Predicting Footbridge Response Using Stochastic Load Models - Topics in Dynamics of Bridges - page 45

5.3 Modelling of Walking Loads

When modelling walking loads, Fourier series are often employed, but for the present studies, the basic excitation frequency
(step frequency f s ) is so close to matching the bridge frequency, that it is not considered necessary to consider super-
harmonic load components. Basically, because the super-harmonic load components are not likely to be able to cause
significant resonant effects.
Therefore the vertical dynamic load induced by a pedestrian is modelled according to ( 5.1 ):

f ð t Þ¼ mg α cos 2 ð π f s t Þ (5.1)

in which t is time, and m is pedestrian weigth (in kg). The parameter g represents acceleration of gravity (for simplicity
g (non-dimensional), and called the dynamic load factor,

has an impact on the amplitude of walking loads. For reference it is mentioned that the modelling approach is in agreement
with [ 7 - 9 ].
When employing the model for walking load introduced above, it is assumed that the pedestrian walks with a constant
step frequency, f s , while crossing the bridge. Nevertheless, the paper will model it so that the step frequency may vary from
one bridge crossing to the next hereby seeking to embrace the random nature of the step frequency known to exist.
On these assumptions, the modal load acting on the bridge, q ( t ), may be computed using ( 5.2 ):

¼ 10 N/kg is assumed), but it is also apparent that the parameter α

q ð t Þ¼ mg α sin 2 ð π f s t Þ Φð t Þ (5.2)

where ( t ) is the value of the mode shape function for the first vertical bending mode of the footbridge at the current position

of the pedestrian. This value may be computed using ( 5.3 ):

Φ

L t

sin π

f s l s

Φð t Þ¼

(5.3)

as the walking velocity v may be derived from the following relationship:

v ¼ f s l s (5.4)

in which l s , represents the stride length of the pedestrian. For later computations, it is assumed that any pedestrian traverses the
bridge using a constant stride length, l s , (step length), and a constant step frequency, f s , and thus with a constant walking speed
v . Nevertheless, the paper will model it so that the step frequency and the stride length may vary from one bridge crossing to
the next hereby seeking to model the random nature of both properties (and thereby a random nature of walking velocity).
Having outlined the basic premises for the studies, more details are now given for the specific sensitivity studies of this paper.

5.4 Study Related to Modelling the Dynamic Load Factor

It is so that there are various ways to model the dynamic load factor. For the sensitivity studies of this paper, three different
potential models are considered and they are outlined in Sect. 5.4.1 . Section 5.4.2 supplements with other study assumptions,
and the last section discusses the results.

5.4.1 Dynamic Load Factor

Model 1 In this model, the dynamic load factor is modelled to depend on step frequency, f s , according to the relationship
given in ( 5.5 )[ 6 ]:

c 1 f s þ c 2 f s þ

α ¼ c 3 f s þ c 4 (5.5)