A New Method for Evaluating the Stability of Retaining Walls

Abstract

The existing stability analysis of Coulomb retaining walls is derived on the basis of four assumptions, and there is no clear description of the interaction mechanism among wedge-shaped bodies, retaining walls, and the two. This article proposes a new method for calculating the stability of Coulomb retaining walls. For soil wedges and retaining walls, numerical theoretical solutions for the stress distribution in soil wedges and retaining walls were obtained on the basis of stress balance differential equations, coordination equations, force boundary conditions, and macroscopic equilibrium. The boundary condition between the soil wedge and the retaining wall is that the resultant force and moment between the two are continuous. Assuming that the soil wedge and retaining wall satisfy the Duncan–Zhang model and the linear elastic Hooke constitutive model, respectively, the strain solutions of the soil wedge and retaining wall are obtained. Assuming that both the peak strength criteria for the soil wedges and the concrete retaining walls satisfy the Mohr–Coulomb criterion, the location of the first point of failure for the soil wedges and retaining walls is determined. Taking the garbage transfer station in Lvcongpo Town, Badong County, as an example, the analysis of the force and displacement of the retaining wall and years of operation show that the proposed method for calculating the stress and strain of the retaining wall and the new method for evaluating the stability of the retaining wall based on the point strength criterion are feasible.

1. Introduction

Retaining walls are structural elements designed to support roadbed fill or hillside soil, thereby preventing deformation and instability. They serve to stabilize embankments and road cut slopes, reduce the height of excavation slopes, minimize earthwork excavation volumes and land occupation, protect the toes of roadbed slopes, prevent scouring of the roadbed by water flow, inhibit slippage of slope coverings, and increase the overall resistance of slopes to damage. In addition, they are frequently employed to address disasters, such as collapses and landslides [1,2,3].

The calculation method of the active earth pressure on retaining walls is a classic topic in the field of soil mechanics [4,5,6]. To date, accurately calculating the active earth pressure is still difficult. For example, determining the magnitude and position of active earth pressure and the deformation and failure mode of retaining walls requires assumptions about the slip surface, so established calculation models are generally limited in applicability [7,8,9]. Since the establishment of the Coulomb and Rankine soil pressure theories, scholars have developed various improved theories. Lai F et al. [10] proposed an analytical framework for estimating the active soil pressure of narrow fill retaining walls. By combining the arch-shaped differential element method and the sliding wedge method, this framework considers the friction and cohesion of the soil, the soil arch effect, and the shear stress between adjacent differential elements. Yang D et al. [11] adopted an analytical method to address the issue of soil pressure behind retaining walls, providing a new perspective and solution for research in this field. Cao W et al. [12] used the curved thin-layer element method to solve the active soil pressure of non-cohesive soil behind inclined retaining walls, providing a new analytical solution for the calculation of the active soil pressure of this type of retaining wall. Chen RP et al. [13] established a theoretical model for the lateral soil arching effect in flexible retaining walls in sandy soil, filling the gap in theoretical research in related fields and providing a theoretical basis for studying the lateral soil arching effect in the interaction between flexible retaining walls and sandy soil. Gong Q et al. [14] proposed an approximate numerical solution for the seismic response of transversely isotropic porous elastic soil behind rigid retaining walls, overcoming the limitations of traditional methods and providing a new computational method for studying the response of such soil under earthquake action. Que Y et al. [15] conducted spatial soil pressure analysis on cohesive fill behind retaining walls in gully terrain, filling the gap in spatial soil pressure research on cohesive fill behind retaining walls under specific terrain conditions. Chen et al. [16] combined finite element limit analysis with neural network models to predict the stability of cantilever retaining walls comprehensively, overcoming the limitations of a single method and providing a more comprehensive and accurate evaluation method for the stability analysis of cantilever retaining walls. Yang Tao [17] applied the point safety factor to the analysis of landslides and retaining walls.

With the development of numerical analysis methods, different calculation methods have been proposed. Li Zhaoying et al. [18] expanded and improved the existing soil pressure theory, combined it with seismic mechanics principles, and constructed a more suitable theoretical framework for calculating the soil pressure of cantilever retaining walls under earthquake action. Ruirui et al. [19] established a quantitative relationship model between the displacement mode of rigid retaining walls and soil pressure based on experimental data. This model can provide a theoretical tool for engineers to quickly estimate the magnitude of soil pressure based on the estimated displacement pattern when designing retaining walls. Deng Bo et al. [20] proposed a new active soil pressure calculation method based on the research results of unsaturated soil failure modes, which comprehensively considers various factors, such as matric suction and the moisture content of unsaturated soil, and improved the accuracy and practicality of active soil pressure calculations. Du Haoyuan et al. [21] established a mechanical relationship model between the soil pressure and various influencing factors (such as the spacing between reinforcement materials, strength, and soil properties) on the characteristics of gravity reinforced soil retaining walls. This model provides a theoretical basis for a deeper understanding of the mechanical behavior of gravity-reinforced soil retaining walls. On the basis of elastic theory, Party Funing et al. [22] conducted research on soil pressure under finite displacement conditions, enriching the content of soil pressure theory under different displacement conditions and improving the theoretical system of soil pressure. Considering the effects of different boundary conditions, several other authors [23,24,25] have proposed a new numerical calculation method with broad applicability.

Building upon traditional studies on earth pressure and anti-sliding and anti-overturning stability analysis for retaining walls, this paper proposes a new method for analyzing the force and displacement of soil wedges and retaining walls. The following innovations are realized: (1) Under the corresponding boundary conditions, the stress distribution solutions at various points within the soil wedges and retaining walls are obtained. (2) Based on these results, Duncan–Chang and Hooke’s constitutive models are employed for the soil wedges and retaining walls, respectively, to derive the corresponding strain solutions. (3) The proposed theoretical framework facilitates the analysis of various retaining wall failure modes, including overturning, foundation sliding, and tensile and bulging failures, and can be extended to related fields. This study provides a theoretical foundation for slope prevention and control project design. By selecting the form and material of retaining walls according to the stress characteristics, more economical, rational, and effective prevention and control strategies can be developed.

2. Coulomb Retaining Walls

The Coulomb soil pressure theory was developed by French scholar Coulomb in 1976 and is based on four basic assumptions to obtain the formula for the principal passive earth pressure of a soil wedge.

2.1. Active Earth Pressure

The active earth pressure is calculated by

$$E_a={\displaystyle \frac{1}{2}}\gamma K_a{H}^2$$

(1)

where:

$K_{\alpha }$—Coulomb’s active earth pressure coefficient, $K_a={\displaystyle \frac{{\cos }^2(\phi -\alpha )}{{\cos }^2\alpha \cdot \cos (\alpha +\delta )\cdot {\left[1+\sqrt{\frac{\sin (\phi +\delta )\cdot \sin (\phi -\beta )}{\cos (\alpha +\delta )\cdot \cos (\alpha -\beta )}}\right]}^2}}$;

$\gamma$—Unit weight of the soil mass, unit (kN/m³);

$\alpha$—Inclination angle of the wall back (°);

$\phi$—Internal friction angle of the soil mass (°);

$\beta$—Inclination angle of the backfill surface behind the wall (°);

$\delta$—Friction angle of the soil with respect to the back of the retaining wall (°);

H—Height of the retaining wall (m).

When the wall back is vertical, smooth, and has a horizontal filling surface ($\alpha =0, \beta =0, \delta =0$), Coulomb’s active earth pressure can be formulated as:

$$E_a={\displaystyle \frac{1}{2}}\gamma H^2{\tan }^2({45}^{\circ }-{\displaystyle \frac{\phi }{2}})$$

(2)

The distribution of Coulomb’s active earth pressure is shown in Figure 1.

2.2. Passive Earth Pressure

The passive earth pressure is calculated by

$$E_p={\displaystyle \frac{1}{2}}\gamma K_p{H}^2$$

(3)

where:

$K_p$ is Coulomb’s passive earth pressure coefficient, $K_p={\displaystyle \frac{{\cos }^2(\phi +\alpha )}{{\cos }^2\alpha \cdot \cos (\alpha -\delta )\cdot {\left[1+\sqrt{\frac{\sin (\phi +\delta )\cdot \sin (\phi +\beta )}{\cos (\alpha -\delta )\cdot \cos (\alpha -\beta )}}\right]}^2}}$.

When the wall back is vertical, smooth, and has a horizontal filling surface ($\alpha =0, \beta =0, \delta =0$), the expression for Coulomb’s passive earth pressure is:

$$E_p={\displaystyle \frac{1}{2}}\gamma H^2{\tan }^2({45}^{\circ }+{\displaystyle \frac{\phi }{2}})$$

(4)

Equations (2) and (4) are special cases of the Coulomb’s earth pressure theory.

The distribution of the Coulomb passive earth pressure is shown in Figure 2.

It can be inferred from the above derivation that the active earth pressure assumes that the first and third principal stresses of the backfill soil wedge lie in the vertical and horizontal directions, respectively. In contrast, with respect to the passive earth pressure, the first and third principal stresses of the backfill soil wedge are assumed to be in the horizontal and vertical directions, respectively, with a failure angle of (${45}^0+\phi /2$) relative to the third principal stress. For the BC and AB surfaces, the radial and tangential stresses are assumed to be in critical states, and the friction angles are $\phi$ and $\delta$.

2.3. Scientific Problems of Retaining Walls

First, as shown in Figure 1b, under active soil pressure, the equilibrium equation of the soil wedge in the X and Y directions is as follows:

$$R\sin (\theta -\phi )=E_a\cos (\delta +\alpha )$$

(5)

$$R\cos (\theta -\phi )+E_a\sin (\delta +\alpha )=G$$

(6)

Figure 2b shows that under passive earth pressure, the equilibrium equation of the soil wedge in the X and Y directions is as follows:

$$R\sin (\theta +\phi )=E_p\cos (\delta -\alpha )$$

(7)

$$R\cos (\theta +\phi )=E_p\sin (\delta -\alpha )+G$$

(8)

Generally, the friction angle of a soil wedge is known. Under active soil pressure, two other unknown solutions (R and $\delta$) can be obtained via Equations (1), (5) and (6). There are similar issues with passive earth pressure. If the value of $\delta$ is given, there is an imbalance in the soil wedge force.

Second, when the wall back is vertical, smooth, etc., if Equation (2) or (4) is converted into stress, the expressions for the horizontal and vertical stresses of Coulomb’s principal passive earth pressure on the soil facing surface are as follows:

The expression of active earth pressure stress is as follows:

$${\sigma }_{xx}^a=\gamma H{\tan }^2({45}^{\circ }-{\displaystyle \frac{\phi }{2}})\cos \delta$$

(9)

$${\tau }_{xy}^a=\gamma H{\tan }^2({45}^{\circ }-{\displaystyle \frac{\phi }{2}})\sin \delta$$

(10)

The expression of the passive earth pressure stress is as follows:

$${\sigma }_{xx}^p=\gamma H{\tan }^2({45}^{\circ }+{\displaystyle \frac{\phi }{2}})\cos \delta$$

(11)

$${\tau }_{xy}^p=-\gamma H{\tan }^2({45}^{\circ }+{\displaystyle \frac{\phi }{2}})\sin \delta$$

(12)

The above expression shows that the stress does not satisfy the stress differential equilibrium equation (note that the shear stresses of the active and passive soil pressure stresses at the contact surface are positive and negative, respectively).

Third, traditional soil pressure calculations for retaining walls do not consider the boundary conditions of the free surface of the retaining wall, so it is necessary to conduct research on this topic.

In response to the above issues, this paper applies the sliding surface boundary method [23] and proposes a new numerical theoretical solution [24,25] to investigate the stress–strain distribution characteristics of the backfill soil wedge and retaining wall under different rotation axis conditions. Based on this approach, the stress and strain solutions for the backfill soil wedge and retaining wall are derived, and their characteristics are analyzed. A point failure criterion is subsequently introduced to evaluate the stability of the soil wedge and retaining wall. This method is applicable not only to non-cohesive soils but also to cohesive soils.

3. Numerical Theoretical Solution of the Backfill Soil Wedge and Retaining Wall

3.1. Assumptions of the Soil Wedge and Retaining Wall

According to the ideas in references [24,25], when any material or object under the influence of boundary stress is studied, once its shape is determined, its stress theoretical solution is clear, and as the boundary conditions change, its shape also changes, and the results of the stress solution also change accordingly.

Based on the characteristics of the soil wedge and the retaining wall as research subjects, stress continuity can be assumed. Under their respective stress boundary conditions, equilibrium differential equations, and compatibility requirements, the corresponding stress distribution solutions can be derived. This method is applicable for solving the stress distribution of any geometric shape, including two-dimensional and three-dimensional problems.

3.2. Boundary Conditions

Based on the characteristics of the stress interaction between the soil wedge and the retaining wall, the horizontal and vertical resultant forces are continuous, and the sum of the moments is likewise continuous.

