Download - 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
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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.
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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.
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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.
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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
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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
3.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
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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
4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
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
5.11 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6 Evaluation of the response of a conventional cable-stayed footbridge
under serviceability conditions 157
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
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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
6.13 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
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
7.4.1 Articulation of the deck . . . . . . . . . . . . . . . . . . . . . . . . 209
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
7.4.5 Section of the steel girders . . . . . . . . . . . . . . . . . . . . . . . 212
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Contents
7.4.6 Concrete slab section . . . . . . . . . . . . . . . . . . . . . . . . . . 213
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
7.6 Comfort appraisal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
7.7 Additional dissipation of the serviceability movements: inherent or external
movement control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
7.8 Serviceability limit state of deflections . . . . . . . . . . . . . . . . . . . . 231
7.9 Deck normal stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
7.10 Deck shear stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
7.11 Normal stresses at the pylon . . . . . . . . . . . . . . . . . . . . . . . . . . 235
7.12 Performance of the stay cables . . . . . . . . . . . . . . . . . . . . . . . . . 236
7.13 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
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
8.5.1 Articulation of the deck . . . . . . . . . . . . . . . . . . . . . . . . 245
8.5.2 Area of backstay cable . . . . . . . . . . . . . . . . . . . . . . . . . 246
8.5.3 Area of main span stays . . . . . . . . . . . . . . . . . . . . . . . . 248
8.5.4 Section of the steel girders . . . . . . . . . . . . . . . . . . . . . . . 250
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
8.5.9 Shape of the pylon . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
8.5.10 Cable system: transverse inclination of cables . . . . . . . . . . . . 257
8.5.11 Cable system: anchorage spacing . . . . . . . . . . . . . . . . . . . 257
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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
8.6.1 Articulation of the deck . . . . . . . . . . . . . . . . . . . . . . . . 261
8.6.2 Area of backstay cables . . . . . . . . . . . . . . . . . . . . . . . . . 262
8.6.3 Area of main span stays . . . . . . . . . . . . . . . . . . . . . . . . 264
8.6.4 Section of the steel girders . . . . . . . . . . . . . . . . . . . . . . . 264
8.6.5 Concrete slab section . . . . . . . . . . . . . . . . . . . . . . . . . . 265
8.6.6 Pylon section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
8.6.7 Pylon height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
8.6.8 Longitudinal inclination of the pylon . . . . . . . . . . . . . . . . . 268
8.6.9 Shape of the pylon . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
8.6.10 Cable system: transverse inclination of cables . . . . . . . . . . . . 270
8.6.11 Cable system: anchorage spacing . . . . . . . . . . . . . . . . . . . 271
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
8.8 Comfort appraisal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
8.9 Additional dissipation of the serviceability movements: inherent or external
movement control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
8.10 Serviceability limit state of deflections . . . . . . . . . . . . . . . . . . . . 280
8.11 Deck normal stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
8.12 Deck shear stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
8.13 Normal stresses at the pylon . . . . . . . . . . . . . . . . . . . . . . . . . . 284
8.14 Performance of stay cables . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
8.15 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
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
6.17 Dynamic and equivalent static vertical deflections caused by pedestrian
flows with 0.6 ped/m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.18 Dynamic and equivalent static vertical deflections caused by pedestrian
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
6.20 Dynamic and equivalent static lateral deflections caused by pedestrian flows
with 0.6 ped/m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.21 Dynamic and equivalent static lateral deflections caused by pedestrian flows
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
6.24 Dynamic and equivalent static bending moments caused by pedestrian flows
with 0.6 ped/m2 along the length of the deck. . . . . . . . . . . . . . . . . 177
6.25 Dynamic and equivalent static bending moments caused by pedestrian flows
with 1.0 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
6.33 Maximum stress variations of the backstay and main span cables generated
by medium-density pedestrian flows. . . . . . . . . . . . . . . . . . . . . . 181
6.34 Maximum stress variations of the backstay and main span cables generated
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
7.10 Vertical accelerations felt by users amax,P i compared to amax,P . Curves
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:
(a) peak and 1s-RMS vertical accelerations and (b) comparison of maxi-
mum absolute peak and 1s-RMS accelerations to those of the benchmark
case acc 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7.13 Vertical accelerations felt by users amax,P i compared to amax,P . Curves
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
7.16 Vertical accelerations felt by users amax,P i compared to amax,P . Curves
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:
(a) peak and 1s-RMS vertical accelerations and (b) comparison of maxi-
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
7.20 Vertical accelerations felt by users amax,P i compared to amax,P . Curves
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)
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
