# Identification of a Human-Structure Interaction Model on an Ultra-Lightweight FRP Footbridge
## 4 Numerical Study
### Moving Force Models  
Several non-interacting MF models have been proposed over the years to represent the vertical walking action of a single pedestrian [12; 15; 17]. This human load may be modelled as a periodical function defined by a Fourier series, which typical expression (disregarding the static load term) in the time domain is as follows  
\[F(t)=W_{h}\sum_{r=1}^{N_{r}}\mathrm{DLF}_{r}\cdot\sin\left(r\cdot 2\pi f_{a}t+ \phi_{r}\right) \tag{5}\]  
with \(r=1\), \(2\), \(\dots\)\(N_{r}\), where \(N_{r}\) is the total number of harmonics considered, \(r\) is the harmonic number, \(W_{h}\) is the static weight of a person usually taken as 700 N, \(\mathrm{DLF}_{r}\) is the dynamic loading factor associated to the \(r\)th harmonic, \(f_{a}\) is the frequency component of the repetitive loading, and \(\phi_{r}\) is the phase or lag angle of the \(r\)th harmonic. From the second harmonic onwards, \(\phi_{r}\) can be taken equal to \(\pi/2\) as a conservative approach [15]. Even though traditional guidelines and codes imply that footbridges with fundamental vertical frequencies above to 5 Hz are unlikely to be excited by higher harmonics of a human activity (Figure 1), significant vertical vibrations were seen when the fourth harmonic of a single pedestrian walking was synchronised with the fundamental frequency of the lightweight FRP facility studied hereby (Figure 6c). The only guideline that accounts for MF models with four harmonics to describe a walker is ISO-10137 [15]. Thus, the numerical response of the FRP structure was calculated applying its recommendations even though ISO-10137 [15] sets design guidance against building floor vibrations. Table 7 presents the values of \(\mathrm{DLF}_{r}\) for a step frequency (\(f_{a}\)) of 1.9 Hz accounting for four harmonics (\(N_{r}=4\)). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters Updated** & **Initial Values** & **Updated Values** & **Change (\%)** \\ \hline \(\rho_{GFR}\) (kg/m\({}^{3}\)) & 1800.0 & 1759.44 & \(-2.3\) \\ \(E_{1-St}\) (GPa) & 24.0 & 25.91 & 8.0 \\ \(G_{12-St}\) (GPa) & 3.0 & 3.18 & 6.0 \\ \(E_{2-D}\) (GPa) & 8.0 & 8.71 & 8.9 \\ \(G_{12-D}\) (GPa) & 3.0 & 3.27 & 9.0 \\ \(E_{1-C}\) (GPa) & 139.0 & 151.22 & 8.8 \\ \(E_{c}\) (GPa) & 34.5 & 37.04 & 7.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters of the FRP footbridge before and after updating. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{**Description**} & \multicolumn{2}{c}{**Frequency (Hz)**} & \multicolumn{2}{c}{**Error (\%)**} & \multicolumn{2}{c}{**MAC Values**} \\ \cline{2-7}  & **OMA** & **Initial** & **Updated** & **Before** & **After** & **Before** & **After** \\  & **Data** & **FEM** & **FEM** & **Updating** & **Updating** & **Updating** & **Updating** \\ \hline Mode 1 & 7.66 & 7.25 & 7.63 & 5.4 & 0.4 & 0.99 & 0.99 \\ Mode 2 & 10.96 & 10.01 & 10.49 & 8.7 & 4.3 & 0.97 & 0.97 \\ Mode 3 & 15.01 & 14.67 & 15.35 & 2.3 & \(-2.3\) & 0.99 & 0.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of the modal analysis before and after updating. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{**Description**} & \multicolumn{2}{c}{**Displacement (mm)**} & \multicolumn{2}{c}{**Error (\%)**} & \multicolumn{2}{c}{**After**} \\ \cline{2-7}  & **Test** & **Initial** & **Updated** & **Before** & **After** \\  & **Data** & **FEM** & **FEM** & **Updating** & **Updating** & **Updating** \\ \hline Midspan & 12.8 & 13.9 & 12.7 & \(-8.6\) & 0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results of the static analysis before and after updating. Considering the updated FE model and the MF from ISO-10137 [15] travelling at 1.3 m/s, the numerical response of the footbridge was obtained through the FE Method using Abaqus [36].