Proceedings of the IMAC-XXVII
February 9-12, 2009 Orlando, Florida USA
©2009 Society for Experimental Mechanics Inc.
Influence of Walking and Standing Crowds on Structural Dynamic
Properties
S. Živanović1, I. M. Díaz2, A. Pavić1
1
Vibration Engineering Section
Department of Civil and Structural Engineering, The University of Sheffield
Sir Frederick Mappin Building, Mappin Street, Sheffield, S1 3JD, UK
2
Escuela Técnica Superior de Ingenieros Industriales, Universidad de Castilla-La Mancha
Av. Camilo José Cela s/n, Ciudad Real, 13071, Spain
ABSTRACT
Civil engineering structures that accommodate pedestrians, such as footbridges and floors, could be exposed to
excessive vibrations under walking-induced dynamic excitation. Since humans are quite sensitive vibration
receivers, this situation leads to pedestrians’ interaction with the structure. For laterally moving structure, the
interaction often forces synchronisation of crowd’s walking frequency with the structural oscillating frequency,
leading to a dramatic increase of vibration response; a phenomenon that made the Millennium Bridge in London
and Solférino Passerelle in Paris famous, although for unintended reasons. While there is a general agreement
that crowd of pedestrians could act as negative dampers when walking across laterally moving surface, the
interaction of the same crowd with the vertically moving surface is far less understood. This paper aims to shed
some light on the latter issue by conducting the modal testing of a footbridge for three different setups: 1) empty
structure, 2) bridge occupied by a passive (i.e. standing) crowd and 3) the bridge occupied by an active (i.e.
walking) crowd. The dynamic properties of the three systems have been identified and compared. The results
have shown that the intrinsic damping of the human-structure system, occupied either by passive or active crowd,
was much larger than for the empty structure. Therefore, the human presence on the structure increased the
dampening potential of the vibrating system and mitigated the vibration response of the structure.
1
INTRODUCTION
Problems with excessive vibration that have occurred on some high profile structures in recent years while they
were exposed to human-induced dynamic loading – the Millennium Bridge in London [1], Solférino passerelle in
Paris [2], the Millennium Stadium in Cardiff [3], Brooklyn Bridge in New York during the blackout [4] are only some
examples – triggered a lot of research into modelling the human-induced loads (such as walking, jumping,
swaying, etc.) and estimation of the responses they generate. Some advances in this area have already been
made [1, 5-13]. However, what is still under researched and not well understood is the interaction between people
and perceptibly moving structure. This interaction occurs because humans are quite sensitive vibration receivers
and highly complex dynamic systems [14]. As such, they tend to change their behaviour when they perceive
strong structural vibrations in both lateral and vertical directions.
There are at least two approaches to analysing the human-structure interaction (HSI) phenomenon. The first
approach investigates the changes in the dynamic force that occur due to people’s either conscious or
unconscious changes in behaviour as reaction to being supported by a perceptibly oscillating structure. This
approach was used by Yao et al. [15] for studying the jumping force. They found that people have difficulties in
timing their jumping if the structure moves perceptibly and that the magnitude of the jumping force drops in the
region of the resonant and near resonant frequencies. Similarly, it was found that when people perceive strong
vertical vibration while crossing footbridges, they could ‘lose’ their natural step which again leads to reduction in
the magnitude of the walking force [16].
The second approach to HSI looks into the influence of people’s presence on the structure on dynamic properties
of the human-structure system, and consequently on the structural response. For example, strong lateral vibration
of the Millennium Bridge was explained as a consequence of negative damping induced into the structure by
people walking in a manner to synchronise themselves with the structure [1]. On the other hand, it is well known
now that the influence of people standing on a vertically vibrating structure could be accounted for as additional
damping to the system [17]. However, when considering people that walk over a vertically oscillating structure,
there are only some indications that these pedestrians, too, have potential to increase the damping of the system
[18-20], and this has not yet been investigated in detail.
This paper is probably the first study that investigates the interaction of walking people with the vertically moving
structure in a systematic manner. For this, the second approach to HSI was employed and the contribution of
walking people to the increase in the system damping was quantified on a simple footbridge structure. For
comparison, the influence of standing people was also studied. The effort of the investigators was concentrated
on the fundamental vibration mode only. Namely, this mode at around 4.5 Hz was the only structural mode of
vibration below 15 Hz. As such, this mode is to be most influenced by peoples’ presence due to its low frequency
and largest contribution to the total vibration response.
The first part of the paper describes the structure investigated and its dynamic properties identified in a modal
testing exercise. Then the nonlinear behaviour of the fundamental vibration mode is investigated. After this,
testing of the structure occupied by either walking or standing people is described and results are presented.
Finally, the results are summarised and discussed.
