Predicting Footbridge Response Using Stochastic Load Models - Topics in Dynamics of Bridges - page 47

Table 5.4 Ratio k ( y )

Bridge A B C D E

k (1) 1 1 1 1 1

k (2) ~1 ~1 ~1 ~1 ~1

k (3) ~1 ~1 1.1 1.15 1.15

5.4.3 Computations

A Newmark time integration scheme was employed to determine a bridge acceleration time history at bridge midspan each
time a pedestrian crossed the bridge. MonteCarlo-simulations ensured that the random nature of the load was modelled
according to the assumptions, and it was ensured that the number of simulation runs (bridge crossings) were higher than what
was actually required to give sound estimates of the bridge acceleration property in focus.
The acceleration property in focus was the peak acceleration in the time series (such a value was extracted for each bridge
crossing) and focus was on the 95 % quantile of all identified peak accelerations. This value was extracted from
the probability distribution function that is the first result of the simulations. For simplicity, the 95 % quantile will hereafter
be denoted a , and will be referred to as the acceleration property. This property (the 95 % quantile) has previously been the
focus of interest in studies dealing with bridge response in a stochastic manner [ 4 , 10 ], and is therefore of interest to monitor.
By definition the value, a, is expected to be exceeded in 1 out 20 bridge crossings.

5.4.4 Results

Values of the acceleration property a were computed for all five bridges on the three different assumptions for modelling the
dynamic load factor (for model y , where y represent the model number being either 1, 2 or 3). For the presentation of results
for the five bridges, it is useful to introduce the ratio k defined in ( 5.10 ).

k ð y Þ¼ a ð y Þ= a ð 1 Þ (5.10)

The ratio relates acceleration properties calculated assuming model y ( a ( y )) to those obtained when employing model 1 ( a (1)),
in which the dynamic load factor was modelled as a random variable.
Results in term of the ratio k , is given in Table 5.4 for the five bridges (A-E). For ease of overview only approximate
values for the ratio k is given.
In Table 5.4 , the notation ~1 indicates that there is a marginal difference from the value of 1 (at least less than a 3 %
difference). Of cause and by definition, the ratio k (1) attains a value of 1, but it is interesting to observe that the ratios k (2), and for
all bridges, attain values very close to 1. The only difference between model 1 and model 2 is that in model 2, the standard
deviation of the dynamic load factor is set to zero. Hence, the results suggest that the acceleration property a is fairly insensitive
to whether the standard deviation is modelled. In other words the acceleration property a is fairly insensitive to whether the
dynamic load factor is modelled as a random variable (with non-zero standard deviation) or as a deterministic property.
Another thing that can be seen in Table 5.4 is that for the bridges C, D, and E, model 3 results in a higher value of a than
model 2. Both model 2 and 3 are characterised by having the standard deviation of the dynamic load factor set to zero. Hence
the only difference between the two models is the mean value curve for the dynamic load factor [( 5.5 ) and ( 5.8 ),
respectively]. If these were plotted, it would appear that for frequencies above 1.75 Hz, the dynamic load factor is highest
in model 3. This directly explains the observation in that the bridges C, D and E have frequencies higher than 1.75 Hz.
Overall the results suggest that the acceleration property a is fairly insensitive to the standard deviation of the dynamic
load factor but it is sensitive to the mean value (curve) of the dynamic load factor. Hence, it is important to employ a
representative mean value (curve) for the dynamic load factor for bridge response calculations.

5.5 Study Related to Modelling Pedestrian Weight

Some uncertainty exists when it comes to modelling the weight of the pedestrian (as already indicated in the previous
section). In that section a Gaussian distribution was assumed, and this approach is also employed here as a first estimate.
However, in the previous section only a single value was assumed for the mean value of pedestrian weight and a single
accompanying value (non-zero value) was assumed for the standard deviation of pedestrian weight.