## 2 Analytical Solution of Multiple Contact Pairs for the MLTWC
### Analytical Solution of Bi-layer Thick-walled Cylinders
Derived from the displacement coordination in Equation (6), the contact pressure \(P_{i}\) can be obtained as shown in Equation (7):  
\[P_{i}=\frac{\delta_{i}}{\frac{r_{2,out}\left(-r_{2,out}^{2}\left(-1+v_{1} \right)+r_{1}^{2}\left(1+v_{1}\right)\right)}{E_{1}\left(r_{1}^{2}-r_{2,out}^{ 2}\right)}+\frac{r_{2,in}\left(r_{2,in}^{2}\left(-1+v_{2}\right)-r_{3}^{2} \left(1+v_{2}\right)\right)}{E_{2}\left(r_{3}^{2}-r_{2,in}^{2}\right)}} \tag{7}\]  
Approximately regarding \(r_{2,out}\) is equal to \(r_{2,in}\), and denoting both of them as \(r_{2}\), the simplified pressure is expressed in Equation (8). \[P_{i}=\frac{\delta_{i}}{r_{2}\left(\frac{r_{2}^{2}\left(-1+v_{1}\right)}{E_{1 }\left(-r_{1}^{2}+r_{2}^{2}\right)}+\frac{r_{1}^{2}\left(1+v_{1}\right)}{E_{1 }\left(r_{1}^{2}-r_{2}^{2}\right)}-\frac{r_{2}^{2}\left(-1+v_{2}\right)}{E_{2} \left(r_{2}^{2}-r_{3}^{2}\right)}+\frac{r_{1}^{2}\left(1+v_{2}\right)}{E_{2} \left(r_{2}^{2}-r_{3}^{2}\right)}\right)}\]fraction is regarded as the stiffness discount factor \(\phi\), so that the contact pressure between the bi-layer thick-walled cylinders is:  
\[P=\frac{\delta}{r_{2}\left(\frac{r_{2}^{2}(-1+v_{1})}{E_{1}\varphi_{1}(-r_{1}^{2 }+r_{2}^{2})}+\frac{r_{1}^{2}(1+v_{1})}{E_{1}\varphi_{1}(r_{1}^{2}-r_{2}^{2})}- \frac{r_{2}^{2}(-1+v_{2})}{E_{2}\varphi_{2}(r_{2}^{2}-r_{3}^{2})}+\frac{r_{2}^{ 2}(1+v_{2})}{E_{2}\varphi_{2}(r_{2}^{2}-r_{3}^{2})}\right)} \tag{9}\]  
Equations (8) and (9) indicate that the contact pressure is in direct proportion to the interference, the proportionality coefficient of which depends on the geometrical characteristic and material property of the contact pair. Material with larger Young modulus has larger proportionality coefficient, resulting in higher contact pressure. Given the contact pressure of the bi-layer cylinders \(P_{i}\), the stress of the inner and outer cylinder are shown in Equations (10) and (11), respectively, from the theoretic solution of space axisymmetric problems of thick-walled cylinder. Stress of the inner cylinder,  
\[\left\{ \begin{array}{c}\sigma_{r,in}=\frac{-P_{i}r_{2}^{2}}{r_{2}^{2}-r_{3} ^{2}}+\frac{1}{r^{2}}\frac{r_{3}^{2}r_{1}^{2}P_{i}}{r_{2}^{2}-r_{3}^{2}}\\ \sigma_{\theta,in}=\frac{-P_{i}r_{2}^{2}}{r_{2}^{2}-r_{3}^{2}}-\frac{1}{r^{2}} \frac{r_{3}^{2}r_{1}^{2}P_{i}}{r_{2}^{2}-r_{3}^{2}}\\ \sigma_{z,out}=0\\ \tau_{zr,out}=0\end{array} \right. \tag{10}\]  
Stress of the outer cylinder,  
\[\left\{ \begin{array}{c}\sigma_{r,out}=\frac{P_{i}r_{2}^{2}}{r_{1}^{2}-r_{2} ^{2}}-\frac{1}{r^{2}}\frac{r_{2}^{2}r_{1}^{2}P_{i}}{r_{1}^{2}-r_{2}^{2}}\\ \sigma_{\theta,out}=\frac{P_{i}r_{2}^{2}}{r_{1}^{2}-r_{2}^{2}}+\frac{1}{r^{2}} \frac{r_{2}^{2}r_{1}^{2}P_{i}}{r_{1}^{2}-r_{2}^{2}}\\ \sigma_{z,out}=0\\ \tau_{zr,out}=0\end{array} \right. \tag{11}\]
Figure 4: Calculation flow diagram for interference algorithm for tri-layer thick-walled cylinders (TTC) with different assembly sequence. (**a**) installation from inside to outside: proceeding from the interference of inner contact pair _ii_, the contact pressure of outer pair \(i\) can be calculated by updating the interference of outer contact pair \(i\); (**b**) installation from outside to inside: proceeding from the interference of outer contact pair \(i\), the contact pressure of inner pair _ii_ can be calculated by updating the interference of inner contact pair _ii_. Figure 5: Superposition principle of tri-layer thick-walled cylinders installed from inside to outside.