# Fatigue Reliability Analysis of Rib-To-Deck Joints Using Test Data and In-Situ Measurements
## 3 Derivation of Probability-Stress-Life Curves  
### Statistics Method for P-S-N Derivation  
Based on the above analysis, the fatigue strength of TEU specimens and CU specimens are compared in terms of the mean value. However, in the fatigue design, it is risked and unreasonable to directly use the mean value, which stands for a failure rate of 50%. Instead, the stress-life curve established under a specific survival probability is generally applied, which is also called as the probability-stress-life (P-S-N) curve. To derive the P-S-N curve, the uncertainties in fatigue strength should be fully considered through the statistics analysis on the test data. In the fatigue design, [PERSON]'s equation is generally applied as the basic model for the stress-life relation [36], as shown in Equation (3) (power form) and Equation (4) (log-log form). \[\Delta\sigma^{m}\times N=C\] \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \multirow{2}{*}{**CODE**} & \multicolumn{2}{c}{**Measured Stress Range (MPa)**} & \multicolumn{2}{c}{**Equivalent Fatigue Strength (MPa)**} & \multicolumn{2}{c}{**Average Equivalent Fatigue Strength (MPa)**} \\ \cline{2-7}  & **Nominal** & **Hot Spot** & & **Nominal** & **Hot Spot** & **Nominal** & **Hot Spot** \\ \hline CU1\({}^{*}\) & 81 & 98 & 269 & 89 & 108 & & \\ CU2\({}^{*}\) & 103 & 121 & 255 & 112 & 131 & & \\ CU3\({}^{*}\) & 123 & 149 & 91 & 95 & 115 & 103 & 128 \\ CU4\({}^{*}\) & 58 & 81 & 824 & 93 & 130 & (13.0)\({}^{1}\) & (12.5) \\ CU5\({}^{*}\) & 81 & 113 & 314 & 94 & 131 & & \\ CU6 & 115 & 135 & 206 & 116 & 136 & & \\ CU7 & 111 & 132 & 263 & 121 & 145 & & \\ \hline TEU1\({}^{*}\) & 114 & 144 & 147 & 103 & 130 & & \\ TEU2\({}^{*}\) & 119 & 146 & 162 & 111 & 136 & & \\ TEU3\({}^{*}\) & 125 & 154 & 234 & 132 & 162 & & \\ TEU4\({}^{*}\) & 129 & 153 & 193 & 127 & 151 & & \\ TEU5\({}^{*}\) & 78 & 113 & 648 & 115 & 167 & 125 & 155 \\ TEU6\({}^{*}\) & 106 & 131 & 223 & 110 & 136 & & (14.1) & (14.9) \\ TEU7 & 114 & 144 & 333 & 135 & 171 & & \\ TEU8 & 199 & 160 & 204 & 140 & 161 & & \\ TEU9 & 151 & 187 & 129 & 130 & 162 & & \\ TEU10 & 147 & 170 & 210 & 149 & 173 & & \\ TEU11 & 174 & 211 & 75 & 125 & 152 & & \\ \hline \hline \end{tabular} \({}^{*}\): The specimen previously tested in [10; 18]. \({}^{1}\) The value in brackets is the standard deviation.  
\end{table} 
Table 2: Stress–life data measured from fatigue tests. \[m\times\log\Delta\sigma+\log N=\log C \tag{4}\]  
where \(\Delta\sigma\) stands for the applied stress range; \(N\) is the fatigue life; \(m\) and \(C\) are the material-related power index and constant, respectively. For welded joints in steel structures, the power index \(m\) can be treat as a fixed value of 3 while the constant \(C\) accounts for the uncertainty in fatigue strength [35]. On this end, the P-S-N can be established based on the characteristic value of the constant \(C\), or more generally its logarithm \(\log C\), under a specific survival rate. In serval codes of practice [20; 21], the value \(\log C\) is assumed to be normal distributed and a survival probability of 97.7% is considered. As a result, the characteristic value of \(\log C\) can be calculated as the sample mean value minus two times the sample deviation. Obviously, the 97.7% survival rate-based method is straightforward and easy to use. However, the uncertainty in the estimators, i.e., the sample and deviation, is totally ignored in the above method, which may cause errors and even risks in the fatigue design. One this end, the statistics approach proposed by IIW [36] is also considered, in which the uncertainty in estimators is included using the tolerant interval, as shown in Equations (5) and (6).