## Appendix A Solving Steps of the Interactions in Section 4.3
\begin{array}{l}\varphi_{2}^{(2)}(z_{2})=\frac{p_{R}R_{1}^{2}}{( \overline{c})^{2}}\bigg{[}\frac{-2R_{1}^{6}-3R_{2}^{4}\overline{c}_{2z}}{ \overline{c}(\overline{c}_{2z}+\overline{c}_{2z})^{2}}\bigg{]}-\frac{p_{R}R_ {2}^{2}}{z_{2}}+\frac{p_{A}R_{1}^{2}\overline{c}_{2z}^{2}}{\overline{c}^{2} +\overline{c}_{2z}}-\frac{B_{R}R_{1}^{2}\overline{c}_{2z}^{2}}{(R_{2}^{2}+ \overline{c}_{2z})^{2}}+\frac{p_{R}R_{1}^{4}\overline{c}_{2z}^{3}}{(R_{2}^{2}+ \overline{c}_{2z})^{3}}\\ \psi_{2}^{(2)}(z_{2})=-\frac{P_{A}R_{2}^{2}}{z_{2}}+\frac{P_{A}R_{1}^{2}z_{2}} {\overline{c}_{2}^{2}+\overline{c}_{2z}}-\frac{P_{B}R_{1}^{2}\overline{c}_{2z }^{2}}{z_{2}c^{2}}-\bigg{(}\frac{R_{2}^{2}}{z_{2}}-\overline{c}\bigg{)} \varphi_{2}^{(1)^{\prime}}(z_{2})\end{array} \right.\] (A8)  
**Step (d):**  
The complex stress functions \(\varphi_{1}^{(2)}(z_{1})\) and \(\psi_{1}^{(2)}(z_{1})\) in step (d) can be obtained by the coordinate transformation. \[\left\{ \begin{array}{l}\varphi_{1}^{(2)}(z_{1})=\varphi_{2}^{(2)}(z_{1}-c) \\ \psi_{1}^{(2)}(z_{1})=\psi_{2}^{(2)}(z_{1}-c)-\overline{c}\varphi_{2}^{(2)^{ \prime}}(z_{1}-c)\end{array} \right.\] (A9)  
Thus, the first iterative process is complete. The complex stress around section AT and FT-I is given by  
\[\left\{ \begin{array}{l}\varphi_{1}(z_{1})=\varphi_{1}^{(1)}(z_{1})+\varphi_ {1}^{(2)}(z_{1})\\ \psi_{1}(z_{1})=\psi_{1}^{(1)}(z_{1})+\psi_{1}^{(2)}(z_{1})\\ \varphi_{2}(z_{2})=\varphi_{2}^{(1)}(z_{2})+\varphi_{2}^{(2)}(z_{2})\\ \psi_{2}(z_{2})=\psi_{2}^{(1)}(z_{2})+\psi_{2}^{(2)}(z_{2})\end{array} \right.\] (A10)  
**Step (e):**  
Repeat the above steps to obtain all the complex stress solutions for the 3 rd iteration. The complex stress functions \(\varphi_{1}^{(3)}(z_{1})\) and \(\psi_{1}^{(3)}(z_{1})\) in step (e) can be expressed as follows:  
\[\left\{ \begin{array}{l}\varphi_{1}^{(3)}(z_{1})=\frac{P_{A}R_{2}^{2}z_{1}}{ E}-\frac{P_{B}R_{1}^{2}}{F}+\frac{P_{B}R_{1}^{2}R_{2}^{2}}{(\overline{c})^{2}E}+F \bigg{[}\frac{P_{B}R_{1}^{2}}{(\overline{c})^{2}}\bigg{(}\frac{3R_{1}^{4}}{ \overline{c}^{2}C^{2}}-\frac{2R_{2}^{6}}{\overline{c}^{3}C^{3}}\bigg{)}\\ +\frac{P_{B}R_{2}^{2}}{\overline{c}^{2}}+\frac{P_{A}R_{1}^{2}R_{2}^{2}}{ \overline{c}^{2}F^{2}}+\frac{3P_{B}R_{1}^{4}R_{2}^{2}}{\overline{c}^{2}F^{2} }-\frac{2\overline{c}P_{B}R_{1}^{2}R_{2}^{2}}{\overline{c}^{2}F^{2}}\bigg{]}- \frac{3P_{B}R_{1}^{2}H^{2}}{\overline{c}^{4}R_{1}^{2}}\bigg{(}\frac{H^{3}}{ \overline{c}^{2}}+\frac{2H^{2}}{\overline{K}}\bigg{)}\\ -\frac{2P_{B}R_{1}^{6}H^{2}}{\overline{c}^{3}R_{1}^{4}}\bigg{(}\frac{H^{4}}{ \overline{K}^{3}}+\frac{4H^{3}}{\overline{K}^{2}}+\frac{6H}{\overline{K}} \bigg{)}-\frac{P_{B}R_{2}^{2}}{(\overline{c})^{2}}\bigg{(}\frac{R_{1}^{6}}{( \overline{c})^{2}E^{2}}+\frac{3R_{1}^{4}}{(\overline{c})^{2}E}\bigg{)}\\ -\frac{P_{A}R_{2}^{2}H^{2}}{\overline{c}^{2}R_{1}^{2}}\bigg{(}\frac{H^{3}}{ \overline{K}^{2}}+\frac{3H^{2}}{\overline{K}}\bigg{)}+\frac{3P_{B}R_{2}^{2}H^{ 4}}{\overline{c}^{4}R_{1}^{4}}\bigg{(}(\overline{c})^{2}z_{1}-\frac{z_{1}^{ 2}(R_{1}^{2}-\overline{c}_{1})^{2}}{\overline{K}^{4}}\bigg{)}\\ +\frac{2\overline{c}P_{B}R_{2}^{2}H^{3}}{\overline{c}^{3}R_{1}^{4}}\bigg{(} \frac{\overline{c}_{1}^{4}-R_{1}^{3}\overline{c}_{3}^{3}}{\overline{K}^{3}}- \overline{c}_{2}1\bigg{)}\\ \Psi_{1}^{(3)}(z_{1})=\frac{P_{B}R_{1}^{2}}{(\overline{c})^{2}}\bigg{(}\frac{3R_ {1}^{4}}{\overline{c}^{2}C^{2}}-\frac{3R_{1}^{4}}{\overline{c}^{2}C^{2}} \bigg{)}+\frac{P_{B}R_{1}^{2}}{\overline{c}}-\frac{P_{A}R_{1}^{2}}{\overline{F}} +\frac{P_{B}R_{1}^{4}}{\overline{F}^{3}}\\ +\frac{P_{B}\overline{5}R_{1}^{2}}{\overline{F}^{2}}-\frac{R_{1}^{2}}{ \overline{c}_{1}}\varphi_{2}^{(2)^{\prime}}(0)-\frac{\overline{c}_{1}^{2}}{ \overline{c}_{1}}\varphi_{1}^{(3)^{\prime}}(z_{1})\end{array} \right.\] (A11)where \(E=\frac{R_{1}^{2}}{z_{1}}-\overline{c}\), \(F=\frac{R_{2}^{2}z_{1}}{R_{1}^{2}-z_{1}^{2}}+c\), \(G=\frac{R_{1}^{2}}{z_{1}}-\overline{c}+\frac{R_{2}^{2}}{c}\), \(H=\frac{R_{1}^{2}c}{c\overline{c}-R_{2}^{2}}\), and \(K=z_{1}-\frac{R_{1}^{2}c}{c\overline{c}-R_{2}^{2}}\), respectively.