2
EMPTY STRUCTURE
This section describes the structure used in the experiments and its modal properties as identified in modal
testing programme designed for measuring frequency response function (FRF) of the system. To check if the
structural dynamic properties of the first vibration mode depend on the response level, the free decay of the
response in this vibration mode was also analysed.
2.1
Description of the structure
The structure investigated resides in a laboratory at Sheffield University. It is a small prestressed reinforced
concrete footbridge spanning 10.8 m with 2.0 m wide deck. The bridge is a simple beam structure, with knife edge
supports along its edges (Figure 1). The bridge weighs about 15000 kg.
10.8m
11.2m
Elevation
0.275m
2.0m
Plan view
Figure 1: The footbridge structure.
2.2
FRF-based modal testing
FRF-based modal testing was conducted using random excitation covering frequency content up to 50 Hz. The
force was generated using an APS electrodynamic shaker, model 400, operated in the inertial mode. A grid of 27
response points was used, as shown in Figure 2. The acceleration response at these points was measured using
force balanced QA accelerometers. The shaker was placed at test point (TP) 7 to allow for identification of both
vertical and torsional modes. The force induced by the shaker was measured indirectly, by measuring
acceleration of the reaction masses (using a piezoelectric ENDEVCO accelerometer) and multiplying this by the
moving mass of the shaker (30.6 kg).
A view (from TP 18, Figure 2) of a measurement setup including TPs 1-8 and 10-17 is shown in Figure 3.
The modal properties of the structure (mode shapes, modal damping ratios, natural frequencies and modal
masses) were obtained using a circle fit procedure available in MEscope software. The results are shown in
Figure 4. There are five modes in the frequency range up to 40 Hz. Only the first one will be investigated
regarding the HSI.
2x1.0m=2.0m
For further tests that include people on the structure, it was preferable to excite the structure to high-level
vibration clearly perceptible by people. To achieve this, it was necessary to concentrate the excitation energy in a
relatively narrow frequency range. This influenced the decision to concentrate the investigation on a single mode.
Since the low-frequency modes are most interesting in vibration serviceability of structures occupied and
dynamically excited by humans, the first mode, i.e. the frequency range around this mode, was chosen for further
study in this paper.
19
20
21
22
23
24
25
26
10
11
12
13
14
15
16
17
1
2
3
4
5
6
7
8
8x1.35m=10.8m
Figure 2: Measurement grid for FRF-based modal testing and the walking path.
Figure 3: Modal testing setup.
27
18
9
Mode 1 (vertical): 4.5 Hz, 0.64%
Mode 2 (vertical): 16.8 Hz, 0.36%
Mode 3 (rocking-torsion): 26.1 Hz, 0.88%
Mode 4 (torsion): 28.7 Hz, 1.42%
Mode 5 (vertical): 37.7 Hz, 0.93%
Figure 4: Modal properties of the structure.
2.3
Nonlinear behaviour
As already explained, this paper investigates the properties of the human-structure system in a narrow frequency
range (3.5-5.5 Hz) relevant for the fundamental vibration mode. However, before studying the human-structure
system, it is useful to have a detailed insight into the properties of the empty structure, for example its behaviour
under different levels of the acceleration response. These properties (modal damping ratio and natural frequency)
were estimated using a simple free decay analysis. The free decay was obtained by jumping on the structure in
the way that the second harmonic of the jumping force matches the natural frequency of 4.5 Hz. Jumping was
performed until strong response of the structure was developed, and then the test subject jumped off the
structure. Every four consecutive cycles of the free vibration decay were curve-fitted to produce an estimate of the
modal properties for different response levels. The results were obtained using MODAL software, developed by
Prof. Brownjohn of Sheffield University and they are shown in Figure 5.
It can be seen that the damping ratio is around 0.45% for low vibration magnitude of 0.2 m/s2, and that it goes up
to 0.65% for highest vibration response of almost 1.8 m/s2 achieved in the test. This result is expected, since
damping usually increases with increasing vibration magnitude due to engagement of additional damping
mechanisms, such as the friction at the supports. On the contrary, the natural frequency dropped slightly, from
4.45 Hz for low vibration levels to around 4.385 Hz for high vibration levels. Figure 5 also presents a polynomial fit
of the two parameters. Therefore, the footbridge structure studied exhibits amplitude-dependent nonlinear
behaviour, with damping being up to 0.65% when vibration amplitudes are quite high. Since the curve fitting
procedure used for identifying the FRFs in the previous section and later for identification of human-structure
system properties is based on the assumption of linearity, the procedure results in best (linear fit) estimates of the
modal properties.
