## 2 Method to Evaluate the Lateral Vibration of Footbridges Including the Postlock-in Behaviour
### Footbridge Response Due to the Auto-Induced Component of the Pedestrian Load
In a real case assessment, there is a crowd of pedestrians walking randomly with different frequencies \(f_{p}\). Hence, using the probability distribution function of the step frequency \(P\)(_f_\({}_{p}\)), and applying the superposition principle, the contribution of each possible value of \(f_{p}\) included in the interval (\(\mu\)-3\(\sigma\), \(\mu\)+3\(\sigma\)) in the auto-induced load exerted by one pedestrian is expressed through the coefficient \(c_{p}\)(_f_\({}_{b}\)) (Nsm\({}^{-1}\)), as follows:  
\[c_{p}(f_{b})=\ \int_{\mu-3\sigma}^{\mu+3\sigma}c_{p}\bigg{(}\frac{f_{p}}{f_{b}} \bigg{)}p\big{(}f_{p}\big{)}df_{p};\ \ \ 0.4\leq f_{b}\leq 1.3 \tag{5}\]  
Figure 4 shows the value of the coefficient \(c_{p}\)(_f_\({}_{b}\)) calculated through Equation (5). It can be seen that between 0.42 and 1.23 Hz, the resonant harmonic of the auto-induced pedestrian force is positive, limiting the footbridge natural frequency interval of possible instability. The next step is to calculate the spectral value of the auto-induced resonant harmonic exerted by one single pedestrian \(|F\)(_f_\({}_{b}\)) | expressed in Newton (N), which can be computed by the product of the coefficient \(c_{p}\)(_f_\({}_{b}\)) (Equation (5)) and the amplitude of the footbridge velocity (\(y_{0}\)'= 2\(\pi\)\(f_{b}\)\(y_{0}\) (m/s)), as follows:  
\[\big{|}F(f_{b})\big{|}=\ c_{p}(f_{b})2\pi f_{b}y_{0} \tag{6}\]  
In the real case of \(N\) pedestrians uniformly distributed on a span of length \(L\) (m), assuming that the studied vibration mode is the first one (mode shape sine lateral wave of effective length \(L_{d}\) (m)),  
Figure 4: The damping coefficient _cp(f)_ independent of the pedestrian step frequency \(f_{p}\) obtained through Equation (5). Figure 3: Statistical characterisation of damping coefficient _cp(f)(f)_ lateral amplitude dependent [17] vs. polynomial best fit lateral amplitude independent [42]. the equivalent spectral value of the auto-induced resonant harmonic exerted by \(N\) pedestrians \(|F_{N}(f_{b})|\) in Newton (N) is expressed in modal coordinates, as follows:  
\[\big{|}F_{N}(f_{b})\big{|}=\frac{N}{L}\bigg{(}\int_{0}^{L_{d}}\sin\frac{\pi x}{L _{d}}dx\bigg{)}c_{p}(f_{b})2\pi f_{b}u_{0,N}=N\frac{L_{d}}{L}4f_{b}c_{p}(f_{b})u _{0,N} \tag{7}\]  
in which \(u_{0,N}\) is the modal displacement amplitude of the footbridge due to \(N\) pedestrians before the HSI develops. Consequently, the spectral value of the auto-induced displacement response, \(|Y_{N}(f_{b})|\), is computed by the product of the spectral value of the auto-induced resonant harmonic exerted by \(N\) pedestrians, \(|F_{N}(f_{b})|\) (Equation (7)), and the resonant amplitude of the complex response function, \(|H(f_{b})|\) (Equation (1) particularised in the natural frequency \(f_{b}\)), resulting:  
\[\big{|}Y_{N}(f_{b})\big{|}=\big{|}F_{N}(f_{b})\big{|}\big{|}H(f_{b})\big{|}=N \frac{L_{d}}{L}4f_{b}c_{p}(f_{b})u_{0,N}\big{|}H(f_{b})\big{|} \tag{8}\]  
Considering Equation (8), the amplitude of the resonant displacement response in the time domain can be computed doubling the spectral value \(|Y_{N}(f_{b})|\). Consequently, the amplitude of the resonant acceleration response due to the auto-induced component \(a_{v,N}\) (m/s\({}^{2}\)) can be expressed in terms of amplification ratio to the modal acceleration amplitude \(a_{0,N}\) (m/s\({}^{2}\)) without pedestrian-structure interaction, as follows:  
\[\frac{a_{v,N}}{a_{0,N}}=N\bigg{[}\frac{L_{d}}{L}8f_{b}c_{p}(f_{b})\big{|}H(f_{ b})\big{|}\bigg{]} \tag{9}\]  
The right side of Equation (9) has two factors. One is the number of pedestrians and the other depends on the modal properties of the structure, which are constant for a specific mode of vibration.