## 2 Pedestrian Load Model
### Multi-Person Foot Force Model  
The correlation between pedestrians and the coupling effect between pedestrians and bridge structures led to difficulties in calculating and simulating the multi-person load. Hence, for the convenience of calculation, multi-person loads were classified into three types: small groups walking in pairs, low-density crowds walking freely, and high-density crowds moving [36]. The dynamic foot force load of a group of pedestrians can be estimated by multiplying the first-order harmonic component of a single foot force load by the number of people in the group:  
\[F_{p}(t)=W\cdot\alpha\cdot\sin(2\pi f_{p}t)\cdot n \tag{3}\]  
where \(n\) is the number of people in the group. For the low-density crowd walking freely, it is assumed that there is a uniformly distributed crowd flow with speed of \(1.5\) m/s, and the total number of people maintained on the bridge, \(n\), is constant. The step frequency \(f_{p}\) of these \(n\) people obeys normal distribution, and its mean value is exactly equal to the frequency of the \(i\)-th order mode of the bridge \(f_{i}\). Simultaneously, the phase difference \(\psi\) consisting of \(n\) individuals is uniformly distributed between 0-2\(\pi\), so that the step frequency and phase of the \(j\)-th person on the bridge can be expressed as follows:  
\[f_{j}=f_{i}+\sigma\cdot\mu_{j} \tag{4}\]  
\[\psi_{j}=2\pi\cdot v_{j} \tag{5}\]  
where \(\mu_{j}\) is the standard normal distribution random number; \(v_{j}\) is a uniformly distributed random number in the interval (0, 1). Due to the difficulties in synchronizing the second-, third- and higher-order components of the vertical load, only the first-order component of the vertical load is considered. Then the load of the \(j\)-th individual at time \(t\) is as follows:  
\[F_{j}(t)=\left.W\cdot\alpha\cdot\sin(2\pi f_{j}t-\psi_{j})\right|_{x=1.5t} \tag{6}\]  
Assuming the \(i\)-th order vibration function is \(\varphi_{i}(x)\) and the damping of the vibration mode is \(\xi\), the time history curve of the maximum acceleration response can be calculated by the theory of structural dynamics. Applying the Monte Carlo method, a sufficient number of time history sample curves can be obtained and graded according to the maximum acceleration. The number of samples at each level are counted to ensure that the maximum acceleration of the bridge is excited. The total length of the calculated time course is twice as long as the single pedestrian passing the bridge at time \(T\) (equal to the bridge length divided by the speed 1.5 m/s), i.e., 2 \(T\). Then, it is assumed that there are \(n_{p}\) (\(n_{p}<n\)) pedestrians uniformly distributed on the same bridge of the same order of vibration, whose step frequency is equal to \(f_{i}\), and the phase difference between each other, that is, \(\psi_{i}=0\), and the direction of the applied force always take the direction of increasing displacement. In this way, the action of each pedestrian is to produce the maximum acceleration of the bridge, if the maximum acceleration of this \(n_{p}\) pedestrian is equal to the maximum acceleration produced by the n pedestrians walking randomly in front, then \(n_{p}\) is the equivalent number of the \(n\) pedestrians walking randomly and completely synchronized. According to a large number of numerical simulations using the random variable method, the equivalent crowd size for a low-density crowd walking freely proposed by the French Guide is calculated as below:  
\[n_{p}=10.8\sqrt{n\cdot\xi_{i}}\left(\text{when the crowd density }<1.0\text{ person}/\text{m}^{2}\right) \tag{7}\]  
This formula assumes that at the i-th order of vibration of a bridge, there are \(n_{p}\) pedestrians uniformly distributed on the bridge; not only is the step frequency equal to \(f_{i}\), no phase differences occur between each other, and the direction of the applied force always takes the direction of increasing displacement.