The presence of a concealed karst cave above a deep highway tunnel may cause the collapse of the rock mass between the karst cave and tunnel during excavation. Rock mass collapse threatens the safety of tunnel construction personnel. A prediction method of the collapse region induced by a concealed karst cave above a deep highway tunnel is proposed on the basis of the upper bound theorem of limit analysis. An analytical expression of the collapse surface is derived from the variational principle. Using the analytical expression of the collapse surface, the shapes of the collapse surfaces are plotted for different rock mass parameters. Moreover, the minimum safe distance between the karst cave and tunnel is defined, and the computational equation of the minimum safe distance is derived. The proposed method is applied in a highway tunnel excavated in a karst terrain as a case study. Based on geological survey report parameters, the shape of the collapse surface and the minimum safe distance between the karst cave and tunnel are obtained. Finally, the collapse surface of the rock mass provided by the proposed approach is compared with that provided by numerical simulation, and the favorable result comparison shows that the proposed method is valid.

Karst caves are widely present in karst terrains, so karst caves are common in the adjacent regions of planned tunnels or even in the excavated regions of planned tunnels. Most highway tunnels are excavated by drilling and blasting methods. If the karst cave is too close to the planned tunnel, the rock mass between the karst cave and the tunnel may collapse under the effect of the blasting shock. Moreover, karst caves can be filled, and the karst cave fill material would rush into the tunnel instantaneously, which would threaten the personal safety of construction personnel. Even if there is no fill material or the volume of fill material in the karst cave is small, the collapse of the rock mass between the karst cave and the tunnel may damage the construction equipment, which would cause delays in the construction period and economic losses. Figure

Collapse of the roof induced by a karst cave located above a highway tunnel.

Because of the lack of understanding of karst characteristics, great difficulties are encountered in tunnel construction in karst terrains. Understanding the features of karst is a key factor for tunnel construction in karst terrains, and many investigators have realized the aforementioned, and great attention has been paid to this issue. Taking the Yichang–Wanzhou railway tunnel as the engineering background, Fan et al. [

Although the stability of a tunnel excavated in a karst area has drawn the attention of many scholars, previous studies were carried out on the basis of the numerical simulation technique and model experiments. However, theoretical studies of the collapse mechanism and range of the surrounding rock induced by a concealed karst cave in tunnel construction are rare. Fraldi and Guarracino [

Considering that the variational principle is an effective method to determine the collapse region of the surrounding rock in tunnel construction, investigators used this method to study the collapse region induced by a karst cave in a tunnel excavated in a karst area. If a karst cave is present beneath a deep highway tunnel, there is a risk of collapse of the tunnel floor. In view of this problem, Huang et al. [

Presently, studies of the collapse region of the surrounding rock induced by a concealed karst cave focus on the rock mass located beneath the tunnel floor. However, if a concealed karst cave exists above the tunnel roof, the collapse of the rock mass above the tunnel roof occurs much more easily. In actual engineering, the pressure of the filler material in the karst cave, the gravity of the rock mass, and the blasting shock are three adverse factors that can cause the collapse of the rock pillar between the karst cave and the tunnel roof. Under extreme conditions, the three factors may work simultaneously, thereby enhancing the rock mass collapse. Therefore, it is necessary to develop a theoretical method to study the collapse mechanism of the rock mass induced by a concealed karst cave that is present above the tunnel roof.

This paper aims to investigate the collapse mode of the rock mass between the concealed karst cave and tunnel roof within the framework of the upper bound theorem. A failure mechanism is proposed based on the collapse characteristics of the rock mass induced by the concealed karst cave. Using the failure mechanism and variational principle, the analytical equation of the collapse surface of the rock mass is derived, and the range of the collapse region is obtained. To apply this method in an actual project, the Wuzhishan highway tunnel, excavated in a karst area, is selected as a case study. Based on a geological survey report and the position of the karst cave encountered during tunnel excavation, the failure mechanism in this actual engineering project is established. An analytical equation of the collapse surface of the rock mass above the Wuzhishan tunnel is derived. Furthermore, to validate the presented method, the obtained result is compared with that derived from the numerical simulation technique.

A schematic diagram of the collapse mode of the rock mass under consideration is illustrated in Figure _{i} and _{i} are the tangential and normal directions of a random point along the collapse surface direction, respectively.

Collapse mechanism of a highway tunnel roof induced by a karst cave.

Because the collapse range is unknown, the collapse surface is assumed to consist of two symmetrical arbitrary curves

The karst cave is assumed to be a circle for mathematical convenience and the equation of the circle is

Standard internal contours and construction clearance of a highway tunnel.

Because of the relative sliding movement between the collapsing block and the rest of the surrounding rock, energy dissipation would occur along the collapse surface. The upper bound theorem states that the upper bound solution of the equation of the collapse surface can be derived from the relation between the rate of total energy dissipation and the rate of external work. Therefore, the energy dissipation of a random point on the collapse surface should be calculated first. According to Chen [

According to Fraldi and Guarracino [

The external forces in this failure mechanism include the gravity of the rock mass, supporting force, and filler material pressure. The total rate of the work in this mechanism is the sum of the rates of work performed by these external forces. The rate of work produced by the self-weight of the half-collapsing block is given by

Substituting equations (

Because the virtual work equation includes the first derivative of

Substituting (

By solving this differential equation, the analytical expression of the collapse surface can be written as

Furthermore, the values of _{1} and _{2} are required to derive the final expression of the collapse surface

Substituting equations (_{1} is obtained as follows:

