Civil Engineering Reference
In-Depth Information
where
F(t)
is the dynamic load from the hydraulic actuator,
v(t)
it the vertical velocity of the footbridge at mid-span, and
T
is
the period of one cycle. The modified footbridge damping, expressed as an equivalent viscous damping ratio as a percentage
of critical, is then found as [
5
]:
E
D
ξ
0
¼
(4.5)
m
0
ω
0
D
2
X
2
2
π
Where
X
is the maximum mid-span displacement of the footbridge. It can be seen from (
4.5
) that the equivalent viscous
damping ratio is a function of the modified mass and, as such, the modified modal mass of the footbridge cannot be
determined without employing an iterative approach.
4.2.1 Pedestrian Mass and Damping Coefficient
To generalize the findings, the individual contributions of each pedestrian to any potential change in footbridge mass and
damping are determined. The change in the footbridge's damping ratio attributable to the pedestrians will be:
ξ
A
¼ ξ
0
ξ
(4.6)
The cumulative damping coefficient for all pedestrians can be written as:
2
m
0
ξ
A
ω
0
D
C
A
¼
(4.7)
Assuming a uniform distribution of the pedestrian population on the footbridge,
c
A
can also be written as:
Z
L
2
dy
c
A
¼
c
p
N
ϕð
y
Þ
(4.8)
0
where
c
p
is the damping coefficient of the individual pedestrian,
N
is the number of pedestrians on the footbridge, and
(y)
is
the mode shape of the eigenmode under examination. From (
4.7
)to(
4.8
), the damping coefficient of the individual
pedestrian is found as:
ϕ
2
m
0
ξ
A
ω
0
D
N
R
c
p
¼
(4.9)
L
0
ϕð
2
dy
y
Þ
Similarly, it can be found that the mass contribution of the individual pedestrian is:
m
0
m
m
p
¼
(4.10)
N
R
L
0
ϕð
2
dy
y
Þ
4.2.2 Experimental Setup
A simply-supported 16 m-long steel double U-beam footbridge, located in the Structures Laboratory of the Department of
Civil Engineering at DTU (Figs.
4.1
and
4.2
) was used as the basis structure for the experiments. The longitudinal beam
profiles were UNP 350, with UNP 200 crossbeams placed at 1,400 mm intervals. For the experiments described herewith,
masses were added to the footbridge crossbeams and to the center of the footbridge to increase the modal mass and thus
decrease the footbridge's frequency to a level close to the expected mean pacing frequency of the pedestrians. The total mass
of the footbridge was 5,224 kg.
