## 5 Model Updating for the Segmental Beam
### Prestress Force Quantification under the Dynamic Load  
In practice, dynamic testing is relatively easier to carry out than static loading tests. In this section, numerical study is conducted to identify the prestress force by using the strain responses at the web of the bridge model subjected to a dynamic load. This investigation is conducted to simulate the realistic situation to predict the prestress force of the segmental beam from ordinary dynamic tests after the post-tensioning force has been applied, such as hammer impact tests. Figure 13 shows the hammer impact load applied on the middle of the second segment with a sampling rate of 2048 Hz and a sampling duration of 0.5 s. It is noted that this hammer impact force is measured from a real vibration test with an instrumented hammer, and this force is used as the external load applied in this numerical simulation to calculate the strain response data. The true applied prestress force of the numerical model is 107 kN. The strain responses calculated from the FEM at the top and bottom locations at the middle of the segment S2, i.e., II_t and II_b described in the above experimental tests, under the applied dynamic load, are shown in Figure 14.  
The prestress force is identified by using the proposed method based on Equations (3)-(5), with the above-extracted strain values at the same locations as those in the experimental test. The identified prestress force from strain measurements, and the relative error between the applied prestress force in the FEM and the identified prestress force during the dynamic testing period, are shown in Figure 15a,b, respectively.  
Figure 12: (**a**) The stress distribution in the longitudinal direction of the cross-section of strain gauge locations in S3. (**b**) Stress distribution over the cross-section with strain gauges in S3.  
Figure 13: Dynamic load applied on the numerical model.  
It can be observed from Figure 15 that the identified prestress forces have relatively large fluctuations within the first 0.05 s, owing to the influence of the applied external force acting on the beam at the beginning, which in fact changes the prestress force. However, as the vibration level is small, the change of the prestress force level is also small. The largest error of the identified prestress force level is less than 3%. In the free-vibration phase, the vibration level is smaller, the identified prestress force is relatively stable, that is, it oscillates between 108-109 kN. The relative error between the identified average prestress force and the actual applied prestress force in the numerical model in the free-vibration phase from 0.4-0.5 s is given in Table 12, and it is 1.32%, indicating that using the strain measurements in segment S2 in standard vibration test, which is relatively easy to carry out, gives good prestress force identification. These results demonstrate the feasibility and accuracy of using the proposed method in practical applications for quantifying the prestress force.  
Figure 14: The strain responses calculated from the FEM at the locations of II_t and II_b.  
Figure 15: Prestress force identification results: (**a**) The identified prestress force from strain measurements. (**b**) The relative error between the simulated prestress force in the FEM and the identified prestress force with strain responses.