From Figure

By substituting (

Therefore, we have to establish another equation to determine these unknowns. By equating the external rate of work due to external forces to the total internal energy dissipation along the collapse surface, the last equation, which is used to calculate unknowns

Combining (

The aim of this paper is to study the collapse mode of the rock mass between the concealed karst cave and highway tunnel roof. If the shape and range of the collapse surface are obtained, the collapse mode of the rock mass in this region can be examined intuitively. Based on the analytical equation of collapse surface ^{−3} are illustrated in Figures

The shape and range of the collapse surface for different values of

The shape and range of the collapse surface for different values of

The shape and range of the collapse surface for different values of

The shape and range of the collapse surface for different values of

As mentioned above, the distance between a concealed karst cave and the proposed nearby highway tunnel is a major cause that induces the collapse of the rock mass in this region. Numerous investigators have used various methods to study the minimum safe distance between a karst cave and tunnel. However, presently, no theoretical equation can calculate the minimum safe distance between a karst cave and tunnel accurately. To solve this problem, the shapes of the collapse surfaces for different distances between the karst cave and tunnel are investigated. Figure

The shape and range of the collapse surface for different values of

Because (

Based on the derivation procedure of the minimum safe distance

The influence of the different parameters on

To apply the proposed method in actual engineering, a highway tunnel located in a karst area of Guangxi Province is selected as a case study. The engineering geological conditions of this tunnel are described as follows. The Wuzhishan tunnel is a highway tunnel for the Leye-Baise highway in China. The maximum buried depth of the tunnel is 191 m, and the total length is 1230 m. This tunnel is excavated in an area where the surrounding rock has been eroded by tectonic denudation and karst is developed. The principal lithology penetrated by the Wuzhishan tunnel is intermediary weathered limestone and intermediary weathered sandstone. By taking advantage of geophysical prospecting, it was deduced that a karst cave exists above the tunnel roof in the section of KD119 + 319. The engineering geological conditions of this cross section are presented in Figure

Engineering geological conditions for a certain cross section of the Wuzhishan tunnel.

Based on the engineering geological and geometric conditions, an upper bound calculation model of this case study is constructed. The shape of the karst cave is simplified as a circle, and the radius of this circle is 3 m. Using the analytical equation of collapse surface

The parameters of rock mass provided by geological survey report of Wuzhishan tunnel.

^{3}) | ||||
---|---|---|---|---|

0.5375 | 0.7816 | 0.3 | 1 | 24 |

The shape and range of the collapse surface for the Wuzhishan tunnel.

To verify the proposed method, the results obtained by the upper bound calculation are compared with those provided by numerical analysis using finite difference software FLAC3D. Because the internal contour of the Wuzhishan highway tunnel is multicircular, the tunnel model cannot be constructed by the modeling tool of FLAC3D directly. The tunnel model is constructed by using finite element software ANSYS, and the final model is obtained by importing the data into FLAC3D. The geometrical dimensions of this model are determined by the engineering geological conditions of the Wuzhishan tunnel mentioned above. The whole model is illustrated in Figure

The numerical model of a karst cave above the Wuzhishan tunnel.

The fill material pressure and supporting pressure in the numerical model.

Because the H–B failure criterion is used in the upper bound calculation, the H–B failure criterion was invoked in the simulation process to compare the upper bound calculation and numerical simulation under the same conditions. Furthermore, the rate of energy dissipation along the collapse surface is associated with the normal stress strain and shear stress strain, and the H–B failure criterion represented by the normal and shear stresses was employed in the upper bound calculation. However, the H–B failure criterion embedded in FLAC3D is represented in terms of the major and minor principal stresses. As the parameters of the H–B failure criterion for the two forms are different, the comparison of the collapse surfaces obtained by the upper bound calculation and numerical simulation cannot be achieved under the same conditions. To solve this discrepancy, Hoek and Brown [

Equivalent parameters for the two forms of the H–B failure criterion.

0.5375 | |

0.7816 | |

1 MPa | |

0.575 | |

1.5 | |

0 | |

1 MPa | |

15 |

Using the model illustrated in Figure

The parameters used in the numerical simulation.

^{3}) | ||||||
---|---|---|---|---|---|---|

24 | 0.4 | 0.3 | 1 | 0.575 | 1.5 | 0 |

Comparison of the collapse surface of a highway tunnel roof provided by the upper bound theorem and the numerical simulation.

For a tunnel excavated in a karst terrain, excavation may induce collapse of the surrounding rock, which poses a threat to the personal safety of construction personnel. A possible prediction method of the collapse region induced by a concealed karst cave above a deep highway tunnel is presented within the framework of the upper bound theorem of limit analysis. A new failure mechanism composed of arbitrary curves is proposed to describe the collapse mode of the rock mass between the karst cave and tunnel. Based on the failure mechanism, an objective function including the equation of the collapse surface is derived from the relation between the rates of total energy dissipation and external work. The analytical expression of the collapse surface is obtained on the basis of the variational principle.

Using the analytical expression of the collapse surface, the shapes of the collapse surfaces for different parameter values are plotted. By studying the change law of the collapse surface as a function of these parameters, it is found that the collapse surface increases with increasing

Based on the geological survey report of a highway tunnel excavated in Guangxi Province, the collapse surface of the rock mass above this tunnel is calculated to verify the practicality of the proposed method. Moreover, the computed result of the proposed approach is compared with that derived from the numerical simulation technique under the same conditions. The similarity between the collapse surface results obtained by the upper bound solution and numerical simulation indicates that the proposed method is valid.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This study was supported by the National Natural Science Foundation of China (Grants 51878074 and 51678071). Their financial support is greatly appreciated.