## 4 The Predicted Stiffness
### Comparison of the Results  
The stiffness and their reduction coefficient at the mid-span position from different methods are compared in Figures 12 and 13, respectively.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Load** & \multicolumn{3}{c}{**Short-Term Bending Stiffness (10\({}^{16}\) N\(\cdot\)mm\({}^{2}\))**} & \multicolumn{3}{c}{**Reduction Coefficients**} \\ \cline{2-10} **(kN)** & **Ref. [3]** & **Ref. [4]** & **Ref. [5]** & **Ref. [6]** & **Ref. [3]** & **Ref. [4]** & **Ref. [5]** & **Ref. [6]** \\ \hline
630 & 3.62 & 4.01 & 4.22 & 4.22 & 1.00 & 1.00 & 1.00 & 1.00 \\
870 & 3.62 & 4.01 & 4.22 & 4.22 & 1.00 & 1.00 & 1.00 & 1.00 \\
1030 & 3.62 & 4.01 & 4.22 & 4.22 & 1.00 & 1.00 & 1.00 & 1.00 \\
1110 & 3.24 & 3.45 & 3.91 & 3.60 & 0.90 & 0.86 & 0.93 & 0.85 \\
1190 & 2.73 & 2.79 & 3.40 & 2.87 & 0.75 & 0.70 & 0.81 & 0.68 \\
1300 & 2.30 & 2.31 & 2.89 & 2.35 & 0.64 & 0.58 & 0.68 & 0.56 \\
1500 & 1.88 & 1.89 & 2.31 & 1.91 & 0.52 & 0.47 & 0.55 & 0.45 \\
1580 & 1.77 & 1.79 & 2.15 & 1.81 & 0.49 & 0.45 & 0.51 & 0.43 \\
1700 & 1.65 & 1.68 & 1.97 & 1.70 & 0.46 & 0.42 & 0.47 & 0.40 \\
1900 & 1.50 & 1.56 & 1.75 & 1.57 & 0.41 & 0.39 & 0.41 & 0.37 \\
2100 & 1.40 & 1.49 & 1.62 & 1.49 & 0.39 & 0.37 & 0.38 & 0.35 \\
2220 & 1.36 & 1.46 & 1.56 & 1.46 & 0.38 & 0.36 & 0.37 & 0.35 \\
2500 & 1.27 & 1.40 & 1.45 & 1.40 & 0.35 & 0.35 & 0.34 & 0.33 \\
2540 & 1.26 & 1.39 & 1.44 & 1.39 & 0.35 & 0.35 & 0.34 & 0.33 \\
2550 & 1.26 & 1.39 & 1.44 & 1.39 & 0.35 & 0.35 & 0.34 & 0.33 \\ \hline \end{tabular}
\end{table}
Table 4: The short-term stiffness and their reduction coefficients at the mid-span position.  
Figure 12: Comparison of the stiffness values at the mid-span position using different methods.  
In Figure 12, it can be observed that the stiffness of the pre-stressed concrete T-beams remains nearly constant in the completely linear stage, as the beam body does not crack. However, the equivalent bending stiffness was higher than what was calculated using short-term formulas. As the beam entered the incomplete stage, the equivalent bending stiffness decreased rapidly due to gradual damage and continuous crack propagation. Equation (1) considers the entire beam as having a constant stiffness and replaces the actual stiffness with the average stiffness of the cracked beam. This leads to a calculated stiffness at the mid-span position that is higher than the actual value, as the stiffness of the entire beam is replaced by the stiffness at the mid-span position. This method fails to reflect the situation at the cracking section. In contrast, the short-term calculation method specific for damaged and cracked sections calculates the stiffness at the section where cracks occur. This approach accurately reflects the presence of cracks at specific sections. Therefore, compared to the equivalent stiffness method, the short-term calculation method is more reasonable for evaluating damaged and cracked sections.  
Therefore, the calculation of short-term stiffness provides a more accurate evaluation of the stiffness value for a cracked beam. As the beam transitions into the non-linear stage, the stiffness decreases at a faster rate. To accurately predict the stiffness of the beam throughout the destructive process, including the complete, incomplete linear, and non-linear stages, a new method is proposed. The piecewise stiffness, back-calculated by system identification, is suggested as follows.