# Complex Variable Solution for Stress and Displacement of Layered Soil with Finite Thickness
## 2 Problem Statement and Method Presentation
\tag{19}\]  
In Equation (17)  
\[\omega_{n}(\sigma_{n})=\frac{H_{n}}{\pi}\ln\left(\frac{-\sin\theta_{n}}{1-\cos \theta_{n}}\right)-\frac{H_{n}}{2}i,\theta_{n}\in[\pi,2\pi] \tag{20}\]  
By combining boundary conditions and continuity conditions, a system of linear equations for \(a_{jk},\overline{a_{jk}},\)\(b_{j,k}\) and \(\overline{b_{j,k}}\) can be obtained  
\[2R_{e}\left(\sum\limits_{k=1}^{n_{1,1}}c_{1,ks}a_{1,k}\right)+\sum\limits_{k=1} ^{n_{1,1}}d_{1,ks}a_{1,k}-\sum\limits_{k=1}^{n_{1,1}}e_{1,ks}a_{1,k}+\sum\limits _{k=1}^{n_{1,2}}f_{1,ks}b_{1,k}=f_{s} \tag{27}\]  
\[ \begin{array}{l}\sum\limits_{k=1}^{n_{j,1}}\delta_{j,ks}a_{j,k}-\sum\limits_ {k=1}^{n_{j,1}}h_{j,ks}\overline{a_{j,k}}-\sum\limits_{k=1}^{n_{j,2}}l_{j,ks} \overline{b_{j,k}}+\frac{\kappa_{j}}{C_{j}}a_{j,0}-\frac{1}{C_{j}}\overline{b_ {j,0}}-\\ \sum\limits_{k=1}^{n}g_{j+1,ks}a_{j+1,k}+\sum\limits_{k=1}^{n_{j,1}}h_{j+1,ks} \overline{a_{j+1,k}}+\sum\limits_{k=1}^{n_{j+1,2}}l_{j+1,ks}\overline{b_{j+1, k}}-\frac{\kappa_{j+1}}{C_{j+1}}a_{j+1,0}+\frac{1}{C_{j+1}}\overline{b_{j+1,0}}=0 \end{array}  \tag{28}\]  
\[ \begin{array}{l}2\text{Re}\left(\sum\limits_{k=1}^{n_{j,1}}c_{ijk}a_{j,k} \right)+\sum\limits_{k=1}^{n_{j,1}}d_{j,ks}a_{j,k}-\sum\limits_{k=1}^{n_{j,1} }e_{j,ks}a_{j,k}+\sum\limits_{k=1}^{n_{j,2}}f_{j,ks}b_{j,k}-\\ 2\text{Re}\left(\sum\limits_{k=1}^{n_{j+1,1}}c_{j+1,ks}a_{j+1,k}\right)-\sum \limits_{k=1}^{n_{j+1,1}}d_{j+1,ks}a_{j+1,k}+\sum\limits_{k=1}^{n_{j+1,1}}e_{ j+1,ks}a_{j+1,k}-\sum\limits_{k=1}^{n_{j+1,2}}f_{j+1,ks}b_{j+1,k}=0 \end{array}  \tag{29}\]  
\[ \begin{array}{l}\sum\limits_{k=1}^{n_{n,1}}g_{n,ks}a_{n,k}-\sum\limits_{k=1 }^{n_{n,1}}h_{n,ks}\overline{a_{n,k}}-\sum\limits_{k=1}^{n_{n,2}}l_{n,ks} \overline{b_{n,k}}+\frac{\kappa_{n}}{G_{n}}a_{n,0}-\frac{1}{G_{n}}\overline{b_ {n,0}}=0\end{array}  \tag{30}\]  
In Equation (27), \(s=1,\ldots,m_{1}+1\). In Equations (28) and (29), for the \(j\)-th layer \(s=m_{j}+2,m_{j}+3,\ldots,m_{j}+m_{j+1}+2\), for the \(j+1\)-th layer \(s=m_{j+1}+1,m_{j+1},\ldots,1\) (\(j=1,\ldots,n-1\)). In Equation (30), \(s=m_{n}+2,\ldots,m_{n}+m_{n+1}+2\). The coefficients \(c_{j,ks},\)\(d_{j,ks},\)\(e_{j,ks},f_{j,ks},g_{j,ks},\)\(h_{j,ks},\)\(l_{j,ls},\)\(f_{j}\) in Equations (27)-(30) are provided in Appendix A.  
Equations (31) and (32) can be obtained from the real and imaginary parts of Equation (27). \[ \begin{array}{l}\sum\limits_{k=1}^{n_{1,1}}\text{Re}[2c_{1,ks}+d_{1,ks}-e_{ 1,ks}]a_{1,k,R}+\sum\limits_{k=1}^{n_{1,1}}\text{Im}[-2c_{1,ks}-d_{1,ks}+e_{1, ks}]a_{1,k,l}+\\ \sum\limits_{k=1}^{n_{1,2}}\text{Re}[f_{1,ks}]b_{1,k,R}-\sum\limits_{k=1}^{n_{1,2}}\text{Im}[f_{1,ks}]b_{1,k,l}=\text{Re}[f_{s}]\end{array}  \tag{31}\]  
\[ \begin{array}{l}\sum\limits_{k=1}^{n_{1,1}}\text{Im}[d_{1,ks}-e_{1,ks}]a_{1, k,R}+\sum\limits_{k=1}^{n_{1,1}}\text{Re}[d_{1,ks}-e_{1,ks}]a_{1,k,l}+\\ \sum\limits_{k=1}^{n_{1,2}}\text{Im}[f_{1,ks}]b_{1,k,R}+\sum\limits_{k=1}^{n_{1,2}}\text{Re}[f_{1,ks}]b_{1,k,l}=\text{Im}[f_{s}]\end{array}  \tag{32}\]  
Equations (33) and (34) can be obtained from the real and imaginary parts of Equation (28). \[ \begin{array}{l}\sum\limits_{k=1}^{n_{j,1}}\text{Re}\Big{[}g_{j,ks}-h_{j,ks} \Big{]}a_{j,k,R}-\sum\limits_{k=1}^{n_{j,1}}\text{Im}\Big{[}g_{j,ks}+h_{j,ks} \Big{]}a_{j,k,l}-\\ \sum\limits_{k=1}^{n_{j,2}}\text{Re}\Big{[}l_{j,ks}\Big{]}b_{j,k,R}-\sum\limits_ {k=1}^{n_{j,2}}\text{Im}\Big{[}l_{j,ks}\Big{]}b_{j,k,R}+\frac{\kappa_{j}}{G_{j} }a_{j,0,R}-\frac{1}{C_{j}}b_{j,0,R}-\\ \sum\limits_{k=1}^{n_{j+1,1}}\text{Re}\Big{[}g_{j+1,ks}-h_{j+1,ks}\Big{]}a_{j+1, k,R}+\sum\limits_{k=1}^{n_{j+1,1}}\text{Im}\Big{[}g_{j+1,ks}+h_{j+1,ks} \Big{]}a_{j+1,k,l}+\\ \sum\limits_{k=1}^{n_{j+1,2}}\text{Re}\Big{[}l_{j+1,ks}\Big{]}b_{j+1,k,R}+\sum \limits_{k=1}^{n_{j+1,2}}\text{Im}\Big{[}l_{j+1,ks}\Big{]}b_{j+1,k,R}-\frac{ \kappa_{j+1}}{C_{j+1}}a_{j,0,R}+\frac{1}{C_{j+1}}b_{j+1,0,R}=0\end{array}  \tag{33}\]  
\[