## 3 Real-Time Hybrid Simulation (RTHS)  
### Numerical Substructure (NS): Reference Footbridge Model  
The time-history dynamic response of the reference footbridge, described in Section 2.1, under interaction with human-induced gait loads, was assessed using an analytical continuous model represented by a fourth-order partial differential equation (PDE), Equation (2), which constitutes the numerical substructure (NS). In this equation, \(u(x,t)\) represents the deflection of any point along the \(x\)-axis of the structure for all the simple times evaluated. Likewise, \(p(x,t)\) includes human-induced gait loads as dynamic perturbations in the system that estimates the reference footbridge, where \(m(x)\) and \(c(x)\) are constants that represent the mass and viscous damping coefficient of the dynamic system, respectively. \[\frac{\partial^{2}}{\partial x^{2}}\bigg{[}EI(x)\frac{\partial^{2}u(x,t)}{ \partial x^{2}}\bigg{]}+m(x)\frac{\partial^{2}u(x,t)}{\partial x^{2}}+c(x) \frac{\partial u(x,t)}{\partial t}=p(x,t) \tag{2}\]  
The variable separation method was used to resolve the posed equation, which describes the dynamic behavior of a reference footbridge under interaction with human gait. This solution technique incorporates modal decomposition, as shown in Equation (3), which employs vibration models represented by sinusoidal functions as shown in Equation (4). The dynamic motion (\(Y_{\hat{n}}(t)\)) of the \(n\)-th in modal coordinates (\(\phi_{\hat{n},\hat{i}}(x)\)) is determined using Equation (5), where the natural frequencies (\(\omega_{\hat{n},\hat{i}}\)) and structural damping ratio (\(\xi_{i}\)) correspond to the parameters described in Section 2.1 (Table 1). The mass (\(M_{\hat{n}}\)) and modal loads (\(P_{\hat{n}}(t)\)) were calculated using Equations (6) and (7), respectively. \[u(x,t)=\sum_{\hat{n}=1}^{\infty}\phi_{\hat{n}}(x)\cdot Y_{\hat{n}}(t) \tag{3}\]  
\[\phi_{\hat{n}}(x)=sin\bigg{(}\frac{\hat{n}\pi}{L}x\bigg{)} \tag{4}\]  
\[\ddot{Y}_{\hat{n}}(t)+2\cdot\xi\cdot\omega_{\hat{n}}\cdot\dot{Y}_{\hat{n}}(t)+ \omega_{\hat{n}}{}^{2}\cdot Y_{\hat{n}}(t)=\frac{P_{\hat{n}}(t)}{M_{\hat{n}}} \tag{5}\]  
\[M_{\hat{n}}=\int_{0}^{L}\big{[}\phi_{\hat{n}}(x)\big{]}^{2}\cdot m(x)\;dx \tag{6}\]  
\[P_{\hat{n}}(t)=\int_{0}^{L}\phi_{\hat{n}}(x)\cdot p(x,t)\;dx \tag{7}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**ID** & **Age [yr]** & **Height [cm]** & **CM [cm]** & **BM [kg]** & **S** \\ \hline
1 & 22.0 & 169.0 & 100.0 & 68.3 & M \\
2 & 25.0 & 170.0 & 95.0 & 85.6 & M \\
3 & 27.0 & 153.0 & 97.0 & 66.5 & f \\
4 & 33.0 & 170.0 & 110.0 & 65.2 & M \\
5 & 22.0  
\[p(x,t)=\left[\sum_{i=1}^{n}\!m_{i}\!\cdot\!\left(\tilde{a}_{i}(t)-g+\tilde{a}_{s,j}( t)\right)\right]\!\cdot\!\delta\!\left(x-x_{fb}(t)\right) \tag{8}\]  
Human-induced gait loads in generalized coordinates (\(p(x,t)\)) were formulated as the sum of the inertial load of the \(n\)-th human body segment affected by the dynamic interaction of the footbridge structural acceleration effects (\(\tilde{a}_{s,j}(t)\)), which act in the \(x_{fb}(t)\) position on the reference structure for all times evaluated, as expressed in Equation (8). In this equation, \(g\), \(x\), and \(x_{fb}\) represent the gravity acceleration (only in the \(y\)-axis), longitudinal axis of the reference structure, and current position of the anthropic loads on the footbridge, respectively. The multi-axial structural accelerations generated by the human-induced gait loads were calculated in modal coordinates using Equation (9), which was based on the position of the anthropic load on the structure and \(n\)-th vibration modes assessed. The human-induced gait loads expressed in modal coordinates were obtained by replacing Equations (4) and (8) in Equation (7) in order to obtain Equation (10), which depends on the human-induced gait load (\(F_{human}(t)\)), modal information, crowd modal mass, and multi-axial structural acceleration (\(\tilde{a}_{s,j}(t)\)).