# A Study on Karst Cave Collapse Based on Improved Terzaghi Theory and Upper Limit Analysis
## 2 Basic Theories
### Limit Analysis Method
\tag{17}\]  
(1) Groundwater table below karst cave  
Evidently, the failure zone and half of the karst cave simultaneously constitute two frustum of cones. The radii of the bottom and top surfaces of the circular frustum of cones are expressed as follows:  
\[\left\{ \begin{array}{l}r_{1}=\frac{d}{2}\\ r_{2}=\frac{d}{2}(1+\tan\alpha)\\ r_{3}=\frac{d}{2}+h\tan\alpha\end{array} \right. \tag{18}\]  
The total gravitational power \(P_{g}\) is the sum of gravitational power acting on the circular frustum of cone \(ABDC\) and the irregular region \(CDFGE\):  
\[P_{g}=\frac{\gamma v_{0}\pi}{3\tan\alpha}\left[{r_{3}}^{3}-{r_{1}}^{3}(1+2\tan \alpha)\right] \tag{19}\]  
Similar to the force method in [PERSON] theory, the power \(P_{l}\) generated by the surface load (\(F_{1}\)) and the vacuum absorption erosion force (\(F_{2}\)) is as follows:  
\[P_{l}=\left(\sigma_{1}+\sigma_{p}\right)\pi{r_{1}}^{2}v_{0} \tag{20}\]  
For the entire failure zone, the velocity discontinuity surface is the side of the circular frustum of cone \(ABFE\). The internal dissipation power, \(P_{i}\), is the sum of the energy dissipation power on the velocity discontinuity surface. \[P_{i}=\frac{1}{3\cos\alpha}\pi K_{0}\gamma h^{2}v_{0}\tan\varphi(2r_{1}+r_{3}) +\frac{c\pi hv_{0}}{\cos^{2}\alpha}(r_{1}+r_{3}) \tag{21}\]  
(2) Groundwater table above karst cave  
As previously noted, the failure zone and half of the karst cave form three frustum of cones. The radii of these frustum cones can be described as  
\[\left\{ \begin{array}{l}r_{1}=\frac{d}{2}\\ r_{4}=\frac{d}{2}(1+\tan\beta)\\ r_{5}=(h-z_{1})\tan\beta+\frac{d}{2}\\ r_{6}=(h-z_{1})\tan\beta+z_{1}\tan\alpha+\frac{d}{2}\end{array} \right. \tag{22}\]  
If the gravitational power of \(ABDC\) is \(P_{ABDC}\), the gravitational power of \(CDFE\) is \(P_{CDFE}\), the gravitational power of irregular region \(EFHIG\) is \(P_{EFHIG}\), and the total gravitational power is \(P_{g}\), the following formulas are obtained:  
\[\left\{ \begin{array}{l}P_{g}=P_{ABDC}+P_{CDFE}+P_{EFHIG}\\ P_{ABDC}=\frac{\gamma v_{0}\pi}{3\tan\alpha}\left({r_{3}}^{3}-{r_{3}}^{3}\right) \\ P_{CDFE}=\frac{\gamma v_{0}\pi}{3\tan\beta}\left({r_{3}}^{3}-{r_{4}}^{3}\right) \\ P_{EFHIG}=\frac{\gamma v_{0}\pi}{3\tan\beta}\left[{r_{4}}^{3}-{r_{1}}^{3}(1+2 \tan\beta)\right]\end{array} \right. \tag{23}\]The power \(P_{l}\) conducted by surface load \(F_{1}\) is  
\[P_{l}=\sigma_{1}\pi r_{1}{}^{2}v_{0} \tag{24}\]  
The velocity discontinuity surfaces are defined by the circular frustum of cones \(ABDC\), \(CDFE\), and \(EFHG\), as well as the circle centered at \(l\) with a given radius \(r_{5}\). The internal dissipation power \(P_{l}\) is the sum of the energy dissipation on the velocity discontinuity surfaces. Let the energy dissipation powers of these four areas be \(P_{1}\), \(P_{2}\), \(P_{3}\), and \(P_{4}\). \[\left\{ \begin{array}{l}P_{l}=P_{1}+P_{2}+P_{3}+P_{4}\\ P_{1}=\frac{1}{3\cos\pi}\pi K_{0}\gamma z_{1}^{2}v_{0}\tan\varphi(2r_{5}+r_{6}) +\frac{\sqrt{r_{1}}\tan}{\cos^{2}\theta}(r_{5}+r_{6})\\ P_{2}=\frac{1}{3\cos\theta}\pi K_{0}z_{2}v_{0}\tan\varphi^{\prime}[3\gamma z_{ 1}(r_{4}+r_{5})+\gamma^{\prime}z_{2}(2r_{4}+r_{5})]+\frac{c^{\prime}\pi z_{2} \tan}{\cos^{2}\beta}(r_{4}+r_{5})\\ P_{3}=\frac{1}{3\cos\theta}\pi K_{0}r_{1}v_{0}\tan\varphi^{\prime}[3(\gamma z_{ 1}+\gamma^{\prime}z_{2})(r_{1}+r_{4})+\gamma^{\prime}r_{1}(2r_{1}+r_{4})]+ \frac{c^{\prime}\pi m_{1}\tan}{\cos^{2}\beta}(r_{1}+r_{4})\\ P_{4}=\pi r_{5}^{2}v_{0}(\tan\beta-\tan\alpha)(\gamma z_{1}\tan\varphi^{\prime} +c^{\prime})\end{array} \right. \tag{25}\]  
According to virtual power theory, at the point of limit equilibrium in the failure zone, the power of the external forces equals the sum of the internal dissipation power. The stability coefficients can be defined to evaluate the failure zone as follows:  
\[K=\frac{P_{i}}{P_{g}+P_{l}} \tag{26}\]  
When \(K=1\), the karst cave is in a state of limit equilibrium. When \(K>1\), the karst cave is stable. When \(K<1\), the karst cave fails.