## 5 Model Updating for the Segmental Beam
### Discussions  
From the above results, the prestress force calculated based on the strains measured at locations away from the segment joint is closer to the real values, while there is a relatively larger difference by using the strains near the joint of the beam. In this section, the stress distributions of the beam at the corresponding locations from segments S2 and S3 are extracted \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Cases**} & \multicolumn{3}{c}{**Neural Axis Location**} \\ \cline{2-4}  & & **MNAL2** & **MNAL3** \\ \hline \multirow{2}{*}{RE (\%)} & Original & \(-1.16\) & \(-1.90\) \\  & Updated & 0.99 & 1.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Defined error indices for the updated model under the prestressing force of 62 kN. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Applied Prestress Force (kN)** & **TF2 (kN)** & **RE (\%)** & **TF3 (kN)** & **RE (\%)** \\ \hline
62 & 59.95 & \(-3.31\%\) & 65.10 & 5.00\% \\
70 & 69.10 & \(-1.29\%\) & 77.95 & 11.36\% \\
80 & 80.21 & 0.26\% & 91.18 & 13.98\% \\
90 & 92.52 & 2.80\% & 103.39 & 14.88\% \\
107 & 109.18 & 2.04\% & 118.27 & 10.53\% \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Comparison between the applied and predicted prestress forces from the proposed approach. from the ANSYS model. Figure 10 shows the stress distributions when the prestress force is 107 kN. As shown in the stress contour, the stress over the cross-section in S3 close to the joint obviously does not distribute linearly. The assumption of plane-remains plane therefore is not satisfied at this section close to the joint, which results in the large errors in prestress estimation from neutral axis location. To further observe this, the stress values along the height of the cross-section in S2 and S3 are extracted and shown in Figures 11 and 12, respectively. As shown in Figure 10, the green and yellow areas represent the compressive stress of the beam induced by poststening force, while the tension area of the beam is in the red region. The greener the area, the greater is the compressive stress in the beam. The stress values are extracted at different locations along the height of the cross-section. Figure 11 shows the stress distribution at the section with strain gauges in the segment S2, the compressive zone is at the bottom of the beam, the tensile stress is observed at the top of the beam, and the stress is distributed almost linearly over the cross-section. These observations indicate that plane-remains-plane assumption is satisfied, therefore the above derived relation of neutral axis location and the prestress force amplitude is valid. The stress distribution in the section in the segment S3 near the joint location is shown in Figure 12. Large compressive stresses closer to the bottom of the section are observed. This may be caused by the influence of squeezing two shear keys under the effect of the prestress tendon. Stress concentration is also observed at shear keys along this section. Therefore, the stress and hence the strain are not linearly distributed along the cross-section, indicating the plane-remains-plane assumption is not satisfied. As a result, the above relation between neutral axis location and the prestress force amplitude based on plane-remains-plane assumption leads to larger errors. From the above discussions, the strains should be measured at locations away from the joints in segmental beam structure for estimating the prestress force. The above results also indicate that even the proposed method based on neutral axis location may not give a very accurate estimation of the prestress force if the fundamental Bernoulli beam plane-remains--plane assumption is not satisfied, it is very sensitive to prestress force change, therefore the proposed method can still be used to identify the loss of prestress force, and also approximately estimate the prestress force amplitude.