## 5 Back-Calculation of Piecewise Stiffness
### Comparison of the Calculation Results  
Once the modulus of a section has been back-calculated, the bending stiffness of each element can be determined by combining the inertia moment and dimension at mid-span. This allows for the calculation of the back-calculated bending stiffness in the mid-span position under various load levels and three different working condition cases. The bending  
Figure 16: Schematic diagram of the stepped flexural stiffness after beam cracking.  
Figure 17: Beam elements’ division diagram.  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Condition Category** & **Description** \\ \hline Case1 & Considering the beam as a whole, the entire beam has the same modulus. \\  & Divide the beam into two sections, with units 1–13 as the first section \\ Case2 & including mid-span and elements 14-25 as the second section. During the \\  & back-calculation, output two moduli of each section separately. \\  & Divide the beam into three sections, with elements 1–8 as the first section, \\ Case3 & elements 9–17 as the second section, and elements 18–25 as the third one. \\  & During the back-calculation, output three moduli of each section separately. \\ \hline \hline \end{tabular}
\end{table}
Table 5: Calculation conditions.  
two numerical curves being 3.67%. This is because the entire bridge is considered as a single modulus for back-calculation. The equivalent stiffness method only uses mid-span deflection to calculate the stiffness, while the system identification method uses multiple point deflections to back-calculate the stiffness. These two methods are essentially similar. In Case 2, the entire beam is divided into two sections, and the results shown in Figure 18 are only for the first section, which includes the mid-span position. According to Figure 18, the back-calculated results are also very close to the equivalent values, with the average percentage error between the two numerical curves being 2.44%. The principle behind this approach is similar to that used in Case 1.  
The back-calculated results shown in Figure 19 are for the second section, which includes the mid-span position. As illustrated in Figure 19, it can be observed that the back-calculation results are close to the equivalent values in the complete linear stage. However, when the load level reaches 1030 kN (crack initiation), as the load level increases, the back-calculation value appears to be lower than the equivalent value, and gradually approaches the value calculated from short-term stiffness. As the beam enters the non-linear stage, the back-calculation program can more accurately simulate the decrease in stiffness after yielding when compared to the short-term values.  
In Case 3, the back-calculated results are for the second section, which includes the mid-span position. However, the length of the second section defined in Case 3 is not adjusted as the cracks develop towards the two ends of the beam. During the early stages of crack development, the actual length of the damaged section is smaller than that defined in Case 3. As the load increases, the cracks continue to expand towards the beam ends, and the actual length of the damaged section gradually approaches the defined value.  
Based on the above analysis, we can conclude that, using a system identification-based segmented stiffness reverse calculation method, the segment length corresponding to the stiffness inversion area is set as the actual length of crack propagation, which can more accurately depict the changes in the stiffness of each segment section of the 50 m T-beam during the entire failure process.