## 3 Case Study
### Modelling and Parameter Identification  
When thin layer elements are used to model bolted structures, the role of bolts on the structure is replaced by thin layer elements established on the contact surface. A schematic diagram of the method is shown in Figure 10.  
However, the pressure distribution on the bolt joint surface is not uniform. The pressure gradually decreases along the radial direction, as shown in Figure 11. The pressure distribution of the joint surface is shown in Equations (19)-(21). \[P_{\text{max}}=\frac{3F_{b}}{\pi(r_{m}^{2}+r_{m}r_{i}+r_{i}^{2})} \tag{19}\]  
\[P(r)=-\frac{P_{\text{max}}}{r_{m}-r_{i}}(r-r_{m}) \tag{20}\]  
\[r_{m}=r_{b}+h\cdot\tan\alpha \tag{21}\]  
where \(F_{b}\) is the bolt pre-tightening force, \(r_{i}\) is the radius of the bolt hole, \(r_{m}\) is the maximum contact radius, \(r_{b}\) is the radius of the bolt head, and \(h\) is the thickness of the connected piece. Figure 11: Schematic diagram of pressure distribution in bolted joint. Figure 10: Parameterization of contact interface. It is important to note the setting of the initial value of the identification parameter. The parameters (\(E_{1}\), \(G_{1}\)) of the thin layer element are of the same order of magnitude as the material parameters of the bolt where close to the bolt. However, the parameter selection (\(E_{2}\), \(G_{2}\)) is an order of magnitude lower than the material parameters where away from the bolt. The initial value as well as the identification results are listed in Table 6. The iterative convergence curves of the identification parameters are as shown in Figure 14, when the pre-tightening torque is \(T_{N}=20\) N\(\cdot\)m.  
Table 7 lists the modal frequency parameters before and after identification as well as the error between the above parameters and the experimental frequency. It can be seen from the identification results that the maximum updated error is no more than 2.5%. The results have comparatively higher identification accuracy. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Identified Parameters (N/m\({}^{2}\))** & \(E_{1}\) & \(G_{1}\) & \(E_{2}\) & \(G_{2}\) \\ \hline Initial value & \(3.00\times 10^{9}\) & \(1.15\times 10^{9}\) & \(3.00\times 10^{6}\) & \(1.15\times 10^{6}\) \\ Identified value & \(8.63\times 10^{8}\) & \(3.42\times 10^{8}\) & \(9.33\times 10^{5}\) & \(4.46\times 10^{5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Initial value of the identified parameters. Figure 14: Iterative convergence curves of parameter selection. Figure 13: FEM of the connection structure. The elastic modulus \(E\) is greater than the shear modulus \(G\), which reflects the characteristics of normal contact stiffness more than the tangential stiffness. The identification results (\(E_{1}\), \(G_{1}\)) near the bolt area are far greater than that of the bolt area (\(E_{2}\), \(G_{2}\)), which better reflects the actual situation. Therefore, the connection performance of the overlapping structure with multiple bolts can be well simulated by the of isotropic thin layer element. Similarly, the material parameters of the thin layer element under different pre-tightening torque are obtained by the iterative solution method. Finally, the relationship between the identification parameters and the different pre-tightening torque is achieved, as shown in Figure 15. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Order** & **Experiment (Hz)** & **Initial (Hz)** & **Error \%** & **Updated (Hz)** & **Error \%** \\ \hline
1 & 424.91 & 437.55 & 2.97 & 427.01 & 0.49 \\
2 & 1012.50 & 1044.80 & 3.19 & 1037.05 & 2.42 \\
3 & 2277.04 & 2314.00 & 1.62 & 2261.19 & \(-\)0.69 \\
4 & 3153.12 & 3209.10 & 1.78 & 3136.38 & \(-\)0.53 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of modal frequency errors. Figure 15: The relation curves of identification parameters change with pre-tightening torque.