0.65
4.46
0.60
Nat. Freq.=
Natural frequency [Hz]
Damping ratio [%]
0.55
0.50
0.45
0.40
Damping=
0.35
128.8A+32.76
3
2
A -5.423A +183A+91.55
4.44
1213A+393.2
2
A +275.8A+88.18
4.42
4.40
0.30
0.25
(a)
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6 1.8
2
Acceleration amplitude A [m/s ]
(b)
4.38
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6 1.8
2
Acceleration amplitude A [m/s ]
Figure 5: (a) The damping ratio and (b) natural frequency of the fundamental vibration mode as a function of
response amplitude.
3
STRUCTURE OCCUPIED BY PEOPLE
To identify modal properties of the human-structure system, two series of tests were designed. The first series
involved modal testing while groups of 2, 4, 6 and 10 people were standing on the structure for about 5 minutes.
The positions of the test subjects in these tests are shown as circles in Figure 6. The numbers within the circles
designate the number of people participating in the test, with each circle representing one person. For example,
four circles with number 4 within correspond to positions of four people taking part in the test. The second series
of tests involved the modal identification with the same groups of people walking at their usual pacing rate in a
circle, entering the structure between test points 18 and 27 (Figure 2) and then walking along the path shown in
Figure 2. The walking tests lasted about 15 minutes each.
0.5m
10
2,4,6,
10
6,10
6,10
0.5m
4
10
4
0.5m
2,4,6,
10
6,10
1.35m
2.70m
5.40m
10
6,10
10
0.5m
1.35m
2.70m
5.40m
Figure 6: Position of human test subjects in standing tests.
3.1
Modal testing setup
For these experiments all equipment was removed from the footbridge (Figure 7: left). This was done to allow
people to move naturally across the bridge without any obstacles in walking tests. For this reason the number of
points to measure response was reduced to eight (Figure 8) and all eight accelerometers were glued to the
concrete surface beneath the slab. The accelerometer glued at TP106 is shown in Figure 7: right as well as the
shaker operating in fixed-armature mode placed at the same test point. For this particular configuration, the
reaction masses used in the previous test for generating the force (see Figure 3) were removed and the shaker
was placed in a pit, specially constructed for this type of tests, beneath the slab. The shaker was then attached to
the soffit of the structure using a stinger through which the force was induced to the structure.
Since the reaction masses and suspension bands were removed, shaker current could be considered as
proportional to the generated force. The generated force was approximately equal to the force delivered since the
shaker mass was very small compared with the modal mass of the test structure. Therefore, the force induced
into the structure was measured by monitoring the instantaneous current.
The structure was excited by a chirp signal with frequency content between 3.5 and 5.5 Hz. For tests with
standing people five data blocks, each lasting 64 s, were collected. For tests with walking people 15 data blocks
were acquired. In each data block the excitation lasted for 51.2 s. After 51.2 s the excitation stopped allowing the
response signal to die out before the acquisition of the next data block started.
A typical representation of the force generated during each 64 s long data block is shown in Figure 9a together
with its frequency content given in Figure 9b. The time and frequency domain responses at TP106 ars shown in
Figures 9c and 9d.
Figure 7: Footbridge ready for crowd tests (left) and setup with an accelerometer and a shaker at TP106 beneath
the slab (right).
4.48m
105
106
107
108
101
102
103
104
0.92m 0.92m
1.78m
2.0m
5.4m
1.0m
5.4m
2.7m
Figure 8: Test grid for modal testing of human-structure system.
400
30
300
Fourier amplitude [N]
Force [N]
200
100
0
-100
-200
20
10
-300
-400
(a)
0
10
20
30
40
Time [s]
50
60
70
0
(b)
2
6
4
Frequency [Hz]
8
10
0
2
4
6
Frequency [Hz]
8
10
0.12
1.0
Fourier amplitude [m/s2]
Acceleration at TP106 [m/s2]
1.5
0
0.5
0.0
-0.5
0.08
0.04
-1.0
-1.5
(c)
0
10
20
30
40
Time [s]
50
60
70
0.00
(d)
Figure 9: (a) Time domain and (b) frequency domain representation of the chirp excitation and the corresponding
(c) time and (d) frequency domain representation of the acceleration response at TP106 during a 64s data block.
The data were acquired in a test with six walking people.
3.2
FRFs acquired during tests with standing and walking crowd
For each test (either with active or passive people) eight FRFs (one per response point) were acquired. However,
in the process of identification of modal properties only the point mobility (FRF between force at TP106 and the
acceleration response at the same point) was used. An example of acquired (averaged) point FRFs for tests with
six people walking and six people standing is shown in Figure 10 (dashed lines). Then a series of analytical FRF
functions was generated for different combinations of modal mass, damping ratio and natural frequency. The
resolution used for these three parameters was 100 kg, 0.02% and 0.01 Hz, respectively. The parameters of best
analytical fit to measured FRF magnitude were identified as those producing the least square error. The best fits
for the measured data presented in Figure 10 are shown in the same figure as solid lines. It can be seen that the
analytical model matches the experimental one quite well, even for data representing the phase of FRF, although
these data were not employed during the modelling process. This gives confidence that the methodology used is
robust, and could be used for identification of modal properties.
