# Complex Variable Solution for Stress and Displacement of Layered Soil with Finite Thickness
## 2 Problem Statement and Method Presentation
\begin{array}{l}\sum\limits_{k=1}^{n_{j,1}}\text{Im}\Big{[}g_{j,ks}-h_{j,ks} \Big{]}a_{j,k,R}+\sum\limits_{k=1}^{n_{j,1}}\text{Re}\Big{[}g_{j,ks}+h_{j,ks} \Big{]}a_{j,k,l}-\\ \sum\limits_{k=1}^{n_{j,2}}\text{Im}\Big{[}l_{j,ks}\Big{]}b_{j,k,R}+\sum \limits_{k=1}^{n_{j,2}}\text{Re}\Big{[}l_{j,ks}\Big{]}b_{j,k,l}+\frac{\kappa_{j}}{ C_{j}}a_{j,0,l}+\frac{1}{C_{j}}b_{j,0,l}-\\ \sum\limits_{k=1}^{n_{j+1,1}}\text{Im}\Big{[}g_{j+1,ks}-h_{j+1,ks}\Big{]}a_{j+1, k,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,l}+\\ \sum\limits_{k=1}^{n_{j+1,2}}\text{Im}\Big{[}l_{j+1,ks}\Big{]}b_{j+1,k,R}-\sum \limits_{k=1}^{n_{j+1,2}}\text{Re}\Big{[}l_{j+1,ks}\Big{]}b_{j+1,k,l}-\frac{ \kappa_{j+1}}{G_{j+1}}a_{j+1,0,l}-\frac{1}{C_{j+1}}b_{j+1,0,l}=0\end{array}  \tag{34}\]Equations (35) and (36) can be obtained from the real and imaginary parts of Equation (29). \[ \begin{array}{l}\sum\limits_{k=1}^{n_{i,1}}\mathrm{Re}\Big{[}2c_{j,ks}+d_{j,ks} -e_{j,ks}\Big{]}a_{j,k,R}+\sum\limits_{k=1}^{n_{i,1}}\mathrm{Im}\Big{[}-2c_{j, ks}-d_{j,ks}+e_{j,ks}\Big{]}a_{j,k,I}+\\ \sum\limits_{k=1}^{n_{i,2}}\mathrm{Re}\Big{[}f_{j,ks}\Big{]}b_{j,k,R}-\sum \limits_{k=1}^{n_{i,2}}\mathrm{Im}\Big{[}f_{j,ks}\Big{]}b_{j,k,I}-\\ \sum\limits_{k=1}^{n_{i,1}}\mathrm{Re}\Big{[}2c_{j+1,ks}+d_{j+1,ks}-e_{j+1,ks} \Big{]}a_{j+1,k,R}-\sum\limits_{k=1}^{n_{i+1,1}}\mathrm{Im}\Big{[}-2c_{j+1,ks}-d _{j+1,ks}+e_{j+1,ks}\Big{]}a_{j+1,k,I}-\\ \sum\limits_{k=1}^{n}\mathrm{Re}\Big{[}f_{j+1,ks}\Big{]}b_{j+1,k,R}+\sum \limits_{k=1}^{n_{i+1,2}}\mathrm{Im}\Big{[}f_{j+1,ks}\Big{]}b_{j+1,k,I}=0\\ \sum\limits_{k=1}^{n_{i,1}}\mathrm{Im}\Big{[}d_{j,ks}-e_{j,ks}\Big{]}a_{j,k,R} +\sum\limits_{k=1}^{n_{i,1}}\mathrm{Re}\Big{[}d_{j,ks}-e_{j,ks}\Big{]}a_{j,k,I }+\\ \sum\limits_{k=1}^{n_{i,2}}\mathrm{Im}\Big{[}f_{j,ks}\Big{]}b_{j,k,R}+\sum \limits_{k=1}^{n_{i,2}}\mathrm{Re}\Big{[}f_{j,ks}\Big{]}b_{j,k,I}-\\ \sum\limits_{k=1}^{n_{i,1}}\mathrm{Im}\Big{[}d_{j+1,ks}-e_{j,ks}\Big{]}a_{j+1, k,R}-\sum\limits_{k=1}^{n_{i,1}}\mathrm{Re}\Big{[}d_{j+1,ks}-e_{j+1,ks}\Big{]}a_{j+1, k,I}-\\ \sum\limits_{k=1}^{n_{i+1,2}}\mathrm{Im}\Big{[}f_{j+1,ks}\Big{]}b_{j+1,k,R}- \sum\limits_{k=1}^{n_{i+1,2}}\mathrm{Re}\Big{[}f_{j+1,ks}\Big{]}b_{j+1,k,I}=0 \end{array}  \tag{36}\]  
