## 7 Conclusions  
In this study, three-dimensional FEMs of a regular bolted joint and seven anti-loosening bolted connections were established, and the processes of tightening, vibration-induced loosening, and fatigue were simulated via FEA. A MATLAB program based on a rainflow counting method was developed to perform cycle counting and accumulate the fatigue damage of bolted joints. According to the results of the simulation, the tightening, anti-loosening, and fatigue resistance characteristics of a typical bolted joint and anti-loosening  
Figure 31: Stress time series and stress range histograms of bolted joints installed with a double-thread bolt: (**a**) stress time series at the root point; (**b**) stress range histogram. Figure 32: Stress time series and stress range histograms of bolted joints installed with an eccentric double nut: (**a**) stress time series at the root point; (**b**) stress range histogram. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Preload Assigned to the Upper Nut** & **5 kN** & **10 kN** & **15 kN** & **20 kN** & **25 kN** & **30 kN** \\ \hline Regular double nuts & 0.0812 & 0.0674 & 0.0676 & 0.0585 & 0.0370 & 0.0298 \\ Double-thread bolt connection & 0.0438 & 0.0378 & 0.0292 & 0.0248 & 0.0243 & 0.0099 \\ Eccentric double nut & 0.1041 & 0.0878 & 0.0752 & 0.0655 & 0.0651 & 0.0416 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Fatigue damage indexes of three bolted joints installed with double nuts. -bolted connections were evaluated comprehensively. The main conclusions are summarized as follows. 1. The results of the tightening simulation indicate that variable-diameter nuts and wedge-locking nuts are more difficult tighten than regular nuts and that the application of plain and wedge washers does not affect the tightening characteristics. The simulation results are the same as the experimental results reported in [18]. In terms of bolted joints with double nuts, it was found that the interference and eccentricity in an eccentric double nut make it more difficult to tighten than regular and double-thread bolted joints. 2. The results of vibration-induced loosening simulation verify that the researched bolted connections installed with different washers or nuts all possess anti-loosening capacities under transversal vibration compared to a regular bolted joint. Specifically, the anti-loosening performance of a wedge-locking nut is not as strong as that of other bolted connections. Additionally, a small drop in the preload of regular double nuts was observed after the initial vibration cycle. This phenomenon might be caused by the effect of stress redistribution. 3. The results of fatigue simulation demonstrate that the application of anti-loosening bolted connections improves fatigue resistance to varying degrees. Moreover, the fatigue resistance performances of bolted joints installed with a variable-diameter nut and wedge-locking nut are the best among all the anti-loosening bolted connections. It is worth mentioning that although the bolted joint installed with eccentric double nut showed superior anti-loosening ability, its fatigue resistance performance under transversal vibration was the worst among the three bolted joints with double nuts. 4. The effects of preload distribution on the two nuts on tightening, anti-loosening, and fatigue resistance performances were discussed. The results show that, for different bolted joints with double nuts, the tightening performance is greatly influenced by the preload distribution applied to the upper and lower nuts. We recommend increasing the proportion of preload assigned to the upper nut in order to achieve better anti-loosening performance and extend the fatigue life of bolted connections. The importance of reasonably distributing the preload of the two nuts should be emphasized in engineering applications. Overall, the results obtained in this work can provide technical assistance and guidance for engineers in terms of selecting appropriate anti-loosening bolted connections in critical applications. Resources, J.L.