## 5 Conclusions  
A novel simplified analysis method of equivalent fatigue stress-strain relationship is proposed by introducing fatigue damage evolution into the concrete damaged plasticity model (CDPM), extending the practical application of CDPM to high-cycle fatigue simulation. The following conclusions can be drawn:  
(1) A damage variable was constructed based on the maximum fatigue strain, and a nonlinear fatigue damage evolution equation, Equation (20), was proposed to characterize the three-stage fatigue damage development. The effects of three parameters on the evolution equation were explored. The stability factor \(a\) influences the proportion of stages I and III, the development rate factor \(c\) affects the development rate of stage II, and the composite factor \(b\) is determined by \((a^{c}+1)\). These parameters could be fitted to experimental fatigue data, and the value range of each parameter was recommended as \(a\in\) (0.2, 0.7), \(c\in\) (3, 7), and the corresponding \(b\in\) (1.0, 1.3). Fatigue damage evolution equations are affected mainly by the maximum stress level and less affected by concrete strength grade, loading frequency, or specimen size. (2) The fatigue damage evolution equation was incorporated when calculating critical CDPM parameters, and the equivalent fatigue constitutive model after multiple cycles was derived. Changing the shape function \(f(n,N_{\text{f}})\) from cycle ratio \(n/N_{\text{f}}\) or relative logarithmic life \(\text{lg}n/\text{lg}N_{\text{f}}\) to fatigue damage evolution equation \(D_{\text{f}}(n/N_{\text{f}})\), we obtained an improved estimate of fatigue residual strength. The ascending branch of the equivalent constitutive model follows the yield hardening rule, and the residual strain before fatigue failure can be calculated from the fatigue residual secant stiffness. The residual strain for any given number of fatigue cycles is then estimated based on the damage evolution equation. By modeling cyclic loading at discrete cycle ratios, the degradation process of fatigue residual strength, residual stiffness, and residual strain accumulation in high-cycle fatigue could be simulated with low computation cost. (3) The proposed equivalent fatigue constitutive model was validated with experimental data, showing acceptable agreement with observed behavior for fatigue tests under axial compression and flexural tension stress conditions. Our work demonstrates for the first time that the fatigue residual strength envelope curve also applies to bending fatigue behavior analysis. The simplified analysis method is not restricted to any specific equation of concrete constitution; both empirical formulas in the codes and stress-strain relationship directly measured from experiments have derived adequate prediction results. However, the matching between assumed constitutive forms and the measured maximum fatigue strains affected the accuracy of predicted strains. The proposed fatigue damage evolution equation and equivalent fatigue constitutive model laid the foundation for subsequent applications in analyzing the fatigue response of reinforced concrete structures. However, the proposed method belongs to the deterministic analysis method category and only applies to particular cases. Further research is needed to consider randomness and general cases of three-dimensional and variable amplitude loading fatigue. Conceptualization, H.C. and W.Z. ; Data curation, Z.S. ; Formal analysis, Z.S. and X.Z. ; Funding acquisition, H.C. and W.Z. ; Investigation, Z.S. and X.Z. ; Methodology, H.C. and X.Z. ; Project administration, H.C.; Resources, H.C. and W.Z. ; Software, Z.S. ; Supervision, H.C.; Validation, H.C., X.Z., and W.Z. ; Visualization, Z.S. ; Writing--original draft, H.C. and Z.S. ; Writing--review and editing, H.C., Z.S., X.Z., and W.Z. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China, grant number 52278137.