# Identification of a Human-Structure Interaction Model on an Ultra-Lightweight FRP Footbridge
## 4 Numerical Study
### Identification of the HSI Model  
Based on the experimental 1s RMS acceleration, presented in Section 3.3, the dynamic parameters of the human body and the GLFs coefficients for the first four harmonics of a walking pedestrian are identified herein. Again, an optimization problem was formulated employing a single-objective approach, so the employed expression is as follows  
minimize: \[J_{2}(\mathbf{\eta})\] (13) subjected to: \[\mathbf{\eta}_{l}\leq\mathbf{\eta}\leq\mathbf{\eta}_{u}\]  
being \(J_{2}(\mathbf{\eta})\) the RMSE between the experimental and numerical 1s running RMS, \(\mathbf{\eta}\) the parameters to be obtained, and \(\mathbf{\eta}_{l}\) and \(\mathbf{\eta}_{u}\) the lower and upper limits, respectively.  
For the identification, the model of the structure was assumed to present an equivalent modal mass (\(m_{s1}\)) of 405 kg and a natural frequency (\(f_{s1}\)) of 7.63 Hz, based on the results from the calibrated FE model (Section 4.3). Also, 1.55% was considered as damping ratio of the structure (\(\zeta_{s1}\)), considering the experimental result from Section 3.2. The velocity of the MSDA system, which represents the pedestrian, was set equal to 1.3 m/s (Section 3.3). The lower and upper limits for the properties of the human body were based on the mean mass value obtained among the test subjects shown in Table 2 (70.8 kg) and values reported in literature ([PERSON], 1966). Thus, the search domain for the seven \(\mathbf{\eta}\) parameters to be determined were:  
* Human body: \(m_{h}\in[0.9,1.0]\cdot 70.8\) kg, \(f_{h}\in[1.5,3.0]\) Hz, and \(\zeta_{h}\in[0.1,0.6]\).
* Load factors: \(\mathrm{GLF}_{1}\in[0.01,0.30]\), \(\mathrm{GLF}_{2}\in[0.01,0.30]\), \(\mathrm{GLF}_{3}\in[0.01,0.20]\) and \(\mathrm{GLF}_{4}\in[0.01,0.20]\).  
The GA, available in MATLAB (MattLab, 2017), was used to solve the optimization problem. An initial population size of 20, a fitness limit of \(10^{-6}\), and a maximum number of generations of 100 were considered. Taking advantage of the simple HSI model based on the closed-loop TF \(GH_{CL}\), one hundred runs were performed (GA1-GA100). From the results, 25 solutions were disregarded as valid outcomes given the achieved value of the objective function \(J_{2}\) was greater than 5%. Figure 11 displays the RMSE obtained in each run. Accounting for the remaining 75 outcomes, a single solution was calculated by averaging the values of the obtained \(\mathbf{\eta}\) parameters. Hence, the SDOF MSDA system was defined using the following information: \(m_{h}=0.93\cdot 70.8\) kg, \(f_{h}=1.88\) Hz, \(\zeta_{h}=23.4\%\), \(\mathrm{GLF}_{1}=0.107\), \(\mathrm{GLF}_{2}=0.172\), \(\mathrm{GLF}_{3}=0.101\), and \(\mathrm{GLF}_{4}=0.032\).  
Figure 12 displays the experimental outcomes and the numerical outputs obtained from the identified HSI model (\(GH_{CL}\)). The RMSE between both curves was 3.4%. In comparison with the results calculated using the MF model (Figure 9), the prediction of the real response of the footbridge was clearly improved after adopting the SDOF MSDA system to represent a walking pedestrian.