20
List of Figures
7.23 Vertical accelerations felt by users amax,P i compared to amax,P . Curves
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
vertical and torsional modes. . . . . . . . . . . . . . . . . . . . . . . . . . . 201
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 maxi-
mum absolute peak and 1s-RMS accelerations to those of the reference case
acc0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
7.26 Vertical accelerations felt by users amax,P i compared to amax,P . Curves
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)
peak and 1s-RMS vertical accelerations and (b) comparison of maximum
absolute peak and 1s-RMS accelerations to those of the benchmark case acc0.203
7.29 Vertical accelerations felt by users amax,P i compared to amax,P . Curves
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
7.32 Vertical accelerations felt by users amax,P i compared to amax,P . Curves
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
maximum absolute peak and 1s-RMS accelerations to those of the bench-
mark case acc0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7.35 Vertical accelerations felt by users amax,P i compared to amax,P . Curves
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
maximum absolute peak and 1s-RMS accelerations to those of the bench-
mark case acc 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.38 Vertical accelerations felt by users amax,P i compared to amax,P . Curves
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
vertical and torsional modes. . . . . . . . . . . . . . . . . . . . . . . . . . . 208
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
7.41 Vertical accelerations felt by users amax,P i compared to amax,P . Curves
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
7.44 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the CSF deck according to backstay area and (b) lateral accel-
erations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
7.45 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the CSF deck according to main span stay area and (b) lateral
accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
7.46 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the CSF deck according to bottom flange thickness and (b)
lateral accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . 213
7.47 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the CSF deck according to concrete slab depth and (b) lat-
eral accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . 214
7.48 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the CSF deck according to pylon diameter and (b) lateral ac-
celerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
7.49 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the CSF deck according to pylon height and (b) lateral ac-
celerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
7.50 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the CSF deck according to pylon longitudinal inclination and
(b) lateral accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . 216
7.51 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the CSF deck according to pylon shape and (b) lateral ac-
celerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
7.52 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the CSF deck according to pylon transverse inclination and
(b) lateral accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . 218
22
List of Figures
7.53 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the CSF deck according to deck width and (b) lateral acceler-
ations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
7.54 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the CSF deck according to side span length and (b) lateral
accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
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
7.60 Peak vertical (a) and lateral (b) accelerations recorded at benchmark CSF
or CSF with larger stays (2.5AS,0), with depths Lm/100 and Lm/200. . . . 226
7.61 Peak vertical (a) and lateral (b) accelerations recorded at benchmark CSF
or CSF with slab depth 2tc,0, with depths Lm/100 and Lm/200. . . . . . . 226
7.62 Peak vertical (a) and lateral (b) accelerations recorded at benchmark CSF
or CSF with shorter pylon (0.25Lm), with depths Lm/100 and Lm/200. . . 227
7.63 Peak vertical (a) and lateral (b) accelerations recorded at benchmark CSF
or CSF with inclined 20o towards the side span, with depths Lm/100 and
Lm/200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
7.64 Peak vertical (a) and lateral (b) accelerations recorded at benchmark CSF
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
7.66 Comfort assessment of CSF according to the measures implemented to
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
maximum static deflections umax and (b) frequencies [Hz] of vertical and
torsional modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
8.7 (a) Absolute peak vertical, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to backstay area and (b) vertical
accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
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
static deflections umax and (b) frequencies [Hz] of vertical and torsional
modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.9 (a) Absolute peak vertical, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to stays area and (b) vertical
accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
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
8.11 (a) Absolute peak vertical, and relative peak and 1s-RMS accelerations
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
8.13 (a) Absolute peak vertical, and relative peak and 1s-RMS accelerations
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
maximum static deflections umax and (b) frequencies [Hz] of vertical and
torsional modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
8.15 (a) Absolute peak vertical, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to pylon diameter and (b) vertical
accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
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
static deflections umax and (b) frequencies [Hz] of vertical and torsional
modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
8.17 (a) Absolute peak vertical, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to pylon height and (b) vertical
accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
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
8.19 (a) Absolute peak vertical, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to pylon longitudinal inclination
and (b) vertical accelerations felt by users. . . . . . . . . . . . . . . . . . . 255
25
List of Figures
8.20 (a) Absolute peak vertical, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to pylon shape and (b) vertical
accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
8.21 (a) Absolute peak vertical, and relative peak and 1s-RMS accelerations
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
8.23 (a) Absolute peak vertical, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to distance of cable anchorages
and (b) vertical accelerations felt by users. . . . . . . . . . . . . . . . . . . 258
8.24 Dynamic behaviour of the 2T-CSF according to deck width: frequencies
[Hz] of vertical and torsional modes. . . . . . . . . . . . . . . . . . . . . . . 259
8.25 (a) Absolute peak vertical, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to deck width and (b) vertical
accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
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
8.27 (a) Absolute peak vertical, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to side span length and (b) vertical
accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
8.28 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck with alternative support schemes and (b)
lateral accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . 262
8.29 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to backstay area and (b) lateral
accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
8.30 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to main stays area and (b) lateral
accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
8.31 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to bottom flange steel girder thick-
ness and (b) lateral accelerations felt by users. . . . . . . . . . . . . . . . . 265
8.32 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to concrete slab thickness and
(b) lateral accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . 266
8.33 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to pylon diameter and (b) lat-
eral accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . 267
26
List of Figures
8.34 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to pylon height and (b) lateral
accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
8.35 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to pylon longitudinal inclination
and (b) lateral accelerations felt by users. . . . . . . . . . . . . . . . . . . . 269
8.36 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to pylon shape and (b) lateral
accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
8.37 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to lateral pylon inclination and
(b) lateral accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . 270
8.38 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to cable anchorage spacing and
(b) lateral accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . 271
8.39 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to deck width and (b) lateral
accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
8.40 (a) Absolute peak lateral, and relative peak and 1s-RMS accelerations
recorded at the 2T-CSFs deck according to side span length and (b) lateral
accelerations felt by users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
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
8.45 Comfort assessment of CSF according to the measures implemented to
modify vertical accelerations. . . . . . . . . . . . . . . . . . . . . . . . . . . 278
8.46 Comfort assessment of CSF according to the measures implemented to
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)
60
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).
61
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
62
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).
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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
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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-
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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.
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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
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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
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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).
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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
3. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
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-
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3. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
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
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3. Methodology: modelling and basic assumptions
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
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3. Methodology: modelling and basic assumptions
(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
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3. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
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.
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3. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
• 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.
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3. Methodology: modelling and basic assumptions
• 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
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3. Methodology: modelling and basic assumptions
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
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3. Methodology: modelling and basic assumptions
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. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
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. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
(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
3. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
(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
3. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
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. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
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
3. Methodology: modelling and basic assumptions
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
3.8 Concluding remarks
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
4. Relevance of stochastic representation of reality
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
4. Relevance of stochastic representation of reality
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
4. Relevance of stochastic representation of reality
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
4. Relevance of stochastic representation of reality
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
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4. Relevance of stochastic representation of reality
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.
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4. Relevance of stochastic representation of reality
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
4. Relevance of stochastic representation of reality
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
4. Relevance of stochastic representation of reality
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
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4. Relevance of stochastic representation of reality
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
4. Relevance of stochastic representation of reality
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).
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4. Relevance of stochastic representation of reality
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
4. Relevance of stochastic representation of reality
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
4. Relevance of stochastic representation of reality
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.
4.5 Concluding remarks
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.
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4. Relevance of stochastic representation of reality
• 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
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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:
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5. Girder footbridge design
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.
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5. Girder footbridge design
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
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5. Girder footbridge design
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
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5. Girder footbridge design
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
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5. Girder footbridge design
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-
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5. Girder footbridge design
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
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5. Girder footbridge design
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.
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5. Girder footbridge design
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,
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5. Girder footbridge design
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)
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5. Girder footbridge design
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
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5. Girder footbridge design
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. Girder footbridge design
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
5. Girder footbridge design
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
5. Girder footbridge design
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).
5.11 Concluding remarks
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
5. Girder footbridge design
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
6. Conventional cable-stayed footbridge
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
6. Conventional cable-stayed footbridge
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
6. Conventional cable-stayed footbridge
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
6. Conventional cable-stayed footbridge
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
Step frequencies [Hz]
0.6 ped/m2
1.5 2 2.50
100
200
300
Step frequencies [Hz]
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
Step frequencies [Hz]
1.0 ped/m2
1 1.5 2 2.50
50
100
150
Step frequencies [Hz]
0.6 ped/m2
1.5 2 2.50
50
Step frequencies [Hz]
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
6. Conventional cable-stayed footbridge
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
0.6 ped/m2 − Commuter
1 2 3 4 5 6 7 8 9 10
100
Frequency [Hz]
Am
plit
ude
0.6 ped/m2 − Leisure
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
6. Conventional cable-stayed footbridge
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
0.2 ped/m2 − Commuter
0 10 20 30 40 50 600
0.10.20.30.40.5
0.2 ped/m2 − Leisure
0 10 20 30 40 50 600
0.10.20.30.40.5
Late
ral accele
ration [
m/s
2]
0.6 ped/m2 − Commuter
0 10 20 30 40 50 600
0.10.20.30.40.5
0.6 ped/m2 − Leisure
0 10 20 30 40 50 600
0.10.20.30.40.5
Structure length [m]
1.0 ped/m2 − Commuter
0 10 20 30 40 50 600
0.10.20.30.40.5
1.0 ped/m2 − Leisure
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
6. Conventional cable-stayed footbridge
1 2 3 4 5 6 7 8 9 10
100
Am
plit
ude
0.6 ped/m2 − Commuter
1 2 3 4 5 6 7 8 9 10
100
Frequency [Hz]
Am
plit
ude
0.6 ped/m2 − Leisure
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
6. Conventional cable-stayed footbridge
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
6. Conventional cable-stayed footbridge
(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.
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6. Conventional cable-stayed footbridge
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
6. Conventional cable-stayed footbridge
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
6. Conventional cable-stayed footbridge
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
6. Conventional cable-stayed footbridge
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
6. Conventional cable-stayed footbridge
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
6. Conventional cable-stayed footbridge
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.
Structure length [m]
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
Structure length [m]
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
Figure 6.17: Dynamic and equivalent static vertical deflections caused by pedestrian flowswith 0.6 ped/m2.
0
Deflection [
mm
]
10
20
40
50
30
Structure length [m]
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
Figure 6.18: Dynamic and equivalent static vertical deflections caused by pedestrian flowswith 1.0 ped/m2.
The contrast of the vertical dynamic deflections with the peak accelerations computed
173
6. Conventional cable-stayed footbridge
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.
Structure length [m]
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.
Structure length [m]
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
Figure 6.20: Dynamic and equivalent static lateral deflections caused by pedestrian flowswith 0.6 ped/m2.
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
6. Conventional cable-stayed footbridge
Structure length [m]
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
Figure 6.21: Dynamic and equivalent static lateral deflections caused by pedestrian flowswith 1.0 ped/m2.
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
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6. Conventional cable-stayed footbridge
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
6. Conventional cable-stayed footbridge
Structure length [m]
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
Structure length [m]
Commuter flow
Leisure flow
Static case
Figure 6.24: Dynamic and equivalent static bending moments caused by pedestrian flowswith 0.6 ped/m2 along the length of the deck.
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
Structure length [m]
Commuter flow
Leisure flow
Static case
Figure 6.25: Dynamic and equivalent static bending moments caused by pedestrian flowswith 1.0 ped/m2 along the length of the deck.
177
6. Conventional cable-stayed footbridge
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
Structure length [m]
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
6. Conventional cable-stayed footbridge
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
Structure length [m]
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
Structure length [m]
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. Conventional cable-stayed footbridge
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
6. Conventional cable-stayed footbridge
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
0Backstay CB1 CB2 CB4 CB5CB3
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
0Backstay CB1 CB2 CB4 CB5CB3
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
6. Conventional cable-stayed footbridge
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
6. Conventional cable-stayed footbridge
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
6.13 Concluding remarks
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
6. Conventional cable-stayed footbridge
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
7. Design of cable-stayed footbridges with a single pylon
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
7. Design of cable-stayed footbridges with a single pylon
(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
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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
7. Design of cable-stayed footbridges with a single 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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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
Cable distance Dc [m]
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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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
7. Design of cable-stayed footbridges with a single 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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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.
7.4.1 Articulation of the deck
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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
‘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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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.
7.4.5 Section of the steel girders
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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.
7.4.6 Concrete slab section
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
Peak v
ert
ical
acc.
[m/s
2]
0 5 10 150
1
2
3
Lateral inclination α [º]
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. Design of cable-stayed footbridges with a single pylon
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.
7.4.14 Side span length
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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
7. Design of cable-stayed footbridges with a single pylon
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
Cable No. Strands Cable No. StrandsBS
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
7. Design of cable-stayed footbridges with a single pylon
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
7. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
Peak V
ert
.
acc.
[m/s
2]
0 20 40 60 80 100 1200
0.5
1
1.5
Structure length [m]
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. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
Peak v
ert
.
acc.
[m/s
2]
0 20 40 60 80 100 1200
0.5
1
1.5
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
0 20 40 60 80 100 1200
0.5
1
1.5
2
Structure length [m]
Peak v
ert
.
acc.
[m/s
2]
0 20 40 60 80 100 1200
0.5
1
1.5
Structure length [m]
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
Structure length [m]
Peak v
ert
.
acc.
[m/s
2]
0 20 40 60 80 100 1200
0.5
1
1.5
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
0 20 40 60 80 100 1200
0.5
1
1.5
2
Structure length [m]
Peak v
ert
.
acc.
[m/s
2]
0 20 40 60 80 100 1200
0.5
1
1.5
Structure length [m]
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
Structure length [m]
Peak v
ert
.
acc.
[m/s
2]
0 20 40 60 80 100 1200
0.5
1
1.5
Structure length [m]
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.
7.6 Comfort appraisal
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
7. Design of cable-stayed footbridges with a single pylon
0 20 40 60 80 100 1200
0.5
1
1.5
2
Structure length [m]
Peak v
ert
.
acc.
[m/s
2]
0 20 40 60 80 100 1200
0.5
1
1.5
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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. Design of cable-stayed footbridges with a single pylon
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
Structure length [m]
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
7. Design of cable-stayed footbridges with a single pylon
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.
7.8 Serviceability limit state of deflections
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
7. Design of cable-stayed footbridges with a single pylon
Deflection [
mm
]
Structure length [m]
(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.
