## 4 Applications
### Concrete Fatigue in Flexural Tension
#### 4.2.1 Equivalent Fatigue Damage Constitutive Curve  
Literature [47] reported a four-point bending fatigue test of 100 mm \(\times\) 100 mm \(\times\) 400 mm specimens. Due to experimental condition limitations, only the stress-strain curve's ascending branch under monotonic bending was measured [47]. The test confirmed that the maximum fatigue strain at failure is inversely proportional to the applied maximum fatigue stress. Thus, the descending branch, also the hypothetical fatigue residual strength envelope curve, was obtained by fitting Equation (22). Parameter \(\alpha_{\tau}\) was determined based on the measured maximum fatigue strain at fatigue failure and the applied maximum fatigue stress. Specimens C39A5 from the literature [47] were selected for analysis. The specimens were made from a concrete batch with a mix ratio of 1:0.39:1.7:3.2, where C39 represents a water-cement ratio of 0.39, and A5 represents a 5% air content in the concrete. The monotonic constitutive model of specimen C39A5 is shown in Figure 24, which exhibits a good agreement with the experimental values at two critical positions, representing the initial and ultimate loading cycles, respectively.  
The equivalent fatigue damage constitutive equations under two stress levels are derived based on the initial constitutive model and fatigue damage evolution equation from Figure 9. Figure 25 clearly shows the fatigue residual strength, fatigue residual strain,  
Figure 23: Comparison between simulated and experimental values of \(\varepsilon_{\text{max}}^{\text{n}}/\varepsilon_{\text{p},c}\) for fatigue tests under axial compression from the literature [42]: (**a**) Specimen F1; (**b**) Specimen F3.  
Figure 24: Monotonic constitutive curve of bending specimens C39A5 in fatigue tests under flexural tension from the literature [47].  
and stress-strain development curve at every tenth of the cycle ratio. The equivalent fatigue constitutive curves could reasonably reflect the fatigue strains at minimum and maximum stress levels. Compared to experimental values of fatigue strains, theoretical values of the maximum fatigue strain are higher than the experimental results during stage I of fatigue damage development, gradually approaching the experimental observation by the end of stage III. This discrepancy could be because the fatigue residual strength envelope curve based on the descending branch of the monotonic stress-strain curve is obtained from experiments under axial loading. As we know, a difference in stress distribution exists between these two tensile stress states. Because of the stress gradient, the probability of crack initiation in a beam specimen under bending would be lower than in axial tension. The stress reduction after peak strain under flexural tension is expected to be slower than under axial tension due to the redistribution of bending stress. The initial rapid decreasing rate of the envelope curve leads to a lower simulated fatigue residual strength calculated by Equation (24), which would cause faster strain development. The equivalent fatigue constitutive equations under two stress levels are then integrated into the CDPM in Abaqus to predict the residual fatigue behavior of concrete in flexural tension.