Stress Distribution Characteristics of Retaining Walls Considering Different Rotation Points

Lu Lier¹,²*, Liu Zhijun¹, Sun Guanhua², Shen Haokun³, Lu Yingfa³

¹ School of Civil Engineering and Mechanics, Key Laboratory of Mechanics on Disaster and Environment in Western China, The Ministry of Education of China, Lanzhou University, Lanzhou, Gansu 730000, China
² Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, Hubei 430071, China
³ School of Civil Engineering and Environment, Hubei University of Technology, Wuhan, Hubei 430068, China

Abstract

Based on the characteristics of small deformation theory, this paper proposes a novel method for stability analysis of slope retaining walls under different rotation axis conditions (small deformation theory solution). For the slope body, theoretical solutions for stress distribution are derived based on satisfaction of the stress equilibrium differential equations, compatibility equations, force boundary conditions, and macroscopic equilibrium. For the retaining wall, theoretical solutions for stress distribution under moment equilibrium conditions at different rotation points are obtained using the same methodology, and a point failure criterion is established. A case study of the waste transfer station in Jinguoping Town, Badong County demonstrates that the proposed stress calculation method can determine respective stress solutions for both the slope and retaining wall. The new stability evaluation approach based on point strength criteria can correctly assess the stability state of each point. Years of successful operation of this retaining wall confirm the feasibility of the proposed design analysis method.

Keywords: small deformation theory solution; stress differential equilibrium equation; retaining wall; rotation axis; stress distribution

Introduction

Retaining walls are structures designed to support roadbed fill or hillside soil masses, preventing deformation and instability of the fill or soil body. They can stabilize road embankments and cut slopes, and are frequently used to remediate landslides and other geological hazards. Coulomb's earth pressure theory assumes a planar slip surface, cohesionless backfill soil, and a triangular earth pressure distribution to derive the active earth pressure formula. Rankine's earth pressure theory is based on semi-infinite soil mass assumptions with a rigid wall, vertical and smooth wall back, horizontal fill surface, and triangular pressure distribution, from which theoretical solutions are obtained using limit equilibrium principles. Terzaghi argued that active earth pressure on rigid retaining walls depends on wall movement, with significant differences observed among three distinct movement modes: translation, rotation about the wall top, and rotation about the wall toe. Mao Yisheng explained the fundamental defects in Coulomb's earth pressure theory from a mechanical perspective. Kezdi investigated the rotation mode of retaining walls about the toe. Handy applied the soil arching effect principle combined with finite element methods to study stress distribution in retaining walls.

Current research on retaining walls primarily focuses on earth pressure magnitude and application points, as well as deformation and failure modes of retaining walls. Stability evaluation of retaining walls currently relies mainly on safety factors, which have been applied in engineering practice for many years. However, some retaining walls still fail despite meeting shear and overturning strength requirements. Huang Yuewen noted that traditional theories are overly simplified. With the development of numerical analysis methods, various computational approaches have been proposed. Recently, the partial strength reduction method for analyzing progressive slope failure has been widely applied. References [9-10] propose a new numerical calculation method that can consider different boundary conditions and be generalized for application.

2.1 Basic Method

For any material considering boundary effects, when the geometry is determined, its theoretical solution is definite. As boundary conditions change, the geometry also changes, and the solution changes accordingly. Assuming a continuous medium, the solution must satisfy stress boundary conditions, equilibrium equations, and compatibility equations. When stress distributions are unequal, discontinuous stress solutions can be obtained, thereby addressing stress discontinuity issues during failure processes. The methodology proceeds as follows:

1. Determine the macroscopic geometric characteristics of the research object through precise measurement, and establish geometric characteristic description equations associated with the research object.
2. Analyze the unit weight distribution of the research object and establish unit weight distribution equations.
3. Analyze the stress characteristics of the research object, and establish corresponding boundary condition stress equations for different situations.
4. When the research object satisfies the corresponding equilibrium equations, stress boundary condition equations, and compatibility equations, select the stress equation representation form and calculate the corresponding constant coefficients.
5. Conduct specific analysis of the force characteristics of the research object, and determine its failure characteristics in combination with current strength criteria.

2.2 Two-Dimensional Slope Retaining Wall Example

Applying the above basic concepts, a two-dimensional plane strain retaining wall theoretical solution is studied as an example. For step 1, precise measurement determines the macroscopic geometric characteristics, and geometric characteristic description equations associated with the research object are established as shown in [FIGURE:1]. These can be used to represent the geometric characteristic description equations for boundaries AD, DC, BC, BF, FH, HG, GE, and EA.

