Mathematical Problems in Engineering

Volume 2018, Article ID 1096513, 11 pages

https://doi.org/10.1155/2018/1096513

## Impact of Crack on Stability of Slope with Linearly Increasing Undrained Strength

^{1}Key Laboratory of Concrete and Prestressed Concrete Structures of the Ministry of Education, Southeast University, No. 2, Sipailou, Nanjing 210096, China^{2}Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University, No. 1, Xikang Road, Nanjing 210098, China

Correspondence should be addressed to Fei Zhang; nc.ude.uhh@gnahzief

Received 13 September 2017; Revised 22 February 2018; Accepted 18 March 2018; Published 30 April 2018

Academic Editor: J.-C. Cortés

Copyright © 2018 Bing 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

This paper presents a procedure for assessment of the impact of tension crack on stability of slope in clays with linearly increasing undrained strength. The procedure is based on the limit equilibrium method with variational extremization. The distribution of the normal stress over slip surface is mathematically obtained for slopes in clays with the linearly increasing undrained strength and then used to determine the tension crack for clays with zero tensile strength. The seismic effect is also included using the pseudostatic approach. Closed-form solutions to the minimum safety factor and the maximum crack depth can be derived and given in the form of chart for convenient use. The results demonstrate a significant effect of the tension crack on the stability of steep slopes, especially for strong seismic conditions. In this situation, neglecting the impact of tension crack in traditional analyses may overestimate the slope safety. The most adverse location of the tension crack can be also determined and presented in the charts, which may be useful in designing reinforcements and remedial measures for slope stabilization.

#### 1. Introduction

Slopes in clays are usually under short-term undrained conditions, such as slopes at the end of construction, slopes subjected to earthquakes, and rapid excavations. In these situations, the traditional limit equilibrium (LE) analysis is carried out to assess the stability of slopes. The pore water pressure is not considered in the total stress analysis and using the undrained strength of clays can determine the factor of safety of slopes. Taylor [1] first adopted an average constant shear strength for clays () into the undrained LE analysis and presented a classical stability chart for slope stability assessment. Through experimental observations, the normal consolidated clays exhibit a linear increase of the undrained shear strength with depth aswhere is undrained strength at the ground surface and is gradient at which undrained strength increases with depth . Gibson and Morgenstern [2] adopted the linear increasing undrained strength with into the analysis and then established an expression to calculate the factor of safety. Hunter and Schuster [3] then extended this expression to account for . Similar problems for stability assessment were further addressed by many investigators [4–8]. Based on the theory of soil plasticity, Booker and Davis [9] and Chen et al. [10] proposed kinematical approach of limit analysis (LA) to obtain the upper-bound solution for this slope problem. Yu et al. [11] used finite-element LA method to calculate the lower- and upper-bound solutions and compared with LE results from the analysis. Their solutions were given in the form of charts for convenient use in practice. The method is also used to assess stability of embankments over soft ground. Leshchinsky and Smith [12] and Low [13] adopted the LE method to calculate the factor of safety of embankments constructed on the soft clay. The undrained strength of the clay is linearly increasing with the depth. Chai et al. [14, 15] employed the finite-element (FE) method to back analyze the failure of an embankment on clay deposit.

Based on the analysis of slope stability, Nakase [17] extended it into stability analysis of low embankment on cohesive soil stratum. A vertical tension crack is involved to obtain more critical circular slip surface, but the depth of the crack is assumed. Actually, many investigations after slope failures indicate that tension cracks often occur at the crest of slopes in clays. Baker and Leshchinsky [18] utilized the safety map proposed by Baker and Leshchinsky [19], to explore the spatial distribution of safety factors in in a vertical purely cohesive cut. Their derived results demonstrate that the tension crack would be formed in vertical cut with zero tensile strength of soil and result in a significant decrease on the factor of safety by 70%. Some efforts have been made by Baker [20], Utili [21], and Michalowski [22], to investigate the impacts of the tension crack on slope stability. However, their studies are focused on stability of slopes under drained conditions. The purpose of this paper is to include the effects of tension crack into stability analyses of undrained slopes. The undrained strength of clays is assumed to linearly increase from the slope crest. Based on the variational analysis of slope stability by Baker [20], a procedure is developed to obtain a closed-form solution for evaluating the effects of tension crack on the slope stability.

#### 2. Variational Analysis of Clay Slope Stability with Tension Crack

##### 2.1. Definition and Formulation of Problem

Figure 1 illustrates a slope in clay with linearly increasing undrained strength and its corresponding potential the slip surface. The LE formulation follows the notations presented by Baker [20] and Leshchinsky and San [23]. A brief description of the relevant formulae is given here for clarity. For convenience of presentation of results, the following nondimensional parameters are introduced first (see Figure 1):where , , and represent the equation of the slip surface, the slope surface, and the top surface; and are the depth of the tension crack and its horizontal distance on the crest from the slope; is normal stress along the slip surface; is tensile strength; is unit weight; is slope height; is cohesion coefficient; is reduction factor of soil strength. Using these nondimensional parameters, the force or moment equilibrium equations can be obtained for the soil mass bounded by the slip surface and the soil surface. To account for the seismic effects, the pseudostatic approach is adopted here. Similar to the procedure of Baker [20], the horizontal force, vertical force, and moment equilibrium equations are given for a slope in clay with linearly increasing undrained strength aswhere and are the horizontal and vertical seismic acceleration coefficient, respectively. There are an unknown parameter and two unknown functions and in these equations. The unknown parameter can be explicitly expressed through any one of the three equilibrium equations ((3a), (3b), (3c)), but the two equations left must be satisfied simultaneously. Such a problem can be solved as a standard isoperimetric problem, to render the minimum value of (i.e., the factor of safety ). As presented by Baker and Garber [24], using the variational calculus principles can find a pair of functions and to determine the factor of safety. The details of the variational derivation can be found elsewhere [20, 24, 25]. To find the functions, an auxiliary functional is defined here:where parameters and are Lagrange’s undetermined multipliers. Based on the variational theorem of the isoperimetric problem, the functions and which minimize the functional must satisfy the following two Euler differential equations: