An analytical method for determining the stresses and deformations of landfills contained by retaining walls is proposed in this paper. In the proposed method, the sliding resisting normal and tangential stresses of the retaining wall and the stress field of the sliding body are obtained considering the differential stress equilibrium equations, boundary conditions, and macroscopic forces and moments applied to the system, assuming continuous stresses at the interface between the sliding body and the retaining wall. The solutions to determine stresses and deformations of landfills contained by retaining walls are obtained using the Duncan-Chang and Hooke constitutive models. A case study of a landfill in the Hubei Province in China is used to validate the proposed method. The theoretical stress results for a slope with a retaining wall are compared with FEM results, and the proposed theoretical method is found appropriate for calculating the stress field of a slope with a retaining wall.
Retaining walls are typically used to support subgrade or sloped fills, stabilize embankments, and prevent deformation failures, or reduce the height of sloped excavations. In order to mitigate the failure risk of a sliding slope supported by a retaining wall, an effective protection solution must be employed to ensure the stability of the slope [
The slope failure mechanism and stability analysis are traditional topics in geotechnical engineering. Previous studies have proposed many equilibrium limit stability calculation methods, such as the Fellenius method, simplified Bishop method, Spencer method, Janbu method, transfer coefficient method, Sarma method, wedge method, and finite element strength reduction method (SRM) [
The current studies on slopes and retaining walls mainly focus on the earth pressure calculation and stability evaluation under some assumptions. Based on the classical theories of earth pressure, the sliding and overturning stability, and safety evaluation of retaining walls and slopes, a new method of force and displacement analysis of landfills contained with retaining walls is proposed in this article. The method has the following characteristics:
A theoretical solution for the stress at any point of the landfill confined with a retaining wall can be obtained, considering the corresponding boundary conditions, but ignoring the critical state assumptions. The Duncan-Chang and Hooke constitutive models are used to obtain the strains and displacements in landfills confined with retaining walls. The proposed method can not only be used to study the overturning and sliding failure modes but the tensile and bulging failure modes of retaining walls and landfills as well. The overall internal failure evaluation of retaining walls and landfills can be carried out. The normal and tangential stresses produced by the lateral earth thrust acting on the retaining wall can be taken into account. The method provides a theoretical basis for slope design. Different retaining wall forms and materials can be adopted for different stress distributions, which can lead to economic, rational, and effective designs.
The conventional method for calculation of lateral stress for gravity retaining walls with rigid foundations and homogeneous cohesionless fills is introduced first.
The active earth pressure acting on the retaining wall can be calculated for a homogeneous cohesionless backfill in
When the backfill top surface behind the wall is inclined (see
The stress at the base of a retaining wall can be calculated in
The stresses of the retaining wall founded on hard rock should meet the following criteria:
The maximum retaining wall base stress should not be greater than the allowable bearing capacity of the hard rock; and Except for the construction period and under an earthquake excitation, there must be no tensile stress on the base of the retaining wall.
The safety factor against sliding of the retaining wall along the rock base is calculated according to
When the retaining wall back surface is inclined towards the direction of the backfill, the safety factor along the surface between the retaining wall and rock can be calculated according to
The factor of safety against overturning of a retaining wall is calculated according to
According to the above formulas, the sliding stress is produced by the active earth pressure, and the resisting stresses include the compressive and shear strength at the wall base. However, the above results do not consider the tensile failure of the retaining wall. This paper presents a novel method of calculating the stresses and strains for a retaining wall.
When the shape of an object is determined, the stress solution must be clear and be updated with the boundary condition changes. Assuming the stresses are continuous, the solutions must satisfy the differential stress equilibrium equations and deformation boundary conditions. This method can be used to find the stress distribution for any geometry, including two-dimensional (2D) and three dimensional (3D) cases. When the boundary conditions and stress field are discontinuous, the discontinuous stress and displacement solutions can also be obtained. The landfill with a retaining wall is taken as an example to illustrate the basic ideas and approaches.
In this study, a theoretical solution for an arbitrary polygon under a 2D plane strain problem is studied. The landfill and the retaining wall are the polygon ABCDP and the quadrilateral element BCFE, respectively. The analysis steps are as follows:
(1) The boundaries of the analyzed system are assumed (see
(2) The specific gravity (
(3) Using the stress field characteristics, the stress boundary conditions can be formed. The normal stresses at the transition between the landfill (i.e.,
(4) The landfill must satisfy the equilibrium equations, the stress boundary conditions, and the deformation equations. The expressions for the stress equilibrium equations are written and their coefficients are calculated. The stress expressions are assumed for a 2D landfill (note the stress expressions can be changed for different conditions) in
The number of constant coefficients in
The following differential stress equilibrium equations,
The corresponding coefficients are zero at any point with a constant unit weight (
Boundary segment AB of polygon ABCDP can be expressed mathematically in
The stress conditions on boundary AB are in
Once the landfill is in a balanced and stable state, the force equilibrium of polygon ABCDP in X- and Y-direction are in
The expressions for
The equation of boundary segment CD is in
The expressions for
By substituting
The normal force acting on segment CD can be obtained by integration in
The normal forces acting on segments BC and PD can be derived in a similar way. The equation for segment BC is in
The resulting normal forces acting on segments BC and PD are in
The shear stresses between the landfill and the sliding bed are discontinuous. The frictional stresses along segment CD can be taken as the residual stress due to the sliding body with the landfill and expressed in
By integrating
Simplifying
The equation for segment PD is
The stresses between the landfill and the retaining wall are continuous. A similar method is adopted to calculate the shear stress along segment BC in
The weight (
The equation of boundary segment AP is in
The shear stress resultant along segment AP can be assumed zero according to the Saint Venant principle shown in
If coefficients
The stresses acting on segments BC and CF of the retaining wall have been obtained. This section derives the stresses for the retaining wall assuming stress continuity between the landfill and the retaining wall. The coordinates used to analyze the retaining wall are shown
Stress equilibrium is assumed along segments BC and CF based on the continuity of stresses between the landfill and the retaining wall and between the retaining wall and its base.
