Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1985458, 11 pages

https://doi.org/10.1155/2017/1985458

## Upper-Bound Multi-Rigid-Block Solutions for Seismic Performance of Slopes with a Weak Thin Layer

^{1}School of Civil Engineering, Tianjin University, Tianjin 300072, China^{2}Key Laboratory of Coast Civil Structure Safety of Ministry of Education, Tianjin University, Tianjin 300072, China^{3}State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China

Correspondence should be addressed to Haizuo Zhou; moc.361@ybborzhz

Received 17 July 2017; Accepted 22 October 2017; Published 26 November 2017

Academic Editor: Giovanni Garcea

Copyright © 2017 Gang Zheng 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 presence of a weak layer has an adverse influence on the seismic performance of slopes. The upper-bound solution serves as a rigorous method in the stability analysis of geotechnical problems. In this study, a multi-rigid-block solution based on the category of the upper-bound theorem of limit analysis is presented to examine the seismic performance of nonhomogeneous slopes with a weak thin layer. Comparison of the static factors of safety is conducted with various solutions (i.e., limit analysis with a different failure mechanism, limit equilibrium solution, and numerical method), and the results exhibit reasonable consistency. An analytical solution in estimating the critical yield acceleration coefficient is derived, and the influence of slope angle, slope height, and soil strength on the critical yield acceleration coefficient and failure mechanism is analyzed. Subsequently, Newmark’s analytical procedure is employed to evaluate cumulative displacement with various real earthquake acceleration records as input motion. Results show that the strength and geometric parameters have a remarkable influence on the critical yield acceleration coefficient, and the cumulative displacement increases with the increasing slope angle.

#### 1. Introduction

Extensive investigations have been conducted for the stability of slopes with homogeneous soil. However, the presence of a weak thin layer occurs often in practical engineering. This characteristic requires special attention from engineers with regard to the low shear strength of the weak layer, which has an adverse influence on the performance of slopes. Varying methodologies have been employed to estimate the static stability of slopes with a thin layer, including the limit equilibrium method (Fredlund and Krahn [1]), the upper-bound solution method (Huang and Song [2]), and finite element analysis (Griffiths and Marquez [3]; Ho [4]). However, few studies have focused on the seismic stability of nonhomogeneous slopes.

Many catastrophic slope failures have occurred because of earthquakes, such as the 2004 Chuetsu earthquake and the 2008 Wenchuan earthquake, highlighting the importance of addressing the seismic stability of slopes in geotechnical engineering practices. Once the damage of slopes occurs in urban area, a great number of economic losses may occur. The factor of safety of a slope, estimated using a pseudostatic approach, and the cumulative displacement, determined by adopting Newmark’s sliding block method [5], are two commonly used tools to evaluate the seismic stability of slopes. The former provides a simple solution to evaluate the static stability (e.g., Seed et al. [6], Seed [7], and Chen [8]). The latter commonly adopts Newmark’s sliding technique in estimating the cumulative displacement, which provides detailed information for the earthquake process. Compared with the pseudostatic approach, which underestimates the seismic stability (Ling et al. [9]; Michalowski [10]), Newmark’s sliding technique helps to accurately evaluate the stability of slopes and can precisely determine effective reinforcements. Therefore, this method has recently been widely employed to slopes with a single layer of homogeneous soil, with and without reinforcements (Chang et al. [11]; Ling et al. [9]; Ling and Leshchinsky [12]; Michalowski and You [13]; Li et al. [14]; He et al. [15]).

The extensions of the upper-bound solution to solving geotechnical problems have been explored by Chen [8]. The multi-rigid-block upper-bound solution is advantageous because it is conceptually clear and easily adopted and it satisfies the Mohr-Coulomb yield criterion (Michalowski [16]; Huang and Qin [17]; Huang and Song [2]). In this study, a three-rigid-block “classroom example” is first presented to show how to calculate the cumulative displacements, and a multi-rigid-block failure mechanism is then developed. The proposed failure mechanism is validated using previous studies with respect to the static factor of safety. The critical yield acceleration coefficient of an earth slope is evaluated by employing the pseudo-static method within the limit analysis framework. The influence of strength and geometric parameters on the critical yield acceleration coefficient is discussed. Subsequently, Newmark’s analytical approach is employed to assess the cumulative displacement by considering different real earthquake acceleration records as input motion.

#### 2. Upper-Bound Theorem

The method adopted is based on the kinematical theorem of limit analysis. The upper-bound theorem states that the rate of work done by external forces is less than or equal to the rate of energy dissipation in any kinematically admissible velocity field (Drucker et al. [18]). It can be expressed asThe first term on the left-hand side in (1) is the work rate of the unknown distributed load on the loaded boundary moving with the given velocity . The second term on the left-hand side represents the work rate of the given distributed forces in the kinematically admissible velocity . The right-hand side is the rate of the internal work integrated over the entire volume of the collapse mechanism.

Based on upper-bound solution, Michalowski and Drescher [19] proposed a class of three-dimensional rotational failure mechanism for the static stability of homogeneous slopes. This failure mechanism was further adopted for analyzing the static stability of slopes with reinforcement (Gao et al. [20]; Zhang et al. [21]; Yang et al. [22]). Different from the rotational failure mechanism for homogeneous slopes, the slip surface passes along the weak thin layer when a weak layer exists (Griffiths and Marquez [3]; Ho [4]) because the weak layer governs the failure mechanism. Huang et al. [23] originally developed a rotational-translational mechanism, which contains three rigid blocks, for a slope with a weak layer, as shown in Figure 1. The velocity of block is , and the angular velocities of block and block are and , respectively. An elaborate effort is required as the velocity discontinuities are bent at the interface between adjacent blocks. Zhou et al. [24] proposed a translational failure mechanism to examine the same issue. The rigid rotational blocks and in Huang et al.’s study were replaced by continuous deformation regions and , including a sequence of rigid triangles.