# Identification of a Human-Structure Interaction Model on an Ultra-Lightweight FRP Footbridge
## 4 Numerical Study
### Human-Structure Interaction Model  
Based on [PERSON] et al. [29], a SDOF MSDA system was adopted to address the interaction phenomenon when a person crosses the footbridge. Therefore, the HSI model was comprised of a simply supported beam and the human walking. A schematic representation of the model is shown in Figure 10a considering that the pedestrian is located at midspan. Additionally, Figure 10b displays the corresponding block diagram. \begin{table}
\begin{tabular}{l c c c} \hline \hline \(r\) & **Range (Hz)** & **DLF** & \(f_{a}=1.9\) Hz \\ \hline
1 & 1.2–2.4 & 0.37\((f_{a}-1.0)\) & 0.333 \\
2 & 2.4–4.8 & 0.10 & 0.100 \\
3 & 3.6–7.2 & 0.06 & 0.060 \\
4 & 4.8–9.6 & 0.06 & 0.060 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Vertical MF model of a single pedestrian walking according to ISO-10137 [15]. Figure 9: Comparison of 1s RMS acceleration using a non-interacting MF model. From the graph, the closed-loop TF \((GH_{CL})\) of the coupled human-structure system in the Laplace domain can be derived as follows  
\[GH_{CL}(s)=\frac{G_{S}(s)\cdot G_{H}(s)}{1+G_{S}(s)\cdot G_{HSI}(s)} \tag{7}\]  
where \(s=j\omega\) is the Laplace variable, \(\omega\) is the circular frequency in rad/s, \(G_{S}(s)\) is the TF of the structural system, \(G_{H}(s)\) is the TF between the actuator force (human driving force) and the contact force of the human with the structure, and \(G_{HSI}(s)\) is the TF related to the interaction phenomenon. Generally, the TF between the structure acceleration at midspan and a force applied at a point \(x\), considering only the fundamental vibration mode, is defined as follows  
\[G_{S}(s)=\frac{s^{2}Y_{s}(s)}{\phi(x)\cdot F_{h}(s)}=\frac{1/m_{s1}\cdot s^{2}} {s^{2}+2\omega_{s1}\xi_{s1}s+\omega_{s1}^{2}} \tag{8}\]  
being \(\omega_{s1}=2\pi f_{s1}\) the circular natural frequency of the 1 st vibration mode of the structure in rad/s, \(F_{h}(s)\) the Laplace transform of the applied force acting on the structure, and \(s^{2}Y_{s}(s)\) the Laplace transform of the structural acceleration at midspan, where \(Y_{s}(s)\) denotes the Laplace transform of the structural vertical displacement. Additionally, \(\phi(x)\) is the mode shape of a simply supported beam that considers the variation of the pedestrian position \(x\), so the following expression is employed  
\[\phi(x)=\sin\left(\frac{\pi x}{L}\right) \tag{9}\]  
For the model of the person, mass \((m_{h})\), natural frequency \((f_{h})\) and damping ratio \((\zeta_{h})\) of the human body are considered. Also, the human driving force \((F_{a})\) is accounted for via a pair of action-reaction forces acting simultaneously on both the footbridge and the pedestrian at a location \(x\). Thus, the ground reaction force is equal to the sum of the actuator force and the forces caused by the spring and damper of the human system. The TF between \(F_{ha}\), which is the force generated by the humans without including the force transmitted to them due to the structure movement, and the human driving force is presented below  
\[G_{H}(s)=\frac{F_{ha}(s)}{F_{a}(s)}=\frac{-s^{2}}{s^{2}+2\omega_{h}\zeta_{h}s+ \omega_{h}^{2}} \tag{10}\]  
where \(\omega_{h}=2\pi f_{h}\) is the circular frequency of the body in rad/s, and \(\zeta_{h}\) is the associated damping ratio of the human. \(F_{a}\), which is the input force to the \(GH_{CL}\) model, can be defined as follows  
\[F_{a}(t)=W_{h}\sum_{r=1}^{N_{r}}\mathrm{GLF}_{r}\cdot\sin\left(r\cdot 2\pi f_{a }t+\phi_{r}\right) \tag{11}\]  
where the values of GLF\({}_{r}\) are not the standard DLF\({}_{r}\) as \(F_{a}\) affects both, the human body and the structure. The TF between the human interacting force \((F_{bisi})\) applied at location \(x\)  
Figure 10: HSI model: (**a**) Schematic representation. (**b**) Block diagram.