Advances in Civil Engineering

Volume 2018, Article ID 3862974, 11 pages

https://doi.org/10.1155/2018/3862974

## Upper Bound Limit Analysis for Seismic Stability of Rock Slope with Tunnel

^{1}College of Civil Engineering, Central South University of Forestry and Technology, Changsha, Hunan 410004, China^{2}Rock and Soil Engineering Research Institute, Central South University of Forestry and Technology, Changsha, Hunan 410004, China

Correspondence should be addressed to Jiayong Niu; moc.qq@3872757511

Received 23 November 2017; Accepted 11 January 2018; Published 4 April 2018

Academic Editor: Yixian Wang

Copyright © 2018 Xueliang Jiang 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

The rock slopes with tunnels appear widely in the actual project, but there is no executable basis for the seismic stability calculation of the rock slope with tunnel. According to the upper bound theorem of plastic limit analysis and pseudostatic method, the upper bound solution of the safety factor of the rock slope with tunnel was rigorously derived under earthquake loading. This upper solution takes into account the design parameters of the slope and the tunnel, the horizontal and vertical seismic loads, and the physical and mechanical parameters of the rock mass. Comparing the calculated results with the existing results, the validity of the proposed method was verified. The sensitivity and influence of different parameters on the seismic stability of the slope were analyzed. The results show that the three factors such as the horizontal seismic force coefficient, the slope height, and the internal friction angle are the three key factors that influence the sensitivity of the safety factor and have a great effect on it.

#### 1. Introduction

In 2017, the Jiuzhaigou earthquake with a moment magnitude of 7.0 triggered a large number of slope failures in Sichuan Province, China. The earthquake-induced slope failures led to serious damage to the highway and tunnel system. The stability of rock and soil slope [1, 2] under earthquake has become a hot research topic at present [3–5]. Based on the quasi-static method, the limit equilibrium analysis and upper bound approach of limit analysis have been widely utilized to evaluate the seismic stability of the slope. Deng et al. [6] applied the limit equilibrium method to analyze the stability of the slope under earthquake considering three kinds of the slip surface (line, circular, and arbitrary curves). Lu et al. [7] extended Newmark’s method to three dimensions and proposed a new method for evaluating the seismic permanent displacement of 3D slope. Liang and Knappett [8] presented an improved Newmark sliding block procedure for predicting the seismic slip of a vegetated slope. Although the limit equilibrium analysis is simple and has certain accuracy, the solution obtained by the limit equilibrium method is not the upper or lower limit of the real solution, which has some limitations in theory [9]. The upper bound method of limit analysis has made great progress in slope engineering [10] since Chen [11] introduced the plastic limit analysis into the stability analysis of the soil slope. Zhao et al. [12] applied the upper bound limit analysis theorem and the shear strength reduction technique to define the safety factor and corresponding critical failure mechanism of a layered soft-rock slope. Liu et al. [13] presented a new approach for determining the factor of safety and the corresponding critical slip surface of a layered rock slope under seismic excitations, which were obtained by the limit equilibrium method and pseudostatic approach. A 3D rotational mechanism was adopted by He et al. [14] to analyze seismic displacement of slopes reinforced with piles using limit analysis theory and Newmark’s analytical procedure. Ausilio et al. [15, 16] used the kinematic approach of limit analysis to analyze the seismic stability of slopes reinforced with geosynthetics and slopes reinforced with piles, respectively.

However, most of the published works of slope limit analysis were mainly concentrated at the research of a slope without tunnel. Few scholars have carried out theoretical research on the slope with tunnel [17]. The rock slope with tunnel, as a composed slope structure, may have a complex interaction between the tunnel and slope. The stability calculation theory under natural or earthquake condition is far from keeping pace with the engineering construction of the slope with tunnel. In the present study, the failure mechanism of the rock slope with tunnel is constructed, and the upper bound solution of the safety factor of the rock slope is deduced based on the upper bound limit analysis theorem and the pseudostatic approach. In addition, the solutions presented in this study are compared with those obtained by the pseudostatic approach. The effects of physical and mechanical parameters [18, 19] of the rock slope and design parameters of the tunnel slope with tunnel are investigated.

#### 2. Upper Bound Theorem of Plastic Limit Analysis

The upper bound theorem of plastic limit analysis is a useful method in the stability limit analysis, which has attracted extensive attention to solving geotechnical problems [20–22]. In the stability analysis of the rock and soil slope, it is necessary to understand the damage load of rock and soil when it begins to produce unrestricted plastic flow; it is not necessary to know the change process of stress and strain with the external load. The upper limit theorem of the plastic limit analysis assumes that the rock mass moves in the form of a rigid plastic body during failure. It is required that the rate of work of force on the slope surface and the physical force is not greater than the energy dissipation in the permissible velocity field for arbitrary maneuver admissible damage mechanism, which means that the external rate of work is not greater than the internal energy dissipation rate. The external rate of work generally includes the rate of work induced by the rock mass, the pore water pressure, the slope overload, and the seismic load. The internal energy dissipation rate includes the internal dissipated rate of the failure surface and the rate of work induced by the resistance of the supporting structure.

##### 2.1. Basic Assumptions

In the stability analysis of the system of the tunnel and slope, in order to simplify the research object, the rock and soil mass and the tunnel structure are usually treated separately as two separate structures, which is called a structured method [23]. The pseudostatic method [24] is often used to analyze the dynamic effect of earthquake in engineering practice.

Based on the relevant research results [25, 26], the following basic assumptions are made for the convenience of analysis. (1) The rock mass of the slope is considered to be soft rock which is homogeneous and isotropic. (2) The tunnel surrounding rock is regarded as an ideal elastomeric body following the Mohr–Coulomb yield criterion. (3) The failure surface is a log-spiral failure surface passing through the slope toe and tunnel. (4) The stability analysis of the slope with tunnel is simplified as a plane strain problem. (5) The tunnel is simplified as a circular cross section for calculation. The vertical surrounding rock pressure acting on the tunnel vault is simplified as a linear uniform load *q*, and the horizontal surrounding rock pressure acting on the side wall is simplified as a linear uniform load *e*. (6) The shear strength parameters of the rock material do not change with the action of the earthquake, and the quasi-static method is used to analyze the seismic effect.

##### 2.2. Failure Mechanism

Constructing reasonable failure mode is the premise and key of upper bound limit analysis. The strength reduction analysis module of MIDAS GTS/NX was used to simulate the failure mode of the slope with tunnel. The elasticity modulus *E* of rock mass was 0.6 GPa, the cohesion *c* was 0.15 MPa, the internal frictional angle *φ* was 20°, and the rock unit weight *γ* was 19 kN·m^{−3}. The equivalent plastic strain is a measure to analyze the comprehensive deformation state of the material under the condition of complex stress compared with uniaxial tension. Figure 1 shows the equivalent strain distribution of the slope with tunnel. The sliding surface of the slope passes through the left sidewall and the right spandrel of the tunnel from the slope toe and extends upward to the top of the slope. It is similar to the most unfavorable position of the goaf in the slope, which is proposed by the literature [27]. When the geometric center of the goaf lies on the sliding belt, the stability of the slope is much lower than that of the geometric center of the goaf which lies in the interior of the sliding belt or lies in the outside of the sliding belt. Referring to related literatures [28], the log-spiral failure surface is consistent with the actual failure surface; the failure mode of kinematically permissible velocity field is close to the actual failure mode, so the log-spiral failure mechanism is used in this paper.