# A Review of Prediction Methods for Ultra-Low Cycle Fatigue Damage of Steel Piers Under Earthquakes
## 2 Research Status and Development Trend at Home and Abroad
### Empirical Formula Method  
In 1954, [PERSON] [28] and [PERSON] [42] proposed the Coffin-Manson Formula (1) to predict low cycle fatigue life based on the relationship between fatigue life and plastic strain amplitude.  
\[\Delta\varepsilon_{p}/2=\varepsilon_{f}^{\prime}(2N_{f})^{c} \tag{1}\]  
Figure 2: Electron microscopy scanning of steel fracture surface. (**a**) Ductile fracture [20]. (**b**) Brittle fracture [21].  
In the formula, \(\Delta\varepsilon_{p}\) is the plastic strain amplitude; \(N_{f}\) is the fatigue life of the material; \(\varepsilon^{\prime}_{f}\) is the fatigue ductility coefficient; and \(c\) is the fatigue ductility exponent.  
The Coffin-Manson formula remains the primary method for predicting low cycle fatigue life in steel. Its accuracy has been confirmed by numerous experimental results [43; 44; 45]; however, it tends to overestimate ultra-low cycle fatigue life in steel [34; 46; 47]. To enhance the prediction accuracy, researchers have developed empirical formulas suitable for both low and ultra-low cycle fatigue predictions, including the Kuroda model [48], Tateishi model [33], and Xue model [25]. Nonetheless, these models are limited to uniaxial stress states and do not account for stress triaxiality, making them inadequate for predicting ultra-low cycle fatigue life in welded joints under complex stress conditions. Additionally, the parameters in these empirical formulas vary with material properties and welding methods, necessitating extensive parameter calibration experiments [49].  
Beyond predicting fatigue life in steel and welded joints, researchers have also investigated methods for predicting ultra-low cycle fatigue damage in steel piers, typically based on the Coffin-Manson formula [38; 50; 51]. For instance, [PERSON] et al. [38] proposed a prediction method for ultra-low cycle fatigue damage in steel bridge piers, which combines the Coffin-Manson formula with the Miner cumulative damage criterion, as shown in Formula (2):  
\[D=C^{\prime}\sum_{i=1}^{n}\left(\Delta\varepsilon_{pi}\right)^{m} \tag{2}\]  
In the formula, \(D\) is the damage index; \(\Delta\varepsilon_{pi}\) is the plastic strain amplitude; and the material parameters \(m=1/k\) and \(C^{\prime}=\left(1/C\right)^{m}\) are determined according to the test results at fracture (\(D=1\)). However, in the modeling process of steel bridge piers, the length of \(2B\) (\(B\) is the width of the flange plate) or \(3l_{d}\) (\(l_{d}\) is the spacing of the diaphragm) is shown according to Figures 3 and 4, and the element division criteria are not established. It is relatively difficult to promote the application. To enhance the prediction accuracy and stability, [PERSON] et al. [50] proposed a non-local damage variable in 2015 using an exponential weight function, which led to the development of a non-local damage method.  
Figure 3: Analytical model of steel single-column piers [38].  
The aforementioned research indicates that the empirical formula method is only applicable to uniaxial stress states. It is unable to consider the effects of stress triaxiality and cannot predict the ultra-low cycle fatigue life of welded joints under complex stress states. This method requires numerous parameter calibration experiments.