Mathematical Problems in Engineering

Volume 2016, Article ID 1527659, 10 pages

http://dx.doi.org/10.1155/2016/1527659

## Displacement of Pile-Reinforced Slopes with a Weak Layer Subjected to Seismic Loads

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

Received 30 May 2016; Accepted 14 August 2016

Academic Editor: Giovanni Garcea

Copyright © 2016 Haizuo Zhou 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 in a slope requires special attention because it has a negative impact on slope stability. However, limited insight into the seismic stability of slopes with a weak layer exists. In this study, the seismic stability of a pile-reinforced slope with a weak thin layer is investigated. Based on the limit analysis theory, a translational failure mechanism for an earth slope is developed. The rotational rigid blocks in the previous rotational-translational failure mechanism are replaced by continuous deformation regions, which consist of a sequence of rigid triangles. The predicted static factor of safety and collapse mechanism in two typical examples of slopes with a weak layer compare well with the results obtained from the available literature and by using the Discontinuity Layout Optimization (DLO) technique. The lateral forces provided by the stabilizing piles are evaluated using the theory of plastic deformation. An analytical solution for estimating the critical yield acceleration coefficient for the pile-reinforced slopes is derived. Based on the proposed translational failure mechanism and the corresponding critical yield acceleration coefficient, Newmark’s analytical procedure is employed to evaluate the cumulative displacement. Considering different real earthquake acceleration records as input motion, the effect of stabilizing piles and varying the spacing of piles on the cumulative displacement of slopes with a weak layer is investigated.

#### 1. Introduction

Many catastrophic slope failures have been reported in the past due to earthquakes. Estimation of the stability of slopes subjected to seismic loads is a very important task in geotechnical engineering. There are two practical estimation methods that exist for this problem: the first one is calculating the factor of safety of slopes by considering pseudostatic earthquake body forces within a soil mass (e.g., Seed et al. [1]; Seed [2]; Chen [3]). The concept of factor of safety has been widely used because it is simple and straightforward extension of static considerations. However, it provides no details regarding the shaking process. On the other hand, the technique of slopes stabilized by reinforcement is widely employed by geotechnical engineers to enhance the stability of slopes. The pseudostatic approach may underestimate the stability of reinforced slopes for large earthquake acceleration in engineering design (Ling et al. [4]; Michalowski [5]).

The second one calculates the cumulative displacement subjected to seismic loads. The most common approach is Newmark’s [6] sliding block method, which calculates the cumulative displacement of slopes by integrating earthquake acceleration in a one-block translational mechanism. This approach has the advantage of providing information during an earthquake and being less time consuming; it has been further extended to the rotational mechanism (e.g., Chang et al. [7]; Li et al. [8]) and multiblock mechanism (e.g., Michalowski [9]) of slope through limit analysis. Moreover, this approach has been employed for the seismic displacements of reinforced slopes by other researchers (e.g., Ling et al. [4]; Ling and Leshchinsky [10]; Michalowski and You [11]; He et al. [12]).

In engineering practice, the existence of a weak layer in slopes requires special attention because the low shear strength of a weak layer has an adverse effect on the performance of a slope. The previous studies on slopes with a weak layer were performed under the static condition. Fredlund and Krahn [13] compared several methods for analyzing the stability of a nonhomogeneous soil slope with a weak layer. Moreover, finite element (FE) analysis, incorporated with the shear strength reduction technique, was applied to investigate the stability of a slope with a weak layer (Griffiths and Marquez [14]; Ho [15]). Based on the upper-bound method, Huang et al. [16] proposed a rotational-translational collapse mechanism to assess the factor of safety of slopes with a weak layer. Their analytical results were verified using the finite element method. However, few studies have been conducted regarding evaluating the seismic performance of slopes with a weak layer.

In this study, a translational failure mechanism is developed to evaluate the stability of slopes with a weak layer, which is validated using other solutions with respect to the static factor of safety. Furthermore, a pseudostatic method is employed within the limit analysis framework to calculate the critical yield acceleration coefficient of a reinforced slope. Finally, considering different real earthquake acceleration records as input motion, Newmark’s analytical approach is used to assess the cumulative displacement of two typical cases of slopes with a weak thin layer.

#### 2. Critical Yield Acceleration Coefficient for Pile-Reinforced Slopes with a Weak Layer

In limit analysis theory, soil is assumed to deform plastically according to the normality rule associated with the Mohr-Coulomb yield criterion (e.g., Lu et al. [17]). For homogeneous slopes, the rotational log-spiral failure mechanism has been found to be the most adverse for the stability of slopes (e.g., Chen [3]; Chang et al. [7]; Li et al. [8]; Gao et al. [18]; Gao et al. [19]). When a weak layer exists (the strength of a layer is relatively weak) in a slope, the slip surface of the soil slides along the weak layer (Griffiths and Marquez [14]; Huang et al. [16]). Therefore, the weak layer governs the failure mechanism and the conventional rotational log-spiral failure mechanism is not suitable. Farzaneh et al. [20] developed a rotational-translational mechanism to solve the bearing capacity problems. Recently, Huang et al. [16] proposed 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 , while the angular velocities of block and block are and , respectively. Every point on discontinuous surfaces FN and O’M should satisfy the kinematically admissible velocity. Thus, computational effort is needed to determine those discontinuous surfaces, which might become an issue when deriving upper-bound limit solution in seismic analysis.