1. sports
2. walking

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)
ISSN: 2311-9020; ISBN: 978-972-752-165-4
Analytic – numerical model for walking person – footbridge structure interaction
1
Michèle Pfeil1, Natasha Amador 1, Roberto Pimentel2, Raimundo Vasconcelos3
Instituto COPPE, Univesidade Federal do Rio de Janeiro, POBox68506, 21941-972, Rio de Janeiro, Brasil
2
Department of Civil Eng., Universidade Federal da Paraíba, João Pessoa, 58051-900, Brasil
3
Department of Civil Eng., Universidade Federal do Amazonas, Manaus, Brasil
email: [email protected], [email protected], [email protected], [email protected]
ABSTRACT: In traditional dynamic analysis of footbridges the pedestrians are considered as moving time varying loads in
vertical and horizontal directions. This approach is appropriate for most footbridges systems but for lightweight structures such
as composite footbridges a high degree of human – structure interaction may be expected. This interaction mechanism, which
has been reported in the literature,
leads to a reduction in the footbridge response as compared to the response under the moving load model when there is one
walking pedestrian in resonance with the structure. This paper presents an analytic – numerical model to address this feature in
vertical vibrations by considering the coupling of a biodynamic model to a one degree of freedom system representing the
structure. The walking person is simulated by a single degree of freedom mechanical model whose mass is set into vertical
motion by heel´s elevations in forward steps. The model is applied to the Aberfeldy cable-stayed composite lively footbridge
and the results are shown to be in good agreement with the experimental measurements performed under the passage of a single
known pedestrian in resonance with the structure.
KEY WORDS: footbridges, human-induced vibrations, pedestrian-structure dynamic interaction.
1
INTRODUCTION
The current approach recommended in most codes of
practice to estimate human-induced vibration amplitudes of
vertical bending in footbridges is based on a moving load
model whose magnitude is given by a Fourier series with 3 or
4 terms [1]. This procedure is appropriate for most situations
but except for lightweight structures being crossed by
pedestrians whose pacing frequency is close to a natural
frequency of the system. In these cases it overestimates the
maximum amplitudes [2] showing that the pedestrian –
structure dynamic interaction has to be taken into account.
References can be found in which this interaction is
considered by coupling the bridge model to a moving springmass-damper (SMD) model to represent the vertical action of
a walking person. The numerical model used by Fanning et al.
[2] consisted in a moving mass connected to the upper node of
a spring-damper element whose lower node was attached to a
contact element through which the Fourier series load was
applied to the numerical model of the bridge. The results
focused on the vibration amplitudes given by the moving load
model and the interaction model as compared to results from
measurements of pedestrian crossings of Aberfeldy composite
footbridge. Caprani et al. [3] addressed the same issue by
developing a 2 degrees of freedom analytical-numerical model
of the coupled system (bridge + pedestrian) and testing the
influence of a range of parameters in the bridge response due
to single pedestrian in addition to address crowd modelling.
In a similar manner this paper presents a two degrees of
freedom analytical-numerical model of the coupled system in
which the walking person is simulated by a SMD model. The
model is applied to the Aberfeldy cable-stayed composite
lively footbridge. The theoretical results obtained from the
moving load model and from the interaction model are
compared to experimental results obtained by Pimentel [4]
from two measurements of the structure under the passage of
a single known pedestrian in resonance with the structure in
order to excite the first and the second vibration modes.
Correlations between the results focus on vibration amplitudes
and also on the general aspect of time history responses.
In relation to the parameters of the SMD model to represent
a walking person in vertical direction there are few proposals
regarding pedestrian – structure interaction studies. Kim et al.
[5] proposed a two - degrees of freedom (2-DOF) biodynamic
model whose parameters are, otherwise, applicable to standing
people. For a single DOF model Caprani et al. [3] proposed
values for the spring stiffness and damping coefficient within
ranges taken from the human body biomechanic literature. In
the present paper the single DOF model proposed by Silva
and Pimentel [6] is used to simulate the walking person –
structure interaction. This model was developed on the basis
of vertical accelerations measurements of 20 tests subjects
walking along a rigid surface and has the merit to allow
application to each walking person as a function of its body
mass and pacing frequency.
ANALYTIC – NUMERICAL MODEL
2
2.1
Pedestrian walking on a rigid surface
The walking person is represented by the single degree of
freedom (SDOF) mechanical model with mass mp which is set
into vertical motion by heels movements represented by
displacement ur as illustrated in Fig. 1. The elastic and
damping components of the contact force, Fep and Fap ,
between the pedestrian and the surface can be written as
F t   Fep  Fep

k p (u pr  ur )  c p (u pr  u r )
(1)
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
where u pr and u pr are respectively the vertical displacement
and velocity of the subject center of mass while walking in a
rigid surface; k p and c p are the pedestrian stiffness and
damping respectively. The resultant force of eq. (1) is equal to
the ground reaction force which can be represented as a
Fourier series with 3 or 4 terms [1]:

