Shock and Vibration

Volume 2016 (2016), Article ID 4158785, 10 pages

http://dx.doi.org/10.1155/2016/4158785

## Study on the Seismic Active Earth Pressure by Variational Limit Equilibrium Method

^{1}College of Civil Engineering, Chongqing University, Chongqing 400045, China^{2}Key Laboratory of New Technology for Construction of Cities in Mountain Area, Chongqing University, Ministry of Education, Chongqing 400045, China

Received 8 January 2016; Revised 19 April 2016; Accepted 8 May 2016

Academic Editor: Sergio De Rosa

Copyright © 2016 Jiangong Chen 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

In the framework of limit equilibrium theory, the isoperimetric model of functional extremum regarding the seismic active earth pressure is deduced according to the variational method. On this basis, Lagrange multipliers are introduced to convert the problem of seismic active earth pressure into the problem on the functional extremum of two undetermined function arguments. Based on the necessary conditions required for the existence of functional extremum, the function of the slip surface and the normal stress distribution on the slip surface is obtained, and the functional extremum problem is further converted into a function optimization problem with two undetermined Lagrange multipliers. The calculated results show that the slip surface is a plane and the seismic active earth pressure is minimal when the action point is at the lower limit position. As the action point moves upward, the slip surface becomes a logarithmic spiral and the corresponding value of seismic active earth pressure increases in a nonlinear manner. And the seismic active earth pressure is maximal at the upper limit position. The interval estimation constructed by the minimum and maximum values of seismic active earth pressure can provide a reference for the aseismic design of gravity retaining walls.

#### 1. Introduction

The magnitude and distribution of active earth pressure on the retaining wall under the seismic loading are the theoretical premises of the aseismic design for the retaining wall and play a vital role in evaluating the stability of the retaining wall in the seismic area. The common calculation method of seismic active earth pressure is the Mononobe-Okabe (M-O) theory which is based on the Coulomb theory and believes that the sliding soil wedge is a rigid body under the seismic loading. The pseudo-static approach is adopted to simplify the seismic force into an inertia force acting on the sliding soil wedge and transcribe the dynamic problem as a static problem [1]. The M-O theory assumes that the backfill behind the wall is cohesion-less soil and the slip surface is a plane, and the theory cannot obtain the real action point of resultant force without considering the equation of moment equilibrium; thus, restrictions are formed on this theory. The M-O theory has been improved by a number of scholars to expand its application scope [2–6].

Both the M-O theory and the improvement method based thereon are on the basis of the assumption that the slip surface is a plane, which does not conform to the practical situation. One rigorous method in math is the variational limit equilibrium method, by which the seismic active earth pressure on the retaining wall is attributed as the functional extremum problem of two undetermined functions. One undetermined function is expressed by the shape of the slip surface, while the other is the function of the normal stress distribution on the slip surface. They are numerically solved by the variational method.

The variational limit equilibrium method was first proposed by Kopáscy [7–9]. Then it was introduced to the stability analysis of slope and foundation [10–22]. And some scholars utilized the variational limit equilibrium method to study the lateral earth pressure on the retaining wall [23–26]. According to the variational method, Shaojun studied the shape of the slip surface of the sliding soil wedge behind the retaining wall and obtained the analytical solution of the shape of the slip surface and the magnitude of seismic active earth pressure [27]. However, only the cases where the retaining wall is vertical and the backfill surface is horizontal without surcharge are considered in the calculation models, while the influences of wall-movement modes of the retaining wall on the magnitude and the action point position of seismic active earth pressure fail to be taken into account. In fact, the distribution of earth pressure on the retaining wall is nonlinear. The magnitude and the action point position of seismic active earth pressure depend on the coordinated deformation of soil-wall contact surface and vary with the change of wall-movement modes of the retaining wall [26]. However, in the design process, it is often hard to accurately estimate the wall-movement modes of the retaining wall. For the static and dynamic ultimate load acting on the retaining wall, a reasonable approach is to contain the seismic active earth pressure under different wall-movement modes in a certain range as possible for the engineering designers to select and use. In this paper, the variational limit equilibrium method is used to study the seismic active earth pressure on the gravity retaining wall under general conditions (the retaining wall is inclined and coarse; the backfill is cohesive soil; the backfill surface is a curved surface with nonuniform surcharge). The interval of the seismic active earth pressure under different wall-movement modes can be effectively estimated by the proposed approach.

#### 2. Variational Analysis of Seismic Active Earth Pressure

##### 2.1. Basic Assumption

(1) The research problem is a plane strain problem; (2) the soil behind the wall is Coulomb material, which can be represented by the intensity parameter cohesion and the internal friction angle ; (3) when the backfill soil is in the critical active state, a sliding soil wedge is formed and its slip surface passes through the wall heel; (4) the retaining wall is rigid and its motion forms are unconstrained, wherein the motion displacement can be ignored compared with the wall height; (5) the seismic action is simplified to static load acting on the sliding wedge, with horizontal seismic coefficient and vertical seismic coefficient .

##### 2.2. Limit Equilibrium Equation of Sliding Soil Wedge

The calculation model of active earth pressure under the seismic loading is shown in Figure 1, wherein the height of the retaining wall is ; the retaining wall is inclined and coarse; the slope angle of the wall to vertical is ; the friction angle between soil and wall is ; the unit weight of soil is ; the cohesion is ; the internal fraction angle is , the expression of the backfill surface is , and the expression of the slip surface is ; the expression of the vertical surcharge distribution on the backfill surface is , the expression of the tangential stress distribution on the slip surface is , and the expression of the normal stress distribution on the slip surface is ; the seismic soil pressure on the wall is .