Limit analysis is a practical and meaningful method to predict the stability of geomechanical properties. This work investigates the pore water effect on new collapse mechanisms and possible collapsing block shapes of shallow tunnels with considering the effects of surface settlement. The analysis is performed within the framework of upper bound theorem. Furthermore, the NL nonlinear failure criterion is used to examine the influence of different factors on the collapsing shape and the minimum supporting pressure in shallow tunnels. Analytical solutions derived by functional catastrophe theory for the two different shape curves which describe the distinct characteristics of falling blocks up and down the water level are obtained by virtual work equations under the variational principle. By considering that the mechanical properties of soil are not affected by the presence of underground water, the strength parameters in NL failure criterion can be taken to be the same under and above the water table. According to the numerical results in this work, the influences on the size of collapsing block different parameters have are presented in the tables and the upper bounds on the loads required to resist collapse are derived and illustrated in the form of supporting forces graphs that account for the variation of the embedded depth and other factors.

With the development of cities, it is necessary to use a large amount of the underground area for the construction of transportation infrastructures. The ground movements are caused inevitably during the construction of the shallow tunneling in soft ground. Due to the adverse impact of the presence of underground water, as the occurrence of collapse of the tunnel exerts great threats to people’s lives and engineering loss [

To estimate the face tunnel stability problem is a hot topic in the tunnel engineering. The shallow tunnels are often chosen in the urban subway projects. Among a large number of methods which are suitable for solving the stability problem of shallow tunnels, limit equilibrium method, numerical simulation, and the limit analysis method are widely used. Owing to the ignorance of the relationship between stress and strain, the equilibrium method only considering stress balance has great defect. Due to the limitations of the limit equilibrium method and other methods, some scholars adopted the limit analysis approach to predict the stability and failure modes of the face and crown of tunnels, which shows an extreme simplicity and great effectiveness. For instance, the rigorous bounds of supporting pressure were obtained by Sloan and Assadi [

With the development of the limit analysis method, the linear criterion [

To choose appropriate failure criteria is one of the key steps in predicting the stability of the tunnel roofs. Numerous failure criteria have been used for rock failure analysis, but there is no common agreement of which failure criterion to select. The Mohr-Coulomb failure criterion was recommended for stability analysis because of the more realistic results compared with the different forms of Drucker-Prager [

Owing to a big difference in the tunnel roof collapse mechanism between shallow tunnels and deep tunnels, a new curved failure mechanism should be proposed to reflect the identities more appropriately. Yang and Wang [

The catastrophe theory has been proved to be effective in analyzing the stability problems in geology and geomechanics. Many scholars have applied this theory in prediction of the stability in the engineering. A fold catastrophe model of a tunnel rock burst was established by Pan et al. [

According to the introduction of the previous works which focus on the predictions of the stability of the tunnel buried in shallow soil layer, this paper establishes a failure mechanism of shallow tunnels with regard to surface settlement. Referring to the NL failure criterion, research on failure is conducted due to the presence of varying water table in the limit analysis. Moreover the functional catastrophe theory is employed to investigate the mechanisms of tunnel roof collapse. The analytical solution of the collapsing block shape curve is obtained and the effects of different parameters on the collapsing block shape are also discussed in this work. Furthermore the upper bound supporting pressure is obtained to ensure the safety of the roof of the shallow tunnels during the constructions.

The catastrophe theory was put forward by Thom in 1972 [

Potential functions used in elementary catastrophe theory.

