# Identification of a Human-Structure Interaction Model on an Ultra-Lightweight FRP Footbridge
## 4 Numerical Study
### FE Model Updating
#### 4.3.2 Calibration  
Among the several approaches available to update a FE model [39], an iterative method was used hereby considering an objective function that quantifies the difference between the experimental and numerical results. The FE model updating procedure using a single-objective approach may be formulated as follows  
Figure 8: Spearman correlation matrix between the parameters of the FE model and the responses of the FRP footbridge.  
\[\text{minimize: }J_{1}(\theta) \tag{1}\] \[\text{subjected to: }\left\{\begin{array}{l}\theta_{l}\leq \theta\leq\theta_{u}\\ w_{f}+w_{m}+w_{\delta}=1\end{array}\right.\]  
being \(\theta\) the parameters to be updated, \(\theta_{l}\) and \(\theta_{u}\) the corresponding lower and upper limits, and \(w\) the weights of the objective function \(J_{1}(\theta)\), which is defined as follows  
\[J_{1}(\theta)=\frac{1}{3}\left[\sum_{i=1}^{N_{\text{s}}}w_{f}\cdot r_{f,i}( \theta)^{2}\right]^{\frac{1}{2}}+\frac{1}{3}\left[\sum_{i=1}^{N_{\text{s}}}w_{ m}\cdot r_{m,i}(\theta)^{2}\right]^{\frac{1}{2}}+\frac{1}{3}\left[\sum_{i=1}^{N_{ \text{s}}}w_{\delta}\cdot r_{\delta,i}(\theta)^{2}\right]^{\frac{1}{2}} \tag{2}\]  
The residuals \(r_{f,i}\), \(r_{m,i}\) and \(r_{\delta,i}\) associated to natural frequencies, mode shapes and static displacements, respectively, are defined by the following expressions  
\[r_{f,i}(\theta) =\frac{f_{num,i}(\theta)-f_{exp,i}}{f_{exp,i}}\qquad i=1,\,2,\, \ldots N_{\text{s}} \tag{3}\] \[r_{m,i}(\theta)^{2} =\frac{\left(1-\sqrt{MAC_{i}(\theta)}\right)^{2}}{MAC_{i}(\theta )}\qquad i=1,\,2,\,\ldots N_{\text{s}}\] \[r_{\delta,i}(\theta) =\frac{\delta_{num,i}(\theta)-\delta_{exp,i}}{\delta_{exp,i}} \qquad i=1,\,2,\,\ldots N_{\delta}\]  
where \(N_{\text{s}}\) is the number of considered vibration modes of the structure and \(N_{\delta}\) is the number of measured static displacements. Subscripts \(num\) and \(exp\) indicate numerical and experimental values, respectively. The parameter \(MAC(\theta)\) gives the orthogonality between two vibration modes and is defined as  
\[MAC_{i}(\theta)=\frac{\left(\phi_{num,i}^{T}(\theta)\cdot\phi_{exp,i}\right) ^{2}}{\left(\phi_{num,i}^{T}(\theta)\cdot\phi_{num,i}(\theta)\right)\cdot \left(\phi_{exp,i}^{T}\cdot\phi_{exp,i}\right)} \tag{4}\]  
To solve the optimization problem, the Genetic Algorithm (GA) was applied in this study involving the use of two software: Abaqus [36] and Matlab [40]. Considering the seven experimental results (Table 1 and the static sag) and the seven \(\theta\) parameters selected previously, the following values were adopted for the updating process of the FE model of the FRP footbridge: \(w_{f}=0.6\), \(w_{m}=0.1\), \(w_{\delta}=0.3\), \(N_{\text{s}}=3\), and \(N_{\delta}=1\). In addition, the search domain for the parameters is presented in Table 3, and the following factors were considered for the GA: initial population of 20 individuals, fitness limit of \(10^{-6}\), and maximum number of generations of 100.  
Four updating runs (GA1-GA4) were carried out with different initial population, and values obtained from the updating run GA2 were adopted as solution for the posterior dynamic analyses of the composite structure. This result was selected considering that the updated values showed the least dispersion from the corresponding initial values. The calibrated natural frequencies are presented in Table 4. A clear reduction of the relative error is achieved while the MAC values remain in good agreement. Also, Table 5 reflects the good compromise between the numerical and experimental displacement at mid-span. In this case, the relative error is below 1%. Finally, Table 6 presents the material properties before and after the calibration procedure.