## 3 Numerical Applications
### Numerical Response Evaluation  
Following, the numerical assessment of the maximum acceleration response at the mid-span was performed by using the method proposed in this paper. The modal properties of both structures, which constitute the method data, are summarised in Table 2, where \(\xi\) (-) is the damping ratio. The parameters of Equations (18-20) that establish the analytical relation between the maximum acceleration response and the number of pedestrians were calculated by using Equations (1,5,11,16,17). These parameters are summarised in Table 3. The saturation level or acceleration value, beyond which some pedestrians feel uncomfortable and choose to stop, was assumed to be 1.2 m/s\({}^{2}\), consistent with the recommendations of some authors [7; 21] and the values recorded in the experiments on both footbridges. Equations (18-21) are represented in Figure 10 (the Pedro e Ines footbridge) and Figure 11 (the Lardal footbridge). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Footbridge** & \(f_{b}\) **(Hz)** & \(M\) **(kg)** & \(\xi\) **(-)** & \(L\) **(m)** & \(L_{d}\) **(m)** & \(b\) **(m)** \\ \hline Pedro e Inés & 0.91 & 165,880 & \(5.80\times 10^{-3}\) & 144.00 & 88.00 & 4.00 \\ Lardal & 0.83 & 18,000 & \(2.50\times 10^{-2}\) & 91.00 & 80.00 & 2.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numerical response simulation. Data [18; 21]. Figure 9: The Lardal footbridge. (Picture after [[http://broer-vesfold.blogspot.com/2011/08/bro-v-kjrra-fosespark-i-lardal.html](http://broer-vesfold.blogspot.com/2011/08/bro-v-kjrra-fosespark-i-lardal.html)]([http://broer-vesfold.blogspot.com/2011/08/bro-v-kjrra-fosespark-i-lardal.html](http://broer-vesfold.blogspot.com/2011/08/bro-v-kjrra-fosespark-i-lardal.html)).). Additionally, the numerical evaluation of the lateral response of the two structures was calculated following the Hivoss guideline [12]. The assessment was also performed in the frequency domain. The structural response was evaluated by using an equivalent SDOF system, in which the structure was characterised through its modal properties (Table 2). The stochastic stream of \(N\) pedestrians was defined through the idealised stream of \(N\)' perfectly synchronised pedestrians modelled as a harmonic deterministic load. As a result, the analysis provided the maximum resonant acceleration  
Figure 11: The Lardal footbridge. Analysis of the results. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Footbridge** & \
\begin{tabular}{c} \(\left|H\left(\
icefrac{{f_{p}}}{{p}}\right|\) \\ **(mN\({}^{-1}\))** \\ \end{tabular} & \begin{tabular}{c} \(\mathbf{c_{p}}\) \\ **(Nsm\({}^{-1}\))** \\ \end{tabular} & \begin{tabular}{c} \(\mathbf{G}\) (\(\mathbf{\cdot}\)) \\ **(m/s\({}^{2}\))** \\ \end{tabular} & \begin{tabular}{c} \(\mathbf{a_{0,mean}}\) \\ **(m/s\({}^{2}\))** \\ \end{tabular} & \begin{tabular}{c} \(\mathbf{a_{0,max}}\) \\ **(m/s\({}^{2}\))** \\ \end{tabular} & \
\begin{tabular}{c} \(\mathbf{N_{crit}}\) (\(\mathbf{\cdot}\))** \\ \end{tabular} &
\begin{tabular}{c} \(\mathbf{N_{sat}}\) (\(\mathbf{\cdot}\)) \\ \end{tabular} \\ \hline Pedro e Inés & \(1.59\times 10^{-5}\) & \(170.09\) & \(3.20\times 10^{-2}\) & \(1.64\times 10^{-3}\) & \(2.53\times 10^{-3}\) & \(75\) & \(129\) \\ Lardal & \(4.09\times 10^{-5}\) & \(177.36\) & \(5.5\times 10^{-2}\) & \(9.52\times 10^{-3}\) & \(15.00\times 10^{-3}\) & \(13\) & \(36\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Numerical response simulation. Results obtained by applying the proposed method. Figure 10: The Pedro e Inés footbridge. Analysis of the results. response at the centre of the footbridge.