This paper investigates the collapse mechanisms and possible collapsing block shapes of shallow unlined tunnels under conditions of plane strain. The analysis is performed following the framework from a branch of catastrophe theory, functional catastrophe theory. First, the basic principles of functional catastrophe theory are introduced. Then, an analytical solution for the shape curve of the collapsing block of a shallow unlined tunnel is derived using functional catastrophe theory based on the nonlinear HoekBrown failure criterion. The effects of the rock mass parameters of the proposed method on the shape and weight of the collapsing block are examined. Moreover, a critical cover depth expression to classify deep and shallow tunnels is proposed. The analytical results are consistent with those obtained by numerical simulation using the particle flow code, demonstrating the validity of the proposed analytical method. The obtained formulas can be used to predict the height and width of the collapsing block of a shallow unlined tunnel and to provide a direct estimate of the overburden on the tunnel lining. The obtained formulas can be easily used by tunnel engineers and researchers due to their simplicity.
Shallow tunneling in soft ground inevitably results in ground movements. These movements may deform, rotate, distort, and irrevocably damage adjacent surface buildings in urban environments. In general, there are five different categories of tunnel collapses according to different factors as follows: daylight collapse, underground collapse, rock burst, inrush of water, and portal collapse [
Some examples of daylight collapse induced by shallow tunneling.
Munich in Germany, 1994
Taegu in South Korea, 2004
Lausanne in Switzerland, 2005
Chengdu in China, 2007
Tunnel stability analyses have been investigated by several scholars using various analytical approaches. The limit equilibrium method is widely used in the theoretical analysis of tunnel stability. Based on the failure mechanism of a twodimensional logspiral shaped slip plane, Murayama et al. [
Because the deformation and failure of tunnel surrounding rock are characterized by nonlinear and discontinuous phenomena, catastrophe theory is suitable for investigating the behaviors of tunnel collapse. Catastrophe theory arises from the mathematical work of Thom [
Although catastrophe theory has been proven to be a promising analysis tool for discontinuous phenomena, the current research on tunnel collapse is generally based on elementary catastrophe theory (ECT). In ECT, the potential function of the system is one of the eleven elementary functions defined by polynomial functions, whose independent variables are the state variable and control variable. Moreover, ECT can be used to solve discontinuous problems, where the state variable is a changing value
This paper investigates the collapse shape and mechanism of a shallow unlined tunnel under the conditions of plane strain in the FCT framework. First, the basic principles of FCT are introduced in Section
The research objects of catastrophe theory are the potential functions
In ECT, the potential function of the system is one of the eleven elementary functions
Potential functions used in ECT.
Name  Potential function  State variable  Control variable 

Fold 



Cusp 



Swallowtail 



Butterfly 



Indian tent 



Elliptic umbilic 



Hyperbolic umbilic 



Parabolic umbilic 



Symbolic umbilic 



Second hyperbolic umbilic 



Second elliptic umbilic 



If the potential function of the system is defined by a functional
To obtain the nonMorse critical point
Importantly, the functional
The comparison of ECT and FCT is shown in Table
Comparison of ECT and FCT.
ECT  FCT  

Fundamental theories  ① Implicit function theorem  
② Morse lemma 

③ Thorn splitting lemma  
Research content  The nonMorse critical point (or singular) of the potential function  
Potential function  Elementary function (polynomial functions) 
Functional 
State variables 


