Complexity

Volume 2018, Article ID 2829873, 10 pages

https://doi.org/10.1155/2018/2829873

## Rigorous Solution of Slopes’ Stability considering Hydrostatic Pressure

Department of Civil and Architecture Engineering, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu 212003, China

Correspondence should be addressed to Chengchao Li; moc.361@5102ccltsuj

Received 10 December 2017; Revised 23 April 2018; Accepted 8 May 2018; Published 12 June 2018

Academic Editor: Rafał Burdzik

Copyright © 2018 Chengchao Li 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

According to characteristics of soils in failure, a sliding mechanism of slopes in limit state is divided into five parts, for building a slip line field satisfying all possible boundary conditions. An algorithm is built to obtain the rigorous solution approaching upper and lower bound values simultaneously, which satisfies the static boundary and the kinematical boundary based on the slip line field, while stress discontinuity line and velocity discontinuity line are key points. This algorithm is copared with the Spencer method to prove its feasibility with a special example. The variation of rigorous solution, including an ultimate load and a sliding belt the rigid body sliding along rather than a single slip surface for friction-type soils, is achieved considering hydrostatic pressure with soil parameters changing.

#### 1. Introduction

The stability of slopes has been regarded as a classic and difficult problem for engineers because of less boundary constraints, compared with the earth pressure of retaining wall and the bearing capacity of foundation. Over the past few years, many investigators have evaluated slope stability, thereby developing many methods for meeting engineering requirements, such as the limit equilibrium method (LEM) [1, 2], the finite element method (FEM) [3, 4], and the limit analysis method (LAM) [5–7]. The LEM captures the static equilibrium of rigid blocks on a particular slip surface, while not considering the plastic deformation of soils. The equilibrium equation accounts for the whole slice but not guarantees each point in soils. The strength reduction method (SRM) [8–10] is the main finite element slope stability method currently employed, by which stress field and displacement of soils in slopes can be calculated with an elastic-plastic constitutive model to get the safety factor. Although the displacement mutation, the numerical calculation of nonconvergence, and other criteria can determine the slope instability, the slope displacement calculated by the SRM is not the actual displacement of soils and the safety factor required is an approximation. For the slope stability, it is sometimes not necessary to get the variation of stress field and strain field, only to get the ultimate load. Based on the extremum principal [11], the lower bound (LB) [12, 13] solution can be got by static analysis for limit equilibrium problems and the upper bound (UB) [14–17] solution can be got by dynamic analysis. If the lower bound solution satisfies all the kinematic conditions or the upper bound solution satisfies all the static conditions, the solution will be the rigorous one. Compared with obtaining the safety factor, the solution calculated by the upper and lower bound theorems is closer to the real condition because the ultimate load approaches the upper bound solution and the lower bound solution at the same time, satisfying all possible boundary conditions.

When a slope is on the verge of collapse, for type, a sliding belt is emerged within the slope due to the friction between soils. And the sliding belt is not a single slip surface, but a thin shear zone. This paper builds the slip line field satisfying the static boundary condition and the velocity boundary condition according to the characteristics of the stress discontinuity line and the velocity discontinuity line and compiles an algorithm to gain the distribution of the sliding belt and the ultimate capacity of the slope considering the hydrostatic pressure. It also indicates the variation of the ultimate bearing capacity and the sliding belt with different parameters. The application of this algorithm is proved by comparing with the Spencer’s method.

#### 2. Slip Line Field

To analyze the slope stability based on the LAM, the slip line field is constructed according to the stress boundary conditions; one of which is a noncharacteristic line stress boundary with normal stress and shear normal stress, and the other one is the interface of a rigid region and a plastic region or a stress discontinuity surface [18]. Under the limit state, only the noncharacteristic line boundary can be used to construct the slip line field, while the other is unknown. Basic coordinate and noncharacteristic line stress boundary is shown in Figure 1. is the angle between the direction of the maximum principal stress , and the -axis and is the noncharacteristic line stress boundary. is the angle between the tangential direction of the boundary and the -axis, so the angle between and is . According to and , two Mohr’s circles tangent to the Coulomb failure line can be drawn (Figure 2). When soils are in the extreme state, , , and on the boundary are expressed, respectively, as follows: where , , and are normal stress, shear stress, and tangent normal stress on the boundary , respectively. is average stress; is radius of Mohr’s circle; is soil cohesion; and is internal friction angle. is cohesive internal stress, which is given by: .