1 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
CSHM CSH M – 2 Works Workshop hop,, 28th Se Sept ptem embe berr – 1st October 2008, Taormina
Pedestrian Loads and Dynamic Performances of Lively Live ly Foot Footbrid bridges ges:: an Overvie Overview w
Fiammetta Venuti Luca Bruno
Politecnico Politecnico di Torino Tori no (Italy) Department of Structural Engineering and Geotechnics
2 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Introduction PEDESTRIAN BRIDGES
Increasing strength of materials Increase of slenderness
Critical performances of new structures reduced serviceabilit high costs for dynamic assessment after construction
ROAD BRIDGES
Increase of traffic Increase of vehicles weight
Critical performances of existing structures re redu duce ced d safe safett an and d stab stabili ilitt
The dynamic behaviour should be considered in a very early design stage
Need for comfort criteria Need for suitable and predictive load models Need for practical design rules
3 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Introduction Human-induced vibration problems on footbridges were discovered in the 19th century collapse of a footbridge in Broughton due to marching soldiers
Attention focused on vertical vibrations in the 20th century From 2000, with the closure of the London Millennium Bridge, the attention is focused on lateral vibrations due to synchronisation phenomena (a few episodes had been already reported from the Seventies) Auckland Harbour bridge, 1975
London Millennium Bridge opening day, July 2000
4 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Introduction In the last decade, increasing attention to human-induced vibrations on footbridges testified by:
Specific international conference
International reseach projects and guidelines FIB Federation International du Beton. Guidelines for the design of footbridges, fib Bulletin No. 32, Lausanne, 2006. SETRA/AFGC. Passerelles piétonnes – Evaluation du comportement vibratoire sous l’action de piétons. Guide méthodologique. Paris, 2006
European Project SINPEX BUTZ C. et al., Advanced load models for synchronous pedestrian excitation and optimised design guidelines for steel footbridges (SYNPEX), Final report, RFS-CR 03019, Research Fund for Coal and Steel, 2007
5 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Introduction Objective state-of-the-art about human-induced vibrations on footbridges Summary
Phenomenological analysis of pedestrian loading
pedestrian on a rigid surface
pedestrian on a vibrating surface
Comfort criteria
Pedestrian load models
single pedestrian
groups of pedestrians
crowds
Experimental tests
laboratory tests
field tests
human-structure interaction
Pedestrian loads and dynamic performances of lively footbridges: an overview F. Venuti,, L. Bruno, CSHM-2, 28 Sept. – 1 Oct. 008, Taormina
P H E N O M E N O L O G I C
6 / 3 3
Pedestrian loads and dynamic performances of lively footbridges: an overview F. Venuti,, L. Bruno, CSHM-2, 28 Sept. – 1 Oct. 008, Taormina
Andriacchi et al. (1997)
W a l k i n g f r e q u e n c y r a n g e F s F L L f o r d i f f e r e n t a c t i v i t i
F
F
L
V
p l
F
H
F
H
W a l k i
F
F
V
V
Number of people 5 0
1 0 0
7 / 3 3
P e d e s t r i a n w a l k i n g o n a r i g i d s
8 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Pedestrians walking on a vibrating surface Human-structure interaction Modification of the footbridge dynamic properties Change in natural frequencies due to pedestrians mass Change in damping (the effect of moving people is still unexplored) Synchronisation between the pedestrians and the structure The phenomenon is much more probable in the horizontal direction
Synchronous Lateral Excitation (SLE)
Auckland Harbour New Zealand 1975
Groves Bridge Chester (UK) 1977
T-bridge Japan 1993
Passerelle Solferino Paris 2000
Millennium Bridge London 2000
“[..] the phenomenom could occour on any bridge with a lateral frequency below about 1.3 Hz loaded with a sufficient number of pedestrians.” (Dallard et al., 2001)