Equations (37) and (38) can be obtained from the real and imaginary parts of Equation (30). \[\begin{array}{l}\sum\limits_{k=1}^{n_{n,1}}\mathrm{Re}[g_{n,ks}-h_{n,ks}]a_{ n,k,R}-\sum\limits_{k=1}^{n_{n,1}}\mathrm{Im}[g_{n,ks}+h_{n,ks}]a_{n,k,I}-\\ \sum\limits_{k=1}^{n_{n,2}}\mathrm{Re}[h_{n,ks}]b_{n,k,R}-\sum\limits_{k=1}^{n_ {n,2}}\mathrm{Im}[h_{n,ks}]b_{n,k,R}+\frac{\kappa_{n}}{\alpha_{n}}a_{n,0,R}- \frac{1}{\alpha_{n}}b_{n,0,R}=0\\ \sum\limits_{k=1}^{n_{n,1}}\mathrm{Im}[g_{n,ks  
The displacement and stress of any point in the \(j\)-th layer can be calculated by Equations (41) and (42), respectively.  
\[2G_{j}(u_{j}+i\eta_{j})=\kappa_{j}\varphi_{j}(\zeta_{j})-\frac{\omega_{j}(\zeta_{ j})}{\omega_{j}^{\prime}(\zeta_{j})}\overline{\phi_{j}^{\prime}(\zeta_{j})}- \overline{\psi_{j}(\zeta_{j})} \tag{41}\]  
\[\left\{ \begin{array}{l}\sigma_{j\psi}+\sigma_{j,x}=4\text{Re}\bigg{[}\frac{ \varphi_{j}^{\prime}(\zeta_{j})}{\omega_{j}^{\prime}(\zeta_{j})}\bigg{]}\\ \\ \sigma_{j\psi}-\sigma_{j,x}+2i\gamma_{j,xy}=2\frac{\omega_{j}(\zeta_{j})}{ \big{[}\omega_{j}^{\prime}(\zeta_{j})\big{]}^{2}}\bigg{[}\varphi_{j}^{\prime }(\zeta_{j})-\frac{\omega^{\prime\prime}(\zeta_{j})\varphi_{j}^{\prime}(\zeta_ {j})}{\omega_{j}^{\prime}(\zeta_{j})}\bigg{]}+2\frac{\psi_{j}^{\prime}(\zeta_ {j})}{\omega_{j}^{\prime}(\zeta_{j})}\end{array} \right. \tag{42}\]  
From Equations (41) and (42), we have  
\[\left\{ \begin{array}{l}u_{j}=\frac{\text{Re}\bigg{(}\kappa_{j}t_{j,x}-\frac{ t_{j,1}}{t_{j,2}}\big{[}\frac{\zeta_{j}-1}{t_{j,2}}\big{)}}{2G_{j}}\\ \\ v_{j}=\frac{\text{Im}\bigg{(}\kappa_{j}t_{j,x}-\frac{t_{j,1}}{t_{j,2}}\big{[} \frac{\zeta_{j}-1}{t_{j,2}}\big{)}}{2G_{j}}\end{array} \right. \tag{43}\]  
\[\left\{ \begin{array}{l}\sigma_{j,x}=(AA_{j}-BB_{j})/2\\ \sigma_{j,y}=(AA_{j}+BB_{j})/2\\ \tau_{j,xy}=CC_{j}/2\end{array} \right.