# A Study on Karst Cave Collapse Based on Improved Terzaghi Theory and Upper Limit Analysis
## 2 Basic Theories
### Improved Terzaghi Theory  
The failure model in the improved Terzaghi theory is plotted in Figure 1, where the diameter of the karst cave is \(d\), the overburden thickness is \(h\), the soil unit weight above and below groundwater table are \(\gamma\) and \(\gamma^{\prime}\), the cohesions are \(c\) and \(c^{\prime}\), and the internal friction angles are \(\varphi\) and \(\varphi^{\prime}\). A uniform surface load, \(F_{1}\), is imposed on the top of the soil column. When the groundwater table is below the karst cave, the vacuum absorption erosion force is one of the reasons for the failure of the karst cave [26]; thus, it is simplified as \(F_{2}\). \(F_{1}\) and \(F_{2}\) can be obtained by  
\[\left\{ \begin{array}{l}F_{1}=\frac{\pi d^{2}}{4}\sigma_{1}\\ F_{2}=\frac{\pi d^{2}}{4}\sigma_{p}\end{array} \right. \tag{2}\]  
where \(\sigma_{1}\) represents the average surface pressure and \(\sigma_{p}\) represents the average vacuum absorption erosion pressure. According to the Terzaghi theory by [PERSON] [27], the improvement in this study includes two key aspects. 1. In practice, diverse relative positional arrangements exist between the groundwater table and karst cave, leading to two distinct scenarios, where the groundwater table can be situated either above or below the karst cave. 2. Equation (3) represents the Coulomb shear strength formula. To facilitate its application within the failure model, the shear strength is defined by Equation (4). \[\tau_{f}=c+\sigma\tan\varphi \tag{3}\]  
\[\tau_{f}=c+K_{0}\gamma z\tan\varphi \tag{4}\]  
where \(K_{0}\) denotes the static earth pressure coefficient. **(1)** **The groundwater table lies beneath the karst cave**  
For the 3D soil column above the karst cave, at a depth of \(z\), a circular thin-layer element with a thickness of \(\mathrm{d}z\) was selected as the analysis object. If the self-weight of the circular thin-layer element is \(\mathrm{d}G\), the pressure of overlying soil body is \(P\), the supporting force of the underlying soil body is \(T\), and the lateral shear strength is \(\mathrm{d}F\), then  
\[\left\{ \begin{array}{l}\mathrm{d}G=\frac{\pi d^{2}\gamma}{4}\mathrm{d}z\\ P=\frac{\pi d^{2}}{4}\sigma_{z}\\ T=\frac{\pi d^{2}}{4}(\sigma_{z}+\mathrm{d}\sigma_{z})\\ \mathrm{d}F=\pi d(K_{0}\gamma z\tan\varphi+c)\mathrm{d}z\end{array} \right. \tag{5}\]  
Figure 1: Failure models developed based on improved Terzaghi theory. (**a**) Groundwater table below karst cave; (**b**) groundwater table above karst cave. where \(\sigma_{z}\) represents the vertical stress of the soil at the depth of \(z\), with the downward direction being positive. Figure 1 demonstrates the directional relationship among the forces. Combined with Equation (5), according to the vertical equilibrium condition, the following can be obtained:  
\[P+\mathrm{d}G-T-\mathrm{d}F=0 \tag{6}\]  
\[\frac{\mathrm{d}\sigma_{z}}{\mathrm{d}z}+\frac{4K_{0}\tan\varphi\gamma z}{d}+ \frac{4c}{d}-\gamma=0 \tag{7}\]  
\[\sigma_{z}=-\frac{2K_{0}\gamma\tan\varphi}{d}z^{2}+\frac{d\gamma-4c}{d}z+C \tag{8}\]  
Due to the existence of \(\sigma_{1}\) and \(\sigma_{p}\), \(C=\sigma_{1}+\sigma_{p}\), which is substituted into Equation (8) as follows:  
\[\sigma_{z}=-\frac{2K_{0}\gamma\tan\varphi}{d}z^{2}+\frac{d\gamma-4c}{d}z+ \sigma_{1}+\sigma_{p} \tag{9}\]  
In the lowest boundary of overburden, the top of karst cave, \(R\) represents the residual collapse resistance upward at the top of the karst cave. At this time, \(z=h\), which is substituted into Equation (9), as follows:  
\[R=-\sigma_{z}=\frac{2K_{0}\gamma\tan\varphi}{d}h^{2}-\frac{d\gamma-4c}{d}h- \sigma_{1}-\sigma_{p} \tag{10}\]  
This force can ensure the soil stability. When \(R=0\), karst cave is in limit equilibrium; when \(R>0\), karst cave is stable; and when \(R<0\), failure of the karst cave can occur. **(2) The groundwater table is above the karst cave. **  
The height of the water table from the ground is \(z_{1}\).