9 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Synchronous Lateral Excitation KEY FEATURES OF THE PHENOMENON 2 kinds of synchronisation:
The deck lateral motion triggers the synchronisation between the pedestrians and the structure
LOCK-IN
The probability of lock-in grows for increasing amplitude of the deck motion
Dallard et al. (2001), Bachmann (2002), Nakamura (2003)
High crowd density causes synchronisation among pedestrians
Venuti et al. (2005), Ricciardelli (2005)
10 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Synchronous Lateral Excitation Self-excitation:
The lateral force grows for increasing amplitude of the deck motion
Dallard et al. (2001)
Pizzimenti (2003)
Self-limitation:
Pedestrians detune or stop walking when vibrations exceed a threshold value
Nakamura (2003)
Pedestrian loads and dynamic performances of lively footbridges: an overview F. Venuti,, L. Bruno, CSHM-2, 28 Sept. – 1 Oct. 008, Taormina
C O M F O R T C
1 / 3 1 3
12 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Comfort requirements The reaction of pedestrians to vibration is very complex:
different people react differently to the same vibration condition
an individual reacts differently to the same vibrations on different days
a pedestrian alone is more sensitive to vibration than in a crowd
a pedestrian who expects vibrations is less sensitive
Comfort requirements:
frequencies
the pedestrian loading frequencies Code/Standard
Vertical [Hz]
Horizontal [Hz]
Eurocode 2
1.6 – 2.4
0.8 – 1.2
Eurocode 5
<5
0.5 – 2.5
Eurocode 1 (UK NA)
< 8 (unloaded bridge)
< 1.5 (loaded bridge)
Seldom fulfilled in new footbridges
Limit values of accelerations
If the limit on frequencies is not satisfied, a dynamic calculation with suitable load models is required
13 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
ISO 10137 – Eurocode 5 ISO 10137 (2007): Bases for design of structures – Serviceability of buildings and
walkways against vibrations. The limit values are obtained by multiplying the base curves of rms accelerations by a factor 60 (pedestrians) or 30 (standing persons) vertical
horizontal
] s / [ S M R -
] s / [ S M R -
2
2
h
v
a
a
f [Hz]
Limit values for pedestrians av,rms
= 0.6 /
av,rms
= 0.3
4 ≤ f ≤ 8
ah,rms
= 0.2
1 ≤ f ≤ 2
f
1 ≤ f ≤ 4 f [Hz]
Eurocode 5:
av,max= 0.7 m/s2
ah,max= 0.2 m/s2
14 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
SETRA Guideline Comfort requirements are not absolute but depend on the comfort level specified by the Owner. Stage 1: determination of the footbridge class Traffic Class
Density d (P=person)
Description
I
d=1.0 P/m2
urban footbridge linking up high pedestrian density areas or that is frequently used by dense crowds, subjected to very heavy traffic
II
d=0.8 P/m2
urban footbridge linking up populated areas, subjected to heavy traffic and that may occasionally be loaded throughout its bearing area
III
d=0.5 P/m2
footbridge for standard use, occasionally crossed by large groups of people but that will never be loaded throughout its bearing area
IV
seldom used footbridge, built to link sparsely populated areas
Stage 2: choice of the comfort level Stage 3: determination of frequencies (risk of resonance)
Comfort Degree of level comfort
Acceleration level Vertical [m/s2]
Acceleration level Horizontal [m/s2]
Lock-in
1
maximum
< 0.5
< 0.1
2
average
0.5 – 1.0
0.15 – 0.3
3
minimum
1.0 – 2.5
0.3 – 0.8
4
discomfort
> 2.5
> 0.8
Stage 4: dynamic calculation (if necessary)
15 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
SYNPEX Guideline
Acceleration checks should be performed if: vertical
1.3 ≤ f v ≤ 2.3 Hz
horizontal 0.5 ≤ f h
≤ 1.2 Hz
Definition of design scenarios, characterised by a traffic class and a comfort level Traffic Class
Density d (P=person)
TC 1
15 P
TC 2
d=0.2 P/m2 Weak traffic: comfortable and free walking
TC 3
d=0.5 P m2 Dense traffic: unresctricted walkin
TC 4
d=1.0 P/m2 Very dense traffic: uncomfortable situation, obstructed walking
TC 5
d=1.5 P/m2 Exceptional dense traffic: crowding begins
Comfort Degree of level comfort
Description Very weak traffic: 15 single persons