The boundary conditions for the soil wedge and retaining wall are as follows:

For the soil wedge, the boundary conditions are as follows:

On the AD side:

$${p_x}^{DA}=l^{DA}{{\sigma }_{xx}}^{DA}+m^{DA}{{\tau }_{yy}}^{DA}$$

(13)

$$p_y^{DA}=m^{DA}{{\sigma }_{yy}}^{DA}+l^{DA}{{\tau }_{xy}}^{DA}$$

(14)

On the DC side:

$${{\sigma }_{xx}}^{DC}=0$$

(15)

$${{\sigma }_{yy}}^{DC}=(y_D-y)\gamma$$

(16)

$${{\tau }_{xy}}^{DC}=0$$

(17)

For the retaining wall, the boundary conditions are as follows:

On the AF side:

$${{\sigma }_{xx}}^{FA}=0,{\sigma }_{yy}^{FA}=0,{{\tau }_{yy}}^{FA}=0$$

(18)

On the EF side:

$${\displaystyle {\int }_{EF}{p}_x}dl=0,{\displaystyle {\int }_{EF}{p}_y}dl=0$$

(19)

where:

$p_x=l{\sigma }_{xx}+m{\tau }_{xy}$, $p_y=m{\sigma }_{yy}+l{\tau }_{xy}$. l, m is the direction cosine, and $\gamma$ is the specific gravity of the retaining wall.

The proposed method yields theoretical solutions for small deformations in both the soil wedge and the retaining wall.

3.3. Solution Steps

On the basis of Refs. [24,25], the new numerical method for the soil wedge and retaining wall is explained in detail as follows:

(1)

Based on the cross-sectional view of the garbage transfer station in Lvcongpo Town, Badong County (see Section 4.2.1 below), a calculation model for the soil wedge and retaining wall is established, as shown in Figure 3. The model consists of two quadrilateral elements. Under a given plane coordinate system, the geometric characteristic equations related to each edge are expressed via $y=kx+b$.

(2)

The specific gravities of the soil wedge and retaining wall are constant. The expressions of the stress equations are selected, and the corresponding constant coefficients when the soil wedge and retaining wall satisfy the stress differential equilibrium equations, stress boundary conditions, and compatibility equations are calculated. The stress expressions are as follows:

$${\sigma }_{xx}=a_{1,0}+a_{1,1}x+a_{1,2}y+a_{1,3}{x}^2+a_{1,4}xy+a_{1,5}{y}^2+a_{1,6}{x}^3+a_{1,7}{x}^{2}y+a_{1,8}x{y}^2+\cdots$$

(20)

$${\sigma }_{yy}=a_{2,0}+a_{2,1}x+(a_{2,2}-{\gamma }_{w,y})y+a_{2,3}{x}^2+a_{2,4}xy+a_{2,5}{y}^2+a_{2,6}{x}^3+a_{2,7}{x}^{2}y+a_{2,8}x{y}^2+\cdots$$

(21)

$${\tau }_{xy}=a_{3,0}+a_{3,1}x+a_{3,2}y+a_{3,3}{x}^2+a_{3,4}xy+a_{3,5}{y}^2+a_{3,6}{x}^3+a_{3,7}{x}^{2}y+a_{3,8}x{y}^2+\cdots$$

(22)

where $a_{1,i},a_{2,i},a_{3,i}$ are constant coefficients where i takes a value of zero and integers; ${\sigma }_{xx},{\sigma }_{yy},{\tau }_{xy}$ denote the stresses in the X and Y directions and the shear stress, respectively; and ${\gamma }_{w,y}$ represents the specific gravity in the Y direction. In the process of theoretical solution, the more high-order the stress expression is fitted and the more constant coefficients there are, the closer the obtained stress is to the actual situation. On the basis of previous studies, this study fits the stress expression in the form of a 5th power and obtains 33 unknown constant coefficients.

(3)

Taking the Y-axis as the perpendicular axis, the following equilibrium equations are satisfied under gravity conditions:

$${\displaystyle \frac{𝜕{\sigma }_{xx}}{𝜕x}}+{\displaystyle \frac{𝜕{\tau }_{xy}}{𝜕y}}=0$$

(23)

$${\displaystyle \frac{𝜕{\tau }_{xy}}{𝜕x}}+{\displaystyle \frac{𝜕{\sigma }_{yy}}{𝜕y}}+{\gamma }_{w,y}=0$$

(24)

(4)

Making the corresponding coefficients zero is a necessary condition for the stress balance equation. Substituting the stress expression (20)–(22) into Equation (23) yields the following relationship:

$$a_{1,1}+a_{3,2}=0$$

(25)

$$2a_{1,3}+a_{3,4}=0$$

(26)

$$a_{1,4}+2a_{3,5}=0$$

(27)

$$3a_{1,6}+a_{3,7}=0$$

(28)

$$2a_{1,7}+2a_{3,8}=0$$

(29)

$$a_{1,8}+3a_{3,9}=0$$

(30)

Similarly, substituting the stress expression into Equation (24) yields:

$$a_{3,1}+a_{2,2}-{\gamma }_{w,y}=0$$

(31)

$$2a_{3,3}+a_{2,4}=0$$

(32)

$$a_{3,4}+2a_{2,5}=0$$

(33)

$$3a_{3,6}+a_{2,7}=0$$

(34)

$$2a_{3,7}+2a_{2,8}=0$$

(35)

$$a_{3,8}+3a_{2,9}=0$$

(36)

When the specific gravities are taken as constants (${\gamma }_{w,x}=0,{\gamma }_{w,y}\ne 0=\gamma$), the corresponding boundary conditions, equilibrium equations, and compatibility equations can be satisfied, and all the constant coefficients can be solved. By substituting these coefficients into the stress expressions, the stress solutions at any point within research objects can be obtained using Equations (20)–(22).

Based on the stress solutions of the soil wedge and retaining wall, their failure characteristics can be determined in conjunction with the applicable strength criterion. The corresponding constitutive model is selected to study the deformation characteristics, which are then compared with field results to elucidate the associated behavioral features. The specific analysis process is as follows: first, the theoretical stress solutions are obtained through the above calculation procedure; second, the corresponding principal stress magnitudes are calculated; finally, the computed stress values are substituted into the strength criterion (e.g., the Mohr–Coulomb criterion) to identify the failure state point.

Figure 3. Calculation model of the soil wedge and retaining wall.

Figure 3. Calculation model of the soil wedge and retaining wall.

4. Case Study

4.1. Division of Computation Units and Equations

This method has formed a complete theoretical system, achieved many research results, and has been widely applied in practical engineering projects, producing significant social impact and economic benefits [26]. This Section illustrates the approach of taking the soil wedge and retaining wall of the garbage transfer station in Lvcongpo Town, Badong County, as an example. The soil wedge and retaining wall model is established, with the AFEBCDA in Figure 3 selected as the research object. When the model is analyzed, the relationship between the boundary conditions and the theoretical solutions is carefully considered. The boundary condition investigation of the model in Figure 3 is as follows:

Taking the stress expression from the constant term to $y^5$ in complete form, 63 constant coefficients are yielded, which can be simplified into 33 constant coefficient expressions through the equilibrium equations in Equations (25)–(36). On the basis of the given stress conditions on different boundaries, the corresponding constant coefficients can be determined. Based on the idea proposed in reference [24,25], the computational model in Figure 3 is divided into two quadrilateral elements.

