## 1 Introduction  
Cyclic loading, a significant factor in the performance degradation of structures like highway bridges, road pavements, railway track slabs, and wind turbine foundations, induces a progressive, permanent, and internal structural change known as fatigue damage [1]. In concrete, the initiation and propagation of microcracks in the cement matrix, mortar, and interfacial transition zones are crucial causes of performance changes due to fatigue action [2]. Structural failure of concrete could occur when microcracks converge into macrocracks of the cross-section. As stress levels in concrete structures increase, fatigue is vital in affecting concrete's durability, crack development, and even failure. This phenomenon underscores the increasing importance of investigating concrete fatigue in structural engineering. Fatigue testing has been the traditional method to investigate the fatigue performance of concrete structures. However, the cost of structural component experiments is exceptionally high, and fatigue test results are usually restricted by experimental requirements such as structural dimensions, loading capacity, and boundary conditions. In recent years, scholars have attempted to propose concrete constitutive models under cyclic loading and combine them with numerical simulation to describe concrete's stress-strain evolution and failure behavior [3]. Models based on plasticity theory and damage mechanics have been developed to evaluate the evolution of concrete's fatigue damage [4; 5; 6]. These theoretical models could accurately simulate concrete's observed behavior; however, predicting engineering structures' fatigue performance requires software redevelopment. The intermediate parameters required to derive theoretical models are difficult to measure through traditional experiments. Employing concrete constitutive models available in commercial software for more accessible applications to practical and complex components is desirable. The constitutive models that incorporate both damage and plasticity have successfully captured the nonlinear behavior of concrete [7]. Among them, the concrete damaged plasticity model (CDPM) proposed by [PERSON] et al. [8] has proven to be particularly effective in reproducing various failure modes, including tensile cracking and compressive crushing. CDPM has been seamlessly integrated into the widely used general-purpose finite element software Abaqus [9]. Its versatility and robustness in successfully simulating monotonic, cyclic, and dynamic behavior is impressive. However, it has been reported that CDPM underestimates the fatigue resistance of concrete structures in some cases [10]. The drawback of high computational cost in a cycle-by-cycle simulation further underscores the urgent need for a simplified analysis method. It is recognized that the fatigue problem experienced by most bridge engineering structures is high-cycle fatigue, where the stress level is relatively low, and the number of cycles before a fatigue failure is longer than \(10^{4}\) cycles. Given the minimal fatigue damage induced in a single loading cycle for high-cycle fatigue problems, a simplified fatigue analysis method that overcomes the high computational cost and complicated simulation process associated with cycle-by-cycle fatigue analysis is practically possible. The cycle-jump acceleration algorithm [11; 12; 13], a method that accelerates the simulation process by skipping some cycles based on the assumption of gradual damage accumulation, can significantly improve the efficiency of the finite element method in simulating the evolution of concrete fatigue damage. Alternatively, the fatigue process can be discretized and analyzed at prescribed cycle ratios to express fatigue damage evolution in a standardized form, making it more convenient for engineering applications. [PERSON] et al. [14] proposed a simplified concrete fatigue model for axial compression. [PERSON] et al. [15] conducted numerical research on the fatigue performance of wind turbine concrete foundations.