# Probabilistic Analysis of Fatigue Behavior of Single Lap Riveted Joints
## 3 Fatigue Tests Results and Discussion
### Critical Crack Size and Location  
Specimens were found to fail according to the two main modes: cracks propagating through-hole and in the proximity of the hole, depending on the level of stress. As higher is the level of stress, as higher is the probability of through-hole cracks.  
More in detail, for the level of stress of 160 MPa, the cracks propagate predominantly through the hole, for the levels of stress of 90 MPa and 120 MPa, cracks were found to propagate mainly closer, but not through, the hole. Figure 4 shows the different fracture types.  
The crack location can be attributed to the clamping pressure, which forces the crack to initiate outside the hole. By increasing the fatigue stress, this clamping pressure is increasingly overruled, and failure is due to the effects of the bearing pressure at the rivet-hole interface. As the bearing pressure increases with the stress level, as more significant is the role played by the notch, because of the complex stress field in this zone. For a lower stress level, at 90 MPa, it can be observed that only 27.2% of the tested specimens showed through-hole cracks, while the remaining 72.8% showed cracks in the proximity of the hole. At 120 MPa, the total of specimens failed at the hole's cross-section increased and reached 36.3% of specimens. At 160 MPa, as anticipated, the 100% of the specimens failed in correspondence of the hole's cross-section.  
Figure 5 shows the morphology for both failure types.  
\begin{table}
\begin{tabular}{c c c} \hline
**Load Level** & **Average Value of Fatigue Life** & **Standard Deviation for Fatigue Life** \\
**[MPa]** & **[10\({}^{3}\) cycles]** & **[10\({}^{3}\) cycles]** \\ \hline
90 & 693.2 & 205.1024 \\
120 & 231 & 34.1735 \\
160 & 70.9 & 16.9139 \\ \hline \end{tabular}
\end{table}
Table 3: Average and standard deviation values for fatigue life.  
Figure 4: Crack types: in the proximity of the hole at 90 MPa (**a**) and at 120 MPa (**b**); through-hole at 160 MPa (**c**).  
As can be seen from Figure 5, the crack starts on the faying surface and in general on several macro zones. The macro zones close to the rivet hole (Figure 5a) or in the middle of the specimen cracked in proximity of the hole (Figure 5b) are initiation areas rather than initiation sites. For the through-hole cracked specimens, the cracks did not initiate as corner cracks. This initiation behavior agrees with that described in [PERSON] et al. [23].  
Another discrimination can be done by considering the crack location on the joint, as shown in Figure 6.  
For the levels stress of 90 MPa and 120 MPa, it was observed that 93.7% and 96% of specimens, respectively, failed in correspondence of the zone, indicated in Figure 6a. Percentage changes for the level stress of 160 MPa, where a smaller percentage of specimens, 61.2%, failed in correspondence of such zone, while the remaining part, 38.8%, was closer to the zone shown in Figure 6b.  
This difference can be attributed to the clamping pressure, which forces the crack to initiate outside the hole. Such clamping pressure can be different at the location shown in Figure 6a, under the rivet head, and at the location shown in Figure 6b, under the squeezer head. Such an aspect can be addressed also to a not uniform rivet expansion along the hole. By increasing the fatigue stress, this clamping pressure is increasingly overruled, and the failure is induced by the bearing pressure at the rivet-hole, leading the specimens to fail in correspondence of the hole's cross-section.  
Figure 7 shows the critical fatigue crack length values at the three different levels of stress. The critical crack size decreases as the level of stress increases. Additionally, the average and the standard deviation values of critical crack sizes per each stress level are reported in Table 4. It is worth to note that the variability decreases as