For Unit I, if the stress boundary condition on the DA side is zero, 12 equations can be established under strong constraint conditions. If the stress boundary conditions on the DC side satisfy Equations (15)–(17), 18 equations can be formulated under strong constraint conditions. Considering both the DC and DA boundary conditions, a total of 30 equations can be obtained. In this study, the soil wedge consists of backfilled clay, and the inclination angle of the CB is 38°. According to the traditional assumption, the surface has already undergone failure, resulting in discontinuous tangential stress along the surface, whereas the normal stress remains continuous. If the tangential stress follows the principle of strength reduction, the relationship between the tangential stress and the normal stress is given as follows:

$${\tau }_N^{BC}=(C+{\sigma }_N^{BC}\tan \phi )/f$$

(37)

where ${\tau }_N^{BC},{\sigma }_N^{BC}$ represent the tangential stress and normal stress on the BC side and where $C,\phi ,f$ represent the cohesion, friction angle, and strength reduction coefficient of the soil, respectively. The equilibrium equations for the entire soil wedge of the ABCDA are as follows:

Horizontal force balance:

$${\displaystyle \underset{AB}{\int }{p}_x^{AB}dl}+{\displaystyle \underset{BC}{\int }{p}_x^{BC}dl}=F_{x,BC}$$

(38)

Vertical force balance:

$${\displaystyle \underset{AB}{\int }{p}_y^{AB}dl}+{\displaystyle \underset{BC}{\int }{p}_y^{BC}dl}=F_{y,BC+W_{ABCD}}$$

(39)

where $F_{x,BC},F_{y,BC+W_{ABCD}}$ represent the horizontal force and vertical force with gravity from the BC side per unit width of the soil wedge (ABCD), respectively.

The torque balance equation for Unit I around the rotation axis (point) of $O_1$ is as follows:

$${\displaystyle \underset{AB}{\int }{p}_y^{AB}\left|X_{O_1}-x\right|dl}+{\displaystyle \underset{BC}{\int }{p}_y^{BC}\left|X_{O_1}-x\right|dl}+{\displaystyle \underset{AB}{\int }{p}_x^{AB}\left|Y_{O_1}-y\right|dl}+{\displaystyle \underset{BC}{\int }{p}_x^{BC}\left|Y_{O_1}-y\right|dl}+F_{x,BC}\left|Y_{O_1}-x\right|+F_{y,BC+W_{ABCD}}\left|X_{O_1}-x\right|=0$$

(40)

where $X_{O_1},Y_{O_1}$ represents the coordinates of the rotation point $O_1$.

The equilibrium equations for the AFEB retaining wall (unit II) are as follows:

Horizontal force balance:

$${\displaystyle \underset{AB}{\int }{p}_x^{AB}dl}+{\displaystyle \underset{EF}{\int }{p}_x^{EB}dl}=0$$

(41)

Vertical force balance:

$${\displaystyle \underset{AB}{\int }{p}_y^{AB}dl}+{\displaystyle \underset{EF}{\int }{p}_y^{BE}dl}=W_{ABEF}$$

(42)

where $W_{ABEF}$ is the unit width gravity of the retaining wall (AFEB).

The torque balance equation for Unit II around the rotation axis (point) of $O_2$ is as follows:

$${\displaystyle \underset{AB}{\int }{p}_y^{AB}\left|X_{O_2}-x\right|dl}+{\displaystyle \underset{EF}{\int }{p}_y^{EB}\left|X_{O_2}-x\right|dl}+{\displaystyle \underset{AB}{\int }{p}_x^{AB}\left|Y_{O_2}-y\right|dl}+{\displaystyle \underset{EF}{\int }{p}_y^{EB}\left|Y_{O_2}-y\right|dl}+W_{ABEF}\left|X_{O_2}-x\right|=0$$

(43)

4.2. Case Introduction

4.2.1. Overview of the Garbage Transfer Station Project

The garbage transfer station in Lvcongpo Town, Badong County, Hubei Province, is located at an angle of 165° from the county town of Badong, with a direct distance of about 30 km. The project is adjacent to the Lvcongpo market town and is connected by a gravel road. The station occupies an area of 2000.1 m², with a platform elevation of 1670.5 m and a retaining wall bottom elevation ranging from 1669.8 m to 1670.5 m. The slope height is about 6.9 m (see Figure 4 and Figure 5). On the left side of the station, clay has been backfilled to form a section corresponding to the I-I profile (see Figure 5). The lower part of the station consists of high-strength Triassic moderately weathered limestone, which has a uniaxial compressive strength of 45–48 MPa. The angle of the contact surface between the backfilled clay and the underlying rock is 38°. The base of the retaining wall is located above a plain concrete cushion layer (C₂₀) and moderately weathered limestone.

4.2.2. Computational Model

(1)

Computation Model Size

We establish the computation model units on the basis of the I-I profile of the Lvcongpo garbage transfer station as follows. The specific gravity of the backfill clay at the station is 19.6 kN/m³, the friction angle is 18°, and the cohesive force is 20 kPa. The basic dimensions of the ABCDA area of the soil wedge are as follows: the lengths of the sides are as follows: AD = 8.42 m, DC = 0.42 m, CB = 10.45 m, and AB = 5 m. The AFEB retaining wall is constructed with C₂₅ plain concrete, with physical and mechanical parameters as follows: specific gravity of 25 kN/m³, friction angle of 48°, cohesion of 600 kPa, elastic modulus of 28 GPa, and Poisson’s ratio of 0.11. The basic dimensions are AB = 5 m, BE = 4.04 m, EF = 5.22 m, and AF = 1.2 m. The computational model based on the proposed method is illustrated in Figure 3. For comparison with existing approaches, finite element triangular mesh models of the soil wedge and retaining wall are utilized, as shown in Figure 6. The boundary conditions calculated by the finite element model are consistent with those of the calculation model in Figure 3.

(2)

Damage Criterion

This study considers the Mohr–Coulomb criterion as the stress intensity criterion and point failure as the criterion condition. Based on the calculated principal stress results for the soil wedge and retaining wall, the cohesion values at each point are determined using the measured friction angles. If the calculated cohesion value exceeds the material’s tested value, the point is considered to have failed.

