Mathematical Problems in Engineering

Volume 2017, Article ID 3569826, 6 pages

https://doi.org/10.1155/2017/3569826

## A New Calculation Method for the Soil Slope Safety Factor

^{1}College of Material Science and Engineering, Jilin University, Changchun 130022, China^{2}School of Transportation Science and Engineering, Jilin Jianzhu University, Changchun 130118, China

Correspondence should be addressed to Mingzhi Sun; nc.ude.ulj.sliam@41nuszm

Received 27 February 2017; Revised 25 July 2017; Accepted 6 September 2017; Published 9 October 2017

Academic Editor: Dane Quinn

Copyright © 2017 Fu Zhu 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

Based on the unified strength theory, a new method to calculate the plane soil slope safety factor was derived that considers the effect of intermediate principal stress and at-rest lateral pressure coefficient . Calculation examples from the literature were used to compare the new calculation method and the current slice method; the results showed that both provided good consistency. The new method can provide a reference for slope stability evaluation. The new method was used to calculate the soil slope safety factors for different values of intermediate principal stress parameter , double shear stress parameters , and static lateral pressure coefficient . The results showed that the safety factor increased when was increased; first increased and then decreased when was increased; and increased when was increased. These results show that the intermediate principal stress as well as the stress state and its changes cannot be ignored during soil slope stability analysis. The slope soil characteristics and stress state should be considered to determine the unified strength theoretical parameters and static lateral pressure coefficient, maximize the potential of slope soil strength, and effectively reduce the costs of soil slope engineering.

#### 1. Introduction

In roads, bridges, and construction projects, slope stability problems are often encountered during cutting or foundation pit excavation. Slope instability is due to the destruction of the original soil stress state of equilibrium caused by external forces—such as cutting or foundation pit excavation—and the reduction of soil antishear strength by the influence of various external factors—such as rainwater intrusion and soil freeze-thaw. In practical engineering, slope stability is analyzed to test the reasonableness of the soil slope section design. If the slope is too steep, it will slump easily; if the slope is too gentle, it will increase the amount of earthwork needed.

The characteristics of the Ordinary Method of Slices [1], Bishop’s Modified Method [2], Force Equilibrium Methods (e.g., Lowe and Karafiath [3]), Janbu’s Genralized Procedure of Slices [4], Morgenstern and Price’s Method [5], and Spencer’s Method [6] were summarized in most textbooks. Fall et al. [7] have conducted study on the stability analysis of landslides by the finite element method. Cheng and Yip [8] have shown that rigorous method is necessary to give a reliable estimation of stability of landslides in 3D analyses. Zhu and Lee [9] conducted study on the factor of safety based on Bell’s assumption. Bell’s method was improved by Zheng and Tham [10]. Zheng and Tham’ method can be regarded subsequently as the enhancement of Fellenius’ method.

The slope stability safety factor refers to the ratio of the soil shear strength to the shear stress of a possible sliding surface in the slope. The soil stress state and its changes are prerequisites of slope stability; the existing slope circular slipping method (Petterson 1916) and slice method (Fellenius 1927) ignore the impact of the stress state. In reality, the slope stability changes with changes in the stress state. Researchers [11–14] are currently looking for the sliding center and slip surface, supplementing and modifying the basic assumptions of the slice method, and providing a fundamental basis in engineering applications for the slice method. However, defects of slice method and statically indeterminate problem of this method [15] have produced challenges in practical engineering applications.

Based on multislip mechanism and the model of multishear element, Yu established the unified strength theory which takes into consideration the different contribution of all stress components on the yield of failure of materials [16, 17]. The unified strength theory encompasses the twin shear strength theory [18–20] and single strength theory. The excellent agreement between the predicted results by the unified strength theory and the experiment results indicates that the unified strength theory is applicable for a wide range of stress states in many materials (Ma et al. 1985 [21]).

The coefficient of the earth pressure at rest () is defined as the ratio of the in situ horizontal effective stress to the in situ vertical effective stress. The parameter is needed in the interpretation of laboratory and in situ tests and design of retaining structures and excavation support systems. Schnaid and Yu [22] believe that is an important input parameter for the numerical analyses of geotechnical boundary value problems.

In this study, the perspective of a total stress state was considered to derive a new calculation method for the soil slope safety factor based on the unified strength theory. The slope safety factor was defined by considering the effects of the intermediate principal stress and at-rest lateral pressure coefficient. The method was compared and verified with the current slice method and can provide a reference for stability evaluation in soil slope engineering.

#### 2. Basic Theory and Formula Derivation

##### 2.1. Unified Strength Theory

The Mohr–Coulomb strength theory is simple and practical. It is conducive to engineering applications but does not reflect the effect of intermediate principal stress, and the calculated results are relatively conservative. In 1991, Yu suggested the unified strength theory to compensate for the shortcomings of the Mohr–Coulomb strength theory. The unified strength theory can consider the intermediate principal stress effect of the material and can simulate nearly all materials on the partial plane to develop the material strength potentials. There are two equations with a conditional formula for both the mathematical model and the theoretical expression of the unified strength theory, which considers the different contributions of various stress components to the material yield and destruction, reduces the number of material parameters, and makes the limiting surface to reach the outer boundary of the convex criteria; these cannot be achieved by other criteria. The mathematical modeling methods from the two equations can also be used to solve issues with determining the intermediate principal shear stress. Yu derived a mathematical expression of the unified strength theory using the unified twin shear model and a new mathematical model [17]:where and are yield functions; is the ratio of the tensile to compression strengths of the material; is the tensile strength; and are the cohesion and internal friction angle, respectively, of rock and soil; and is the selected failure criterion introduced in the unified strength theory that also reflects the destructive effects of the intermediate principal shear stress and normal stress of the corresponding surface on the material.

The unified strength theory was converted into a formula similar to the Mohr–Coulomb strength theory in order to obtain the friction angle and unified cohesive force ; these are expressed by the internal friction angle and cohesion as follows [23].

When ,When ,where is the twin shear stress parameter.

The unified internal friction angle and unified cohesion can be used to express the Mohr–Coulomb strength theory:

##### 2.2. Basic Assumptions

(1)The excavated soil is simplified as plane soil slopes.(2)Soil is homogeneous.(3)The stress state can be represented by formula (4).(4)The soil antishear strength satisfies formula (3).(5)The horizontal stress of inner points along the depth causes soil slope instability.(6)During excavation of the soil, the static lateral pressure coefficient remains unchanged.(7)The impact of pore water and groundwater is not considered.

##### 2.3. Formula Derivation

The extending direction of the soil slope is taken as the plane stress state, and elastic half-space plane stress analysis is performed for soil in a steady state under gravity stress; the main stress expression of any point is as follows:where is the maximum principal stress, is the minimum principal stress, is the soil gravity, is the static lateral pressure coefficient of the soil, is the distance from the ground surface to any point, is the horizontal stress at any point, and is the vertical stress at any point.

As shown in Figure 1, assuming that the cutting or foundation soil is not excavated, when slope angle , then . In vertical excavation of the soil, when slope angle , then . The horizontal stress of the soil excavation slope angle satisfies the following formula: