## 2 Analytical Solution of Multiple Contact Pairs for the MLTWC
### Analytical Solution of Bi-layer Thick-walled Cylinders
#### 2.5.1 Installation from Inside to Outside  
The installation is split into two steps: the inner contact pair _ii_ goes first, and then the outer contact pair \(i\) is assembled. To calculate contact pressure, the two pairs are analyzed separately, and then the results of them are superposed on the basis of superposition principle, as shown in Figure 5.  
The interference of the inner contact pair \(\delta_{ii}\) can be solved by Equation (3). Using the simplified Formula (8) to calculate the contact pressure between two thick-walled cylinders, the contact pressure of the inner pair _ii_ is derived as:  
\[P_{ii}=\frac{\delta_{ii}}{r_{3}\left(\frac{r_{2}^{2}(-1+\
u_{2})}{E_{2}\left(- \frac{r_{2}^{2}}{2}+r_{3}^{2}\right)}+\frac{r_{2}^{2}(1+\
u_{2})}{E_{2}\left( \frac{r_{2}^{2}}{2}-r_{3}^{2}\right)}-\frac{r_{3}^{2}(-1+\
u_{3})}{E_{3}\left( \frac{r_{3}^{2}}{2}-r_{4}^{2}\right)}+\frac{r_{2}^{2}(1+\
u_{3})}{E_{3}\left( \frac{r_{3}^{2}}{2}-r_{4}^{2}\right)}\right)} \tag{12}\]  
According to the solution of the thick-walled cylinder axisymmetric problem, the contact pair _ii_ will lead to a radial displacement at the exterior of the medial cylinder, as shown in Equation (13).  
\[u_{r}|_{r=r_{2}}=\frac{2P_{ii}r_{2}r_{3}^{2}}{E_{2}\left(r_{2}^{2}-r_{3}^{2} \right)} \tag{13}\]  
Such radial displacement results in an increase of the interference of the outer contact pair _i._ Denoting this increase as \(\delta_{c:ii\to i}\), the value of which is equal to \(u_{r}|_{r=r_{2}}\). \(\delta_{c:ii\to i}\) indicates the  
Figure 6: Superposition principle of tri-layer thick-walled cylinders installed from outside to inside.  
Figure 7: Superposition principle of tri-layer thick-walled cylinders without consideration of assembly order.  
monodirectional effect from contact pair _ii_ to \(i\). Based on the Formula (3), the interference \(\delta_{i}\) of contact pair \(i\) should be expressed as:  
\[\mathbf{\delta_{i}}=\mathbf{\delta_{0}}+\mathbf{\delta_{T}}+\mathbf{\delta_{c}}_{i\gets i} \tag{14}\]  
Applying the simplified computational formula (8) to calculate the contact pressure \(P_{i}\) of the contact pair \(i\), Equation (15) can be derived:  
\[P_{i}=\frac{\delta_{0}+\delta_{T}+\frac{2\gamma_{2}r_{3}^{2}}{E_{2}\left(\frac {r_{3}^{2}}{2}-r_{3}^{2}\right)}\frac{\gamma_{3}^{2}(-1+\gamma_{2})}{\gamma_{2 }\left(\frac{r_{3}^{2}}{2}-\frac{r_{3}^{2}}{2}\right)}+\frac{\delta_{i}}{E_{2} \left(\frac{r_{3}^{2}}{2}-\frac{r_{3}^{2}}{2}\right)}\frac{\gamma_{3}^{2}(-1+ \gamma_{3})}{E_{3}\left(\frac{r_{3}^{2}}{2}-\frac{r_{3}^{2}}{2}\right)}+\frac{ r_{3}^{2}(1+\gamma_{3})}{E_{3}\left(\frac{r_{3}^{2}}{2}-\frac{r_{3}^{2}}{2}\right)} \right)}}{r_{2}\left(\frac{r_{3}^{2}}{2}(-1+\gamma_{1})}\frac{r_{1}^{2}(1+ \gamma_{1})}{E_{1}\left(\frac{r_{3}^{2}}{2}-\frac{r_{3}^{2}}{2}\right)}-\frac{ r_{3}^{2}(-1+\gamma_{2})}{E_{2}\left(\frac{r_{3}^{2}}{2}-\frac{r_{3}^{2}}{2} \right)}+\frac{r_{3}^{2}(1+\gamma_{2})}{E_{2}\left(\frac{r_{3}^{2}}{2}-\frac{ r_{3}^{2}}{2}\right)}\right)} \tag{15}\]  
Such calculation is in accordance with the component installation from inside to outside. The calculated contact pressure of the inner contact pair _ii_ is not the final result. The outer contact pair \(i\) increases the material stiffness of the exterior of the inner contact pair, and thus increases the contact pressure of the inner contact pair. The calculated contact pressure of the inner pair is smaller than the actual value. Besides, the calculation of the contact pressure of the outer pair also does not consider the impact of the innermost material stiffness, as a result, the calculated value is as well smaller than the actual value.