(3)

Computation Results

(1) Stress Calculation Results for the Soil Wedge and Retaining Wall

Based on the new theoretical solution, a calculation model for the soil wedge and retaining wall is established. The coordinates of each point in the model are determined, and the geometric boundary description equations are derived to define the conditions for each boundary. By ensuring that the calculated stress values in the x- and y-directions are consistent with those obtained from Equation (2), the resulting stress distributions for the soil wedge and retaining wall match the values calculated from the active earth pressure. Under the assumption that the soil wedge rotates around its mass center (axis), the correlation coefficients for the soil wedge and retaining wall are determined in the new model (see

Table 1. Stress parameters with respect to the centroid of the unit.

Unit	aij i∈(1,3)	ai,0	ai,1	ai,2	ai,3	ai,4	ai,5	ai,6	ai,7	ai,8	ai,9	ai,10	ai,11	ai,12	ai,13	ai,14	ai,15	ai,16	ai,17	ai,18	ai,19	ai,20
	kPa	kPa	kPa /m	kPa /m	kPa /m2	kPa /m2	kPa /m2	kPa /m3	kPa /m3	kPa /m3	kPa /m3	kPa /m4	kPa /m4	kPa /m4	kPa /m4	kPa /m4	kPa /m5	kPa /m5	kPa /m5	kPa /m5	kPa /m5	kPa /m5
I	σxx	−8.514589832	10.7338006	1.975186322	−4.686005862	−2.4311721	−0.081066929	0.783697159	1.011448983	0.098342413	−6.4374 × 10−15	−0.035729565	−0.147801243	−0.039766443	9.34631 × 10−16	4.18172 × 10−16	0.001208862	0.002318841	0.005360081	−1.1252 × 10−16	−1.12094 × 10−16	0−
	σyy	289.5878581	−25.57832168	−6.284912858	−6.760674299	−9.743081267	−4.686005862	1.783788853	2.472570175	2.351091476	0.337149661	−0.085824316	−0.30484489	−0.214377393	−0.147801243	−0.00662774	0.001113794	0.007179218	0.01208862	0.004637683	0.002680041	−1.1252 × 10−17
	τxy	12.88274782	−13.31508714	−10.7338006	4.871540634	9.372011724	1.21558605	−0.824190058	−2.351091476	−1.011448983	−0.032780804	0.076211222	0.142918262	0.221701864	0.026510962	−2.33658 × 10−16	−0.001435844	−0.00604431	−0.004637683	−0.005360081	5.626 × 10−17	2.24188 × 10−17
II	σxx	−2231.174997	−237.6119698	3290.591566	175.1101396	−754.6661804	−455.0088761	−18.41150807	39.32303445	127.8870528	4.89176 × 10−13	0.499646849	1.664172127	−12.0050015	−2.12887 × 10−13	5.07007 × 10−16	0.00685827	−0.13849182	0.378806868	5.78534 × 10−15	2.40638 × 10−16	0
	σyy	−4074.640752	1346.14285	−729.6690351	−142.4134669	75.15955343	175.1101396	5.264073151	11.25086603	−55.23452421	13.10767815	1.36445 × 10−30	−1.201706004	2.997881096	1.664172127	−2.000833584	0	9.21839 × 1031	0.068582696	−0.276983641	0.189403434	5.78534 × 10−16
	τxy	−2378.01372	704.6690351	237.6119698	−37.57977672	−350.2202793	377.3330902	−3.750288675	55.23452421	−39.32303445	−42.62901762	0.300426501	−1.998587397	−2.49625819	8.003334335	5.32217 × 10−14	−1.84368 × 10−31	−0.034291348	0.276983641	−0.378806868	−2.89267 × 10−15	−4.81276 × 10−17

). The obtained coefficients are substituted into Equations (20)–(22), and the distributions of ${\sigma }_{xx}$, ${\sigma }_{yy}$, and ${\tau }_{xy}$ (see Figure 7, Figure 8 and Figure 9), the principal stresses ${\sigma }_1$ and ${\sigma }_3$ (see Figure 10 and Figure 11), and the c values (see Figure 12) at any point are obtained on the basis of the corresponding coordinate points. For the retaining wall, the correlation coefficients obtained under rotation around the B, B1, B2, and E axes (points) are shown in

Table 2. Stress parameters with respect to the centroid of the retaining wall.

