## 3 Equivalent Bending Stiffness Model of the Joint Connecting the Porcelain Bushing and Flange  
For a porcelain bushing with an elastic modulus of \(E_{c}\), outer diameter of \(d_{c}\), and inner diameter of \(d_{b}\), the bending stiffness, \(K_{c1}\), of the porcelain bushing with a circular section is given by the following:  
\[K_{c1}\ =\ \frac{E_{c}I_{c}}{L_{c}}\ =\ \frac{\pi\big{(}d_{c}^{4}-d_{b}^{4} \big{)}E_{c}}{64I_{c}} \tag{4}\]  
where \(I_{c}\) is the cross-sectional moment of inertia of the porcelain bushing (m\({}^{4}\)), and \(I_{c}\) is the equivalent length (m), which can be taken as 1/20 the length of a single porcelain bushing based on the code [15]. For a flange with an elastic modulus of \(E_{f}\), ring wall outer diameter of \(D_{f}\), and inner diameter of \(d_{f}\), the bending stiffness, \(K_{c3}\), of the flange of the ring section is as follows:  
\[K_{c3}\ =\ \frac{E_{f}I_{f}}{h_{c}}\ =\ \frac{\pi\big{(}D_{f}^{4}-d_{f}^{4} \big{)}E_{f}}{64h_{c}} \tag{5}\]  
where \(I_{f}\) is the upper section moment of inertia of the flange, and \(h_{c}\) is the height of the flange, which can be taken as the height of the cement bond. In the following analysis, we must establish the equivalent bending stiffness of the cement bond, \(K_{c2}\). The porcelain bushing is embedded in the flange cup and bonded to the flange with cement. Owing to the low tensile strength of the cement material, the joint is subjected to a bending moment load. Consequently, cracks will occur in the cement. Therefore, the cement bond is not subject to tensile action, and the load can only be transmitted via extrusion. If we do not consider the effect of the porcelain bushing deformation on the extrusion of the cement bond, its equivalent bending stiffness coefficient, \(K_{\text{c2}}\), can be deduced using the same method as in a previous study [23]. Taking the 1-1 profile in Figure 2c to characterize the extrusion deformation of the cement bond, as shown in Figure 3b, the bond thickness, \(t_{x}(y)\), in the horizontal X-direction is  
\[t_{x}(y)\ =\ \left\{ \begin{array}{l}\sqrt{\left(r_{c}+t_{e}\right)^{2}-y^{2}}- \sqrt{r_{c}^{2}-y^{2}},y\in\left[-r_{c},r_{c}\right]\\ \sqrt{\left(r_{c}+t_{e}\right)^{2}-y^{2}},y\
otin\left[-r_{c},r_{c}\right]\end{array} \right. \tag{6}\]  
When the root of the porcelain bushing rotates around O by an angle \(\theta_{2}\), the horizontal extrusion deformation of the cement bond at height \(z\) can be expressed as follows:  
\[\Delta_{x}(z)\ =\ (z-e)\theta_{2} \tag{7}\]  
Moreover, the elastic stiffness coefficient of the cement \(\text{d}z\) and \(\text{d}y\) units in the X-direction is  
\[k_{x}(y)\ =\ \frac{E_{g}\text{d}z\text{d}y}{t_{x}(y)} \tag{8}\]  
Therefore, the horizontal elastic restoring force, \(f_{x}\)(z,\(y\)), of the \(\text{d}z\) and \(\text{d}y\) units at coordinates (\(z\), \(y\)) is as follows:  
\[f_{x}(z,y)\ =\ k_{x}(z,y)\Delta_{x}(z) \tag{9}\]  
Then, the total elastic stiffness coefficient within the \(\text{d}z\) element at height \(z\) is  
\[k_{z}\ =\ \int_{-r_{c}-l_{e}}^{r_{c}+t_{e}}k_{x}(y)\ =\ \gamma E_{g}\text{d}z \tag{10}\]  
where  
\[\gamma\ =\ \frac{\sqrt{\lambda^{2}-1}+\left(2-\lambda^{2}\right)\text{ arcsin}\frac{1}{\lambda}++\left(2\lambda^{2}-1\right)\text{arcsin}1}{ \lambda^{2}-1} \tag{11}\]  
\[\lambda\ =\ 1+\frac{t_{e}}{r_{c}} \tag{12}\]  
In the above equations, \(E_{g}\) is the elastic modulus of the cement bond (\(\text{N}/\text{m}^{2}\)), \(h_{c}\) is the height of the cement bond in the flange (m), \(t_{e}\) is the thickness of the cement filling between the flange and the porcelain bushing (m), and \(r_{c}\) is the radius of the porcelain bushing (m). Figure 3: Deformation of the cement bond.