# Stability Analysis of the Exploitation System with Room and Pillar by Analytical Methods
## 3 Methods
### Stability Analysis of the Rooms and Pillars Exploitation System through Analytical Method
\begin{array}{l}\left[\frac{9}{8}\pi^{2}+6\left(\pi^{2}-8\right)\frac{\mathrm{h }^{4}}{\mathrm{b}^{4}}+3\left(8+\pi^{2}\right)\frac{\mathrm{h}^{2}}{\mathrm{b} ^{2}}-9\,\mu^{2}\right]\mathrm{A}+\left[2\left(\pi^{2}-8\right)\frac{\mathrm{h }^{2}}{\mathrm{b}^{2}}\times\frac{\mathrm{h}^{2}}{\mathrm{d}^{2}}-\frac{9}{ 8}\pi^{2}\mu+\frac{3}{2}\mu\left(8-\pi^{2}\right)\left(\frac{\mathrm{h}^{2}}{ \mathrm{b}^{2}}+\frac{\mathrm{h}^{2}}{\mathrm{d}^{2}}\right)-9\mu^{2}\right]\\ \times\mathrm{B}+\left[\frac{3}{2}\pi\left(4+\pi\right)\left(\frac{\mathrm{h}^{ 2}}{\mathrm{d}^{2}}-\frac{\mathrm{h}^{2}}{\mathrm{b}^{2}}\right)\right]\! \mathrm{C}=\frac{2\,\mu}{\pi}\mathrm{E}\times\varepsilon_{1}\times\mathrm{h} \\ \left[\frac{9}{8}\pi^{2}+6\left(\pi^{2}-8\right)\frac{\mathrm{h}^{2}}{\mathrm{b }^{4}}+3\left(8\mu+\pi^{2}\right)\frac{\mathrm{h}^{2}}{\mathrm{d}^{2}}-9\,\mu ^{2}\right]\mathrm{B}+\left[2\left(\pi^{2}-8\right)\frac{\mathrm{h}^{2}}{ \mathrm{b}^{2}}\times\frac{\mathrm{h}^{2}}{\mathrm{d}^{2}}-\frac{9}{8}\pi^{2} \mu+\frac{3}{2}\mu\left(8-\pi^{2}\right)\left(\frac{\mathrm{h}^{2}}{\mathrm{b }^{2}}+\frac{\mathrm{h}^{2}}{\mathrm{d}^{2}}\right)-9\mu^{2}\right]\\ \mathrm{A}+\left[\frac{3}{2}\pi\left(4+\pi\right)\left(\frac{\mathrm{h}^{2}}{ \mathrm{b}^{2}}-\frac{\mathrm{h}^{2}}{\mathrm{d}^{2}}\right)-2\,\mu\left(\pi^{ 2}-8\right)\left(\frac{\mathrm{h}^{2}}{\mathrm{b}^{2}}\times\frac{\mathrm{h}^ {2}}{\mathrm{d}^{2}}+3\frac{\mathrm{h}^{4}}{\mathrm{d}^{4}}\right)-6\mu^{2}( \pi+2)\left(\frac{\mathrm{h}^{2}}{\mathrm{d}^{2}}+\frac{\mathrm{h}^{2}}{ \mathrm{d}^{2}}\right)\right]\!\mathrm{C}=\frac{2\,\mu}{\pi}\mathrm{E}\times \varepsilon_{1}\times\mathrm{h}\\ \left[\frac{3}{2}\pi(4+\pi)\left(\frac{\mathrm{h}^{2}}{\mathrm{d}^{2}}-\mu \frac{\mathrm{h}^{2}}{\mathrm{b}^{2}}\right)-2\mu(\pi^{2}-8)\left(\frac{ \mathrm{h}^{2}}{\mathrm{b}^{2}}\times\frac{\mathrm{h}^{2}}{\mathrm{d}^{2}}+3 \frac{\mathrm{h}^{4}}{\mathrm{d}^{4}}\right)-6\mu^{2}(\pi+2)\left(\frac{ \mathrm{h}^{2}}{\mathrm{b}^{2}}+\frac{\mathrm{h}^{2}}{\mathrm{d}^{2}}\right) \right]\!\mathrm{A}+\\ \left[\frac{3}{2}\pi(4+\pi)\left(\frac{\mathrm{h}^{2}}{\mathrm{d}^{2}}-\mu \frac{\mathrm{h}^{2}}{\mathrm{b}^{2}}\right)-2\mu(\pi^{2}-8)\left(\frac{ \mathrm{h}^{2}}{\mathrm{b}^{2}}\times\frac{\mathrm{h}^{2}}{\mathrm{d}^{2}}+3 \frac{\mathrm{h}^{4}}{\mathrm{d}^{4}}\right)-6\mu^{2}(\pi+2)\left(\frac{ \mathrm{h}^{2}}{\mathrm{b}^{2}}+\mu\frac{\mathrm{h}^{2}}{\mathrm{d}^{2}}\right) \right]\!\mathrm{B}+\\ \left[2\pi(3\pi+8)\left(3\frac{\mathrm{h}^{4}}{\mathrm{d}^{4}}+2\frac{\mathrm{h }^{2}}{\mathrm{b}^{2}}\times\frac{\mathrm{h}^{2}}{\mathrm{d}^{2}}+3\frac{ \mathrm{h}^{4}}{\mathrm{b}^{4}}\right)-4(\pi+2)^{2}\mu^{2}\!\left(3\frac{\mathrm{ h}^{4}}{\mathrm{b}^{2}}+2\frac{\mathrm{h}^{2}}{\mathrm{b}^{2}}\times\frac{\mathrm{h}^{2}}{ \mathrm{d}^{2}}+3\frac{\mathrm{h}^{4}}{\mathrm{d}^{4}}\right)\right]\!\mathrm{C }=0\end{array}  \tag{6}\]  
The solution of this system expresses the common three-dimensional working character of the pillar with the entire room-pillar exploitation system. At the same time, it allows the research of the secondary stress state in the pillars in the context of different correlations with their dimensions, both in plan and with their height. 2.2 Analytical Model for Assessing the Stress State of Pillar and Floor Based on the Principle of Complex Variables  
Another analytical approach for establishing the distribution of the secondary stress state is based on the principle of the method of the function of a complex variable. The underground space for which the secondary stress state is quantitatively evaluated is materialized through the salt deposit exploitation system of small rooms and rectangular pillars at different depths (H) from the surface (Figure 4). The shape of the rooms is, therefore, rectangular, having a height (h) identical to the height of the pillars and h << H, and their length far exceeds the two dimensions, width and height. Consequently, the planar approximation can be used to determine the secondary stress state in accordance with the following analytical relations:  
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