## 2 Method to Evaluate the Lateral Vibration of Footbridges Including the Postlock-in Behaviour  
The prediction of the lateral response of footbridges is a complex issue, given that it is difficult to set an analytical expression for the pedestrian action that considers the multiple factors involved. The challenge is to find a procedure sufficiently accurate for evaluating the response and easy to apply in practical engineering applications. Therefore, the evaluation of the lateral dynamic response is performed in the frequency domain, a method especially efficient for representing the pedestrian action as a harmonic load. Furthermore, frequency domain analysis allows superposition, which simplifies the formulation. Another advantage is that the structural response can be evaluated independently for the particular mode of vibration in which the lateral instability is more likely to develop, omitting the possible interaction between modes. The response is calculated using an equivalent single degree of freedom (SDOF) system, where the structure is characterised through the system's complex frequency response function _H(f)_ (mN\({}^{-1}\)), which defines completely the dynamic characteristics of the footbridge for the mode considered, and takes the well-known expression:  
\[H(f)=\frac{1}{K-M(2\pi f)^{2}+iC(2\pi f)} \tag{1}\]  
where \(f\) is the variable frequency (Hz), \(K\) the modal stiffness (Nm\({}^{-1}\)), \(M\) the modal mass (kg) and \(C\) the modal damping (Nsm\({}^{-1}\)). For the particular case of lightly damped footbridges, _H(f)_ presents a sharp peak at \(f\) = _f_\({}_{0}\) (the footbridge's natural frequency) and tends to zero when moving away from this value. Hence, the structural response is amplified only in a very narrow frequency interval centred on _f_\({}_{0}\). The pedestrian flow is modelled by its average properties: the number of pedestrians \(N\) and the pedestrian's lateral step frequency _f_\({}_{p}\) (Hz). The step frequency _f_\({}_{p}\), is a random variable with a normal distribution of probability _P(f_\({}_{p}\))_ (mean \(\mu\) = 0.86 Hz and standard deviation \(\sigma\) = 0.08 Hz [39; 40]). Other factors that also influence the pedestrian action (such as the walking speed or the step length) are considered to be represented by the step frequency and the geometric relations between them [39; 40; 41], and thus are not taken into account for characterising the pedestrian flow. In this paper, the step frequency _f_\({}_{p}\) is taken in the interval (\(\mu\)-3\(\sigma\), \(\mu\)+3\(\sigma\)), which represents a confidence level of 99.7%, which is adequate for the accuracy of this study, since it nearly covers the complete pedestrian's excitation range without compromising the computational cost of the problem. The HSI phenomenon (considered to be the main cause of the onset of high amplitudes of vibrations) is explained using a linear feedback model similar to that proposed by [PERSON] [3]. Figure 2 illustrates the feedback system. In Figure 2, the interval of the footbridge natural frequency is defined according to references [7; 12; 22]. With low pedestrian traffic density (prelock-in stage), the modal displacement response _Y(f)_ is obtained as the algebraic product of the modal pedestrian force _X(f)_ exerted by the pedestrians with no bridge motion and the frequency response function _H(f)_. However, as the number of pedestrians grows above the \"critical number\" (postlock-in stage), the footbridge movement affects the pedestrians' gaits. The function _\(\alpha\)(f)_ is the feedback function by which the modal input force _X(f)_ is amplified by the footbridge response _Y(f)_. The modal displacement response _Y(f)_ is obtained in this stage as the algebraic product of the total modal pedestrian force _X(f)+\(\alpha\)(f)xY(f)_ and _H(f)_. The functions _X(f)_ and _\(\alpha\)(f)_, which define the pedestrian action, are adopted from the results of the extensive experimental campaigns that were performed by [PERSON] et al.