# Complex Variable Solution for Stress and Displacement of Layered Soil with Finite Thickness
## 2 Problem Statement and Method Presentation  
This paper deals with the case of complete contact between layers of multi-layer soil and complete constraint with bedrock below it. When the soil surface is subjected to distributed load, the deformation of bedrock is considered to be negligible compared with the deformation of soil, that is, the displacement component on the interface between the soil and bedrock is equal to zero. Figure 1a shows a problem in which the upper boundary is the stress boundary condition, the lower boundary is the displacement boundary condition, and the contact surface of each layer is the stress and displacement continuity condition. In Figure 1a, the four end points (A, C, D, F) of each layer of soil are points at infinity. \(t\) is any point on the contact surface. There is a vector relation about f:  
\[\overrightarrow{O_{2}O_{1}}+\overrightarrow{O_{1}t}=\overrightarrow{O_{2}t} \tag{1}\]As shown in Figure 1, the \(j\)-th layer (\(j\) = 1,2,\(\cdots\).n) soil is mapped to the inner domain of the unit circle in the \(\zeta_{j}\) plane. The mapping function is:  
\[z_{j}=\omega_{j}\big{(}\zeta_{j}\big{)}=\frac{H_{j}}{\pi}\ln\left(\frac{1+\zeta _{j}}{1-\zeta_{j}}\right) \tag{2}\]  
where \(z_{j}\) is any point in the \(j\)-th layer soil, \(z_{j}=x_{j}+iy_{j}\), \(i=\sqrt{-1}\); \(H_{j}\) is the thickness of the \(j\)-th layer soil. \(\zeta_{j}\) is the point where \(z_{j}\) maps to the image plane. \(\zeta_{j}\) = \(\rho_{j}\mathrm{e}^{i\theta}\), \(\rho_{j}=1\) corresponds to the upper and lower boundaries in the \(j\)-th layer soil, and \(\sigma_{j}\) is used to denote \(\zeta_{j}\) at this time. As shown in Figure 1, in the \(\zeta_{1}\) plane, the circumference of the unit circle (\(\sigma_{1}\) = 1) corresponds to the upper and lower boundaries of the first layer of soil. A\({}_{1}\), B\({}_{1}\), C\({}_{1}\), D\({}_{1}\), E\({}_{1}\), F\({}_{1}\) are mapped to A\({}_{1}^{\prime}\), B\({}_{1}^{\prime}\), C\({}_{1}^{\prime}\), D\({}_{1}^{\prime}\), E\({}_{1}^{\prime}\), F\({}_{1}^{\prime}\), respectively. The upper boundary (\(\overline{A_{1}B_{1}C_{1}}\)) is mapped to the upper semicircle (\(A_{1}^{\prime}\), \(\overline{B_{1}^{\prime}C_{1}^{\prime}}\)). The lower boundary (\(\overline{\mathrm{D_{1}E_{1}F_{1}}}\)) is mapped to the lower semicircle (\(D_{1}^{\prime}\), \(\overline{E_{1}^{\prime}I_{1}^{\prime}}\)). The coordinate origin (O\({}_{1}\)) is mapped to the center of the unit circle (O\({}_{1}^{\prime}\)) of the \(\zeta_{1}\) plane. On the upper and lower boundaries, all points of \(x_{1}\rightarrow+\infty\) are mapped to the same point (\(\sigma_{1}=1\)) of the \(\zeta_{1}\) plane, and all points of \(x_{1}\rightarrow-\infty\) are mapped to the same point (\(\sigma_{1}=-1\)) of the \(\zeta_{1}\) plane. The coordinate axes \(x_{1}\) and \(y_{1}\) are mapped to \(\xi_{1}\) and \(\eta_{1}\) axes. The points \(a\) and \(b\) are mapped to \(a^{\prime}\) and \(b^{\prime}\). The rest of the layers can be deduced by analogy. Figure 1: The multi-layer soil above the bedrock is mapped on to the inner domain of the unit circle in the image plane: (**a**) multi-layer soil subjected to strip load and (**b**) each layer is mapped to the inner domain of the unit circle in the image plane. By substituting Equation (2) into Equation (1), the relationship between \(\theta_{1}\) and \(\theta_{2}\) at the same point \(t\) on the contact surface of the 1 st layer and the 2 nd layer can be obtained:  
\[\theta_{2}=\arccos\left(\frac{{T_{1}}^{2}-1}{{T_{1}}^{2}+1}\right) \tag{3}\]  
where \(T_{1}=\left(-sin\theta_{1}/\left(1-cos\theta_{1}\right)\right)^{H_{1}/H_{2}}\). In the same way, on the contact surface of the \(j\)-th layer and the \(j+1\)-th layer:  
\[\theta_{j+1}=\arccos\left(\frac{{T_{j}}^{2}-1}{{T_{j}}^{2}+1}\right) \tag{4}\]  
where \(T_{j+1}=\left(-sin\theta_{j}/\left(1-cos\theta_{j}\right)\right)^{H_{j}/H_{j+ 1}}\), \(j=1,2,\ldots,n-1\).