For step 2, the unit weight distribution of the research object is analyzed and relevant equations are established. For step 3, the stress characteristics of the research object are analyzed, and stress equations relative to boundary conditions are established for different cases. For plane problems, the horizontal stress ($\sigma_x$) and vertical stress ($\sigma_y$) expressions on boundary AD are:

For step 4, when the research object satisfies the stress differential equilibrium equations and stress boundary conditions, the representation form of the stress equation is selected and the corresponding constant coefficients are calculated. Under two-dimensional conditions, the stress expressions are:

For step 5, when analyzing the force characteristics of the research object, current strength criteria are combined to determine failure characteristics. A constitutive model is selected to study deformation characteristics, which are then compared with field observations to further clarify the associated behavioral features.

3.1 Calculation Unit Division and Equations

Following the derivation of the theoretical stress solution, the slope retaining wall at the Jinguoping Town waste transfer station in Badong County is used as an illustration. A slope model is established, taking ADCBFHGEA in [FIGURE:1] as the research object. Using the complete stress expression form from constant terms to $a_{i,j}$, there are 63 constant coefficients, which can be reduced to 33 constant coefficients through equilibrium equations. The corresponding constants can then be determined based on given different boundary stress conditions.

For Unit I: The stress boundary conditions on DA boundary are zero, and 12 equations can be obtained under strong constraint conditions. The stress boundary conditions on DC boundary are:

On GE boundary, the stress boundary conditions are satisfied, with equation forms identical to those for AE and AD boundaries. However, the EF boundary condition requires equal stress boundary conditions for the sliding mass and retaining wall:

where $\sigma_{xx}^h$, $\sigma_{yy}^h$, $\tau_{xy}^h$ and $\sigma_{xx}^d$, $\sigma_{yy}^d$, $\tau_{xy}^d$ represent the stresses of the sliding mass and retaining wall on the EF boundary, respectively. In this study, the retaining wall is completely connected to the foundation, meaning no failure occurs between the retaining wall and slope mass, and stresses are continuous. The equilibrium equations for retaining wall EFHG are:

Horizontal force equilibrium:
$$
\int_{EF} p_x \, dl + \int_{HF} p_x \, dl = 0
$$

Vertical force equilibrium:
$$
\int_{EF} p_y \, dl + \int_{HF} p_y \, dl + W_{GHEF} = 0
$$

Moment equilibrium equation: For Unit III rotating about its centroidal axis (point $O_3$), where $(X_{O_3}, Y_{O_3})$ are the coordinates of point $O_3$, the moment equilibrium equation under strong constraint conditions is:

Equation (16) provides 18 equations, and the DC and DA boundary conditions provide 30 equations. The slope backfill is clay, and CF is the rock layer dip angle at $47^\circ$. According to traditional assumptions, this surface has already failed, with discontinuous tangential stress but continuous normal stress. The relationship between tangential and normal stresses following the strength reduction method is:

where $\tau_{BC}$ and $\sigma_{BCN}$ are the tangential and normal stresses on BC edge, respectively; $c$, $\phi$, and $f$ are the soil cohesion, friction angle, and strength reduction factor, respectively. Stresses on AB edge are continuous, and the equilibrium equations for sliding mass ABCDA are:

Horizontal force equilibrium:
$$
\int_{AB} p_x \, dl + \int_{BC} p_x \, dl + \int_{CD} p_x \, dl = 0
$$

Vertical force equilibrium:
$$
\int_{AB} p_y \, dl + \int_{BC} p_y \, dl + \int_{CD} p_y \, dl + W_{ABCD} = 0
$$

Moment equilibrium equation: For Unit I rotating about the centroidal axis (point $O_1$) $(X_{O_1}, Y_{O_1})$, the moment equilibrium equation is:

where $(X_{O_1}, Y_{O_1})$ are the coordinates of rotation point $O_1$. For Unit II, the equations are identical to Unit I, but stresses on AB edge must be equal. For the retaining wall (Unit III), a calculation model is established with boundary conditions for the retaining wall model in [FIGURE:1] as follows:

Using the same method as for ADCBA, the corresponding constant coefficients can be obtained.