Unit	aij i∈(1,3)	ai,0	ai,1	ai,2	ai,3	ai,4	ai,5	ai,6	ai,7	ai,8	ai,9	ai,10	ai,11	ai,12	ai,13	ai,14	ai,15	ai,16	ai,17	ai,18	ai,19	ai,20
	kPa	kPa	kPa /m	kPa /m	kPa /m2	kPa /m2	kPa /m2	kPa /m3	kPa /m3	kPa /m3	kPa /m3	kPa /m4	kPa /m4	kPa /m4	kPa /m4	kPa /m4	kPa /m5	kPa /m5	kPa /m5	kPa /m5	kPa /m5	kPa /m5
B	σxx	1057.401297	12.93321823	−901.7984207	−60.84710546	197.8176225	151.0830504	7.693568789	−7.697202933	−42.3634883	5.15296 × 10−13	−0.269465065	−0.816186191	3.936060403	1.58737 × 10−14	−5.39781 × 10−15	0.000776367	0.046335254	−0.118729396	−4.58553 × 10−16	7.72726 × 10−16	0
	σyy	3692.385668	−375.3434249	−132.5773924	0.535922331	−56.80117939	−60.84710546	0.595901458	6.990593268	23.08070637	−2.565734311	0	−0.136035032	−1.616790393	−0.816186191	0.656010067	0	−2.55018 × 10−31	0.00776367	0.092670507	−0.059364698	−4.58553 × 10−17
	τxy	−1790.733007	107.5773924	−12.93321823	28.40058969	121.6942109	−98.90881127	−2.330197756	−23.08070637	7.697202933	14.12116277	0.034008758	1.077860262	1.224279287	−2.624040269	−3.96841 × 10−15	5.10036 × 10−32	−0.003881835	−0.092670507	0.118729396	2.29276 × 10−16	−1.54545 × 10−16
B1	σxx	−2681.606157	550.3246785	1547.407161	−8.249911582	−384.2406183	−217.9893666	−4.65598325	26.96726623	61.30853894	−8.75072 × 10−13	0.332192276	−0.040854443	−5.771058252	1.28719 × 10−13	1.12857 × 10−14	−0.004861229	−0.040614373	0.184239351	9.45318 × 10−16	−1.00888 × 10−15	0
	σyy	−5324.401451	501.0704196	636.8507964	43.73994808	6.369680178	−8.249911582	−3.73124213	−16.21987184	−13.96794975	8.989088744	0	0.851784529	1.993153658	−0.040854443	−0.961843042	0	0	−0.048612289	−0.081228745	0.092119676	9.45318 × 10−17
	τxy	3817.518942	−661.8507964	−550.3246785	−3.184840089	16.49982316	192.1203092	5.406623947	13.96794975	−26.96726623	−20.43617965	−0.212946132	−1.328769105	0.061281665	3.847372168	−3.21797 × 10−14	0	0.024306145	0.081228745	−0.184239351	−4.72659 × 10−16	2.01776 × 10−16
B2	σxx	68517.8476	−18838.22679	−22807.23254	1711.192052	6361.895361	1680.440889	−52.22439244	−586.892854	−471.9583868	6.99976 × 10−12	0.317204083	17.45160699	44.16030183	−4.44711 × 10−13	−9.1815 × 10−14	−0.02005534	0.037415513	−1.374167643	3.45636 × 10−14	−2.338 × 10−14	0
	σyy	−1461.054204	−698.2970009	−4758.585968	246.7227145	574.8492968	1711.192052	−15.39350032	−50.5792929	−156.6731773	−195.6309513	−1.02007 × 10−31	3.514096637	1.903224499	17.45160699	7.360050305	0	1.1157 × 10−32	−0.200553398	0.074831025	−0.687083821	3.45636 × 10−15
	τxy	−26677.40723	4733.585968	18838.22679	−287.4246484	−3422.384104	−3180.947681	16.8597643	156.6731773	586.892854	157.3194623	−0.878524159	−1.268816333	−26.17741048	−29.44020122	1.11178 × 10−13	−2.23141 × 10−33	0.100276699	−0.074831025	1.374167643	−1.72818 × 10−14	4.67601 × 10−15
E	σxx	33048.3391	−10789.38026	−6035.154863	1259.831626	2202.733342	−279.2975565	−57.66300158	−300.8469261	78.5299406	−1.43625 × 10−11	0.409520591	18.20442305	−7.383549045	2.22108 × 10−14	−1.08313 × 10−13	0.026114474	−0.415795704	0.234566779	−2.46143 × 10−14	−1.94125 × 10−15	0
	σyy	−59701.97769	13278.26737	4301.825456	−929.8131553	−1476.204359	1259.831626	20.04419616	148.4329786	−172.9890047	−100.2823087	−1.18032 × 10−30	−4.575778143	2.457123548	18.20442305	−1.230591507	0	1.00948 × 10−31	0.26114474	−0.831591408	0.11728339	−2.46143 × 10−15
	τxy	7612.300053	−4326.825456	10789.38026	738.1021796	−2519.663252	−1101.366671	−49.47765953	172.9890047	300.8469261	−26.17664687	1.143944536	−1.638082365	−27.30663457	4.92236603	−5.5527 × 10−15	−2.01896 × 10−32	−0.13057237	0.831591408	−0.234566779	1.23072 × 10−14	3.88249 × 10−16

, and the corresponding distributions of stress, principal stress, and cohesion (C) are shown in the Figure 13 and Figure 14, Figure 15 and Figure 16, Figure 17 and Figure 18, and Figure 19 and Figure 20, respectively.

Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 show that the retaining wall mainly bears thrust from the slope, gradually increasing from top to bottom, and that the stress is concentrated near the bottom of the wall. By analyzing the stress distributions of the soil wedges and retaining walls, it can be concluded that, compared with traditional methods, the stress values and distribution patterns at any point inside the soil wedges and retaining walls provide theoretical support for the design of retaining walls. This study uses the strength values of each point in the soil wedge and retaining wall as a whole for inverse calculation. The inverse calculation method involves keeping the friction angle constant and incorporating it into the Coulomb criterion to calculate the distribution of cohesion C values at each point (Figure 12). When the C value is greater than the material value, the retaining wall will fail at that point. If it is less than the material’s C value, the retaining wall will not fail. This method can intuitively demonstrate the failure characteristics and initial failure locations of retaining walls.

(2) Calculation results of the soil wedge strain

By employing the constitutive relationship, the stress–strain relationship of the soil wedge obtained from experiments satisfies the Duncan–Chang constitutive model, and its basic equations are as follows:

$${\sigma }_1-{\sigma }_3={\displaystyle \frac{{\epsilon }_1}{a_1+b_1{\epsilon }_1}}\Rightarrow {\epsilon }_1={\displaystyle \frac{a_1({\sigma }_1-{\sigma }_3)}{1-b_1({\sigma }_1-{\sigma }_3)}}$$

(44)

$${\sigma }_1-{\sigma }_3={\displaystyle \frac{{\epsilon }_3}{a_2+b_2{\epsilon }_3}}\Rightarrow {\epsilon }_3={\displaystyle \frac{a_2({\sigma }_1-{\sigma }_3)}{1-b_2({\sigma }_1-{\sigma }_3)}}$$

(45)

where ${\epsilon }_1,{\epsilon }_3$ represent the first and third principal strains, respectively. According to the experimental results, $a_1$ is 0.0002 (1/kPa), $a_2$ is 0.00012099 (1/kPa), $b_1$ is −0.000056 (1/kPa), and the value of $b_2$ is 0.0002099 (1/kPa). The distribution characteristics of the principal strains of the soil wedge can be obtained (see Figure 21).

For two-dimensional problems, the strain (${\epsilon }_{ij}$) in any rotation direction with a rotation angle of $\varphi$ can be expressed as:

$${\epsilon }_{xx}={\epsilon }_1{\cos }^2\varphi +{\epsilon }_3{\sin }^2\varphi$$

(46)

$${\epsilon }_{yy}={\epsilon }_1{\sin }^2\varphi +{\epsilon }_3{\cos }^2\varphi$$

(47)

$${\gamma }_{xy}=-({\epsilon }_{xx}-{\epsilon }_{yy})\tan (2\varphi )$$

(48)

where ${\epsilon }_{xx},{\epsilon }_{yy},{\gamma }_{\mathrm{xy}}$ represents the strain.

The rotation angle $\varphi$ is calculated via the following equation:

$$\tan 2\varphi ={\displaystyle \frac{-2{\tau }_{xy}}{{\sigma }_{xx}-{\sigma }_{yy}}}\Rightarrow \varphi ={\displaystyle \frac{1}{2}}\arctan ({\displaystyle \frac{-2{\tau }_{xy}}{{\sigma }_{xx}-{\sigma }_{yy}}})$$

(49)

The calculated strain distributions at each point of the soil wedge are shown in Figure 22.

