## 5 Fatigue Life Calculation of Jiangyin Bridge Based on the Stress Integration Approach
### Fatigue Life Calculation of Jiangyin Bridge Based on the Stress Integration Approach
In Table 4, \(m_{1}\) and \(m_{2}\) are the slope of the S-N curve at non-high cycles \(N\leq 10^{7}\) and high cycles \(N>10^{7}\), respectively. The fatigue life calculated by the stress integration approach is 84 years under the slope \(m_{2}=5\) and 33 years under the slope \(m_{2}=3.125\), respectively, and the fatigue life of the former is 2.5 times that of the latter. Therefore, the slope of the master S-N curve at high cycles significantly affects the actual bridge fatigue life. The fatigue life calculated by the stress integration approach and the extrapolated hot spot stress approach are very similar, at 84 and 83 years respectively. Therefore, the slope \(m_{2}=5\) of the master S-N curve at high cycles is more reasonable. The total damage \(D_{T}\) in Table 4 is the sum of the damage under six main load cases, as shown in Figure 18. A distance of 0.15 m along the transverse direction was assumed as a case of distribution. The wheel load transverse distribution frequencies are shown in Figure 15 of the reference [39]. For simplicity, Figure 18 only shows the load distribution of double wheels. The details of the load distribution under case 4 in Figure 18 refers to Figure 16. The damage \(D_{j}\) calculated by the stress integration approach with \(m_{2}=5\) under six different load cases is given in Figure 18. This shows that the maximum damage occurs under load case 4, followed by case 3 and case 5, and the damage is small under other cases. The damage value under load case 4 is approximately 80% of the total damage. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{3}{*}{**Approach**} & \multirow{3}{*}{**Equation No.**} & \multicolumn{6}{c}{**Step**} \\ \cline{3-8}  & & **(1)** & **(2)** & **(3)** & **(4)** & **(5)** \\ \cline{3-8}  & & **Stress Time** & **History Curve** & **\(\Delta S_{sq}\) (MPa)** & \(n_{eq}\) & \(N\) (\(\times 10^{7}\)) & \(D_{4}\) & \(D_{T}\) & \(N_{a}\) (years) \\ \hline Extrapolated & Equation (15) & Figure 17 & 40.25 & 4 & 6.4852 & 0.0099 & 0.0121 & 83 \\ hot spot stress & \(\begin{array}{c}\text{Equation (13)}\\ \text{$m_{1}=3.125$, $m_{2}=5$}\end{array}\) & Figure 17 & 54.89 & 4 & 6.5136 & 0.0098 & 0.0119 & 84 \\ \cline{2-8} \multirow{3}{*}{
\begin{tabular}{c} Stress integration \\ \text{$m_{1}=m_{2}=3.125$}\end{array}\)} & Equation (14) & Figure 17 & 54.89 & 4 & 3.2258 & 0.0194 & 0.0303 & 33 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fatigue life calculation steps. \(\alpha_{1},\alpha_{2}\) and \(\alpha_{3}\) are the three angles calculated by \(L_{1},L_{2}\) and \(L_{3}\), as shown in Figure 16. The mean, minimum and maximum values of the weld size are shown in Table 6.  
Since the measured actual weld size is discrete, which is mainly caused by weld manufacturing errors, five representative weld types (WTs) were selected in the present study, as shown in Figure 20 and Table 7. In Table 7: (1) The angle \(\alpha_{3}\) remains at 100 degrees. The size of weld type 4 (WT-4) is close to the mean value in Table 6. The length \(L_{1}\) of WT-1 and WT-5 are close to the minimum value and maximum value in Table 6, respectively, and the rest of the values are within the extreme range. (2) A comparison of WT-1, WT-2 and WT-3 shows that the length \(L_{1}\) increases successively when the length \(L_{2}\) remains unchanged. That is, the angle \(\alpha_{2}\) between the deck and rib decreases by 10 degrees successively. (3) A comparison of WT-2, WT-4 and WT-5 shows that the weld angles remain the same, and the length of three sides increases by an equal proportion of 25% successively. That is, the weld area increases by 56% successively. (4) The length \(L_{1}\) of WT-5 is approximately equal to that of WT-3, and the length of \(L_{2}\) is extended by 56%. Figure 21 shows the variation of fatigue life with the weld size under the standard vehicle load in Figure 15. In Figure 21, \(m_{2}\) is the slope of the master S-N curve at high cycles \(N>10^{7}\) using the stress integration approach.