Mathematical Problems in Engineering

Volume 2016, Article ID 4832097, 8 pages

http://dx.doi.org/10.1155/2016/4832097

## Upper Bound Solution of Safety Factor for Shallow Tunnels Face Using a Nonlinear Failure Criterion and Shear Strength Reduction Technique

School of Civil Engineering and Architecture, Changsha University of Science and Technology, Changsha, Hunan 410114, China

Received 5 February 2016; Accepted 26 April 2016

Academic Editor: Yakov Strelniker

Copyright © 2016 Fu Huang 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

A method to evaluate the stability of tunnel face is proposed in the framework of upper bound theorem. The safety factor which is widely applied in slope stability analysis is introduced to estimate the stability of tunnel face using the upper bound theorem of limit analysis in conjunction with a strength reduction technique. Considering almost all geomaterials following a nonlinear failure criterion, a generalized tangential technique is used to calculate the external work and internal energy dissipation in the kinematically admissible velocity field. The upper bound solution of safety factor is obtained by optimization calculation. To evaluate the validity of the method proposed in this paper, the safety factor is compared with those calculated by limit equilibrium method. The comparison shows the solutions derived from these two methods match each other well, which shows the method proposed in this paper can be considered as effective.

#### 1. Introduction

Shallow tunnels are very common nowadays in municipal engineering construction as they make traveling more convenient and also reduce engineering costs. Though shallow tunnels have a number of advantages, accidents due to instability of tunnels face would occur if the supporting pressure cannot resist the earth pressure. Thus, to avoid the occurrence of such accidents, it is necessary to determine the suitable retaining pressure on the tunnel face. The stability analysis of the face for a tunnel excavated in shallow strata has drawn the attention of many investigators [1–4]. On the other hand, the retaining pressure on the tunnel face is constant in practical engineering when the earth pressure balance of tunnel face is achieved by using compressed air or bentonite slurry. Therefore, it is necessary to establish a method to evaluate the stability of tunnel face when the retaining pressure is known. The safety factor (FOS) is a valid index to evaluate the stability of geotechnical structure which is widely used in geotechnical engineering. Presently, the safety factor used in geotechnical stability analysis is mostly calculated by the finite element method with shear strength reduction technique (SSRFEM) [5–9]. Based on this method, the safety factor and critical failure surface can be obtained by reducing the actual shear strength parameters until geotechnical material fails. So the accuracy of FOS derived from SSRFEM relies strongly on the determination of failure of geotechnical structure. However, the definition of failure for geotechnical structure is a disputed issue. Griffiths and Lane [10] chose the nonconvergence of algorithm as the indicator of failure for homogeneous slope. Snitbhan and Chen [11] regarded bulging or progressive loss of ground along the vertical cut as the failure of vertical slope. As there is divergence of definition for failure in the geotechnical engineering field, the different definition of failure would induce different analysis result, which confines the application of SSRFEM in geotechnical engineering.

Large numbers of theoretical researches and practical projects illustrate that the failure of shallow tunnel face can be divided into collapse and blow-out failure mechanisms. The collapse failure mechanism is used to describe the collapse of soil in front of the tunnel face, which is caused by the insufficient retaining pressure on the tunnel face. Contrarily, blow-out failure mechanism occurs when the retaining pressure is so great that soil is heaved in front of the tunnel face. Based on the failure mode and kinematically admissible velocity characteristic of the shallow tunnel face, Leca and Dormieux [12] proposed a three-dimensional failure mechanism composed of solid conical blocks. Due to the slide between the solid conical blocks and surrounding soil, the plastic flow occurs along the velocity discontinuity surface. Using the energy dissipation rate along the surface and the rate of work caused by external force, Leca and Dormieux [12] derived the upper bound solution of retaining pressure for tunnel face. As the mechanism proposed by Leca and Dormieux [12] is supported by centrifuge model tests and well reflects the mechanical characteristics of failure mode for shallow tunnel face, many scholars used this mechanism to analyze the stability of shallow tunnel face under various conditions [13–16].

