## 3 Fatigue Resistance of the Riveted Stringer-to-Cross-Grider Connection
### Determination of the Fatigue Detail Category
Then, the regression line for 50% survival probability can be expressed by the equation  
\[\rm y=\alpha+\beta\cdot x, \tag{5}\]  
in which the parameters \(\alpha\) and \(\beta\) corresponding to the parameters \(\log\) C and \(\rm m\), respectively, in Equation (4) and can be calculated as follows:  
\[\beta=\frac{\sum x_{\rm i}y_{\rm i}-\frac{1}{n}{\left(\sum x_{\rm i}\right)}{ \left(\sum y_{\rm i}\right)}}{\sum x_{\rm i}^{2}-\frac{1}{n}{\left(\sum x_{\rm i }\right)}^{2}};\ \alpha=\frac{\sum y_{\rm i}-\beta\sum x_{\rm i}}{n}, \tag{6}\]  
The stress range \(\Delta\sigma_{\rm P}\) on the regression line corresponding to the number of cycles \(\rm N=2\times 10^{6}\) is  
\[\Delta\sigma_{\rm P}=\left(\frac{2\times 10^{6}}{10^{\alpha}}\right)^{1/\beta}. \tag{7}\]  
The left-sided prediction limit of \(\log\rm N_{P}\) corresponding to the stress range \(\Delta\sigma_{\rm P}\) is given by the formula  
\[\log\rm N_{P}=\log\left(2\times 10^{6}\right)-t_{0.05,n-2}\cdot s_{R}\cdot \sqrt{1+\frac{1}{n}+\frac{\left(\log\Delta\sigma_{P}-\left(\sum x_{\rm i} \right)/n\right)^{2}}{\sum x_{\rm i}^{2}-\frac{1}{n}{\left(\sum x_{\rm i} \right)}^{2}}}, \tag{8}\]  
where \(\rm t_{0.05,n-2}\) is the fifth percentile of the Student's t-distribution with n-2 degrees of freedom (it means 95% survival probability), and \(\rm s_{R}\) is the standard deviation given by the formula  
\[\rm s_{R}=\sqrt{\frac{1}{n-2}{\left(\sum y_{\rm i}^{2}-\frac{1}{n}{\left(\sum y _{\rm i}\right)}^{2}-\beta\cdot\left(\sum x_{\rm i}y_{\rm i}-\frac{1}{n}{ \left(\sum x_{\rm i}\right)}\left(\sum y_{\rm i}\right)\right)\right)}}. \tag{9}\]  
Finally, the characteristic stress range for the determination of the detail category can be calculated from the equation  
\[\Delta\sigma_{\rm C}=\Delta\sigma_{\rm P}\cdot\left(\frac{2\times 10^{6}}{ \rm N_{P}}\right)^{1/\beta}. \tag{10}\]  
Applying the procedure mentioned above and using the data in Table 1, we have determined the values of parameters \(\alpha=12.41\), \(\beta=-3.02\) and the characteristic value of stress range \(\Delta\sigma_{\rm C}=83.9\) MPa. General principles for statistical evaluation of experimental tests according to Eurocode are given in EN 1990, Annex D [33]. A more detailed procedure for evaluating fatigue tests can be found in the background document [34] to EN 1993-1-9 [15]. Application of these procedures can be found in works [35; 36]. In contrast to the procedure above, the slope of the S-N curves is assumed to be of a constant value \(\rm m=3\) for structural details subjected to normal stresses. Then, the second parameter of the regression line for 50% survival probability (here denoted as \(\rm\hat{C}\)) can be derived using Equation (4). \[\log\rm\hat{C}=\frac{1}{n}{\left(\sum\log N_{i}+m\cdot\sum\log\Delta\sigma_{i }\right)}. \tag{11}\]  
The standard deviation \(\rm s\) in terms of \(\log  
fractile estimation [37] depending on whether the standard deviation s (or the coefficient of variation V\({}_{\rm x}\)) is known or unknown. Then, the characteristic value of the intercept log C\({}_{\rm k}\) is defined by the equation  
\[\log\rm{C_{k}}=\log\hat{\rm{C}}-k_{n}\cdot s. \tag{13}\]  
Finally, the characteristic value \(\Delta\sigma_{\rm C}\) of the fatigue strength at 2 \(\times\) 10\({}^{6}\) stress cycles amounts to  
\[\log\Delta\sigma_{\rm C}=\frac{\log 2\times 10^{6}-\log\rm{C_{k}}}{-m}\ \Rightarrow\ \Delta\sigma_{\rm C}=10^{\log\Delta\sigma_{\rm C}}. \tag{14}\]  
Applying this procedure with the value of factor k\({}_{n}\) = 1.92 for a number of samples n = 10 and the unknown value of V\({}_{\rm x}\) from table D.1 in [33] and using the data in Table 1, the parameter log\(\hat{\rm{C}}\) takes on a value 12.35, and the characteristic value of stress range is \(\Delta\sigma_{\rm C}\) = 85.6 MPa. The graphic evaluation of the fatigue test results is shown in \[\rm y=\alpha+\beta\cdot x, \tag{5}\]0  
Both the procedures for processing the fatigue test data provide very similar results, which is understandable since both are based on the same mathematical-statistical principles.