## 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 order to be consistent with the form in the codes, Equation (10) is rewritten as Equation (13):  
\[\Delta S_{s}{}^{m}\cdot N=C \tag{13}\]  
where \(\Delta S_{s}\) is the equivalent structural stress range, \(m=-1/h=3.125\), \(C=13875.8^{m}=8.801\times 10^{12}\). The slope \(m\) of the master S-N curve in Equation (13) given by [PERSON] et al. was 3.125. This is close to the slope \(m=3\) of the S-N curves in the codes [9; 12] under the low cycles. However, in these codes [9; 12], the slope of the S-N curve changes under high cycles. Most experimental data points beyond the 99% survival rate are under high cycles (\(N>10^{7}\)), as shown in Figure 4 of the reference [13]. Therefore, it is not suitable that the master S-N curve adopts a single slope \(m=3.125\) from low to high cycles. Referring to the S-N curve of the hot spot stress approach in IIW [12], the slope of the master S-N curve is set as 5 under high cycles (\(N>10^{7}\)), as shown in Equation (14). \[\left\{ \begin{array}{l}\Delta S_{s}{}^{m_{1}}\cdot N=8.801\times 10^{12} \left(m_{1}=3.125,N\leq 10^{7}\right)\\ \Delta S_{s}{}^{m_{2}}\cdot N=3.246\times 10^{16}\left(m_{2}=5,N>10^{7} \right)\end{array} \right. \tag{14}\]  
The calculation formula of the hot spot stress approach is also given as a comparison, as shown in Equation (15). \[\left\{ \begin{array}{l}\Delta\sigma_{ls}{}^{m_{1}}\cdot N=2\times 10^{12} \left(m_{1}=3,N\leq 10^{7}\right)\\ \Delta\sigma_{ls}{}^{m_{2}}\cdot N=6.851\times 10^{15}\left(m_{2}=5,N>10^{7} \right)\end{array} \right. \tag{15}\]  
The fatigue life calculation steps based on the stress integration approach:  
(1) The principal stress \(\sigma_{x}(y,z)\) of the weld toe section is extracted from FEM. Then, the structural stress value \(\sigma_{s}\) is obtained by substituting \(\sigma_{x}(y,z)\) into Equation (7). Then, the equivalent structural stress \(S_{s}\) is obtained by substituting \(\sigma_{s}\) into Equation (10). The equivalent structural stress time-history curve can be obtained in this step, as shown in Figure 14.  
(2) The equivalent stress range \(\Delta S_{seq}\) and stress cycle times \(n_{eq}\) are calculated by the rain-flow counting method. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Time (s)** & \(\sigma_{m}\) **(MPa)** & \(\sigma_{b}\) **(MPa)** & \(r\) & \(I(r)^{\frac{1}{s}}\) & \(t^{s}\) & \(\sigma_{s}\) **(MPa)** & \(S_{s}\) **(MPa)** \\ \hline
0.56 & \(-0.49\) & \(-20.39\) & 0.98 & 1.32 & 12 & \(-20.88\) & \(-27.39\) \\
0.77 & \(-1.05\) & \(-44.73\) & 0.98 & 1.32 & 12 & \(-45.78\) & \(-60.03\) \\
1.19 & \(-0.94\) & \(-40.25\) & 0.98 & 1.32 & 12 & \(-41.19\) & \(-54.02\) \\
1.27 & \(-0.95\) & \(-40.79\) & 0.98 & 1.32 & 12 & \(-41.74\) & \(-54.74\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Calculation of equivalent structural stress. Figure 17: Stress time-history curve. (3) The equivalent stress range \(\Delta S_{sq}\) is substituted into the master S-N curve using Equation (13) or Equation (14) to obtain the number of cycles \(N\). (4) Damage \(D_{j}\) under a load case \(j\) is calculated, as shown in Equation (16). According to the structural health monitoring results, daily traffic volume \(N_{v,d}\) equals to 2551 [19] and wheel transverse distribution frequency \(f_{j}\) is shown in Figure 15 of the reference [39]. \[D_{j}=\frac{365\cdot f_{j}\cdot n_{\alpha,j}\cdot N_{v,d}}{N_{j}} \tag{16}\]  
(5) The fatigue life \(N_{a}\) is calculated by Equation (17), where \(D_{T}\) is the total damage. \[N_{a}=\frac{1}{D_{T}}=\frac{1}{\sum D_{j}} \tag{17}\]  
Taking the load case in Figure 16 as an example, the fatigue life calculation steps based on the stress integration approach are shown in Table 4. As a comparison, the results of the extrapolated hot spot stress approach are also given, and the extrapolated hot spot stress time-history curve and Equation (15) should be used in step (1) and (3), respectively, in this case.