(3) Calculation Results of the Retaining Wall Strain

The strain problem for the retaining wall can be seen as a planar problem, where ${\epsilon }_z=0$ and ${\sigma }_z\ne 0$. According to the general form of Hooke’s law, we have

$$\begin{gathered} {\epsilon }_{x\mathrm{x}}={\displaystyle \frac{1}{E}}[{\sigma }_{xx}-\mu ({\sigma }_{yy}+{\sigma }_{zz})] \\ {\epsilon }_{\mathrm{y}y}={\displaystyle \frac{1}{E}}[{\sigma }_{yy}-\mu ({\sigma }_{xx}+{\sigma }_{zz})] \\ {\epsilon }_{zz}={\displaystyle \frac{1}{E}}[{\sigma }_{zz}-\mu ({\sigma }_{xx}+{\sigma }_{yy})] \\ {\gamma }_{xy}={\displaystyle \frac{{\tau }_{xy}}{G}},{\gamma }_{yz}={\displaystyle \frac{{\tau }_{yz}}{G}},{\gamma }_{xz}={\displaystyle \frac{{\tau }_{xz}}{G}} \end{gathered}$$

(50)

where:

$$G={\displaystyle \frac{E}{2(1+\mu )}}$$

Here, E denotes the elastic modulus, which is 28 GPa, G represents the shear modulus, and $\mu$ = 0.11 is Poisson’s ratio.

The strain distribution solutions of the retaining wall can be obtained via the above process, as shown in Figure 23, Figure 24, Figure 25, Figure 26, Figure 27, Figure 28, Figure 29 and Figure 30.

4.2.3. Model Comparison

Given that the rotational balance in the finite element method yields an essentially zero moment along the mass center axis (point), a comparison can only be made between the moment balance calculation results at the centroid of the soil wedge and retaining wall using the finite element method and the proposed calculation method. Under conditions of equal torque and identical boundary conditions, the calculation results are obtained using ABAQUS 2020 software. The differences between the stress computed by the finite element method in ABAQUS and the stress calculated using the proposed method are illustrated in Figure 31, Figure 32 and Figure 33. The results show that the deviation between the two is less than 5.05%, indicating that the stress distribution characteristics obtained by the finite element calculation method are basically consistent with those obtained by the theoretical solution. This method is feasible for solving the stress distribution of soil wedges and retaining walls and provides a new theoretical method for practical engineering.

4.3. Analysis of the Calculation Results

This Section analyzes the above calculation results for the soil wedge and retaining wall.

4.3.1. Analysis of Soil Wedge Results

The horizontal stress within the soil wedge is relatively small, and no tensile stress is generated. The vertical stress is less than the product of the specific density and depth, and the shear stress is also minimal, remaining lower than the vertical stress. All three types of stress are minor at the boundaries, satisfying the boundary conditions. The corresponding first and third principal stresses are similarly small, with no observed tensile stress. The cohesion values obtained through reverse calculation are low, and all points within the soil wedge essentially remain within the strength limit. The calculation results indicate that if failure occurs in the soil wedge, it first appears at about two-thirds of the height from the top of the retaining wall. The maximum first and third principal strains, along with the maximum horizontal and vertical strains, are located at the soil–rock interface corresponding to this two-thirds position, which is consistent with the stress distribution results (see Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 21 and Figure 22). Compared with traditional methods, this method can obtain the magnitude and distribution of stress values at any point inside the soil wedge, providing a new theoretical support for the design of retaining walls.

4.3.2. Retaining Wall Stability Analysis

This study obtains solutions when the retaining wall rotates along the mass center axis (point) O₂ and the B, B₁, B₂, and E axes (points). The characteristics of the solutions are as follows:

For the rotation around the central axis of mass, the maximum values of ${\sigma }_{xx}$, ${\sigma }_{yy}$, and ${\tau }_{xy}$ are 194.69, 184.03, and 41.11 kPa, and their minimum values are −60.47, −1.77, and −23.26 kPa, respectively. C achieves a maximum value of 27.94 kPa and a minimum negative value of −78.77 kPa. The maximum values of ${\sigma }_1$ and ${\sigma }_3$ are 201.31 kPa and 75.30 kPa, respectively, and their minimum negative values are 0 kPa and 60.47 kPa, respectively. The maximum values of ${\epsilon }_1$ and ${\epsilon }_3$ are 0.0003 and 0.0002, and their minimum negative values are 0.00004 and 0.00002, respectively. The maximum values of ${\epsilon }_{xx}$, ${\epsilon }_{yy}$, and ${\gamma }_{xy}$ are 0.0006, 0.0006, and 0.0003, respectively, and their minimum negative values are −0.0002, 0.00002, and −0.0001, respectively. The maximum first and third principal strains, as well as the horizontal and vertical strains, are all located at the toe of the retaining wall, which is consistent with the stress distribution results.

Under the condition of B rotation, the maximum values of ${\sigma }_{xx}$, ${\sigma }_{yy}$, and ${\tau }_{xy}$ are 144.59, 184.03, and 52.33 kPa, and their minimum negative values are −67.1, 0, and −14.95 kPa, respectively. C achieves a maximum value of 11.11 kPa and a minimum negative value of −90.41 kPa. The maximum values of ${\sigma }_1$ and ${\sigma }_3$ are 205.61 kPa and 99.7 kPa, and their minimum negative values are 0 kPa and −67.06 kPa, respectively. The maximum values of ${\epsilon }_1$ and ${\epsilon }_3$ are 0.00027 and 0.00016, respectively, whereas their minimum negative values are 0.00007 and 0.00004, respectively. The maximum values of ${\epsilon }_{xx}$, ${\epsilon }_{yy}$, and ${\gamma }_{xy}$ are 0.00042, 0.0006, and 0.00039, and the minimum negative values are −0.00022, 0.00002, and −0.00011, respectively. The maximum first and third principal strains, as well as the horizontal and vertical strains, are located at the toe of the retaining wall; this finding is also consistent with the stress distribution results.

Under the condition of B₁ rotation, the maximum values of ${\sigma }_{xx}$, ${\sigma }_{yy}$, and ${\tau }_{xy}$ are 110.55, 211.24, and 53.73 kPa, and their minimum negative values are −60.38, −80.89, and −36.36 kPa, respectively. C has a maximum value of 11.11 kPa and a minimum negative value of −91.64 kPa. The maximum values of ${\sigma }_1$ and ${\sigma }_3$ are 220.65 kPa and 66.45 kPa, respectively, and their minimum negative values are 0 kPa and −87.6 kPa, respectively. The maximum values of ${\epsilon }_1$ and ${\epsilon }_3$ are 0.00041 and 0.00025, and their minimum negative values are 0.00006 and 0.00004, respectively. The maximum values of ${\epsilon }_{xx}$, ${\epsilon }_{yy}$, and ${\gamma }_{xy}$ are 0.0004, 0.00068, and 0.0004, and their minimum negative values are −0.0002, −0.00031, and −0.00027, respectively. The maximum first and third principal strains, as well as the horizontal and vertical strains, are all located at the toe of the retaining wall; this finding is consistent with the stress distribution results.

