# Design of Conical Foundations with Increased Bearing Capacity in Areas of Undermined Soils
## 1 Introduction
and horizontal stresses \(\sigma_{x}\), \(\sigma_{y}\) by the axes \(x\) and \(y\):  
\[\sigma_{x}=\sigma_{y}=K_{0}\cdot\gamma\cdot z \tag{2}\]  
where \(K_{0}\) = 0.6-1--lateral pressure coefficient. The analysis of the mechanical properties of the undermined soil layer is influenced by cohesion \(c\), modulus of elasticity during soil unloading E\({}_{un}\), tensile strength T, and angle of internal friction \(\varphi\)[5; 9]. The horizontal tensile deformations are considered to be accompanied with curvature deformations of the convexity when tilted at Stage I [14]. Initially, tensile strains will reduce the magnitude of horizontal compressive stresses. Horizontal deformations \(\sigma_{x}\) change by the modulus of elasticity during soil unloading E\({}_{un}\):  
\[\sigma_{x}=K_{0}\gamma z\ -K_{0}\text{E}_{un} \tag{3}\]  
When the tensile strain exceeds the tensile strength of the soil, a crack is formed. The maximum possible depth of development of a rupture crack is limited by the fact that the blocks contoured by these cracks find themselves under conditions of uniaxial compression by gravity. Therefore, the maximum depth of cracks \(z_{max}\) is determined from the condition of equality of vertical stress \(\sigma_{z}\) to uniaxial compression:  
\[\gamma z_{max}=2c\cdot ctg\ (45^{\circ}-\varphi/2)\]  
\[z_{\text{max}}=2c/\gamma\cdot\text{ctg\ (45^{\circ}-\varphi/2)} \tag{4}\]  
where c--cohesion, \(\gamma\)--bulk unit weight, \(\varphi\)--angle of internal friction. A further increase in horizontal tension will be accompanied by the expansion of cracks, and below by plastic shear deformations without the formation of cracks (Figure 2). The stability of foundations in areas with increased horizontal tensile strains, namely in undermined areas, is important for the preservation of buildings and structures, as well as for reducing the destruction and cracks in buildings [14; 15]. According to the Standard in Kazakhstan \"SP RK 2.03-101-2012 Buildings and structures in mined areas and subsidence soils\" [16] the displacement trough values, depending on the relative horizontal deformations and inclination, are divided into the following four groups of territories (Table 1):  
Permission for the construction of buildings and structures in these territories is given on the basis of calculations of allowable deformations, including horizontal tensile deformations, which are measured as relative deformation in mm per meter of distance [17]. During the design, construction and operation process, soil displacement deformations are monitored. Observation geodetic stations, located in the form of profile lines along the strike and across the strike of the layers, record small movements of the soil in the horizontal plane. Observations in the vertical plane are determined by leveling, and in the horizontal plane by measuring the intervals between benchmarks. Figure 2: Deformations of the sediment layer at the contact with bedrock in the extension zone. Considering the operational features of existing traditional forms of foundations (strip, free-standing, prismatic pyramidal), it turned out that under the influence of horizontal displacements in mine workings, there is not a sufficient contact area between the foundation structure and the soil to ensure the stability of the foundation, especially with horizontal tensile deformations [15; 18]. So, the study of the influence of horizontal deformations of the undermined base along the base of strip foundations was carried out by [PERSON] [11], and for pile foundations [19] and the stability of dam on undermined soils by [PERSON] et al. [20]. Full-scale studies of the force effects of horizontal deformations on the lateral surfaces of the foundations of undermined buildings were carried out by [PERSON] and [PERSON] [10; 11]. A case study of building damage risk assessment due to deep excavation was analyzed by [PERSON] et al. [21]. The conical model, which was commonly used for offshore structures, was considered as a basis for the new sustainable models in recent years.