# Complex Variable Solution for Stress and Displacement of Layered Soil with Finite Thickness
## 1 Introduction  
In many places, it is common to see compressible soil overlying rigid bedrock. In this case, the stress and displacement of soil are obviously different from that of a semi-infinite body medium [1]. Natural soils tend to be horizontally layered during deposition, the density and hardness of each layer are generally different, and the deformation characteristics may vary greatly [2]. When the stress in the soil is too large, the deformation of the soil will cause unacceptable settlement of the building and even the instability of the entire soil. Therefore, an accurate calculation of soil stress and displacement is an important basis for the stability analysis of building foundations and geotechnical structures [3]. Finite element method can provide useful solutions to many complex geological conditions. However, extensive computational time is required, especially when complete parametric analyses need to be performed [4]. Analytical solutions provide an efficient and quick approach to gain insight into the nature of the problem [5]; in addition, they enable analyses of a wide range of parameter values so that the physics of the problem can be better understood [4]. In the past, researchers mainly used the integral transformation method to analyze the elastic static problems of layered soil. [PERSON] [6] provided the integral expression of stress and displacement of two layers of soil under a flexible circular foundation based on the integral transformation method, and also established an empirical formula for surface settlement when both layers of soil consist of incompressible materials. In many studies in the literature, the soil surface is subjected to a circular uniformly distributed load, which is considered a spatial axisymmetric problem. Both layers of soilare linear elastomers and in complete contact. The second layer of soil was infinitely deep. Then, smooth contact [7] and three-layer [8] soil with complete contact were considered. Because only the integral transformation equations of stress and displacement could be derived by the integral transformation method, many numerical methods were proposed to solve the integral equations. [PERSON] et al. [9] studied the stress field of two-layer elastomer under elliptical distributed load, in which the second-layer elastomer was a semi-infinite body, and presented a numerical value of the stress on the axis of symmetry. [PERSON] [10] identified the shortcomings of this method, that is, the integral converges slowly or does not even converge at the boundary. [PERSON] [10] improved the method to solve the problem of the stress field and displacement field of three-layer soil under circular and rectangular loads. However, the subsoil was still semi-infinite, which means this method was not suitable for bedrock at the bottom of the soil. Since then, many scholars have used the integral transformation method to study the layered material problem, but the bottom layer was assumed to be a semi-infinite body [11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28]. When there is bedrock under the soil, it is very difficult to obtain an analytical solution regarding the stress and displacement due to the constraint of the bedrock on the soil. At present, only [PERSON] et al. [1] have obtained an explicit analytical expression for the stress and displacement of single-layer finite thickness soil under the action of a strip foundation by a complex variable method. In this paper, the solution in [1] is extended to layered soil, and the complex variable method [29] is used to solve the stress and displacement of multi-layer soil above bedrock under the action of a strip shallow foundation. Under the action of a strip foundation, the soil is subjected to a strip-distributed load. It is assumed that the distributed load does not change along the direction perpendicular to the surface; thus, the problem solved in this paper is a plane strain problem [1].