F (t )  M p g 1 

 .sen i.2 . f
3
i
i 1
pp .t

 i 


(2)
where ( M p g ) is the weight of the pedestrian,  i and  i are
the dynamic load factor and the phase angle associated to each
harmonic i of the load.
mp
upr
kp
displacement at the contact point k is U ek . The pedestrian
model now displays a total vertical displacement equal
to u p  u pr  , where u p is the displacement due solely to the
pedestrian-structure interaction. The sum of the elastic and
damping interaction forces at the contact point k depend on
the relative displacement and velocity between the pedestrian
and the structure and is written as:
 

ur
kp
Figure 1. Analytical model for the interaction between a
walking person and a rigid surface.
k
(3)
The fundamental frequency fpp in eq. (2) is the walking
frequency ranging between 1.5 to 2.4 Hz with a mean value
equal to 2.0 Hz [1]. The magnitude of the dynamic load
factors, particularly 1 , depend mainly on the pacing
frequency f pp , but may display relevant differences among
pedestrians and even for the same person depending for
example on the shoes [7].
According to Zivanovic and Pavic [8] the most extensive
research on the magnitude of dynamic load factor was
conducted by Kerr [9] yielding the following expression for
average values of 1 :
(4)
with 95% of the results falling in the range 1  0.321 .
2.2
The structure modal equation of motion is expressed as


(5)
where yi is the ith mode amplitude and mi , i and i are
respectively the modal mass, damping ratio and frequency
associated to mode i .
In the moving load model the modal force Pi is determined
from the applied force expressed by eqs. (2) and (4).
2.3
Pedestrian-structure interaction model
Figure 2 illustrates the pedestrian model moving with
constant speed along a flexible structure whose vertical
1080
e
deformed
structure


m p up  upr  Fint (t )
(7)
or as the following equation by taking into account eqs. (1),
(3) and (6):




m pup  c p u p  U ek  k p u p  U ek  0
(8)
For the structure discretized in plane frame elements the
vertical displacement and velocity at the contact point k can
be expressed by

U ek  H.Ue and U ek  H.U
e
(9)
 are the
where H is the interpolation vector and U e and U
e
nodal displacement and velocity vectors of element e within
which the contact point is located.
The interaction force Fint t  is applied to the structure
model by means of the equivalent nodal force vector Fe of
element e
Fe  HT Fint
Structure model under human-induced moving load
mi yi  2ii yi  i2 yi  Pi
ur
Uek
The equation of motion of the pedestrian is then
expressed as
The equation of motion of the pedestrian model (Fig. 1)
may then be expressed as
1  0.2649 f p3  1.3206 f p2  1.7597 f p  0.7613
cp
Figure 2. Analytical model for the interaction between a
walking person and a deformed structure.
rigid surface
m pupr   F t 
 (6)
mp
upr +up
undeformed
structure
cp

Fint t   k p u p  u pr  ur  U ek   c p u p  u pr  ur  U ek
(10)
By using the modal superposition method the vertical
displacement at the contact point is expressed as
U ek 
H Φ
ei yi
i


ki yi
(11)
i
where Φei contains the eigenvector amplitudes at the nodes
of element e associated to mode i and ki is the magnitude of
the modal shape at the contact point. The modal force is given
by
Pi  ΦTei Fe
(12)
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
F(t)
Substituting the expression for Pi in equation (5) and taking
into account eqs (1) and (2) leads to

 

mi yi  2i i mi  k2 c p y i  i2 mi  k2 k p yi
 k k p u p  k c p u p  k F t 
ut
y 
X 
 u p 

