# 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}\sigma_{\mathrm{N}}=\frac{\sigma_{x}+\sigma_{x}}{2}+\frac{ \sigma_{x}-\sigma_{x}}{2}\cos 2\beta+\tau_{x\,z}\sin 2\beta\\ \tau_{\mathrm{N}}=\frac{\sigma_{x}-\sigma_{x}}{2}\sin 2\beta+\tau_{x\,z}\cos 2\beta\end{array}  \tag{7}\]  
Figure 4: Quantitative evaluation of the secondary stress state through the operating system with small rooms and rectangular pillars. that are periodic with \(\pi\)[44]. We will consider the cartesian coordinate system as the main system, and therefore, we have:  
\[ \begin{array}{l}\sigma_{\rm z}+\sigma_{\rm x}=\sigma_{1}+\sigma_{3}\\ \sigma_{\rm z}-\sigma_{\rm x}+2\rmi\tau_{x\,x}=(\sigma_{1}-\sigma_{3})\rm{e}^{2 \,i\,\beta}\end{array}  \tag{8}\]  
Thus, the solution of system (8) consists of the summation of two solutions: the first solution is the tensor of the natural stress state, and the second solution is the additional components of the stress state that lead to the occurrence and manifestation of the secondary stress state on the floor of the rooms. In such a context, polynomial functions were used, based on which relations (8) are rendered in the form of functions of a complex variable:  
\[ \begin{array}{l}\sigma_{\rm x}+\sigma_{\rm z}=2\Big{[}\phi^{\prime}\,(c^{ \prime})+\overline{\phi^{\prime}}\,(c)\Big{]}\\ \sigma_{\rm x}-\sigma_{\rm z}+2\rmi\tau_{x\,x}=2\Big{[}c\,\phi^{\prime}\,(c)+ \overline{\phi^{\prime}}\,(c)\Big{]}\end{array}  \tag{9}\]  
where \(c=x+iz\) as a complex variable of the holomorphic function in the domain defined by the salt massif, \(i=\sqrt{-1}\). According to Figure 4, between the operating rooms in sectors \(a_{j}\), \(b_{j}\) (where \(j=1\), \(2\), \(3\),..., \(n\) and \(n\) represent the number of rooms), the pillars of width (\(b_{j}-a_{j}\)) are left. It follows in this case that the number of rooms is (\(n+1\)), and the width of a room (\(j\)) \(-\) (\(j+1\)) can be calculated with the relation (\(a_{j+1}-b_{j}\)). Denoting the width of the entire floor by 2L, the 2L/h ratio leads to a value higher than one, and as a result, in the absence of pillars, the excavation obtained can be analyzed as a gap carried out in the salt massif. Thus, the application of the plane theory is appropriate with the condition that in sectors \(a_{j}\) and \(b_{j}\), at the upper edge, namely, in the ceiling of the excavation, uniformly distributed loads \(P_{j}\) are applied as reactions developed by the pillars (Figure 4). Based on the principles of [PERSON] [26; 47] with reference to the analysis of the loads acting on the pillars and, in the context of the normal stress distribution \(\sigma_{\rm z}\) at the pillar-floor contact, the expression was obtained according to which the reaction \(P_{j}\), developed by the pillars, can be evaluated:  
\[P_{j}=\frac{\gamma_{\rm a}H}{2\rmi p_{j}}\,\Big{(}I_{P_{j}}+I_{j}\Big{)}\,\,- \,\,\xi_{0}\,\gamma_{\rm a}\frac{H}{h}\,\Big{(}I_{P_{j}}+I_{j}\Big{)}\,\,\mu_ {1} \tag{10}\]  
where \(2\,I_{P_{j}}\)--width of pillar \(j\) is equal to \((b_{j}-a_{j})\); \(I_{j}=\frac{1}{2}\,\big{(}a_{j+1}-b_{j-1}\big{)}\) according to Figure 4, \(\mu_{1}\)--coefficient of friction on the contact surface between the floor and the roof (\(\mu_{1}=\rm{tg}\varphi\)); \(\varphi\)--internal friction angle of the salt; \(\xi_{0}\)--coefficient of lateral pressure created by Poisson's ratio (\(\mu\)):  
Based on the clarifications made, the expressions (9) for the considered situation, namely, the room and pillar exploitation system, become:  
\[ \begin{array}{l}\sigma_{\rm z}=\sigma_{\rm z}^{0}+\sigma_{\rm z}^{1}\\ \sigma_{\rm x}=\sigma_{\rm x}^{0}+\sigma_{\rm x}^{1}\\ \tau_{x\,x}=\tau_{x\,x}^{0}+\tau_{x\,x}^{1}\end{array}  \tag{11}\]  
in which \(\sigma_{\rm z}^{0}\); \(\sigma_{\rm x}^{0}\); \(\tau_{x\,x}^{0}\) are the components of the tensor of the natural stress state of the gravitational origin of the undisturbed salt massif and represent the first solution of relations (11).