# Stability Analysis of the Exploitation System with Room and Pillar by Analytical Methods
## 4 Results
In this context, the solution was achieved by considering the real situation of rectangular pillars from a salt mine in operation in Romania at a depth of 100 m in relation to the surface; the input data were those shown in Table 2.  
Figure 8 shows the diagram of the location of the points where the components of the secondary stress state in the pillar were calculated. These points resulted from the intersection of the horizontal planes along the three directions of the coordinate axes (x, y, z). Figure 9 shows the spatial distribution of the vertical component of the secondary stress state (\(\sigma_{z}\)) on the considered horizontal surfaces (z represents the height of the contact surface of the pillar with the floor of the room; z = h/2 is the elevation where the median surface is located in the considered pillar in the study). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**\(\lambda\) = h/b**} & \multirow{2}{*}{**Coefficient**} & \multicolumn{6}{c}{**Ratio h/Day**} \\ \cline{3-8}  & & **0.01** & **0.2** & **0.4** & **0.6** & **0.8** & **1.0** \\ \hline \multirow{3}{*}{0.1} & A\({}_{0}\) & \(-21.31\) & \(-25.76\) & \(-24.64\) & \(-23.44\) & \(-22.61\) & \(-22.07\) \\  & B\({}_{0}\) & \(-28.40\) & \(-20.14\) & \(-13.71\) & \(-8.94\) & \(-5.81\) & \(-3.85\) \\  & C\({}_{0}\) & \(-220. The spatial distribution table of the vertical component of the secondary stress state in the pillar (\(\sigma_{z}\)), evaluated according to the proposed mathematical model, reveals the complicated nature of the manifestation and distribution of the secondary stress state in the pillars. Based on the results obtained from the calculation, the complete distribution diagram of the secondary stress state can be drawn, including all components of the stress tensor: the three normal components \(\sigma_{\mathrm{x}}\), \(\sigma_{\mathrm{y}}\), \(\sigma_{\mathrm{z}}\), and the tangential components \(\tau_{\mathrm{xy}}\), \(\tau_{\mathrm{yz}}\), and \(\tau_{\mathrm{zx}}\). For example, Figure 10 shows the variation in the axial component of the secondary stress state at the points z = 0, z = h/2, and z = 3h/8 in the situation where the pillar has a width of 14 m and a height of 8 m.  
To validate the results obtained by the proposed method, they were compared with the values obtained by FEM and verified with the data obtained from the in situ measurements. These values can be found in the paper [12; 42]. Through the proposed algorithm, the value of the axial component of the secondary stress state at different points along the axis of a pillar located at a depth of 100 m varies between 1.498 and 1.657 MPa, compared to the value obtained by the finite element method and in situ measurements, which was 1.64 MPa [42]. The measurement of the stress state in the pillars was carried out using a cylindrical pressure cell made on the hydraulic principle installed in bore holes. Figure 8: Schema of the location points where the components of the secondary stress state were calculated. Figure 9: The spatial distribution of the vertical component of the secondary stress state. The comparison with the available results for the depth of 100 m obtained by the FEM, laboratory, and in situ measurements further validated the accuracy and consistency of the findings. From the obtained results, it can be seen that the main parameters that determine the character of the distribution of the secondary stress state in the room-pillar system are the dimensions of the geometric elements of the pillar and floor correlated with the value of the Poisson's coefficient. The modulus of elasticity (E) of the salt and the natural state of stress (especially the value of its vertical component \(\sigma_{z}=\gamma_{x}H\)) as stress produced on the exploitation system do not manifest themselves in the character of the distribution of the secondary stress state. However, these parameters influence the absolute size of the components of this stress state.