NonMorse critical points 


Control variables 


Catastrophic conditions 


First, we must obtain the potential function of the system
Second, (
Finally, the unknown constants in
For this study, to estimate the stability of a shallow unlined tunnel, we focus on determining the shape and dimensions of the collapsing block. This block can actually collapse from the roof of the tunnel. By applying the upperbound theorem, Yang and Huang [
Collapse pattern of a shallow unlined tunnel.
Several assumptions are made when solving the proposed problem via FCT. First, we consider only the gravity field and disregard the tectonic stress field for consistency with the relative reference (Yang and Huang [
In Figure
In accordance with the formula presented by Yang and Huang [
By following a purely geometrical line of reasoning and referencing Figure
At an impending collapse, the strain energy of the internal forces on the detaching zone
In the detaching zone, the potential function induced by normal and shear stresses can be expressed as follows:
The key task in catastrophic state analysis is to obtain the specific expression of
By integrating (
By substituting (
Given any value of
Equation (
First, the boundary condition of the ground surface is stressfree, which requires that the shear component of stress vanishes along this plane. According to Fraldi and Guarracino [
Boundary condition of the collapsing block.
By substituting (
Second, Figure
By substituting (
Finally, the boundary point
By substituting (
By simplifying (
By substituting (
The value of
We obtain the final form of the detaching curve
Equation (
We can determine the shape curve of the collapsing block of a shallow unlined tunnel using (
Shapes of the collapsing blocks versus rock mass parameters.
Variations of
Variations of
Variations of
The weight of the collapsing block under various parameters is plotted in Figure
Weight of the collapsing block versus rock mass parameters.
Variations of
Variations of
Variations of
Variations of
Given the significant differences between shallow and deep tunnels in terms of their mechanical characteristics and failure mechanisms, establishing a critical cover depth is important because it can distinguish shallow and deep tunnels.
To express the critical cover depth, we must calculate the value of
Based on this definition, the critical cover depth
Equation (
We plot the influences of the rock mass parameters on the critical cover depth of a shallow tunnel in Figure
Critical cover depth of the shallow unlined tunnel versus rock mass parameters.
Variations of
Variations of
Variations of
Variations of
To validate the method proposed in this study, the analytical solution is compared to the numerical simulation with respect to the shape curve of the collapsing block based on the parameters of the surrounding rock. These results are examined based on the HoekBrown failure criterion. To investigate the stability of a shallow unlined tunnel, we used the twodimensional distinct element program PFC2D [
Table
HoekBrown and equivalent MohrCoulomb parameters (Hoek and Brown [
HoekBrown 






Equivalent MohrCoulomb 



By referencing the selection criteria for microscopic parameters summarized by Zhou et al. [
PFC parameters of the surrounding rock after calibration.
Name  Notation  Value 

Ball density 

2940 
Porosity 

0.15 
Ball minimum radius 

0.05 
Ratio of maximum to minimum radius 

1.5 
Normal contact stiffness of the ball 

5.6 × 10^{8} 
Contact stiffness ratio of the ball 

10 
Ballball contact modulus 

5 × 10^{9} 
Ball friction coefficient 

0.5 
Normal strength of the contact bond 

2 × 10^{4} 
Contactbond strength ratio 

1 
Particle sample from a biaxial test.
Axial stressstrain curves under different confining pressures.
Mohr’s stress circle and envelope curve in the parameter calibration.
To ensure that the analysis results reflect the relevant characteristics on the ground, the numerical model must consider the boundary effects. The width and height of the numerical model are 56 and 30 m, respectively, and its cover depth is 5 m (Figure
Original model for the PFC2D analysis.
To compare the numerical simulation and analytical results in terms of the shape and dimensions of the collapsing block, the fitting curve (the solid line in Figure
Comparison of the numerical simulation and analytical results.
Equation  Results  Gain coefficient  Offset on 
Power  Adj. 


Result of curve fitting 


2.15  0.94  1.93  0.97 
Analytical result 


2.22  1.03  2.00  
Percentage difference  3.15%  8.74%  3.50% 
Shape and fitting curve of the collapsing block in Zone
The potential function based on FCT is presented in the form of a functional, whereas the potential function in ECT is expressed as an elementary function. The former solves complex problems more effectively than the latter. In this study, FCT is employed to investigate the collapse mechanisms and possible collapsing block shapes of shallow unlined tunnels for the first time. The following conclusions can be drawn.
Based on the nonlinear HoekBrown failure criterion and FCT, we derive an analytical solution for the shape curve of the collapsing block of a shallow unlined tunnel. The obtained formula can be used to predict the width of the collapsing block in shallow unlined tunnels and estimate the overburden on tunnel linings directly. The obtained formulas can be easily used by tunnel engineers and researchers due to their simplicity.
According to the FCT based on the nonlinear HoekBrown failure criterion, the outline of the collapsing block in shallow unlined tunnels is parabolic under the assumption in which only the gravity field is considered. Based on the obtained analytical solution, we investigate the influences of rock mass parameters on the shapes and weights of the collapsing blocks of shallow unlined tunnels. The collapsing blocks are small if the values of
We propose a critical cover depth expression to classify deep and shallow tunnels. The critical cover depth of a shallow tunnel increases with increases in
To validate the analytical solution proposed in this study, the analytical result for the shape curve of the collapsing block of an unlined tunnel is compared to that obtained by numerical simulation. The shape and dimensions of the collapsing block determined by numerical simulation are identical to those calculated analytically, thus demonstrating the validity of the proposed method.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors acknowledge the financial support provided by the National Natural Science Foundation of China (no. 51378002), the National Key Technology R&D Program (no. 2012BAJ01B03), and the Program for New Century Excellent Talents in University (no. Ncet120770).