3.2.1 Project Overview of the Waste Transfer Station

The Jinguoping Town waste transfer station in Badong County is located 192° from Badong County city, Hubei Province, at a straight-line distance of approximately 84.2 km. The project is situated beside the Jinguoping town center, connected by a gravel road. The site covers an area of 1,711.50 m². The waste transfer station platform elevation is 612 m, the retaining wall base elevation is 611-613 m, and the slope height is approximately 10 m (see [FIGURE:2]). The right side of the waste transfer station is backfilled with red clay, forming a cross-section corresponding to the I-I profile. Below the transfer station is moderately weathered sandstone of the T2b2 formation with high strength (uniaxial compressive strength of 40-60 MPa) and a rock dip angle of 47°. The retaining wall foundation rests on a plain concrete clay cushion (C20) and the moderately weathered sandstone.

3.2.2 Calculation Model

Based on the I-I profile of the Jinguoping Town waste transfer station, a calculation model is established. The backfill clay unit weight is 19.35 kN/m³, cohesion is 20 kPa, and friction angle is 20°. The basic dimensions of slope mass ADCFEA are: FE = 5 m, AD = 4.28 m, DC = 0.3 m, CF = 12.35 m, EA = 5.91 m. The retaining wall EFHG is C25 plain concrete with basic dimensions: EF = 5 m, FH = 4.04 m, HG = 5.26 m, GE = 0.8 m. For comparison with current methods, the finite element quadrilateral mesh models of the sliding mass and retaining wall are shown in [FIGURE:2].

3.2.3 Model Boundary Conditions and Failure Criteria

For the calculation model, the boundary conditions are: DC boundary stress conditions are given by equation (12); AD, AE, and GE boundary conditions are:

For GH boundary:

Failure criterion: The Mohr-Coulomb criterion is adopted for stress strength, with point failure as the judgment condition. For both slope mass and retaining wall, based on the calculated principal stresses and using the experimental friction angle values, the corresponding cohesion values at each point are back-calculated. When the back-calculated cohesion exceeds the material test value, that point is judged to have failed. The friction angles for slope mass and retaining wall are taken as:

3.2.4 Stress Calculation Results for Slope and Retaining Wall

For the retaining wall, the elastic modulus is 3,000 MPa and Poisson's ratio is 0.11. Based on the theoretical solution, a calculation model is established to determine coordinates of each point and geometric boundary description equations, thereby defining all boundary conditions. Under a slope safety factor ($f = 1.20$) and centroid rotation conditions, the relevant coefficients of the sliding mass are solved. Substituting these coefficients into equations (3-5) yields the stress component distributions (see [FIGURE:3]), principal stress distributions, and cohesion ($c$) value distributions (see [FIGURE:4]). For the retaining wall, stress component ($\sigma_{xx}$, $\sigma_{yy}$, $\tau_{xy}$), principal stress, and cohesion ($c$) distributions under rotation about F and H axes are shown in [FIGURE:5] through [FIGURE:8].

3.2.5 Comparative Study of Calculation Methods

Given that the rotational equilibrium in the finite element method essentially means zero moment about the centroidal axis, only the calculation results for moment equilibrium along the centroid of the sliding mass and retaining wall can be compared between the finite element method and the proposed method. Under equal moment and boundary conditions, using ANSYS software, the calculation results are obtained. The difference between ANSYS finite element calculation stresses and the stresses calculated by this method is shown in [FIGURE:9]: the deviation between the two methods is less than 8.05%, occurring at the retaining wall base as shear stress.

3.3.1 Analysis of Slope Calculation Results

The horizontal stress in the slope mass is small, with no tensile stress generated. The vertical stress is less than the product of unit weight and depth, and shear stress is also small, less than vertical stress. Stresses on boundaries are not large and satisfy boundary conditions. The corresponding first and third principal stresses are small, with no tensile stress observed. The back-calculated cohesion is small. Under a safety factor of 1.2, all points in the slope mass are basically within the strength range. The calculation results indicate that if the slope mass fails first, the failure point would be located along the rock-soil interface at the elevation corresponding to the retaining wall (see [FIGURE:4]).

3.3.2 Retaining Wall Stability Analysis

This study obtained solutions for the retaining wall rotating about centroidal axis $O_3$, F axis, and H axis. Solution characteristics are: under centroidal axis rotation conditions, the maximum values of $\sigma_{xx}$, $\sigma_{yy}$, $\tau_{xy}$ are 124.83 kPa, 156.78 kPa, and 37.98 kPa, respectively; minimum values are -37.06 kPa, 0 kPa, and -9.86 kPa, respectively; the maximum $c$ value is 40.64 kPa and minimum is 2.442 kPa; the maximum $\sigma_1$ and $\sigma_3$ values are 164.04 kPa and 29.49 kPa, respectively, with minimum values of 12.72 kPa and -37.40 kPa, respectively. The maximum first and third principal stresses and $c$ values are all located at the retaining wall toe. Additionally, the minimum $c$ value occurs at the middle of the retaining wall's free face, coinciding with the minimum principal stress, indicating possible bulging failure in the middle of the retaining wall, but this is far below the value for C20 concrete, making failure unlikely.