7.9 Deck normal stresses
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
7. Design of cable-stayed footbridges with a single pylon
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
7. Design of cable-stayed footbridges with a single pylon
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. Design of cable-stayed footbridges with a single pylon
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
Bending Moment [kNm]
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. Design of cable-stayed footbridges with a single pylon
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
7. Design of cable-stayed footbridges with a single pylon
7.13 Concluding remarks
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
7. Design of cable-stayed footbridges with a single pylon
(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.
2. Structural elements:
(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.
3. Structure geometry:
(a) The lateral accelerations increase moderately when heights of the pylon are very
low and remain fairly similar otherwise.
238
7. Design of cable-stayed footbridges with a single pylon
(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
7. Design of cable-stayed footbridges with a single pylon
• 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
Cable No. Strands Cable No. StrandsBS
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. Performance of cable-stayed footbridges with two pylons
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
8. Performance of cable-stayed footbridges with two pylons
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
Structure length [m]
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
Structure length [m]
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
8. Performance of cable-stayed footbridges with two pylons
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
8.5.1 Articulation of the deck
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
8. Performance of cable-stayed footbridges with two pylons
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
Free UxFixed UyFree z
KxKy
KxKy
Free UxFixed UyFree z
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
8. Performance of cable-stayed footbridges with two pylons
0 10 20 30 40 50 60 700
1
2
Structure length [m]
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
8. Performance of cable-stayed footbridges with two pylons
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
Structure length [m]
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.
8.5.3 Area of main span stays
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
8. Performance of cable-stayed footbridges with two pylons
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
Structure length [m]
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
8. Performance of cable-stayed footbridges with two pylons
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.
8.5.4 Section of the steel girders
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
8. Performance of cable-stayed footbridges with two pylons
0 10 20 30 40 50 60 700
0.5
1
1.5
2
2.5
Structure length [m]
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
8. Performance of cable-stayed footbridges with two pylons
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
Structure length [m]
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
8. Performance of cable-stayed footbridges with two pylons
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
Structure length [m]
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
8. Performance of cable-stayed footbridges with two pylons
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
Structure length [m]
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. Performance of cable-stayed footbridges with two pylons
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
Structure length [m]
Peak v
ert
ical
acc.
[m/s
2]
−10 −5 0 5 10 150.5
0.75
1
1.25
1.5
Pylon inclination α [º]
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
8. Performance of cable-stayed footbridges with two pylons
8.5.9 Shape of the pylon
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
Structure length [m]
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
8. Performance of cable-stayed footbridges with two pylons
8.5.10 Cable system: transverse inclination of cables
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
8.5.11 Cable system: anchorage spacing
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
8. Performance of cable-stayed footbridges with two pylons
0 10 20 30 40 50 60 700
0.5
1
1.5
2
Structure length [m]
Peak v
ert
ical
acc.
[m/s
2]
0 5 10 150.5
0.7
0.9
1.1
Lateral inclination α [º]
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
Structure length [m]
Peak v
ert
ical
acc.
[m/s
2]
6 7 8 9
0.8
1
1.2
Cable distance Dc [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
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
8. Performance of cable-stayed footbridges with two pylons
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.
8.5.13 Side span length
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
8. Performance of cable-stayed footbridges with two pylons
0 10 20 30 40 50 60 700
0.5
1
1.5
2
2.5
Structure length [m]
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
8. Performance of cable-stayed footbridges with two pylons
0 10 20 30 40 50 60 700
0.5
1
1.5
2
2.5
Structure length [m]
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.
8.6.1 Articulation of the deck
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
8. Performance of cable-stayed footbridges with two pylons
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
Structure length [m]
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
8. Performance of cable-stayed footbridges with two pylons
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
Structure length [m]
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
8. Performance of cable-stayed footbridges with two pylons
8.6.3 Area of main span stays
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
Structure length [m]
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.
