# A Study on Karst Cave Collapse Based on Improved Terzaghi Theory and Upper Limit Analysis
## 2 Basic Theories
### Limit Analysis Method  
The limit analysis method includes upper and lower limit analyses. The lower limit analysis determines the maximum load that remains static through the construction of a static field, whereas the upper limit analysis identifies the minimum load that induces motion by establishing a velocity field [28]. Compared with the lower limit analysis, the upper limit analysis yields load values that approximate the actual situation more closely [29]. [PERSON] and [PERSON] [29] used the upper limit analysis method to carry out the stability analysis of two-dimensional tunnels. In this study, we employed a similar theory to assess the stability of three-dimensional karst caves. Assuming that the properties of the failure zone satisfy the fundamental assumptions, we can establish the failure models illustrated in Figures 2 and 3 according to the upper limit theorem of the limit analysis method. These models consider spherical karst caves with a diameter denoted as \(d\) and the distance between the surface and center of the karst caves represented as \(h\). As shown in Figure 2b, the groundwater table is below the karst cave and \(G\) and \(H\) are the midpoints of \(CD\) and \(AB\), respectively. The soil unit weight of the failure zone \(ABFGE\) is \(\gamma\), cohesion is \(c\), the internal friction angle is \(\varphi\), and the angle between the perpendicular lines of the surface and the boundary is \(\alpha\). The surface load above the karst caves is \(F_{1}\), and the upper vacuum absorption erosion force of the karst cave is \(F_{2}\). As shown in Figure 3b, when the groundwater table is above the karst cave, \(I\), \(J\), and \(K\) are the midpoints of \(EF\), \(CD\), and \(AB\). \(CD\) is the groundwater table, at a distance from the ground of \(z_{1}\). The soil unit weight of failure zone \(ABDC\) above groundwater is \(\gamma\), cohesion is \(c\), internal friction angle is \(\varphi\), and the angle between perpendicular lines of the surface and the boundary is \(\alpha\). The soil unit weight of failure zone \(CDH\)G below groundwater is \(\gamma^{\prime}\), cohesion is \(c^{\prime}\), internal friction angle is \(\varphi^{\prime}\), and the angle between perpendicular lines of the surface and the boundary is \(\beta\). \(F_{1}\) and \(F_{2}\) have expressions detailed in Equation (16). \[\left\{ \begin{array}{l}F_{1}=\pi{r_{1}}^{2}\sigma_{1}\\ F_{2}=\pi{r_{1}}^{2}\sigma_{p}\end{array} \right. \tag{16}\]  
Figure 3: Failure model and compatible velocity field when groundwater table is above karst cave. (**a**) Three-dimensional failure model; (**b**) failure model profile; (**c**) compatible velocity field. Figure 2: Failure model and compatible velocity field when the groundwater table is below the karst cave. (**a**) Three-dimensional failure model; (**b**) failure model profile; (**c**) compatible velocity field. To ensure the uniqueness of the limit analysis method calculations, according to the associated flow rule, a dilation angle exists between the velocity discontinuity line and the velocity direction. When the failure law of soil adopts the Mohr-Coulomb yield criterion, the dilation angle is equal to the internal friction angle. However, the actual dilation angle of the soil is very small and its influence on the calculation result is negligible. The compatible velocity field corresponds to the failure model (Figures 2c and 3c). The angle is \(\alpha\) when the groundwater table is below the karst cave, while it is \(\beta\) when the groundwater table is above the karst cave. According to the motion permission conditions applicable to a rigid body and taking the groundwater table above the karst cave as an example, the relationship between \(v_{0}\), \(v_{1}\), \(v_{2}\), \(v_{01}\), and \(v_{12}\) can be obtained as follows:  
\[\left\{ \begin{array}{l}v_{1}=\frac{v_{0}}{\cos\alpha}\\ v_{2}=\frac{v_{0}}{\cos\beta}\\ v_{01}=v_{0}\tan\beta\\ v_{12}=v_{0}(\tan\beta-\tan\alpha)\end{array} \right.