## 4 Applications
### Concrete Fatigue in Flexural Tension
#### 4.2.2 Finite Element Analysis  
The finite element model of specimen C39A5 is shown in Figure 26, whose dimensions are identical to the 100 mm \(\times\) 100 mm \(\times\) 400 mm beam in the four-point bending fatigue test from the literature [47]. The loading and supporting device are rigid, and the concrete specimen is defined as deformable solid C3D8R elements. Tangential and normal behavior contact properties are set when defining the interaction between the supporting device and concrete specimen, and \"Tie\" constraints are defined between the loading device and the contacting nodes on the upper surface of the specimen.  
The specimen is monotonically loaded under force control with a 12 kN/min loading rate in finite element analysis. The tensile bending stress at the bottom fiber in the constant moment region is then obtained and displayed in Figure 27. The bending stress increases linearly with the applied load initially. As tensile damage gradually develops due to microcracking, the stress increases more slowly. As shown in Figure 27, the ultimate bending load is simulated as 10.95 kN, beyond which the specimen is completely damaged.  
Figure 25: Equivalent fatigue damage constitutive curves for bending specimens C39A5 from the literature [47], along with experimental values of maximum and minimum fatigue strains measured at various cycle ratios: (**a**) \(S_{\text{max}}\) = 0.75; (**b**) \(S_{\text{max}}\) = 0.80.  
With the obtained ultimate bending capacity, a numerical analysis of the fatigue behavior of specimens C39A5 was conducted at maximum stress levels of 0.80 and 0.75 and a stress ratio of 0.1, with a loading frequency of 5 Hz. Based on the equivalent fatigue damage constitutive model, the simulated strains corresponding to the maximum and minimum stress levels are obtained and compared with test values, as illustrated in Figure 28.  
Figure 28: Comparison between simulated and experimental results of maximum and minimum fatigue strain in specimens C39A5 for fatigue tests under flexural tension from the literature [47]: (**a**) \(S_{\text{max}}=0.75\); (**b**) \(S_{\text{max}}=0.80\).  
Figure 27: Simulated maximum tensile stress from monotonic loading (under force control) for bending specimens from the literature [47].  
In Figure 28, the solid lines stand for the specimen's simulated fatigue stress-strain responses at each predetermined cycle ratio when it is fatigue-loaded between \(S_{\text{min}}\) and \(S_{\text{max}}\); the black dots are the projection of the ends of these space curves into the cycle ratio versus strain plane, representing the maximum and minimum fatigue strains, respectively. Regular and inverted triangles symbolize the measured values of maximum and minimum strains from the fatigue test.  
Consistent with Figure 25, the simulated values of the maximum fatigue strain are slightly higher than the experimental values during the earlier stage of fatigue damage development, gradually approaching each other by the end of the fatigue test. Generally, the equivalent fatigue damage constitutive model presents satisfactory predictions. In Figure 28a, the RMSE between simulated and experimental values of maximum fatigue strain is 0.000017 and 0.000013 for minimum fatigue strain. In Figure 28b, the RMSE between simulated and experimental values of maximum and minimum fatigue strains are 0.000024 and 0.000013, respectively. This application example indicates that the proposed equivalent fatigue damage constitutive model could be applied to simulate fatigue behavior under flexural tension reasonably well.