Name | Potential function | State variable |
---|---|---|

Fold | | |

Cusp | | |

Swallowtail | | |

Butterfly | | |

Elliptic umbilic | | |

Hyperbolic umbilic | | |

Parabolic umbilic | | |

However, the total potential of the system always has a complex mathematical form, such as the state function

Various strength functions (Mohr envelopes) have been proposed to represent nonlinear strength functions for soils. Many publications utilized a simple power law relation of the form. Based on the work of Baker [

The nonlinear strength criterion for rock masses [

By assuming the plastic potential,

So the plastic strain rate can be written as follows:

In order to enforce compatibility, from (

And the normal component of stress can be written as

So, by virtue of the Greenberg minimum principle, the effective collapse mechanism can be found by minimizing the total dissipation; the dissipation density of the internal forces on the detaching surface,

The theory of the upper bound has been widely used to the predictions of the stability of the tunnels. The upper bound theorem of limit analysis can be depicted as follows: when the velocity boundary conditions and consistency conditions for strain and velocity are satisfied by the maneuvering-allowable velocity field which is built, the actual loads should be no more than the values of the calculated loads which is derived from the equation constituted by equating the external rate of work and the rates of the internal energy dissipation.

According to Chen [

Many scholars have studied the failure modes of the deep tunnel roof with arbitrary cross sections in layered rocks. By considering the fact that a large number of shallow-buried tunnels are also constructed in the layered stratums, this work investigates the failure mechanism of shallow tunnels under the condition of varying water table considering the surface settlement. The upper bound solutions derived from this work have general applicability and can be used more widely. Due to the deformation continuity of the failure shape of the collapsing mass, the boundary conditions along the detaching lines should be satisfied to ensure the geometry continuity. In order to get the upper bound solutions to describe the shape of the collapsing block, the first work is to calculate the internal energy dissipation rate produced by the shear stress and normal stress along the two different detaching lines. Furthermore the objective function consisting of the internal and external work should be constructed. Lastly two failure shape curves

Potential falling blocks with consideration of the effects of surface settlement and pore water pressure.

The failure shape of the collapsing block, as illustrated in Figure

From the assumptions above, without considering the geometric difference in different soil layers, the parameters of the NL strength law are assumed to be the same. Due to the presence of velocity detaching line existing in the soil layer of tunnel roofs, the impending failure would slide in a limit state along with the velocity discontinuous surfaces. During the process of the impending collapse, the dissipation densities of the internal forces on the detaching surface are

Curved failure mechanism of shallow circular tunnels.

The work rate of failure block produced by weight can be calculated by integral process

According to the research of Osman [

The distribution of excess pore pressure which is derived from the study of Saada et al. [

Therefore the work rate produced by seepage forces along the velocity discontinuity surface is

Due to the fact that the tunnel is buried in the shallow strata, the supporting structure is unavoidable for the requirement of safety and stability. Therefore, the work rate of supporting pressure in the shallow circular tunnel is

Meanwhile, because of the external force always exerted on the underground structure, the work rate of extra force which puts on the ground surface cannot be ignored. The expressions can be written as

In order to describe the shape and extension of the failure collapsing block, it is essential to obtain the explicit expressions of

So, the effective collapse mechanism can be obtained by minimizing the objective function

Then substituting (

In order to get the extremum of the objective function

By the variational calculation the explicit forms of the two Euler’s equations for (

Obviously, (

From what is mentioned above, the detaching curves are supposed to be symmetric with respect to the

Furthermore, the explicit expressions of the function of velocity discontinuity surface should fulfill other boundary conditions, such as

Substituting (

For the purpose of keeping the whole curve look smooth, another boundary condition should be satisfied:

Building on this result, the expressions of the function of velocity discontinuity surface can be obtained:

On the basis of the expression of the profile of the circular tunnel, the piece of external work can be calculated by integrating

Therefore, the objective function

For the purpose of getting the explicit forms of detaching curve profile consisting of

As mentioned in the previous section, if the potential function of the system is defined by a function

In order to get the non-Morse critical point

Importantly, the function

According to (

The form of catastrophic conditions of function

During the process of the catastrophe analysis of the stability of a shallow tunnel, the specific expressions of

Given any values of

The value of

The explicit failure surfaces of circular tunnel profiles can be drawn according to the analytical solutions of velocity discontinuity surfaces

In order to explore the influence of different parameters such as

The impending roof failure with regard to different parameters.

| | | | | | | | |
---|---|---|---|---|---|---|---|---|

| kN/m^{3} | | m | m | ||||

0.9 | 0.5 | 0.2 | 0.1 | 100 | 16 | 2.5 | 1.0092 | 7.4688 |

0.8 | 0.5 | 0.2 | 0.1 | 100 | 16 | 2.5 | 2.1009 | 8.0279 |

0.7 | 0.5 | 0.2 | 0.1 | 100 | 16 | 2.5 | 3.1228 | 8.5869 |

0.6 | 0.5 | 0.2 | 0.1 | 100 | 16 | 2.5 | 4.0889 | 9.1459 |

0.7 | 0.5 | 0.1 | 0.1 | 100 | 16 | 2.5 | 3.0639 | 9.7330 |

0.7 | 0.5 | 0.2 | 0.1 | 100 | 16 | 2.5 | 3.1228 | 8.5869 |

0.7 | 0.5 | 0.3 | 0.1 | 100 | 16 | 2.5 | 3.1596 | 7.7074 |

0.7 | 0.5 | 0.4 | 0.1 | 100 | 16 | 2.5 | 3.1852 | 6.9660 |

0.7 | 0.5 | 0.6 | 0.1 | 100 | 16 | 2.5 | 3.2173 | 5.7223 |

0.7 | 0.5 | 0.8 | 0.1 | 100 | 16 | 2.5 | 3.2341 | 4.6738 |

0.7 | 0.5 | 0.2 | 0.1 | 100 | 16 | 2.5 | 3.1228 | 8.5869 |

0.7 | 0.5 | 0.2 | 0.2 | 100 | 16 | 2.5 | 2.9906 | 8.5869 |

0.7 | 0.5 | 0.2 | 0.3 | 100 | 16 | 2.5 | 2.8433 | 8.5869 |

0.7 | 0.5 | 0.2 | 0.4 | 100 | 16 | 2.5 | 2.6775 | 8.5869 |

0.7 | 0.5 | 0.2 | 0.1 | 100 | 14 | 2.5 | 2.6359 | 8.3167 |

0.7 | 0.5 | 0.2 | 0.1 | 100 | 16 | 2.5 | 3.1228 | 8.5869 |

0.7 | 0.5 | 0.2 | 0.1 | 100 | 18 | 2.5 | 3.5168 | 8.8107 |

0.7 | 0.5 | 0.2 | 0.1 | 100 | 20 | 2.5 | 3.8430 | 9.0000 |

To endure the surcharge load, soil weight, and other external forces more efficiently, it is meaningful to obtain the minimal upper bound supporting pressure exerted on the tunnel lining within the framework of upper bound theorem with considering the diverse impacts which different parameters have. Parametric analysis is conducted, such as

Effects of different parameters on supporting force of shallow tunnels.

Effects of

Effects of

Effects of

Effects of

According to the figure, the burial depth

The figure shows that the value of maximum surface settlement

On the basis of previous work which has focused the efforts on the collapse mechanism in deep tunnels, a new curved failure mechanism of circular shallow tunnel considering the joint effects of surface settlement and seepage forces is proposed to estimate the stability of tunnel roof under a limit state. Furthermore this work proved the validity of the NL strength function in predicting the stability of the tunnel roof by comparing with HB failure criterion. With NL failure criterion and FCT theory the numerical solution for the shape of collapse mechanism is obtained by setting an objective function consisting of energy dissipation rate and the external work rate. Some conclusions are drawn from above:

With the increase of the shear strength controlling coefficient

The supporting forces tend to decrease with the increase of the radius of the circular tunnels and the surface settlement while the surcharge load and the parameter

The authors declare that they have no competing interests.

Financial support was received from the National Basic Research 973 Program of China (2013CB036004) and National Natural Science Foundation (51378510) for the preparation of this paper. This financial support is greatly appreciated.