03 6.24E-03 3.20E-03 9.39E-03 2.00E-02 2.54E-03

5 9.18E-03 6.36E-03 3.28E-03 9.47E-03 2.03E-02 2.58E-03

5.005 9.06E-03 6.30E-03 3.30E-03 9.25E-03 1.98E-02 2.66E-03

5.01 8.89E-03 6.12E-03 3.23E-03 8.78E-03 1.98E-02 2.69E-03

5.015 8.58E-03 5.81E-03 3.23E-03 8.78E-03 1.89E-02 2.61E-03

5.02 8.10E-03 5.40E-03 3.18E-03 8.11E-03 1.75E-02 2.58E-03

5.025 7.53E-03 5.00E-03 3.03E-03 7.37E-03 1.57E-02 2.50E-03

5.05 4.93E-03 3.61E-03 2.23E-03 4.93E-03 1.01E-02 1.94E-03

5.074 3.86E-03 2.59E-03 1.69E-03 3.60E-03 7.38E-03 1.45E-03

5.099 3.02E-03 2.10E-03 1.46E-03 2.87E-03 6.09E-03 1.06E-03

5.123 2.54E-03 1.79E-03 1.33E-03 2.45E-03 4.88E-03 8.62E-04

5.148 2.29E-03 1.54E-03 1.26E-03 2.18E-03 4.08E-03 7.66E-04

5.172 2.13E-03 1.42E-03 1.25E-03 2.01E-03 3.66E-03 6.29E-04

5.196 1.91E-03 1.32E-03 1.25E-03 1.90E-03 3.29E-03 5.72E-04

5.22 1.74E-03 1.21E-03 1.20E-03 1.80E-03 3.00E-03 5.82E-04

5.244 1.63E-03 1.11E-03 1.14E-03 1.67E-03 2.82E-03 6.37E-04

5.268 1.56E-03 1.03E-03 1.09E-03 1.64E-03 2.65E-03 6.30E-04

5.292 1.51E-03 1.02E-03 1.06E-03 1.58E-03 2.56E-03 5.45E-04

5.315 1.44E-03 9.97E-04 1.06E-03 1.58E-03 2.50E-03 4.77E-04

5.339 1.40E-03 9.63E-04 1.05E-03 1.54E-03 2.42E-03 4.16E-04

5.362 1.37E-03 9.24E-04 1.03E-03 1.51E-03 2.23E-03 3.93E-04

5.385 1.34E-03 9.13E-04 1.02E-03 1.46E-03 2.07E-03 4.03E-04

5.408 1.35E-03 9.19E-04 1.03E-03 1.45E-03 1.99E-03 3.71E-04

5.431 1.33E-03 9.19E-04 1.04E-03 1.48E-03 1.98E-03 3.64E-04

5.454 1.32E-03 8.57E-04 1.02E-03 1.48E-03 1.97E-03 4.13E-04

5.477 1.30E-03 8.31E-04 1.02E-03 1.45E-03 1.89E-03 4.66E-04

5.5 1.29E-03 8.18E-04 1.02E-03 1.44E-03 1.87E-03 4.37E-04

5.523 1.27E-03 8.18E-04 1.06E-03 1.48E-03 1.86E-03 3.64E-04

5.545 1.26E-03 7.86E-04 1.07E-03 1.54E-03 1.85E-03 3.32E-04

5.568 1.27E-03 7.30E-04 1.05E-03 1.61E-03 1.83E-03 3.32E-04

5.59 1.29E-03 7.01E-04 1.08E-03 1.63E-03 1.79E-03 3.21E-04

5.612 1.31E-03 6.95E-04 1.11E-03 1.66E-03 1.79E-03 3.04E-04

5.635 1.36E-03 7.07E-04 1.17E-03 1.69E-03 1.84E-03 3.17E-04

5.657 1.41E-03 7.24E-04 1.25E-03 1.75E-03 1.89E-03 3.54E-04

5.679 1.45E-03 7.38E-04 1.29E-03 1.81E-03 2.07E-03 3.92E-04

5.701 1.52E-03 7.35E-04 1.30E-03 1.84E-03 2.07E-03 4.14E-04

5.723 1.59E-03 7.48E-04 1.31E-03 1.95E-03 2.02E-03 4.09E-04

5.745 1.69E-03 7.86E-04 1.34E-03 2.03E-03 2.02E-03 4.91E-04

5.766 1.82E-03 8.26E-04 1.36E-03 2.20E-03 2.02E-03 5.28E-04

5.788 1.90E-03 8.07E-04 1.49E-03 2.39E-03 2.08E-03 4.83E-04

5.809 2.00E-03 7.77E-04 1.61E-03 2.56E-03 2.19E-03 3.91E-04

5.831 2.20E-03 7.65E-04 1.78E-03 2.71E-03 2.31E-03 3.45E-04

5.852 2.44E-03 7.69E-04 1.91E-03 3.00E-03 2.52E-03 3.14E-04

5.874 2.75E-03 8.02E-04 2.17E-03 3.28E-03 2.70E-03 3.13E-04

5.895 3.01E-03 8.51E-04 2.55E-03 3.88E-03 3.22E-03 3.58E-04

5.916 3.54E-03 9.44E-04 2.85E-03 4.75E-03 3.70E-03 4.19E-04

5.937 4.41E-03 1.09E-03 3.58E-03 5.65E-03 4.43E-03 4.03E-04

5.958 5.81E-03 1.37E-03 4.94E-03 7.80E-03 6.26E-03 3.22E-04

5.979 7.91E-03 1.63E-03 6.17E-03 9.90E-03 8.18E-03 2.89E-04

5.983 8.62E-03 1.83E-03 6.70E-03 9.90E-03 9.03E-03 2.84E-04

5.987 8.62E-03 1.83E-03 6.70E-03 1.09E-02 9.03E-03 2.83E-04

5.992 9.31E-03 1.99E-03 7.15E-03 1.09E-02 9.56E-03 2.84E-04

5.996 9.31E-03 1.99E-03 7.15E-03 1.16E-02 9.56E-03 2.86E-04

6 9.54E-03 2.06E-03 7.26E-03 1.16E-02 9.56E-03 2.87E-04

6.004 9.54E-03 2.02E-03 7.26E-03 1.18E-02 9.84E-03 2.90E-04

6.008 9.26E-03 2.02E-03 7.04E-03 1.18E-02 9.84E-03 2.93E-04

6.012 9.26E-03 2.01E-03 7.04E-03 1.12E-02 9.37E-03 2.98E-04

333

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