# Fatigue Reliability Analysis of Rib-To-Deck Joints Using Test Data and In-Situ Measurements
## 4 In-Situ Measurement-Based Fatigue Reliability Analysis  
### Analysis Method of Fatigue Reliability  
In the reliability analysis, the major task is to solve the possibility that the resistance is lower than the load effect, i.e., the limit state function (LSF) reaches zero or less. In the case of fatigue issue, the LSF can be established, as shown in Equation (7) [3; 24; 26]. \[g(D,t)=\Delta-D(t) \tag{7}\]  
Figure 9: P–S–N curve under the survival rate of 95%: (**a**) nominal stress and (**b**) hot spot stress. where \(\Delta\) is the damage tolerant factor, assumed as 1.0 in this study; \(D(t)\) stands for the time-dependent accumulation of fatigue damage. The fatigue damage can be solved using the linear damage accumulation rule [37], as shown in Equation (8). \[D(t)=\sum\frac{n_{i}(t)}{N_{i}} \tag{8}\]  
where \(n_{i}(t)\) is the number of load cycles corresponding to stress range \(\Delta\sigma_{i}\); \(N_{i}\) is the allowable fatigue life under the stress range \(\Delta\sigma_{i}\). Through incorporating Equation (3), Equation (8) can be rewritten with an equivalent stress range \(\Delta\sigma_{re}\), as shown in Equations (9) and (10). \[D(t)=\Delta\sigma_{re}^{m}N(t)/C \tag{9}\]  
\[\Delta\sigma_{re}=\sqrt[n]{\frac{\sum\Delta\sigma_{i}^{m}\cdot n_{i}(t)}{N(t)}} \tag{10}\]  
where \(N(t)\) is the total number of loading cycles; \(C\) and \(m\) are the material-related constant and power index, respectively; \(\Delta\sigma_{re}\) is the equivalent stress range. In the case of material properties, as aforementioned, the power index \(m\) is usually assumed as a deterministic value of 3 for welded joints in steel structure. Meanwhile, the constant \(C\) is assumed to follow the lognormal distribution, and the distribution parameters can be derived by the test data. According to [38], the damage tolerant factor \(\Delta\) follows the lognormal distribution, with the mean of 1 and coefficient of deviation (COV) of 0.3. The distribution parameters of the material properties are summarized in Table 3. It can be found from Table 3 that, although the COV of the constant \(C\) is smaller in TEU specimens than in CU specimens, the standard deviation of the fatigue strength is almost the same for the two types of specimens. This can be attributed to the improvement in the mean fatigue strength of TEUs specimen, i.e., more than 21% as previously illustrate in Table 2.  
In the case of load effect, the total number of loading cycles \(N(t)\) can be regarded as a time-dependent deterministic value [39]. Meanwhile, the equivalent stress range can be solved using the in-situ measurement data, which will be discussed in detail in the following section. Since the random variables related to the LSF of fatigue issue follows the normal or lognormal distribution, the analytic solution is available for the fatigue reliability \(\beta\), as shown in Equations (11), (12) and (13). \[\beta=\frac{(\lambda_{C}+\lambda_{\Delta})-m\cdot log(\Delta\sigma_{re})-log(N (t))}{\sqrt{\zeta_{C}^{2}+\zeta_{\Delta}^{2}}} \tag{11}\]  
\[\lambda_{C}=log(\mu_{C})-log\big{(}1+\delta_{C}^{2}\big{)}/2;\qquad\zeta_{C}^{ 2}=log\big{(}1+\delta_{C}^{2}\big{)} \tag{12}\]  
\[\lambda_{\Delta}=log(\mu_{\Delta})-log\big{(}1+\delta_{\Delta}^{2}\big{)}/2; \qquad\zeta_{\Delta}^{2}=log\big{(}1+\delta_{\Delta}^{2}\big{)} \tag{13}\] \begin{table}
\begin{tabular}{l l l c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multirow{2}{*}{**Type**} & \multicolumn{2}{c}{**CU Specimens**} & \multicolumn{2}{c}{**TEU Specimens**} \\ \cline{3-6}  & & **Nominal** & **Hot Spot** & **Nominal** & **Hot Spot** \\ \hline \multirow{2}{*}{\(C\)} & Mean (\(\times 10^{12}\)) & \multirow{2}{*}{Lognormal} & 2.271 & 4.299 & 4.068 & 7.586 \\  & COV & & 0.590 & 0.482 & 0.329 & 0.273 \\ \hline \multirow{2}{*}{\(\Delta\)} & Mean & \multirow{2}{*}{Lognormal} & \multirow{2}{*}{} & 1.0 & \multirow{2}{*}{} \\  & COV & & & 0.3 & \\ \hline \(m\) & Mean & Deterministic & & 3.0 & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Distribution of material properties.