# Notch Fatigue Life Research Based on Critical Distance Theory
## 4 Finite Element Analysis
### Fatigue Life Prediction Model  
The critical distance theory was used to estimate the fatigue damage in the critical region, and the critical distance L was first determined for the bridge steel Q345 qD. Where the crack expansion threshold \(\Delta K_{\text{th}}\), given by the empirical Equation (7), was obtained by referring to a series of fatigue crack expansion tests conducted by [PERSON] et al. (2019) on Q345 qD steel at three different stress ratios of \(R\) = 0.05, 0.25 and 0.50. \[\Delta K_{\text{th}}=5.556(1-0.825R)^{1.417} \tag{7}\]  
where \(R\) is the stress ratio; \(\Delta K_{\text{th}}\) unit MPa \(\cdot\) m\({}^{1/2}\). In this paper, a stress ratio of R = 0.02 is used for the fatigue simulation of the notched rectangular plate specimens. Therefore, the crack expansion threshold \(\Delta K_{\text{th}}\) is 5.427 MPa \(\cdot\) m\({}^{1/2}\). The fatigue limit of the orthogonal anisotropic steel bridge panel at the opening is used as the fatigue limit of the critical distance theory analysis, i.e., \(\Delta\sigma_{0}\) = 70 MPa. Substituting the values of \(\Delta K_{\text{th}}\) and \(\Delta\sigma_{0}\) into Equation (6), we obtain L = 1.91 mm (\(R\) = 0.02). Through the finite element solution of three arc notch specimen models with different radii under the tensile load of 50 kN, the respective principal stress distributions were obtained, as shown in Figure 9. From the figure, it can be seen that the first principal stress at the notch decreases with the increase in notch radius: when the radius increases from 10 mm to 20 mm, the first principal stress decreases from 319.8 MPa to 253.7 MPa, with a reduction of 66.1 MPa; when the radius increases from 20 mm to 30 mm, the first principal stress decreases from 253.7 MPa to 225.6 MPa, with a reduction of 28.1 MPa, which is more gentle than before. The first principal stress decreases from 253.7 MPa to 225.6 MPa when the radius is increased from 20 mm to 30 mm, with a decrease of 28.1 MPa, which is flatter than before. Figure 8: Finite element model of curved notch specimens (r = 10 mm). Figure 9: First principal stress nephogram of the cutout (\(\Delta P\) = 50 kN) (Unit: MPa). Since the stress state of the notched rectangular plate specimen is in a plane stress state [24], the characteristic stresses can be calculated using the point method and the line method. Using ABAQUS to set up the path from the peak of the maximum principal stress in the direction of the highest stress gradient, the first principal stress at each point and the distance of its location from the peak of the maximum principal stress are extracted, and the relationship curve is drawn as shown in Figure 10. On the hot spot path, the maximum principal stress is inversely related to the distance. In the range of 1.2 times the critical distance, the smaller the radius of the notch, the higher the stress level. In the region beyond the 2-times critical distance, the maximum principal stress curves of the 20 mm notch radius and 30 mm notch radius tend to converge, indicating that the notch is no longer a key factor affecting the stress level at this time. The extracted Fe-safe lifetime results are presented in Table 4, and the characteristic stresses for the point and line methods are obtained according to the method mentioned in Section 2. It can be seen from Table 4 that the characteristic stress decreases with the increasing radius of the unique variable notch for both the point and line methods. Under the same fatigue load, the prediction accuracy of different radius notch specimens using the point method and the line method are different. The calculation results show that the predicted values of the point method are 7-13% higher than those of the line method. The calculation model of fatigue life with respect to the characteristic stress is obtained by linear fitting as shown in Equations (8) and (9).