These literatures mentioned above all used linear Mohr-Coulomb criterion. However, the stress-strain relation of soils and rocks is nonlinear. This viewpoint has been supported by experiments and some scholars used nonlinear failure criterion to study the stability of tunnel face and other geotechnical structures [17, 18]. Based on the failure mechanism proposed by Leca and Dormieux [12], Huang and Yang [19] calculated the upper bound solution of retaining pressure on the tunnel face using the upper bound theorem in conjunction with nonlinear failure criterion. Senent et al. [20] studied the face stability of circular tunnels excavated in heavily fractured rock masses which are subjected to the nonlinear Hoek-Brown failure criterion. According to their study, the critical retaining pressures computed with limit analysis are very similar to those obtained with the numerical model, which proves their method is valid. Though their studies reflect the influence of nonlinearity on the critical retaining pressures, they failed to propose a method to evaluate the stability of tunnel face when the retaining pressure is known.

In this work, upper bound theorem combined with shear strength reduction technique is used to estimate the face stability of a tunnel excavated in shallow strata following the nonlinear failure criterion. On the basis of upper bound theorem, the rate of external work and the internal energy rate of dissipation for the failure mechanism are calculated. Based on the relationship between the rate of external work and the internal energy rate of dissipation, the convergence of iteration in the strength reduction calculation can be controlled, which avoids the selection of the definition of failure that occurs in the SSRFEM calculation process. To validate the new methodology, the FOS of tunnel face is compared with the result computed by limit equilibrium method. Furthermore, the influence of nonlinear parameter on the stability of tunnel face is investigated.

#### 2. Upper Bound Theorem with Shear Strength Reduction Technique

Shear strength reduction technique was proposed by Bishop [22], whose core content is the reduction of soil shear strength parameters until the soil fails. To achieve this reduction, an important concept is introduced, the shear strength reduction factor . When the actual shear strength parameters and are divided by the shear strength reduction factor, the soil strength parameters and used in upper bound analysis are obtainedWhile shear strength reduction factor increases incrementally, newly reduced soil strength parameters are obtained. The iterative process continues until failure occurs.

As mentioned above, the selection of a suitable definition of failure is a problem in SSRFEM. To overcome this difficulty, the upper bound theorem is used to control the convergence of iterative operation in the strength reduction technique. The upper bound theorem states that when the velocity boundary condition is satisfied, the load derived by equating the rate of external work to the rate of the energy dissipation in any kinematically admissible velocity field is no less than the actual collapse load. Therefore, if the reduced soil strength parameters are introduced in the energy dissipation calculation, the shear strength reduction factor can be obtained on the basis of the relationship between the rate of external work and the rate of the energy dissipation.

#### 3. Upper Bound Solution of FOS for Shallow Tunnel Face

In this work, the failure mechanisms proposed by Leca and Dormieux [12] are used to calculate the rate of external work and the rate of the energy dissipation in the framework of the upper bound theorem of limit analysis. The failure mechanism of Leca and Dormieux [12] is composed of collapse and blow-out failure mechanisms, which is shown in Figure 1. For instance, we use failure mechanism III to illustrate the computational process of the upper bound solution of FOS. As Leca and Dormieux [12] have computed the rate of external work for mechanism III, the expression of can be written aswhere is the tunnel diameter, is the unit weight of the soil, is the friction angle of the soil, is surcharge loading, is the velocity of conical block, is the angle between symmetry axis of conical block and centre line of the tunnel, and the parameters , , are expressed aswhere is the tunnel depth. Moreover, the rate of the energy dissipation produced in kinematically admissible velocity field iswhere is the cohesion of soil. Based on the upper bound theorem, the expression of retaining pressure is obtained by equaling the rate of external work to the rate of the energy dissipationTo calculate the upper bound solution of FOS, the initial cohesion and friction angle of soil are substituted into (1) to obtain the reduced strength parameters and . Then, the reduced strength parameters and are substituted into (2) and (4) to derive the upper bound solution of retaining pressure . Finally, by equaling the practical retaining pressure to the retaining pressure expressed in (5), the objective function of safety factor which includes an angle variable is obtained. However, the objective function is just an expression of numerous upper bound solutions. According to the upper bound theorem, the minimum value of objective function is the real upper bound solution of FOS. Therefore, a sequential quadratic programming is employed to search the minimum value of objective function . As some compatibility relations of velocity should be satisfied in the kinematically admissible velocity field, the optimization calculation is achieved when corresponding constraint conditions are satisfied. Therefore, the expression of mathematical planning for the problem can be written as where is the power of practical retaining pressure.