K
  k k p
2
(14)
m 0 
M

 0 mp 
2 m  k2c p
C
  k c p
m  k2k p
 k c p 

c p 
 k k p 

k p 
 F t 
F k

 0 
The equations are integrated by the Runge-Kutta
method.
The coupling terms in matrix K and C of eqs. (14) may also
be written in terms of the pedestrian modal mass mp by
substituting cp and kp by 2 p p m p and  2p m p respectively
Figure 3 Single degree of freedom model used to develop
the biodynamic model proposed by Silva & Pimentel [6]
The model parameters mp, cp and kp were obtained by
solving a system of three non-linear equations (eq. 16) which
are the expressions of the accelerance frequency response
function of the SDOF model for each the three harmonic of
the forcing function eq.2 (i=1,3)
i2 F (i )
(16)
A(i ) 
k p  i2 m p  ji c p
where j is square root of (-1) and F(i) and A(i) are the input
values, being respectively the amplitudes of the harmonic
components of the ground reaction force F(t) and of the
measured acceleration at waist level for frequency i.
Based on the observed correlations among the parameters of
the model expressions were derived for the biodynamic
parameters so as to take into account these correlations and
allowing the parameters to be expressed as a function of the
total body mass Mp and the pacing rate fpp. These expressions
are given by:
m p  97.082  0.275M p  37.518 f pp
where  p is the damping ratio and  p the natural frequency
of the pedestrian model. It can then be observed that the
occurrence of the dynamic interaction is only to be expected
when the pedestrian mass corresponds to a significant
percentage of the modal mass of the structure.
3
SINGLE DOF BIODYNAMIC MODEL
The biodynamic model proposed by Silva and Pimentel [6]
is used in the present paper as the single DOF model
representing the vertical action of a pedestrian walking along
a rigid surface. It was developed on the basis of vertical
acceleration measurements of the waist of 20 persons walking
with their normal pacing rate on a straight path sufficiently
long so as to ensure a regular movement. The model consists
of a modal mass mp, a spring of elastic constant kp and a
viscous damper whose constant is cp as shown in Fig. 3. These
dynamic parameters of the model were sought in such a way
that the measured accelerations at waist could be reproduced
by the model expressed as follows.
m put  c put  k put  F t 
cp
(13)
Considering a single mode structural response, equations
(8) and (13) form the coupled pedestrian-structure equations
which may be written in matrix form as
  CX
  KX
 F
MX
mp
kp
(15)
where ut is the displacement of the mass degree of freedom
with respect to a fixed reference as shown in Fig. 3 and F(t) is
given by eq. (2) with 1 expressed as Kerr´s expression eq.
(4), 2=0.07 and 3=0.06.
cp 
4
29.041m0p.883
(kg )
(17a)
( Ns / m)
(17b)
k p  30351.744  50.261c p  0.035c 2p
( N / m) (17c)
PRELIMINARY
VALIDATION
PEDESTRIAN-STRUCTURE
MODEL
OF
THE
INTERACTION
Based on the observation made at the end of section 2.3
about the ratio between the modal masses of the pedestrian
and of the bridge models an idealized simply supported 35m
span steel-concrete composite footbridge was tested for the
passage of a single pedestrian (simulated with the biodynamic
model described in section 3) whose body mass Mp is equal to
75 kg. The pace frequency fpp was taken equal to 2Hz,
coincident to the natural frequency of the structure
corresponding to the first vertical bending mode. The ratio
between the pedestrian and the structure modal masses is
0.25% and the structure damping ratio equal to 0.5%. The
acceleration responses at the antinode of mode shape obtained
with the moving load model (section 2.2) and with the
pedestrian-structure interaction model (section 2.3) were
exactly the same [10], as expected for this low ratio between
modal masses.
5
STRUCTURE EXAMPLE AND MEASUREMENTS
The footbridge used as example for pedestrian – structure
dynamic interaction simulation is the known Aberfeldy cablestayed glass reinforced plastic (FRP) footbridge constructed in
1990 [11]. It is a three-span structure, having a main span of
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Table 1. Measured dynamic properties
Mode
m (kg)
f (Hz)

