# Complex Variable Solution for Stress and Displacement of Layered Soil with Finite Thickness
## 2 Problem Statement and Method Presentation
\(\sigma_{j,x}\), \(\sigma_{j,y}\) and \(\tau_{j,xy}\) are the horizontal normal stress, vertical normal stress and shear stress of the \(j\)-th layer soil, respectively, and \(u_{j}\) and \(v_{j}\) are the horizontal displacement and vertical displacement of the \(j\)-th layer soil, respectively. The three stress components \(\sigma_{j,x}\), \(\sigma_{j,y}\), \(\tau_{j,xy}\) and two displacement components \(u_{j}\), \(v_{j}\) of point \(z_{j}\) can be expressed as [21]:  
\[\left\{ \begin{array}{l}\sigma_{j,x}+\sigma_{j,y}=4\text{Re}\left(\varphi^{ \prime}{}_{1,j}(z_{j})\right)\\ \sigma_{j,z}=u_{j}(\sigma_{j,x}+\sigma_{j,y})\end{array} \right. \tag{5}\]  
\[\sigma_{j,y}-\sigma_{j,x}+2i\tau_{j,xy}=2\left[\overline{z_{j}}\varphi^{ \prime\prime}{}_{1,j}(z_{j})+\psi^{\prime}{}_{1,j}(z_{j})\right] \tag{6}\]  
\[u_{j}+iv_{j}=\frac{1}{2G_{j}}\left[\kappa_{j}\varphi_{1,j}(z_{j})-z_{j} \overline{\varphi^{\prime}{}_{1,j}(z_{j})}-\overline{\psi_{1,j}(z_{j})}\right] \tag{7}\]  
where \(\kappa_{j}=3-4\mu_{j}\), and Re\([\)... \(]\) means to take the real part of \([\)... \(]\), the imaginary part represented by Im\([\)... \(]\) will also be involved later. \(G_{j}\) and \(\mu_{j}\) are the shear modulus and Poisson's ratio of the \(j\)-th layer soil, respectively. The single valued analytic functions of the \(j\)-th layer are \(\varphi_{1,j}(z_{j})\) and \(\psi_{1,j}(z_{j})\). The superscripts (\(\cdot\))' and (\(\cdot\))\" denote the first and second derivatives of (\(\cdot\)) with respect to \(z_{j}\). For the problem analyzed in this paper, the stress boundary conditions can be obtained from Equations (5) and (6):  
\[2\text{Re}[\varphi_{1,1}(z_{1})]+\overline{z_{1}}\varphi^{\prime\prime}{}_{1, 1}(z_{1})+\psi^{\prime}{}_{1,1}(z_{1})=\left\{ \begin{array}{l}0,x_{1}\in(x_ {1a},\infty)\\ p(x_{1})+iq(x_{1}),x_{1}\in[x_{1b},x_{1a}]\\ 0,x_{1}\in(-\infty,x_{1b})\end{array} \right. \tag{8}\]  
where \(z_{1}\) is the point on the boundary of \(\overline{A_{1}B_{1}C_{1}}\), \(p_{x}(x_{1})\) is the horizontal load and \(p_{y}(x_{1})\) is the vertical load. The continuity condition of displacement and stress on the contact surface between the lower boundary of the \(j\)-th layer and the upper boundary of the \(j+1\)-th layer can be obtained from Equations (5)-(7). \[\frac{1}{G_{j}}\left[\kappa_{j}\varphi_{1,j}(z_{j})-z_{j}\overline{\varphi^{ \prime}{}_{1,j}(z_{j})}-\overline{\psi_{1,j}(z_{j})}\right]=\frac{1}{G_{j+1} }\left[\kappa_{j+1}\varphi_{1,j+1}(z_{j+1})-z_{j+1}\overline{\varphi^{\prime} {}_{1,j+1}(z_{j+1})}-\overline{\psi_{1,j+1}(z_{j+1})}\right] \tag{9}\]  
\[2\text{Re}\left[\varphi_{1,j}(z_{j})\right]+\overline{z_{j}}\varphi^{\prime \prime}{}_{1,j}(z_{j})+\psi^{\prime}{}_{1,j}(z_{j})=2\text{Re}\left[\varphi_{1,j+1}(z_{j+1})\right]+\overline{z_{j+1}}\varphi^{\prime\prime}{}_{1,j+1}(z_{j +1})+\psi^{\prime}{}_{1,j+1}(z_{j+1}) \tag{10}\]  
In Equations (9) and (10), \(j=1,2,\ldots,n-1\). \(z_{j}\) and \(z_{j+1}\) are the same point on the contact surface, where \(z_{j}\) is on \(\overline{D_{j}E_{j}F_{j}}\), and \(z_{j+1}\) is on \(\overline{A_{j}B_{j}C_{j}}\). Assuming that the bottom boundary is completely constrained, i.e., the horizontal and vertical displacements are equal to zero, the displacement boundary condition on \(\overline{\mathrm{D}_{n}\mathrm{E}_{n}\mathrm{F}_{n}}\) can be obtained from Equation (7):  
\[\kappa_{n}\varphi_{1,n}(z_{n})-z_{n}\varphi_{1,n}^{\prime}(z_{n})-\overline{ \psi_{1,n}(z_{n})}=0 \tag{11}\]  
Substituting Equation (2) into \(\varphi_{1,j}(z_{j})\) and \(\psi_{1,j}(z_{j})\), we can obtain \(\varphi_{j}(\zeta_{j})\) and \(\psi_{j}(\zeta_{j})\). In the unit circle, they can be expressed as follows:  
\[\left\{ \begin{array}{l}\varphi_{j}(\zeta_{j})=\sum\limits_{k=0}^{\infty}a_{ jk}\zeta_{j}^{k}\\ \psi_{j}(\zeta_{j})=\sum\limits_{k=0}^{\infty}b_{jk}\zeta_{j}^{k}\end{array}  \right.\left(j=1,2,\ldots,n\right) \tag{12}\]  
where \(a_{jk}\) and \(b_{jk}\) are the complex coefficients to be solved:  
\[\left\{ \begin{array}{l}a_{jk}=a_{j,k,R}+ia_{j,k,l}\\ b_{jk}=b_{j,k,R}+ib_{j,k,l}\end{array} \right.