## 3 FE Model Updating with Experimental Modal Data
### Sensitivity-Based FE Model Updating  
A sensitivity-based FE model updating approach was applied to the tuned model in order to obtain more accurate numerical results. The updating process was formulated into a numerical optimization problem. It aimed at finding the best model parameters that minimizes the difference between the numerical and experimental modal data. Herein, the following nonlinear least-squares problem had to be solved:  
\[\theta^{*}=\underset{\theta}{\operatorname{argmin}}\mathrm{J}(\theta)= \underset{\theta}{\operatorname{argmin}}\varepsilon_{\mathrm{z}}(\theta)^{T} \mathbf{W}_{\epsilon}\,\varepsilon_{\mathrm{z}}(\theta)=\underset{\theta}{ \operatorname{argmin}}\sum_{\mathrm{i}}\mathrm{w}_{\epsilon,\mathrm{i}}( \varepsilon_{\mathrm{z,i}}(\theta))^{2}, \tag{1}\]  
\(\varepsilon_{\mathrm{z}}\) denotes the residuals between the experimental and numerical modal data z. For the undamped eigenvalue \(\mathrm{z}=\lambda\), the residuals \(\varepsilon_{\lambda,\mathrm{i}}(\theta)\) are given by \begin{table}
\begin{tabular}{c c c c c} \hline
**Components** & **E (MPa)** & **n** & **G (MPa)** & **q (kg/m\({}^{3}\))** \\ \hline
**Box Girder** & 32,373 & 0.2 & 13,489 & 2500 \\
**Slab** & 32,373 & 0.2 & 13,489 & 2500 \\
**Side Barriers** & 16,187 & 0.2 & 6745 & 2500 \\ \hline \end{tabular} Note: E for modulus of elasticity, u for Poisson’s ratio, G for shear modulus and q for density.  
\end{table} 
Table 2: Material properties of the initial FE model. Figure 6: FE models built with 3D solid elements: (**a**) the initial model; (**b**) the manually tuned model. \[\varepsilon_{\lambda,\text{i}}(\theta)=\frac{\lambda_{\text{i}}(\theta)-\widetilde{ \lambda_{\text{i}}}}{\widetilde{\lambda_{\text{i}}}},\text{ for }\lambda_{\text{i}}{=\omega_{\text{i}}}^{2}{=(2\pi\text{f}_{\text{i}})^{2} \text{ and i }\in\{1,\;2,\;\;\cdots,\;\text{n}_{\lambda}\}}, \tag{2}\]  
where the upper tilde denotes the experimental values and f (and w) stands for the (angular) natural frequency. For the mode shape z =\(\Phi\), the residuals \(\varepsilon_{\phi,\text{i}}(\theta)\) are given by  
\[\varepsilon_{\phi,\text{i}}(\theta)=\frac{\boldsymbol{\Phi_{\text{i}}}( \theta)-\widetilde{\boldsymbol{\Phi_{\text{i}}}}}{\|\widetilde{\boldsymbol{ \Phi_{\text{i}}}}\|_{1}},\text{ for i }\in\big{\{}1,2,\ldots,n_{\phi}\big{\}}, \tag{3}\]  
where both the experimental and numerical mode shape vectors (\(\widetilde{\boldsymbol{\Phi_{\text{i}}}}\) and \(\Phi_{\text{i}}\)) are normalized to the maximum unity by considering only the real parts of \(\widetilde{\boldsymbol{\Phi_{\text{i}}}}\). \(\|\mathbf{x}\|_{1}{:=}{\sum\limits_{\text{j}}}\big{|}\mathbf{x}_{\text{j}}\big{|}\) is the sum of the absolute values of the vector \(\mathbf{x}\), according to the 1-norm. In Equation (1), \(\boldsymbol{W}_{\varepsilon}\) is the diagonal weighting matrix, which provides a means to balance the contributions from each individual residual. In detail, the following weighting factors were considered:  
\[\text{w}_{\varepsilon_{\lambda,\text{i}}}=1,\text{ for i }\in\{1,\;2,\;\;\cdots,\;\text{n}_{\lambda}\}\text{ and }\text{w}_{\varepsilon_{\phi,\text{i}}}{=10,\text{ for i }\in\{1,\;2,\;\;\cdots,\;\text{n}_{\phi}\big{\}}}, \tag{4}\]  
in order to provide a heavier weight for the mode shapes. Equation (1) was solved by using the in-built MATLAB function \"Isqnonlin\". It is a trust-region-reflective optimization algorithm, also known as the Levenberg-Marquardt method. The iterative process was terminated when a certain convergence criterion was satisfied, such as the relative change in \(\theta\) being less than the specified tolerance. A preliminary sensitivity study was performed in order to decide the design variables for model updating.