Mode 1
S
2547
1.59
S+P
1.56
0.84
Mode 2
S
3330
1.92
S+P
1.88
0.94
S: isolated structure; S+P: structure + 80kg person
Pimentel [4] also performed pedestrian tests on the cablestayed bridge in which a test subject of 80kg mass, departing
from the south tower, walked with the help of a metronome at
pace rates coincident with the natural frequencies of the
coupled system: 1.56Hz and 1.88Hz, while acceleration
measurements were taken at the antinodes of the
corresponding mode shapes as shown in Figs. 5a and 5b
respectively for the first and second vertical bending modes.
acceleration (m/s²)
63m and two side spans of approximately 25m each. The
towers were designed as A-shaped frames. Two groups of 10
cables connect the 2.12m wide deck to each tower.
The results of the experimental tests campaign conducted by
Pimentel [4] in July 1994 and June 1995 will be used in the
present paper (other experimental assessments are reported in
the literature [12] after reinforcement of the deck). It
comprised a full modal survey - for the determination of the
modal properties of the structure using ambient excitation,
impulsive response and free-decay vibration - and an
investigation of the bridge behavior under pedestrian-induced
loads.
From the ambient vibration tests the modal shapes of nine
vibration modes and their corresponding natural frequencies
were obtained. Fig. 4 illustrates the shapes of the first two
vertical bending modes of the deck.
2,5
2,0
1,5
1,0
0,5
0,0
-0,5
-1,0
-1,5
-2,0
-2,5
acquisition ended while
walking in the main span
25
35
45
55
time (s) 65
Figure 5a. Filtered acceleration time response from
controlled pedestrian test in the footbridge excited at its first
vertical bending mode [4].
Figure 4. First two measured vibration modes of the deck in
vertical direction [4].
Equivalent viscous damping ratios were determined from
the free-vibration decay part of the so called jumping tests in
which the tests subjects were asked to bounce instead of
jumping, as this was found to be an easy way to excite the
structure at a pre-defined frequency [4]. The 80 kg mass test
subject bounced in positions corresponding to the antinodes of
the first and second mode shapes as illustrated in Fig. 4. There
was no significant change in damping values along the freevibration decay records, which is an indication of linear
behavior of the structure under normal usage.
The results for damping ratios  and frequencies of
vibration obtained from the jumping tests are shown in Table
1 in the lines corresponding to the S+P (structure + person)
indication. This table also shows the natural frequencies of the
isolated structure as measured in the ambient vibration tests
(see S lines) together with the modal mass m of each mode as
obtained from a calibrated numerical model of the structure
[4]. It can be noted that the person´s mass (80kg) is 3.1% the
generalized mass of the first mode and 2.4% of that of the
second mode. This is consistent with the observed lower
frequencies of the coupled S+P system in relation to the
isolated structure.
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Acceleration (m/s²)
1,5
1,0
0,5
0,0
-0,5
-1,0
-1,5
23
28
33
38
43
48
53
58
63
time (s)
Figure 5b. Filtered acceleration time response from
controlled pedestrian test in the footbridge excited at its
second vertical bending mode [4].
6
THEORETICAL RESULTS
The moving load theoretical model (section 2.3) was applied
to the cable-stayed footbridge with the properties shown in
Table 2.
Table 3 presents the parameters of the isolated bridge
associated to the first two vibration modes and that of the
biodynamic model of the walking person as adopted in the
analysis of the dynamic interaction model (section 2.3). The
properties of the biodynamic model were obtained with eqs
(17) for Mp equal to 80kg.
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
4,0
Table 2. Dynamic properties of the coupled system used in the
moving load theoretical model

0.84
0.94
f (Hz)
1.56
1.88
interaction model
acceleration (m/s2)
Mode
m (kg)
Mode 1 S+P
2627
Mode 2 S+P
3410
S+P: structure + 80kg person
Table 3. Parameters of the structure (S) and of the biodynamic
model (P) used in the interaction theoretical model for
analyses of mode 1 and mode 2.
m (kg)
1
S
Mode 1
2547
P fpp=1.56Hz
60.5
2
S
Mode 2
3330
P fpp=1.88Hz
48.6
P: Mp=80kg; fpp= pacing frequency.
f (Hz)
1.59
2.67
1.92
2.65

0.84
91.6
0.94
78.0
moving load
3,0
2,0
1,0
0,0
-1,0
-2,0
-3,0
-4,0
25
35
45
55
time (s) 65
Figure 6. Acceleration time responses at the antinode of
mode 1 of the footbridge under the passage of the pedestrian
with pacing frequency equal to 1.56Hz, obtained with the
moving load model and with the interaction model.
3,0
moving load
acceleration (m/s2)
2,0
1,0
0,0
-1,0
-2,0
-3,0
25
35
45
55
time (s) 65
(a)
3,0
interaction
2,0
acceleration (m/s2)
Although the damping ratios shown in Table 1 have been
obtained from measurements on the coupled system these
values were assigned to the isolated bridge (Table 3, S lines)
in the interaction analyses presented in the present paper. This
is justified by the relatively large spread between the vibration
frequency of the bridge and the first natural frequency of
standing people which is greater than 4Hz [13]. To confirm
this hypothesis a theoretical-numerical analysis of the person
– structure interaction was performed with a model similar to
the one described in section 2 but with a biodynamic model of
a standing person instead of a walking one [10]. This analysis
showed that the 80 kg standing person simulated by a SDOF
model with natural frequency equal to 4.9Hz and damping
ratio equal to 37% [14] did not altered the structural damping
ratio obtained numerically by free-vibration decay analysis.
In the absence of force measurements on a rigid surface due
to the tests subject, the dynamic load factors 1 (eq. 2) were
initially taken as the mean values obtained from eq. (4): 0.224
and 0.360 respectively for modes 1 and 2. With these values
the moving load and interaction models results were tested
against the initial part of the experimental responses where
amplitudes are low and both theoretical models give the same
response eventually adjusting 1 for mode 1 to 0.270.
Figures 6 and 7 show respectively the acceleration time
responses at the antinodes of modes 1 and 2 of the footbridge
under the passage of the pedestrian with controlled pacing
frequency equal to the corresponding natural frequency of the
coupled system obtained with two theoretical models: the
moving load and the interaction model. It is seen in Fig. 6 that
the two responses for mode 1 display the same overall shape
as the experimental response (see Fig. 5a) and differ in terms
of amplitudes, the moving load model yielding maximum
amplitude 25% greater than that of the interaction model.
1,0
0,0
-1,0
-2,0
-3,0
25
35
45
55
65
time (s)
(b)
Figures 7. Acceleration time responses at the antinode of
mode 2 of the footbridge under the passage of the pedestrian
with pacing frequency equal to 1.88Hz, obtained (a) with the
moving load model and (b) with the interaction model.
The comparison between the theoretical responses obtained
for mode 2 (see Figs. 7) and the experimental response shown
in Fig.5b reveals that only the interaction model was able to
reproduce the general aspect of the time response. With the
moving load model the acceleration at the antinode of mode 2
drops to very low amplitudes when the pedestrian has just past
the middle of the center span while the experimental and the
interaction theoretical model results exhibit significant
amplitudes at this same instant. Moreover the largest
acceleration amplitudes are observed in the first part of the
response obtained from the moving load model which is not
the case of the experimental and interaction model responses.
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
7
COMPARISON BETWEEN THEORETICAL AND
EXPERIMENTAL RESULTS
Figure 8 shows the comparison for mode 1 test between the
theoretical result obtained with the interaction model and the
experimental result, the latter shown in terms of envelope
points for the sake of clarity. It can be noticed a very good
comparison until the experimental amplitude displayed a
break from the expected trend of the response envelope
indicated by the tangent lines in Fig.8. This could be due to
the pedestrian losing step while perceiving footbridge
vibrations as already observed by Zivanovic et al. [15].
In order to investigate this possibility a new analysis was
performed in which the pace frequency was changed from
1.56 Hz to 1.57 Hz in the time interval estimated to
correspond to the step losing as shown in Fig. 9. The results
show that the model was able to reproduce the response by
considering an idealized step losing.
3,0
2,5
1,5
0,5
0,0
-0,5
-1,0
-1,5
-2,5
23
28
1,88
1,88 2,5
1,88 2,0
1,88
1,88 1,5
1,88 1,0
1,88
1,88 0,5
1,88 0,0
1,88
-0,5
1,88
1,88 -1,0
1,88
-1,5
1,88
1,88 -2,0
1,88
-2,5
1,88
23
1,88
1,88
-2,0
55
time (s)
Figure 8. Comparison between theoretical and experimental
results for the first vibration mode.
3,0
43
48
53
58
63
interaction
experimental
acceleration ([m/s²)
-1,0
45
38
Figure 10. Theoretical versus experimental results for the
second vibration mode.
0,0
35
33
time (s)
1,0
25
fpp= 1.89 Hz
28
33
38
fpp= 1.89 Hz
43
48
53
58
63
time (s)
Figure 11. Theoretical versus experimental results for the
second vibration mode considering step losing.
interaction model
experimental
2,0
acceleration (m/s2)
1,0
-2,0
-3,0
8
CONCLUSIONS
1,0
0,0