Acceleration level Vertical [m/s2]
overtakin can inhibit
Acceleration level Horizontal [m/s2]
CL 1
maximum
< 0.5
< 0.1
CL 2
medium
0.5 – 1.0
0.1 – 0.3
CL 3
minimum
1.0 – 2.5
0.3 – 0.8
CL 4
discomfort
> 2.5
> 0.8
Lock-in
16 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
UK National Annex to EN 1991-2
Limit on the vertical acceleration:
alim
= 1.0 k 1 k 2
2 k 3 k 4 m/s
0.5 ≤ alim ≤ 2.0 m/s
k 4=1 exposure factor
Comfort criterion on synchronous lateral excitation: Pedestrian excitation mass damping parameter
D =
mbridge ξ m pedestrian
2
17 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Comments
Standard codes and new guidelines has different approaches
Absolute values of comfort requirements
Comfort requirements decided by the owner as a function of the footbridge traffic class and required level of comfort
UK National Annex has a different approach towards the avoidance of SLE mass damping parameter instead of limit on the lateral acceleration
Pedestrian loads and dynamic performances of lively footbridges: an overview F. Venuti,, L. Bruno, CSHM-2, 28 Sept. – 1 Oct. 008, Taormina
L O A D M O
1 / 8 3 3
19 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Classification of load models TIME DOMAIN FORCE MODELS Assumption: both feet produce exactly the same periodic force
Deterministic
general force model for each type of human activity
Probabilistic
take into account that some parameters which influence human force (e.g. frequency, person’s weight) are random variables whose statistical nature should be considered in terms of their probability distribution functions.
FREQUENCY DOMAIN FORCE MODELS
pedestrian loads modelled as random processes
walking forces represented by power spectral densities (PSD)
20 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Single pedestrian load model Framework: Fourier decomposition of the three force components n
G = 700 N pedestrian weight
F vert
=G+
∑ Gα i,vert sin(2π f pt − ϕ i ,vert )
vertical
i =1
α i = Dynamic Load Factor (DLF) of the i th harmonic
n
F lat = ∑ Gα i ,lat sin(π f p t − ϕ i ,lat )
lateral
i =1 n
F long
=
∑ Gα i ,long sin(2π f pt − ϕ i,long )
longitudinal
i =1
Bachmann & Ammann (1987)
l a c i t r e v
l a r e t a l
l a n i d u t i g n o l
Load models in codes and guidelines usually considers only the first harmonic and the resulting sinusoidal force is applied in resonance to the footbridge natural mode of interest
21 /33
Crowd load models: framework
w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Assumption: the action of a group of pedestrians or a crowd is generally modelled by multiplying the action of a single pedestrian by an effective number of pedestrians neff effective number of pedestrians
F (t ) = F 0 sin( 2π ft ) ⋅ neff ⋅ψ reduction coefficient
action of a single pedestrian
F 0
= G ⋅ DLF
F 0 [N] SETRA - SYNPEX UK N.A. EN1991-2
Vertical
280 280 (walk) – 910 (jogging)
Longitudinal
140 -
Lateral
35 -
The action should be applied in resonance with the footbridge natural frequency
22 /33
Effective number of pedestrians
w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
It can be interpreted as a synchronisation factor it represents the percentage of people in the crowd that, by chance, walk in step Matsumoto et al. (1978)
neff
=
n
Uncorrelated pedestrians arriving on the bridge with a Poisson distribution, with resonant frequencies and random phases
ISO 10137
SETRA – SYNPEX
This model is not suitable to model SLE
neff
= 10.8
nξ
for d<=1 P/m2
neff
= 1.85
n
for d>=1.0 P/m2
account for synchronisation due to high density
from probabilistic assumptions: number of pedestrians who, walking in step with the footbridge natural frequency and equally distributed along the deck, produce the 95% fractile of the peak acceleration due to random pedestrian streams.
23 /33
Reduction coefficient
w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Reduction factors to account for the probability of occurrence of step frequencies SETRA – SYNPEX
ψ vert ,long
ψ lat
First harm. econ arm.