The described methodology was used for estimation of modal parameters of human-structure systems exposed to
two, four, six and ten people either walking or standing. The results for the damping ratio and natural frequency,
including those for the empty footbridge, are summarised in Table 1, while FRF models are shown in Figure 11.
Note that modal properties of the empty slab are different from those shown in Figure 4. This is because this time
the parameter identification for the empty slab was done using FRF measurements of the same kind as those for
occupied structure, while the results presented in Figure 4 are based on tests employing random excitation that
generates lower level responses compared with those generated by the chirp excitation.
The most significant observation is that both walking and standing people increased the damping of the system,
with the effect more pronounced for standing people. This effect is not only obvious in the reduction of the FRF
peak, increase of the FRF width (Figure 11a) and the decrease of the slope in FRF phase (Figure 11b) but also in
the reduction of the peak response value compared with the peak response of the empty structure (Table 1). Note
that changes in damping properties when people were present on the structures are an order of magnitude larger
than due to nonlinearities of the empty structure, making it clear that these changes were caused by people’s
presence.
200
measured
model
measured
model
160
3.5
3.0
FRF phase [degree]
6 people
walking
2
FRF magnitude [(m/s )/N]
4.5 x10-3
4.0
2.5
2.0
6 people
standing
1.5
1.0
6 people
walking
120
6 people
standing
80
40
0.5
0.0
3.5
4.0
4.5
Frequency [Hz]
5.0
0
5.5
3.5
4.0
4.5
5.0
5.5
(a)
(b)
Frequency [Hz]
Figure 10: Measured and modelled (a) magnitude and (b) phase of FRF for tests with six people walking and
standing on the footbridge.
Table 1: Modal properties of empty slab and human-structure systems.
Number
of people
STANDING
ζ [%]
f [Hz]
Max acc. [m/s ]
ζ [%]
f [Hz]
Max acc. [m/s2]
0
0.72
4.44
1.92
0.72
4.44
1.92
2
1.58
4.42
1.27
0.96
4.45
1.85
4
2.32
4.34
1.00
1.46
4.45
1.60
6
3.12
4.26
0.70
1.78
4.48
1.46
10
3.62
4.21
0.56
2.86
4.51
1.19
10
200
empty slab
8
empty slab
FRF phase [degree]
6
2 people
standing
4
4 people
standing
6 people
standing
2
2 people
walking
160
2 people
walking
2
FRF magnitude [(m/s )/N]
WALKING
2
4 people
walking
6 people
walking
10 people
walking
2 people
standing
120
4 people
walking
4 people
standing
6 people
standing
80
6 people
walking
10 people
standing
10 people
walking
40
10 people
standing
0
3.5
4.0
4.5
Frequency [Hz]
5.0
5.5
0
3.5
4.0
4.5
5.0
5.5
(a)
(b)
Frequency [Hz]
Figure 11: (a) Magnitude and (b) phase of FRFs for human-structure system including various numbers of people.
4
DISCUSSION
Table 1 shows that the presence of humans on the structure, either in passive or active form, increased the
damping of the new human-structure system compared with the damping of the empty structure. This increase
was significant not only for configurations including people standing on the structure but also for those tests with
walking people (Figure 12). This suggests that it would be useful and cost effective to account for the increased
damping in vibration serviceability design of structures exposed to human occupancy.
4.0
standing
walking
3.5
Damping ratio [%]
3.0
2.5
2.0
1.5
1.0
0.5
0
2
4
6
8
10
Number of people
Figure 12: Damping ratio of the system as a function of the number of people and their activity (standing or
walking).
ACKNOWLEDGEMENTS
The authors would like to thank to students and staff of the University of Sheffield who took part in the testing
programme: Eunice Lawton, Darren Jones, Jonathan Wood, Vitomir Racic, Donald Nyawako, Tuan Norhayati
Tuan Chik, Mohammad Muaz Aldimashki, Alkiviadis Alexakis, Ahmed Babiker, Gbenga Oludotun, Chris Todd and
Mark Foster. Also the authors acknowledge the financial support of the UK Engineering and Physical Sciences
Research Council (EPSRC) under the grant reference GR/T03000/01 (Stochastic Approach to Human-Structure
Dynamic Interaction) as well as Conserjería de Educación y Ciencia of Junta de Comunidades de Castilla-La
Mancha and European Social Fund.
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