## 5 Model Updating for the Segmental Beam
### Model Updating Based on the Measured Static Data  
The objective of the model updating is to minimize the objective function H(\(\theta\)) formulated as the difference in the deflections and neutral axis locations between the experimentally measured and numerical results when the prestress level of the beam is 107 kN for obtaining the optimal value of the candidate parameters. In this paper, the RS method [30] is employed for updating the selected parameters. The second-order polynomial function with interaction effects is used to build the relationship between the physical system parameters (i.e., E\({}_{\text{C}}\), \(\rho_{\text{C}}\), E\({}_{\text{S}}\)) and output response in terms of vertical deflection and neutral-axis location. \[\text{H}(\theta)=\text{ H}_{\text{L}_{\alpha-15}}(\theta)+\text{H}_{\text{L}_{ \alpha-20}}(\theta)+\text{H}_{\text{S}_{2}}(\theta)+\text{H}_{\text{S}_{3}}(\theta) \tag{6}\]  
where  
\[\text{H}_{\text{L}_{\alpha-15}}(\theta)=\sum_{\text{i=1}}^{n}\left(\frac{ \text{RSM}_{\text{L}_{\alpha-15}}(\theta)-\text{EL}_{\alpha\_15}}{\text{EL}_{ \alpha\_15}}\right)^{2} \tag{7}\]  
\[\text{H}_{\text{L}_{\alpha-20}}(\theta)=\sum_{\text{i=1}}^{n}\left(\frac{ \text{RSM}_{\text{L}_{\alpha-20}}(\theta)-\text{EL}_{\alpha\_20}}{\text{EL}_{ \alpha\_20}}\right)^{2} \tag{8}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{**Parameters**} & \multirow{3}{*}{**Changes of Parameters (\%)**} & \multicolumn{3}{c}{**Sensitivity of Displacement Responses with Respect to the**} & \multicolumn{3}{c}{**Sensitivity of Neutral Axis Location (\%)**} \\ \cline{3-7}  & & & **Parameter (\%)** & & **Parameter (\%)** & & **Parameter (\%)** \\ \cline{3-7}  & & & **ML1** & **ML23** & **ML4** & **MNAL2** & **MNAL2** \\ \hline E\({}_{\text{C}}\) & 10 & 8.04 & 8.04 & 8.04 & 0.14 & 0.08 \\ \(\rho_{\text{C}}\) & 10 & 1.38 & 1.40 & 1.37 & \(-0.18\) & \(-0.03\) \\ E\({}_{\text{S}}\) & 10 & \(-10.65\) & \(-10.67\) & \(-10.65\) & 0.41 & 0.25 \\ NSct & 10 & 2.02 & 2.06 & 2.15 & \(-0.01\) & \(-0.02\) \\ TSct & 10 & 0.00 & 0.00 & 0.00 & 0.04 & \(-0.01\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Sensitivity analysis results of the candidate model parameters. \[\mathrm{H}_{\mathrm{S}_{2}}(\theta)=\sum_{\mathrm{i=1}}^{\mathrm{n}}\left(\frac{ \mathrm{RSM}_{\mathrm{S}_{2}}(\theta)-\mathrm{ENAL}_{2}}{\mathrm{ENAL}_{2}} \right)^{2} \tag{9}\]  
\[\mathrm{H}_{\mathrm{S}_{3}}(\theta)=\sum_{\mathrm{i=1}}^{\mathrm{n}}\left(\frac{ \mathrm{RSM}_{\mathrm{S}_{3}}(\theta)-\mathrm{ENAL}_{3}}{\mathrm{ENAL}_{3}} \right)^{2} \tag{10}\]  
in which the vector of parameters to be updated is defined as \(\theta=[\mathrm{E}_{\mathrm{c}},\ \rho_{\mathrm{c}},\ \mathrm{E}_{\mathrm{s}}]\), \(\mathrm{EL}_{\alpha\_15}\) and \(\mathrm{EL}_{\alpha\_20}\) represent the measured static displacements of the segmental beam at location \(\alpha\) corresponding to the four LVDTs during the static loading tests of 15 kN and 20 kN, respectively, and \(\mathrm{RSM}_{\mathrm{L}_{\alpha-15}}(\theta)\) and \(\mathrm{RSM}_{\mathrm{L}_{\alpha-20}}(\theta)\) are the respective predicted static displacements at location \(\alpha\) from the numerical analysis. The term \(\mathrm{RSM}_{\mathrm{S}_{2}}(\theta)\) denotes the predicted neutral axis locations of the second segment during the post-tensioning process, and \(\mathrm{RSM}_{\mathrm{S}_{3}}(\theta)\) denotes the predicted neutral axis locations of the third segment of the tested beam. The estimated RS function can thus be expressed as:  
\[\mathrm{RSM}(\theta)=\ \beta_{0}+\sum_{\mathrm{i}}^{\mathrm{n}}\beta_{\mathrm{i}} \theta_{\mathrm{i}}+\sum_{\mathrm{i}}^{\mathrm{n}}\beta_{\mathrm{ii}}\theta_{ \mathrm{i}}^{2}+\sum_{\mathrm{i<j}}^{\mathrm{n}}\beta_{\mathrm{ij}}\theta_{ \mathrm{i}}\theta_{\mathrm{j}} \tag{11}\]  
where \(\beta_{\mathrm{i}}\) denotes the partial regression coefficient, \(\theta_{\mathrm{i}}\) represents the model parameters, and \(\theta_{\mathrm{i}}\theta_{\mathrm{j}}\) denotes the interaction effect between these parameters.