Under F-axis rotation conditions, the maximum values of $\sigma_{xx}$, $\sigma_{yy}$, $\tau_{xy}$ are 100.59 kPa, 187.15 kPa, and 75.05 kPa, respectively; minimum values are -38.6 kPa, 0 kPa, and -170.7 kPa, respectively; the maximum $c$ value is 89.82 kPa and minimum is 2.443 kPa; the maximum $\sigma_1$ and $\sigma_3$ values are 314.87 kPa and 27.153 kPa, respectively, with minimum values of 12.70 kPa and -53.07 kPa, respectively. The maximum first and third principal stresses and $c$ values are all located at the retaining wall toe, consistent with the stress distribution results.

Under H-axis rotation conditions, the maximum values of $\sigma_{xx}$, $\sigma_{yy}$, $\tau_{xy}$ are 126.98 kPa, 156.84 kPa, and 36.98 kPa, respectively; minimum values are -36.52 kPa, 0 kPa, and -9.54 kPa, respectively; the maximum $c$ value is 40.26 kPa; the maximum $\sigma_1$ and $\sigma_3$ values are 162.25 kPa and 29.68 kPa, respectively, with minimum values of 2.44 kPa, 12.70 kPa, and -36.80 kPa, respectively. The corresponding locations of maximum first and third principal stresses and $c$ values are all at the retaining wall toe.

These calculation results demonstrate that different rotation points produce different stress distributions with significant variations. For the retaining wall under different rotation point conditions, the maximum and minimum $c$ values are 89.82 kPa and 2.29 kPa, respectively, and the minimum principal stress is -53.07 kPa, all of which are smaller than the values for C20 concrete and moderately weathered sandstone. Therefore, local point failure in the retaining wall is impossible, eliminating the possibility of overall failure. Years of operation demonstrate that the retaining wall is stable.

Conclusions

Based on the boundary conditions presented in this paper, a new numerical method is proposed and applied to analyze the stress and strain distribution characteristics of the clay slope and retaining wall at the Jinguoping Town waste transfer station in Badong County. From the solution characteristics, the following conclusions are drawn:

1) This paper analyzes scientific issues regarding boundary conditions in numerical analysis and proposes a small deformation theory solution. This numerical method can consider the effects of different boundary conditions and rotation axes, with calculated stresses and strains showing nonlinear relationships with coordinates.

2) The proposed numerical theoretical solution can obtain stress distributions at different points based on satisfaction of stress differential equations, compatibility equations, and boundary conditions. For soil masses, based on experimental constitutive characteristics of principal stresses and principal strains, strain distributions in any direction can be obtained. Additionally, using current strength criteria, the location of initial failure can be determined, thereby demonstrating the application of point strength design criteria.

3) The numerical method proposed in this paper clearly shows that solutions differ under different working conditions, thereby determining the most unfavorable condition. This calculation method can provide a theoretical basis for anti-sliding design of retaining walls and other structures, while also providing design references for slope control and monitoring. New control and prevention methods can be derived based on different control forms and materials.

4) The numerical calculation results of this paper are compared with finite element calculation results, showing small deviations of less than 8.05%. The case study demonstrates that the proposed method can be applied in engineering practice. This method can be extended to research on dynamic and static loading/unloading analysis and failure processes of related materials in roadbeds, tunnels, dams, and other structures.

References

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Figure Captions

Figure 1 Slope and retaining wall calculation model
Figure 2 Plan view, profile, and finite element mesh of the waste landfill retaining wall
Figure 3 Stress component distributions of sliding mass and retaining wall at $f=1.20$
Figure 4 Principal stress and cohesion ($c$) distributions of sliding mass and retaining wall at $f=1.20$
Figure 5 Stress component distributions of retaining wall rotating about point F
Figure 6 Principal stress and cohesion ($c$) distributions of retaining wall rotating about point F
Figure 7 Stress component distributions of retaining wall rotating about point H
Figure 8 Principal stress and cohesion ($c$) distributions of retaining wall rotating about point H
Figure 9 Difference between finite element and present method calculation results