## 2 Elastic Foundation  
Within the static analysis of the whole structure, it is necessary to pay attention to the solution of foundation structures. In this paper, we deal with the calculation of foundation slabs. From a theoretical point of view, the analysis of the base plate was the solution of the contact problem between the plate and the elastic subsoil. When creating a static model, we had to determine which mathematical model we would use to describe the elastic subsoil. At present, we can opt for the following three subsoil models:  
* single parametric ([PERSON]'s) model of elastic subsoil;
* two parametric ([PERSON]'s) model of elastic subsoil;
* model of elastic half-space ([PERSON]'s). In practice, the first two models, also called surface models, are used quite often. The elastic properties of the subsoil are described by spring constants on the subsoil surface. In this paper, we chose the most exact third subsoil model, i.e., the elastic half-space model. After selecting the subsoil model, we proceeded to solve the problem. We solved the slab in contact with the elastic half-space considering [PERSON]'s assumptions for the slab. In the contact of the slab with the subsoil, we considered the vertical contact stress only. We did not consider the shear stress between the slab and the subsoil. The slab was characterized by the material constants of the concrete \(E_{c}\) (modulus of elasticity of the slab) and \(\mu_{c}\) (Poisson's number of the slab). The half-space was characterized by the constants \(E_{def}\) (deformation modulus of elasticity of the subsoil) and \(\mu\) (Poisson's number of the subsoil). The deflection function of the slab is defined by the partial differential Equation (1). On the right side of the equation is the load acting on the plate, the external load \(q(x,y)\) and the subsoil resistance \(p(x,y)\). The equation of the plate has the shape:  
\[D\
abla^{2}w(x,y)=q(x,y)-p(x,y) \tag{1}\]  
where \(D\) is the stiffness of the slab Equation (2):  
\[D=\frac{E_{c}h^{3}}{12\Big{(}1-\mu_{c}^{2}\Big{)}} \tag{2}\]  
where \(h\) is the depth of the slab. According to the assumption of equality of deflection of the plate and the subsoil, we can substitute the resistance of the subsoil \(p(x,y)\) into the relation (1) and we get the integro-differential equation of the plate rested on the elastic half-space. Its solution using approximate numerical methods are presented in well-known works dealing with these problems. At present, the problem of the slab rested on elastic foundation is solved in most cases by numerical methods, for example, by FEM, in which we start from the minimum potential energy of the system \(\pi\):  
\[\pi=\ \pi d\ +\pi e+\pi p\ =\ min \tag{3}\]  
where the energy of the slab is defined:  
\[\pi_{d}\ =\frac{1}{2}\iint\limits_{0}^{\Omega}\varepsilon^{T}\ \sigma.d\Omega= \frac{1}{2}\iint\limits_{0}^{\Omega}\varepsilon^{T}\ D\varepsilon\,d\Omega \tag{4}\]  
energy of external forces (load) is:  
\[\pi_{e}\ =\frac{1}{2}\iint\limits_{0}^{\Omega 1}q(x.y).w(x,y)\ d\Omega 1 \tag{5}\]  
energy of the subsoil is:  
\[\pi_{p}\ =\frac{1}{2}\iint\limits_{0}^{\Omega}p(x,y).\ w(x,y)\ d\Omega. \tag{6}\]  
When modeling the slab and the elastic subsoil in the FEM using the ANSYS system, we proceeded from the following assumptions. To create the body-block of the elastic half-space, 3D finite elements of the brick type were used (3 degrees of freedom in 8 nodes), the slab was modeled by planar shell elements with slab-wall assumptions (6 degrees of freedom in 4 nodes). The size of the earth body-blocks was based on experience. In horizontal plan, the model was extended to each side by twice the width of the board. In the vertical direction, the size was limited by the depth of the deformation zone. Boundary conditions were introduced so that horizontal displacements in directions of the wall normal lines were captured on all four vertical walls of the block. Vertical displacements were captured at the bottom of the block.