## 2 Method to Evaluate the Lateral Vibration of Footbridges Including the Postlock-in Behaviour
### Footbridge Response without Bridge Motion  
With low traffic density or traffic density that is below the \"critical number\" of pedestrians, the footbridge response does not affect people balance and does not induce HSI. The loads exerted on the footbridge are similar to those measured on a static treadmill. Hence, the function \(X\)(\(f\)) (N) in Figure 2 is expressed through the experimental power spectral density (PSD) of the load that was obtained by [PERSON] and [PERSON] [PERSON] and [PERSON] (1999) in the absence of lateral motion, thus considering the intra-subject variability of the load in the analytical expression. The PSD is based on a Gaussian-shaped function to fit the individual load harmonics, which were previously proposed by [PERSON] and [PERSON] [PERSON] and [PERSON] (1999). Equation (12) is the expression of the mean or the 95% fractile PSD function \(S(f_{p})\) (N\({}^{2}\)Hz\({}^{-1}\)) for the first five harmonics [PERSON] and [PERSON] (1999). In this paper, the lateral step frequency \(f_{p}\) is introduced as a variable, which means that Equation (12) represents the PSD of the load exerted by any pedestrian who walks with any frequency \(f_{p}\):  
\[S\Big{(}f,f_{p}\Big{)}=\sum_{j=1}^{5}\Bigg{(}\frac{2A_{j}\sigma_{j}^{2}W^{2}}{ \sqrt{2\pi}B_{j}f}e^{-2(\frac{\frac{f_{p}}{f_{p}}-1}{B_{j}})}\Bigg{)} \tag{12}\]The parameters \(A_{j}\), \(B_{j}\) and \(\sigma_{j}\) are obtained from the experimental data fit. \(W=700\) N is the mean of people weight. The variance \(\sigma_{j}^{2}\), which represents the energy content in the load around the j-harmonic, can be taken as either the data mean value, in order to represent the mean of the pedestrian load, or the 95% fractile (or that with 5% probability of exceedance), in order to represent the peak value of the pedestrian load. The values \(A_{j}\), \(B_{j}\) and \(\sigma_{j}\) are summarised in Table 1.  
Furthermore, the dependence of the function \(S(f\!f_{p})\) with the step frequency \(f_{p}\) shows the inter-subject variability in the loads. As done before, using the probability distribution function of the step frequency \(P(f\!g_{p})\) and applying the superposition principle, the contribution of each possible value of \(f_{p}\) included in the interval (\(\mu\)-3\(\sigma\), \(\mu\)+3\(\sigma\)) of the load exerted by one pedestrian is expressed through the function \(SF(f)\) (N\({}^{2}\)), which can be expressed independently from the step frequency, as follows:  
\[SF(f)=\int_{\mu-3\sigma}^{\mu+3\sigma}S\big{(}f,f_{p}\big{)}P\big{(}f_{p}\big{)} df_{p} \tag{13}\]  
Figure 5 shows the first and third harmonics determined using Equation (13), centred in \(J\!f_{p}=0.86\) Hz (mean value of the step frequency) and \(3f_{p}=2.58\) Hz. The second, fourth and fifth harmonics are nearly zero as the even harmonics are small because of the asymmetric character of the pedestrian step (Table 1) and \(P(f\!g_{p})\) tends to zero outside of the interval 0.62 Hz and 1.10 Hz. For the case of one pedestrian that is uniformly distributed on the footbridge span, the equivalent PSD of the load \(SX(f)\) (N\({}^{2}\)) is expressed in modal coordinates as follows:  
\[SX(f)=\left(\frac{1}{L}\int_{0}^{L_{d}}\sin\frac{\pi x}{L_{d}}dx\right)^{2}SF( f)=\left(\frac{1}{L}\frac{2L_{d}}{\pi}\right)^{2}SF(f) \tag{14}\]  
The next step is to determine the mean or peak value of the PSD of the lateral response \(SY(f)\) (m\({}^{2}\)) by using the following expression:  
\[SY(f)=\left|H(f)\right|^{2}SX(f) \tag{15}\] \begin{table}
\begin{tabular}{c c c c c c} \hline  & **j = 1** & **j = 2** & **j = 3** & **j = 4** & **j = 5** \\ \hline \(A_{j}\) & 0.900 & 0.020 & 0.774 & 0.0258 & 0.612 \\ \(B_{j}\) & 0.043 & 0.031 & 0.026 & 0.064 & 0.026 \\ \(\sigma\)/W (mean value) & 0.035 & 0.005 & 0.018 & 0.004 & 0.008 \\ \(\sigma\)/W (95\% fractile) & 0.054 & 0.008 & 0.025 & 0.006 & 0.0012 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters for the Gaussian-shape spectrum \(SF(f\!f_{p})\)[42].