-1,0
-2,0
fpp=1.57 Hz
-3,0
25
35
45
55
time (s)
Figure 9. Comparison between theoretical and experimental
results for the first vibration mode considering step losing.
The comparison between experimental and theoretical
results for mode 2 is shown in Fig. 10. Again the theoretical
results give larger amplitudes (in this case much larger) than
the experimental ones. It can be noticed in the experimental
response that step losing events may have taken place now for
small vibration amplitudes and several times as indicated by
color changing in the envelope points and tangent lines. A
similar approach to that used in the case of the first mode
response yielded the result shown in Fig. 11 for the second
mode considering the step losing. As expected, the response is
1084
interaction
experimental
2,0
interaction model
experimental
2,0
acceleration (m/s2)
very sensitive to minor changes in the step frequency. The
correlation between theoretical and experimental results in
Fig. 11 was not successful as for the first mode (Fig.9) but
still the idealized step losing yielded a reasonable theoretical
response as compared to the experimental one.
acceleration ([m/s²)
This is a clear demonstration of the occurrence of the structure
– pedestrian dynamic interaction.
An analytic-numerical model to address the dynamic
interaction between a structure and a single walking person
each represented by a single degree of freedom model was
presented in this paper and applied to the all FRP cable-stayed
Aberfeldy footbridge. The biodynamic model proposed by
29,84375
Silva and Pimentel [6] was used to obtain the parameters of
30,15625
30,46875
the biodynamic model.
The developed equations made clear that the occurrence of
the dynamic interaction is only to be expected when the
pedestrian mass corresponds to a significant percentage of the
modal mass of the structure, also depending on other
parameters. For example, a preliminary test involving a
simply supported steel-concrete composite footbridge whose
modal mass was 0.25% of the modal mass of the pedestrian
model that moved in resonant condition yielded equal
responses from the traditional moving load model and the
interaction model.
The theoretical moving load and interaction models were
applied to the Aberfeldy footbridge under the resonant
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
passage of a single pedestrian in each of two vibration modesthe first and the second vertical bending modes - in order to
compare to experimental measurements [4]. In terms of
maximum vibration amplitudes the moving load model led to
values 30% greater than the interaction model. For the first
mode the time responses obtained from both theoretical
models displayed the same overall shape as the experimental
response but for the second mode the comparison between the
theoretical responses and the experimental one showed that
only the interaction model was able to reproduce the general
aspect of the time response.
Observation of the experimental responses revealed sudden
changes of the tangents to the envelope amplitude which may
be due to pedestrian step losing. In terms of amplitudes over
time the interaction model values for the first mode remained
very close to the experimental ones until the pedestrian
possibly loosed step for high amplitudes. For the second mode
theoretical amplitudes grew beyond the experimental ones due
to more than one step losing events. The correlation between
experimental results and those from the interaction model
considering idealized step losing was very successful for the
first mode and reasonable for the second mode.
In general the results point to the interaction model
including the biodynamic model parameters proposed in [6] as
an appropriate tool to simulate the bridge response under the
passage of a pedestrian in resonance condition and as a
conservative model to predict maximum amplitudes in this
condition, yet not too conservative as the traditional moving
load model.
[11] W. Harvey, A reinforced plastic footbridge, Aberfeldy, UK, Structural
Engineering International 4:229, 232, 1993.
[12] J. Cadei and T. Stratford, The design, construction and in-service
performance of the all-composite Aberfeldy footbridge, In Shenoi, A.,
Moy, S., & Holloway, L. (Eds.), Advanced Polymer Composites for
Structural Applications in Construction. (pp. 445-453). ICE Publishing.,
2002.
[13] R.Sachse, A.Pavic, P. Reynolds, Human-structure dynamic interaction
in civil engineering dynamics: a literature review, The Shock and
Vibration Digest 35(351):3-18, 2003.
[14] J.M.W.Brownjohn, Energy dissipation in one-way slabs with human
participation, Proceedings of the Asian-Pacific Vibration Conference
99, vol.1:155-60, Singapore, 1999.
[15] S.Zivanovic, A Pavic, P. Reynolds, Human-structure dynamic
interaction in footbridges, Bridge Engineering 158 (BE4), 2005.
ACKNOWLEDGMENTS
The authors would like to acknowledge the brazilian agency
for scientific and technological development Conselho
Nacional de Desenvolvimento Científico e Tecnológico –
CNPq for supporting this research.
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