UK N.A. EN1991-2
Population factor k ( f v )
Only for vertical vibration
f v
24 /33
Load distribution along the deck
w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Single pedestrian or group: Pulsating force F [N] moving across the span at constant speed v Crowd: The distributed oscillating loading should be applied in order to obtain the most unfavourable effect the amplitude of the load has the same sign as the mode shape configuration
Setra (2006)
Pedestrian loads and dynamic performances of lively footbridges: an overview F. Venuti,, L. Bruno, CSHM-2, 28 Sept. – 1 Oct. 008, Taormina
E X P E R I M E N T A
2 / 5 3 3
26 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Objectives of tests Measurement of:
the intensity of the force exerted by a pedestrian on a rigid surface
the intensity of the force exerted by a pedestrian on a moving surface
the probability that a pedestrian synchronises to the motion of the walking surface
the frequency and velocity of people walking
the crowd characteristic quantities (e.g. density, velocity)
the probability of synchronisation among pedestrians
done partially done to be done
27 /33
Force on a rigid surface: laboratory tests
w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
FORCE PLATE four tri-axial force sensors that measure the force acting between the foot and the ground in 3 axes: transverse (X), anteroposterior (Y) and vertical (Z).
Z X
Y
TREADMILL INSTRUMENTED SHOES Sole with force transducers, allows to measure vertical forces during gait over a great number of steps
28 /33
Force on a moving surface and lock-in: laboratory tests
w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Treadmill laterally moving with different frequencies and amplitudes measure the force on a moving platform and estimate the degree of synchronisation
Pizzimenti, 2005 University of Reggio Calabria
SETRA, 2006 7m-long platform to recreate the same condition of a footbridge
29 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Pedestrian-structure synchronisation: field tests
measure the footbridge dynamic response to different crowd conditions and the triggering of the lock-in
measure the pedestrian lateral motion London Millennium Bridge 2001
Nakamura & Kawasaki, 2003 M-bridge, Japan
Passerelle Simone de Beauvoire, 2006, Paris
30 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Crowd characteristic quantities Available techniques:
Counting: flow measured by counting the number of persons at a specific cross-section in a certain time interval; speed and frequency measured by noting down the number of steps and time taken by randomly selected pedestrians to cross a given length.
GPS:
measure velocity, step frequency, step length
Infrared:
count people moving across a line, extract complete pedestrian trajectories.
Videos:
observation to measure crowd density and velocity.
31 /33
Synchronisation among pedestrians
w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Observation of videos recorded during crowd events
measure the motion of pedestrians’ heads and the motion of the deck
allow the percentage of synchronised pedestrians to be estimated
T-bridge, Fujino et al. 1993
32 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
What has to be done
Measure the probability of synchronisation among pedestrians as a function of the crowd density
Measure the way in which walking velocity (and frequency) are modified by the motion of the walking surface
Measure the forces exerted on real footbridges for different crowd conditions Adaptation of W.I.M. to pedestrian loads?
Critical aspects: Pedestrians do not walk in lanes More than 1 pedestrian in the same deck cross-section Need to measure the lateral force component
33 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
Conclusions
Footbridge serviceability under human-induced excitation is still an open research topic;
Standard codes are still based on outdated assumptions, while design guidelines provide new design methodologies, load models and comfort criteria; research to be deeply understood with contributions from different research fields
Need for experimental tests to
propose and validate load models
statistichally characterise pedestrian walking behaviour (e.g. velocity, frequency, synchronisation, etc.)
34 /33 w e i v r e v o n a : s e g d i r a b t i n o o m f r y o l a e T v , i l 8 f 0 o 0 . e t c c n O a 1 m r – . o t f r p e e p S c 8 i 2 m , a 2 n y M d H S d C n , a o s n d u r a B o . l L n , , a i i r t u t s n e e d V e . P F
A proposal for a different approach for SLE
Description of the synchronous lateral excitation phenomenon through the proposal of a crowd-structure interaction model; model the crowd as a dynamical system instead of as a simple load.
The model is based on: PARTITIONED APPROACH decomposition of the dynamic coupled system into two subsystems
“TWO-WAY” INTERACTION
STRUCTURE Crowd-to-Structure FORCE MODEL action
Structure-to-Crowd action
CROWD
VENUTI F., BRUNO L., BELLOMO N., Crowd dynamics on a moving platform: mathematical modelling and application to lively footbridges, Math. Comput. Model., n. 45, 2007
