Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 7857490, 8 pages

http://dx.doi.org/10.1155/2016/7857490

## Stability Analysis of Anchored Soil Slope Based on Finite Element Limit Equilibrium Method

^{1}Ji’nan Rail Transit Group Co., Ltd., Jinan, Shandong 250101, China^{2}Dalian University Civil Engineering R&D Center, Dalian, Liaoning 116622, China

Received 10 March 2016; Accepted 7 June 2016

Academic Editor: Francesco Tornabene

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

Under the condition of the plane strain, finite element limit equilibrium method is used to study some key problems of stability analysis for anchored slope. The definition of safe factor in slices method is generalized into FEM. The “true” stress field in the whole structure can be obtained by elastic-plastic finite element analysis. Then, the optimal search for the most dangerous sliding surface with Hooke-Jeeves optimized searching method is introduced. Three cases of stability analysis of natural slope, anchored slope with seepage, and excavation anchored slope are conducted. The differences in safety factor quantity, shape and location of slip surface, anchoring effect among slices method, finite element strength reduction method (SRM), and finite element limit equilibrium method are comparatively analyzed. The results show that the safety factor given by the FEM is greater and the unfavorable slip surface is deeper than that by the slice method. The finite element limit equilibrium method has high calculation accuracy, and to some extent the slice method underestimates the effect of anchor, and the effect of anchor is overrated in the SRM.

#### 1. Introduction

As an effective reinforced measure to slope, anchor rod has the advantages of simple construction, being fast, having less quantity of project, and so forth. It is widely used in protective engineering of landslides and other geological disasters. Therefore, the improvement of slope stability needs to be evaluated accurately and efficiently during the design of slope anchored.

Slice method [1, 2] has the advantage of clear concepts, definite physical meaning, and rich experience, but limitation of the method is equally clear: due to the presumption that the potential sliding mass is considered as a rigid body, the anchoring effect of slice method is reflected on the structure of the shear resistance to the balance of force and torque, rather than the actual potential sliding soil mass deformation constraint or the inner force redistribution. Therefore, it is less able to reflect the substance of soil-anchoring structure interaction.

The finite element strength reduction method (SRM), which can meet equilibrium and compatibility conditions automatically, has more rigorous theories system than slice method without assuming the shape and position of the sliding surface. It has received widespread attention since being proposed by Zienkiewicz et al. in 1975 [3], whereafter Matsui and San [4] verified the theoretical and numerical rationality of SRM for the finite element slope stability analysis. Zheng and Zhao [5] have done some research on stability of slope under the action of prestressed anchor cable with SRM based on the reduction of soil strength. However, they did not study the anchor strength reduction. On the basis of previous research, Wei et al. [6, 7] proposed an anchor rod strength model which can be applied in SRM and recommended that while soil strength is reduced, the reduction of anchor rod should be considered. Shi et al. [8] basing their theory on the direct reduction of cohesion and friction angle presented a discount method for tangent modulus. The intersection point of two lines, one of which is the deformation energy integral curve in potential slip area and the other is reduction coefficient curve, is chosen by the safety factor of slope stability. Isakov and Moryachkov [9] established the relationship equation between comprehensive safety factor and strength reduction path and proposed the expression of minimum comprehensive safety factor by the shortest strength reduction path. Bai et al. [10] introduced the classical strength reduction method into the double reduction calculation process and have proved that the safety factor of double reduction method is almost always smaller than that of the classical SRM with theoretical derivation and numerical simulation. Xue et al. [11] based their theory on the assumption of soil strength parameters linear attenuation and introduced the nonproportional relationship between the cohesion reduction coefficient and the friction angle reduction coefficient into the traditional SRM, and the comprehensive safety factor is proposed based on shear strength parameters contributing to the resistant shear force. However, these studies have not mentioned any further research and discussion on some key questions, for example, whether the different soil layers should share the same reduction factor for heterogeneous slopes and whether the reinforcement should reduce structure strength for reinforced slope.

For the first time, Brown and King [12] introduced finite element stress field and homologous sliding surfaces determination method to analyze slope stability. Since then, many domestic and foreign scholars have been making thorough research and developing it. Naylor [13] defined the safety factor on circular slip surface as the ratio of the sum of antislide force to the sum of sliding force for the whole sliding surface. The stress of calculated points is provided by the finite element stress field. Shao and Li [14] who have proposed a proved sufficient and necessary condition to define the safety factor on any sliding surface is using the ratio of shearing resistance integral to shearing stress integral. This method is based on the theoretic foundation for finite element limit equilibrium method. Zhao [15] used an interface element to simulate the interaction between soil nails and surrounding soil and then analyzed the stability of foundation pit soil-nailing supporting engineering with finite element limit equilibrium method. Based on the limit equilibrium principle, Zhu et al. [16] took the anchor load as the analytical elastic stress distribution in an infinite wedge approximating the slope with the anchor load acting on the apex. And then the normal stress on the slip surface for the anchor-reinforced slope is assumed to be the linear combination of two normal stresses, where one exists before the application of anchor and another is induced by the anchor load. Zhuang et al. [17] compared and analyzed the differences between slice method and finite element limit equilibrium method on the shape and position of slice surface, value of safety factor, and anchoring effect, which are based on a detailed study for anchored slope finite element model.

Finite element limit equilibrium method combines the advantages of limit equilibrium method and finite element method organically, avoiding the controversy caused by using SRM. Therefore, it is widely accepted and applied in stability analysis of natural slopes, embankment and excavation slope, tailings dam, reinforced slope, and research on ultimate bearing capacity of soil structure [18–21] in recent years. Many satisfactory engineering results have been obtained by this method. In this paper, the author uses finite element limit equilibrium method to evaluate the stability of anchored slope directly and to explore some of the key issues. The law obtained in this study can provide reference and experience for correlational studies.

#### 2. Anchoring Slope Stability Analysis Approach

##### 2.1. The Limit Equilibrium Method

According to national standards “Construction Side Slope Engineering Technology Standard,” Sweden arc method is suitable for stability analysis of soil slopes, as shown in Figure 1. Contributions made by anchor structures to antislide force (torque) can be expressed as a single variable discrete function:where is the maximum resistance of the first rows of anchor rod section, is the horizontal spacing of the first rows of anchor rod, is the included angle between the first rows of anchor rod and tangent of arch, and is the friction angle of slice . Considering the effect of anchoring structure, we give out the expression of the safety factor calculation formula of anchoring slope:where is soil weight and surface loads of slice , is the length of slip surface, is the intersection angle between tangent of the first slice arch failure surface and horizontal plane, and is the cohesion force of slice .