## 2 Pedestrian Load Model  
### Single-Person Foot Force Model  
The vibration of the pedestrian bridge is caused by the dynamic load of the crowd. The pedestrian bridge will generate a dynamic force of time-range force under the influence of acceleration changes during the crowd movement, which contains three components in three directions: namely, the vertical force along the direction of gravity, the horizontal lateral force perpendicular to the walking direction, and the horizontal longitudinal force along the walking direction. The single person load measured by [PERSON] [32] with the force plate method was selected as the load model in this paper.  
However, the environment of footstep force loads measured on a fixed platform was different from the actual environment of pedestrians crossing a pedestrian bridge. When the self-oscillation frequency of the bridge is similar to the frequency of the pedestrian's pace, a locking phenomenon where pedestrians adjust their pace to synchronize with the bridge vibration will occur. After the vibration event of the Millennium Bridge, experiments and studies on the lateral vibration of pedestrian foot forces were conducted [33; 34], and the results indicated that the synchronization rate of the pedestrian and the main beam is an important factor affecting the lateral response of the main beam. Considering the above factors, the pedestrian foot load in a vibrating environment was investigated, and a time function for the dynamic load of a single human footstep was proposed using the statistical analysis method [35]:  
\[F_{p}(t)=W+W\sum_{i=1}^{n}\alpha_{i}\sin(2\pi if_{p}t-\varphi_{i}) \tag{1}\]  
where \(f_{p}\) is the pedestrian step frequency, which is equal to the total number of steps per second when calculating the vertical load and the lateral and vertical forces. Since there is only one cycle of change for each step of the left and right foot, it is half of the vertical value. \(W\) is the average pedestrian gravity. \(\alpha_{i}\) is called the simple harmonic dynamic load factor of the \(i\)-th order, which is expressed as follows:  
\[\begin{array}{l}\alpha_{1}=0.41f_{p}-0.39\leq 0.56\\ \alpha_{2}=0.069+0.056f_{p}\\ \alpha_{3}=0.033+0.064f_{p}\end{array} \tag{2}\]  
The product \(W\alpha_{i}\) is the magnitude of the first-order dynamic load. (\(\varphi_{i}\), denotes the initial phase of the \(i\)-th order dynamic load which is usually taken as \(\varphi_{1}=0\), thus, \(\varphi_{i}\), \(i=2\), \(3\), \(4\dots\) are the phase difference of the \(i\)-th order dynamic load to the first order dynamic load. The value of Fourier series order \(n\) is related to the required degree of approximation. It is generally considered that the first three orders of vertical loads, the first two orders of longitudinal loads, and the first order of transverse loads have sufficient accuracy. Numerous studies have shown that the dynamic load factor a is an increasing function of the step frequency; the vertical dynamic load coefficient. The values of the above parameters are given in Ref. [36].  
It should be noted that the above mathematical load model of a single human footstep force is the summation of the static load expressed in the Fourier series and several simple harmonic dynamic loads. It is generally considered that the first three orders of vertical loads, the first two orders of longitudinal loads, and the first order of lateral loads have sufficient accuracy.