Under the condition of B₂ rotation, the maximum values of ${\sigma }_{xx}$, ${\sigma }_{yy}$, and ${\tau }_{xy}$ can be seen as 62.91, 287.40, and 102.83 kPa, and their minimum negative values are −308.57, −334.35, and −159.13 kPa, respectively. C has a maximum value of 29.87 kPa and a minimum negative value of −657.72 kPa. The maximum values of ${\sigma }_1$ and ${\sigma }_3$ are 295.10 kPa and 55.22 kPa, respectively, whereas their minimum negative values are −161.81 kPa and −481 kPa, respectively. The maximum values of ${\epsilon }_1$ and ${\epsilon }_3$ are 0.00063 and 0.00039, and their minimum negative values are 0.00011 and 0.00007, respectively. The maximum values of ${\epsilon }_{xx}$, ${\epsilon }_{yy}$, and ${\gamma }_{xy}$ are 0.00014, 0.00094, and 0.00076, whereas their minimum negative values are −0.00091, −0.001, and −0.00118, respectively. The maximum first and third principal strains, as well as the horizontal and vertical strains, all lie at the toe of the retaining wall, and this observation is consistent with the stress distribution results.

Under the condition of E rotation, the maximum values of ${\sigma }_{xx}$, ${\sigma }_{yy}$, and ${\tau }_{xy}$ are 66.74, 374.30, and 115.75 kPa, and their minimum negative values are −73.31, −511.77, and −172.19 kPa, respectively. C has a maximum value of 39.88 kPa, and the minimum negative value is −746.80 kPa. The maximum values of ${\sigma }_1$ and ${\sigma }_3$ are 382.81 kPa and 58.23 kPa, and their minimum negative values are −46.78 kPa and −571.00 kPa, respectively. The maximum values of ${\epsilon }_1$ and ${\epsilon }_3$ are 0.0011 and 0.0007, and their minimum negative values are 0.0001 and 0.00006, respectively. The maximum values of ${\epsilon }_{xx}$, ${\epsilon }_{yy}$, and ${\gamma }_{xy}$ are 0.0001, 0.0012, and 0.0009, and their minimum negative values are −0.0002, −0.0017, and −0.0013, respectively. The maximum first and third principal strains, as well as the horizontal and vertical strains, are located at the toe of the retaining wall; this result is consistent with those of the stress distribution.

By comparing the stress–strain distributions of the retaining walls under different rotation points, it can be concluded that this method can more comprehensively and accurately determine the stress magnitude and distribution at any point within the retaining wall, thereby determining whether the retaining wall meets the point strength design requirements.

The above analysis demonstrates the anti-overturning and anti-shear properties of the retaining wall. The stress characteristics of the bottom side (BE) of the retaining wall, which is in contact with the ground (C₂₀), are analyzed. The friction angle between the retaining wall and the ground is taken as 45°. By calculating the shear and normal stresses along the BE side and applying the Mohr–Coulomb criterion, the cohesive force (C) at different points along the contact surface can be determined. The C value distributions at different points on the contact surface under the five rotation points are shown in Figure 34. The data presented in the figure indicate that the stress varies significantly with changes in the rotation point, and the location of the initial failure point on the retaining wall also shifts accordingly. Specifically, when rotation occurs around the O₂, B, and B₁ axes (or points), the stress distribution along the contact surface remains relatively uniform, and no slip failure occurs along the BE plane. In contrast, when rotation occurs around the B₂ and E axes (or points), the stress distribution becomes uneven, and slip failure is likely to initiate at point E. Additionally, the cohesive strength (C value) of the C₂₀ concrete used for the retaining wall in this study falls within the required anti-slide strength range. Therefore, the retaining wall will not experience overturning or shear failure, and the design satisfies the point strength requirements.

4.4. Discussion

This theoretical calculation method involves the study and exploration of two-dimensional models or plane problems. By changing the boundary conditions of the research object, an equilibrium equation that satisfies the principle of stress continuity is established. The more the stress expression is fitted in high-order form, the closer the stress–strain at any point in the slope obtained is to the actual situation. This article explores the shortcomings of the methods used in this article and future research directions from the following aspects:

(1)

The actual geological conditions of slope engineering are complex, and analyzing and studying only from a two-dimensional plane perspective may result in theoretical calculation results that deviate from the actual situation. Therefore, to fit the actual situation, it is necessary to establish corresponding three-dimensional models to study the corresponding boundary conditions and stress expressions. Further research is needed on the issues of three-dimensional soil wedges and retaining walls

(2)

In addition, the theoretical solution calculation method in this study assumes that the research object is a continuous homogeneous medium, and stress continuity is assumed during the solution process. In practical engineering, the parameters of soil wedges and retaining walls are complex and diverse, and factors, such as the position of the slip surface, rainfall, and groundwater seepage, can affect the calculation results. Therefore, establishing a model that considers discontinuous materials is the next direction of this theoretical calculation method.

5. Conclusions

Based on the boundary conditions, a new numerical method was presented and applied to analyze the stress and strain distribution characteristics of the clay soil wedge and retaining wall at the garbage transfer station in Lvcongpo town, Badong County. The following conclusions are drawn from the characteristics of the solution:

(1)

The proposed numerical method can account for different boundary conditions and rotation axes, and the calculated stress and strain have a non-linear relationship with the coordinates.

(2)

The proposed numerical theoretical solution, which satisfies the stress differential equations, compatibility equations, and boundary conditions, is capable of obtaining stress distributions at different points. The strain distribution in any direction within the soil wedge can be determined on the basis of the constitutive relationship between the principal stress and principal strain observed in experiments. Furthermore, by applying the current strength criterion, the location of the initial failure can be identified, thereby supporting the application of the point strength design criterion.

(3)

The proposed numerical method demonstrates that the solutions vary under different working conditions, allowing the most unfavorable conditions to be identified. Consequently, controlling the rotation point is essential in the design of retaining walls. In addition, the calculation method provides a theoretical basis for the anti-sliding design of retaining walls and other structures and establishes a foundation for slope control and monitoring. Furthermore, new control and prevention methods can be developed on the basis of the form of control and the materials used.

(4)

The numerical calculation results were compared with the finite element calculation results, revealing a small deviation of less than 5.05% between the two. The case study demonstrates that the proposed method can be widely applied in engineering practice, including dynamic and static loading and unloading analyses, as well as failure process investigations of roadbeds, tunnels, and dams.