Dynamic/earthquake analysis - Computational Analysis and Design of Bridge Structures - page 517

and can be treated as a sequence of moving masses. Also, railway traf-
fic provides inherent frequencies due to repetitive characters of wheel or
bogie loads; a more significant resonance might be produced and affects
the bridge durability. Recently, in order to accurately simulate the moving
vehicle-bridge interaction, LS-DYNA (1998) with FEA was used.

17.2.2 Pedestrian bridge vibrations
Resonance has been ignored in the design of pedestrian bridges until recently.
Pedestrian bridges, especially light bridges supported by cables, should be
checked for vibration serviceability due to human activities. Unless the bridge
is supported by flexible substructure or soil condition, only the superstruc-
ture is simplified in the modeling process as a beam linear dynamic analysis
model. Modal analysis is the first step for the pedestrian bridge dynamic
analysis for determining the natural frequencies and mode shapes of a struc-
ture, as well as the responses of individual modes to a given excitation.
Vibration of the pedestrian bridge can be due to two sources, vertical and
lateral vibrations. Lateral vibration is assuming synchronous lateral excita-
tion. This occurs when a large enough group of pedestrians senses a lateral
movement and subconsciously tries to counteract that movement by shifting
their weight in opposition to the perceived movement, in effect creating a
steady driving force. On the other hand, footsteps are the source of vertical
vibration where the force f ( t ) in Equation 17.1 can be represented by

( )

∑

∑

f t ( ) = P 1 + α cos 2 π if t + ϕ (17.8)

i step i

where:
P is the person's weight
α i is the dynamic coefficient for the harmonic force
i is the harmonic multiple (1, 2, 3,…)
f step is the step frequency of activity
t is the time
φ i is the phase angle for the harmonic

In the assumption, f step is commonly assumed at 2 Hz, or 2 steps per second.
Values for alpha are typically taken at 0.5, 0.2, 0.1, and 0.05 for the first
four harmonics of walking. It is when f step matches the frequency of any of
the modes of vibration of the structure that resonance will occur.
Figure 17.6 shows recommended peak acceleration for human comfort
for vibrations due to human activities (Allen and Murray 1993; Murray
et al. 1997). As shown in the figure, the tolerance limits for vibration fre-
quencies between 4 and 8 Hz are lower, whereas outside this frequency
range, people accept higher vibration accelerations. Two sources provide
design-limiting values for bridges: