Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 4820716, 11 pages

http://dx.doi.org/10.1155/2016/4820716

## Functional Catastrophe Analysis of Collapse Mechanism for Shallow Tunnels with Considering Settlement

School of Civil Engineering, Central South University, Hunan 410075, China

Received 26 March 2016; Accepted 4 July 2016

Academic Editor: Oleg V. Gendelman

Copyright © 2016 Rui Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1], it is a great significance to predict the stability of the shallow tunnels in the engineering.

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 [2] with the help of limit analysis theory and finite element technique. In the 1970s the upper bound theorem was proposed by Chen [3]. Then this theorem had great importance in the field of geotechnical engineering because of its great validity in dealing with the stability problems in underground structures. With the extensive use of it, it was improved by many scholars. By introducing a linear multiblock collapse mechanism in 1980, Davis et al. [4] obtained the upper bound solutions of the stability coefficient. In 1994 the accuracy of supporting pressure derived under the three-dimensional collapse mechanism with the centrifugal model test was studied by Chambon and Corté [5]. Yang and Long [6] studied the influences of two angles on the 3D slope stability by limit analysis method and obtained the critical stability number. The upper bound theorem will be illustrated at length in the following.

With the development of the limit analysis method, the linear criterion [7–9] used to evaluate the stability problems has been replaced by the nonlinear criterion [10–15] in terms of the nonlinear mechanical characteristics of geotechnical material in tunnel project. During the process of applying nonlinear failure criterion, a generalized tangential methodology was suggested [16–18] to calculate the energy dissipation rate and the external work rate accurately. In particular on the basis of nonlinear yield criterion [19–24] many researchers put forward the analytical solution for characterizing the collapsing shape. The curved failure mechanism was constructed according to the theory that the energy dissipation rate and external work rate were calculated along the velocity discontinuity [25]. The equation about it will be set in the following.

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 [26]. The Modified-Lade failure criterion was developed by Ewy [27]. Furthermore the Hoek-Brown failure criterion [28] has been widely used in geotechnical analysis and still has been improved by some scholars. This work uses a new nonlinear strength function proposed by Baker [29] to analyze the stability of tunnel roof. The NL failure criterion will be presented at length in the following.

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 [30] put forward a new failure mechanism of shallow tunnels by considering the surface settlement. On the basis of previous work in which a kinematic plastic solution with ground movements is derived by Osman et al. [31], a compatible displacement field to calculate the stability problem of shallow twin tunnels excavated in the soft layer is constructed by Osman [32].

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. [33] to predict the occurrences of a rock burst. Ren et al. [34] studied a cusp catastrophe model to analyze the potential damage mode of the surrounding rock in the tunnels.

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.

#### 2. Overview of Catastrophe Theory

The catastrophe theory was put forward by Thom in 1972 [35]. As a branch of nonlinear theory, catastrophe theory has been developed rapidly and applied widely. From the main starting points which are the basis of the structural stability, singularity theory, and bifurcation theory; the mathematical model is set for the purpose of the discussions of the general laws of the jump change of the state in dynamic system. The catastrophe theory is a good method to illustrate the mechanics of the change from continuous gradient to stable state in nonlinear system. Due to the complexity and diversity of the constitutive relation of the engineering soils and the uncertainty of the boundaries of the elastic and plastic during the process of the tunnel excavation, it is difficult to describe this process with the method of classical mathematical method. The quantitative state of the system can be predicted with a small number of control variables even without considering the constitutive equation and the mechanical properties of the soil mass system, which is the advantage of the catastrophe theory compared with other theories. In order to explain the phenomenon of various mutations, Thom put forward seven different kinds of models; these models are generally based on elementary catastrophe theory (ECT). In ECT, the potential function of the system is one of the seven elementary functions defined by the polynomial functions shown in Table 1.