## 4 Flexural Fatigue Behavior of PB-DA
### Development Pattern of Flexural Stiffness for PB-DA  
During cyclic loading, continuous damage accumulation may lead to a variation in the flexural stiffness of the PB-DA. This section analyzes the evolution law of PB-DA flexural stiffness.  
Initially, taking L3 as an example, the force-displacement curves under different cyclic loading cycles were analyzed, as shown in Figure 12. It is evident from Figure 12 that, with an increase in cyclic loading numbers, the force-displacement curves undergo shifts and tilts. The starting point of the force-displacement curves slightly shifts to the right with increasing cyclic loading numbers, indicating a certain degree of fatigue damage in the PB-DA, which in turn causes the slope of the force-displacement curves to tilt, indicating a slight decrease in PB-DA flexural stiffness. The force-displacement curves for 1 million and 2 million loading cycles almost coincide, demonstrating the PB-DA's good fatigue load resistance capacity when loaded to the maximum number of cycles.  
Figure 11: Influence of D—A content and concrete strength grade on maximum compressive and tensile strain: (**a**) maximum compressive strain; (**b**) maximum tensile strain.  
An expression for flexural stiffness is derived to intuitively analyze the stiffness degradation law of the PB-DA during cyclic loading. When a simply supported beam bridge is subjected to two concentrated forces, the expression for the mid-span displacement (\(S_{\text{max}}\)) is given as follows [35]:  
\[S_{\text{max},N}\frac{Fmn^{2}}{48{(EI)}_{N}}\Big{(}3-4\alpha^{2}\Big{)} \tag{9}\]  
where \(m\) is the distance from the concentrated load to the adjacent support; \(n\) is the distance between the two supports; \(F\) is the resultant force of the two concentrated forces; \(\alpha=m/n\); \(EI\) represents the flexural stiffness; and \(N\) represents the number of load cycles.  
From Equation (9), the expression for flexural stiffness can be derived, as shown in Equation (10).  
\[{(EI)}_{N}=\frac{Fmn^{2}}{48S_{\text{max},N}}\Big{(}3-4\alpha^{2}\Big{)} \tag{10}\]  
For ease of analysis, the flexural stiffness degradation factor \(D\) is defined as the ratio of the flexural stiffness at different load cycles to the initial flexural stiffness, with a smaller \(D\) value indicating more significant stiffness degradation.  
\[D=\frac{{(EI)}_{N}}{{(EI)}_{0}} \tag{11}\]  
Substituting Equation (10) into Equation (11) gives the following:  
\[D=\frac{S_{\text{max},0}}{S_{\text{max},N}} \tag{12}\]  
The variation curve of \(D\) with \(N\) can be obtained by inputting the raw data from the measurement points into Equation (12), as shown in Figure 13. Figure 13 indicates that the PB-DA's flexural stiffness exhibits degradation as the \(N\) increases. \(D\) shows a nonlinear rapid degradation when the \(N\) does not exceed 200,000. After surpassing 200,000 cycles, \(D\) approximates a linear decay. Additionally, increasing the D-A content and the concrete strength grade can increase the \(D\) value overall, indicating that increasing the D-A content and concrete strength grade can enhance the flexural stiffness of the PB-DA under cyclic loading.  
Figure 12: Force–displacement curve of L3 at different cyclic loading numbers.