8.6.4 Section of the steel girders
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
8. Performance of cable-stayed footbridges with two pylons
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
Structure length [m]
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.
8.6.5 Concrete slab section
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
8. Performance of cable-stayed footbridges with two pylons
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
Structure length [m]
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
8. Performance of cable-stayed footbridges with two pylons
0 10 20 30 40 50 60 700
0.2
0.4
Structure length [m]
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
8. Performance of cable-stayed footbridges with two pylons
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
Structure length [m]
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.
8.6.9 Shape of the pylon
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
8. Performance of cable-stayed footbridges with two pylons
0 10 20 30 40 50 60 700
0.2
0.4
Structure length [m]
Peak late
ral
acc.
[m/s
2]
−10 0 10 200.5
0.75
1
1.25
1.5
Pylon inclination α [º]
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
Structure length [m]
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
8. Performance of cable-stayed footbridges with two pylons
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.
8.6.10 Cable system: transverse inclination of cables
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
Structure length [m]
Peak late
ral
acc.
[m/s
2]
0 5 10 15
0.5
1
1.5
Pylon inclination α [º]
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
8. Performance of cable-stayed footbridges with two pylons
8.6.11 Cable system: anchorage spacing
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
Structure length [m]
Peak late
ral
acc.
[m/s
2]
6 7 8 9
0.8
1
1.2
Cable distance Dc [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
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
8. Performance of cable-stayed footbridges with two pylons
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
Structure length [m]
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.
8.6.13 Side span length
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
8. Performance of cable-stayed footbridges with two pylons
0 10 20 30 40 50 60 700
0.25
0.5
0.75
1
Structure length [m]
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
8. Performance of cable-stayed footbridges with two pylons
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.
Characteristics CSFB htot/Lm = 1/100
htot = 1.0
hgird = 0.8
tflange,bot = 0.015
tflange,top = 0.015
tweb = 0.010
Cable No. Strands Cable No. StrandsBS
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)
Characteristics CSFB htot/Lm = 1/200
htot = 0.5
hgird = 0.3
tflange,bot = 0.020
tflange,top = 0.045
tweb = 0.010
Cable No. Strands Cable No. StrandsBS
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
8. Performance of cable-stayed footbridges with two pylons
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. Performance of cable-stayed footbridges with two pylons
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
8. Performance of cable-stayed footbridges with two pylons
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
Structure length [m]
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
Structure length [m]
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
8. Performance of cable-stayed footbridges with two pylons
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.
8.8 Comfort appraisal
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
8. Performance of cable-stayed footbridges with two pylons
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
8. Performance of cable-stayed footbridges with two pylons
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.
8.10 Serviceability limit state of deflections
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
8. Performance of cable-stayed footbridges with two pylons
(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
8. Performance of cable-stayed footbridges with two pylons
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.
8.11 Deck normal stresses
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
8. Performance of cable-stayed footbridges with two pylons
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
Structure length [m]
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
8. Performance of cable-stayed footbridges with two pylons
10.0 20.0 30.0 40.0 50.0 60.0
Structure length [m]
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
Bending Moment [kNm]
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
8. Performance of cable-stayed footbridges with two pylons
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
8. Performance of cable-stayed footbridges with two pylons
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).
8.15 Concluding remarks
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
8. Performance of cable-stayed footbridges with two pylons
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.
2. Structural elements:
(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.
3. Structure geometry:
287
8. Performance of cable-stayed footbridges with two pylons
(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
8. Performance of cable-stayed footbridges with two pylons
2. Structural elements:
(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.
3. Structure geometry:
(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
8. Performance of cable-stayed footbridges with two pylons
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).
Deck Accelerations Trend effect
Parameter adeck,V adeck,L Vertical Lateral
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.
Deck Accelerations Trend effect
Parameter adeck,V adeck,L Vertical Lateral
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.
Deck Accelerations Trend effect
Parameter adeck,V adeck,L Vertical Lateral
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
312
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
326
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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