## 5 Results  
### Tightening Performance Analysis  
The tightening process simulations of all bolted joints mentioned above were accomplished via FEA. First, the discrete data points and curves of tightening torques and the corresponding preloads in five different bolted joints installed with a single nut were obtained, as shown in Figure 19, which shows that it is difficult to differentiate between the discrete data points of regular bolted joints and those of the bolted joints installed with plain and wedge washers due to coincidence. The curve drawn based on discrete data points obtained from the bolted joint installed with a variable-diameter nut is nonlinear and runs above those of the other bolted joints, which are almost linear. To quantitatively evaluate the degrees of tightening difficulties, linear fits were performed for the data points. The slopes of fitted lines for the torsion-tension relationship reflect the tightening performances of different bolted joints. The larger the slope of the fitted line, the more difficult it is to tighten the corresponding bolted joints. The slopes of fitted lines are listed in Table 2. By comparing the slopes shown in Table 2, it is obvious that the slope of variable-diameter nut is as large as 5.498 mm, while the wedge-locking nut has a slope of 2.365 mm. They are increased by 125% and 20%, respectively, compared with the values of a regular bolted joint. However, the slopes of the bolted joints installed with plain and wedge washers are almost the same as those of the regular bolted joint. This indicates that the variable-diameter nut and wedge-locking nut are more difficult to tighten than the normal nut and that the applications of plain and wedge washers do not affect the tightening performance. \begin{table}
\begin{tabular}{l c} \hline
**Bolted Joint** & **Slopes (mm)** \\ \hline Regular bolted joint & 1.983 \\ Plain washer & 2.004 \\ Variable-diameter nut & 5.498 \\ Wedge-locking nut & 2.365 \\ Wedge washer & 1.952 \\ \hline \end{tabular}
\end{table} 
Table 2: Slopes of fitted lines for the torsion–tension relationship in the bolted joints installed with a single nut. Figure 19: Discrete data points and relationship curves of tightening torques and preloads in five different bolted joints installed with a single nut. [PERSON] et al. [18] found that the torque-tension relationships of a regular bolted joint, plain washer, and wedge-locking nut obtained by test machine were similar to the FEA results shown in Figure 19. Specifically, when tightening the wedge-locking nut, a higher torque was required to achieve the same expected preload. Therefore, the results reported in our paper and the available literature prove that a wedge-locking nut is more difficult to tighten than a regular nut and that the application of a plain washer does not affect the tightening performance. Secondly, the discrete data points and curves of tightening torques and the corresponding preloads in three bolted joints installed with double nuts were also obtained, as shown in Figure 20. In this study, the total preload was set to approximately 30 kN, and the upper and lower nuts were both assigned values of 15 kN. It can be observed from the figure that there are two distinct stages in these relationship curves. In the first stage, the lower nut was tightened to obtain the discrete data and curves of torques and preloads. When tightening the second nut, we found that the preload existed but that the torque increased from 0. Consequently, there was a sudden change in the relationship curves. Subsequently, the preload and torque increased together as tightening proceeded. In the first stage, we found that the curve slope of the eccentric double nut was the largest, while the other two bolted joints had a similar curve slope. In the second stage, the curve slope of the double-thread bolt connection was the smallest, while those of the other two bolted joints were close.