## 2 Basic Theory
### Thin-Layer Element
\right] \tag{11}\]  
By using the two-node Gaussian integral method, the stiffness matrix \(K\) of the thin layer element is expressed as follows:  
\[\mathbf{K}=\sum_{i=1}^{2}\!\sum_{j=1}^{2}\!\sum_{k=1}^{2}\!\mathbf{B}^{T}\big{(}\xi_{ i},\eta_{j},\xi_{k}\big{)}\mathbf{C}\mathbf{B}\big{(}\xi_{i},\eta_{j},\xi_{k}\big{)} det\big{(}\mathbf{J}\big{(}\xi_{i},\eta_{j},\xi_{k}\big{)}\big{)}w_{\xi_{i}}w_{\eta _{j}}w_{\xi_{j}} \tag{12}\]  
where \(w_{\xi}\), \(w_{\eta}\), and \(w_{\xi}\) are the Gaussian integral weight function. The thickness selection of thin layer elements affects the calculation results for the mechanical properties of structures. When the thickness is too large, there are too many components on the contact interface, which is inconsistent with the actual performance of the contact surface. When the thickness is too small, the Jacobian matrix tends to be a singular matrix, and the error is too large to be used to calculate the displacement strain  
Figure 3: Isoparametric transformation of the thin layer element. relationship. The ratio coefficient is used as the selection condition for the thickness of thin layer element modeling, as shown below:  
\[R=\frac{max(l_{1},l_{2})}{e} \tag{13}\]  
[PERSON] et al. [24] proposed that when the range of \(R\) is 10-100 we can get an exact result, as studied in the context of the static contact problem where the thin layer element is based on a linear constitutive. [PERSON] and [PERSON] [22] think reasonable results can be obtained when \(R\) is 5. In this paper, \(R\) in the case study is 10. The bolted joints fix the interface; thus, the determination of the contact stiffness is the core of the dynamic analysis of the bolted structure. [PERSON] and [PERSON] obtain the normal contact stiffness and tangential contact stiffness by testing the relationship between the stress and the displacement of the contact interface:  
\[\left\{ \begin{array}{c}k_{\tau}=\frac{c\tau}{e\mu_{r}}\\ k_{n}=\frac{c\sigma_{n}}{e\
u_{r}}\end{array} \right. \tag{14}\]  
where \(k_{n}\) is the normal contact stiffness, \(k_{\tau}\) is the tangential contact stiffness, \(e\) is thickness, \(u_{r}\) and \(v_{r}\) are the normal displacement and tangential displacement of the thin layers, respectively, and \(\sigma\) and \(\tau\) are normal stress and tangential stress. The interface is generated by the Hexahedron element with linear constitutive \(C\), which is as follows:  
\[C=\frac{E}{(1+\
u)(1-2\
u)}\left[ \begin{array}{ccc}1-\
u&\
u&0\\ v&1-\
u&\
u\\ 0&0&(1-2\
u)/2\end{array} \right] \tag{15}\]  
where \(E\) is the elastic Modulus, \(\
u\) is the Poisson's ratio. [PERSON] and [PERSON] [25] obtained the relationship between the equivalent shear modulus of the contact surface and the tangential stiffness of the contact surface by testing tangential force of the bolted joints' structure:  
\[G=\frac{ke}{A} \tag{16}\]  
where \(A\) is the real contact area and \(k\) is the tangential stiffness of the connection which correlates with the surface pressure and friction properties of the contact interface. The approximate elastic parameters of the thin layer element can be obtained by the above two methods of testing the contact stiffness. The thin layer element can provide the computation, distribution, and concentration of stresses and strain within the thin finite zones, and hence, can permit evaluation of progressive damage and failure as they occur in many engineering problems. In the case of this paper, the first four order bending modal data are used. The parameter identification method is used to determine the parameters of the thin layer element.