The equation for segment BC is
In
By combining
The stresses acting on the retaining wall are defined in the
The forces acting on the retaining wall include the normal and tangential forces along segment BC (
In
The calculation statements are presented in the following form:
The normal force acting on boundary segment BC of the retaining wall is obtained by integrating
The normal stress on boundary segment CF is in
In
The normal force acting on boundary segment CF of the retaining wall is obtained by integrating
A similar approach is adopted for calculating the tangential forces along segments BC and CF of the retaining wall in
The tangential force acting on boundary segment BC can be achieved by integrating
The shear stress acting on segment CF is in
The tangential force acting on boundary segment CF is found by integrating
The barycentric coordinates of triangles BEC and EFC are denoted as I and J, respectively. The barycentric coordinates (
The weight per unit thickness of the retaining wall is
Point
Moments
The moment balance equation for the retaining wall can be written in
The lever arms of moments
The stresses at point E
Therefore, we have
The 18 coefficients
The Duncan-Chang constitutive model is employed to describe the strain distribution in the landfill. The basic equations are
The expressions for strains (
The retaining wall can be assumed to be a plane strain problem, i.e.,
A case study of a landfill project located in Fengjiadagou of Guandukou Town of Badong County in Hubei Province of China is studied. The national road No. 209 passes to the west side of the landfill. The landfill area is about 2.1 × 104 m2 and the effective waste storage capacity of the landfill is 10.5 × 104 m3. The daily average processing capacity of the landfill is 230 kN/d for 5 years. Currently, the landfill is closed (see
The elevation of the landfill back side is 262 m and the elevation of its front side at the top surface of the retaining wall is 225 m. The sloped length of the landfill is about 54.3 m and its width is 37 m with a slope angle of about 20° (see
The analytical model was established according to profile I-I of the Guandukou Town landfill (see
The dimensions and angles of the landfill ABCDP are as follows: AB = 54.3 m, AP = 1.3 m, PD = 26.9 m, DC = 28.8 m, BC = 4.3 m,
The dimensions and angles of the retaining wall are as follows: EB = 1.2 m, BC = 4.3 m, CF = 2.4 m, FE = 4.1 m and
Eighteen coefficients were determined for the landfill solutions under the condition that the force boundary conditions and stress differential equilibrium equations are satisfied. Their values for the stress field (see
The parameters of the Duncan-Chang constitutive model are as follows:
The unit weight of the retaining wall was assumed as
The stress and principle stress fields in the retaining wall are presented in
The obtained stress and strain fields in the landfill are logical. Stresses
The conventional factor of safety against sliding (i.e.,
From the results of the proposed method, it can be seen that the maximum tensile stress is located at point B (78.2 kPa). This value is lower than the strength of M15 mortar and block stone used in the project. The maximum compressive stress (1005 kPa) is located at point F of the retaining wall, which is less than the strength of the retaining wall material. At any point in the retaining wall, the cohesion intercept (C) value is less than 910 kPa (see
The ANSYS finite element software was also employed to study the stresses and strains within the landfill and the retaining wall (see
Based on the traditional analysis of the retaining wall (the limited rigid body balance method and FEM) and the proposed methods in this paper, the studied retaining wall is in a stable condition.
The stress and strain field characteristics of the landfill and the retaining wall have been determined. The following conclusions can be drawn: The stress fields in the landfill and the retaining wall are non-linear within the domain. The analytical method proposed in this article can provide a theoretical basis for the control design and displacement prediction of a slope and a retaining wall. Based on the geometry and material of the retaining wall, novel methods for preventing retaining wall failures can be proposed and validated. The analytical solutions presented in this article are based on the assumption that the stresses are continuous or discontinuous. It is acceptable that the results of the proposed method in this paper are comparable to that of the finite element method under the given boundary conditions. In this article, the limit equilibrium state hypothesis was ignored for the retaining wall. The stresses acting on the retaining wall included normal stress and the shear stress. The design methodology of the retaining wall was explained using the results of the numerical analysis. According to the analytical results, the tensile and bulging failure characteristics of the retaining wall and the landfill can be determined. The internal failure of the retaining wall and the landfill can be conducted at any point within the domain, and therefore a new stability analysis method for the anti-sliding design was proposed.
The moment analysis is presented below; for force lever arms see
Segment BC:
Any point Q is chosen, X’ coordinate is between X’C and X’G, and the lever arm of
Any point S is chosen, X’ coordinate is between X’G and X’B, and the lever arm of
The lever arm of
Segment CF:
(1) Any point V is chosen, X’ coordinate is between X’C and X’P, and the lever arm of
The equation of straight line CF is
(2) Any point X is chosen, X’ coordinate is between X’P and X’F, and the lever arm of
The equation of straight line CF is
The lever arm of