A New Calculation Method of Force and Displacement of Retaining Wall and Slope

1. Introduction

2. Problem formulation

2.1. Side Slope Problems

For a long time, the finite unit method has been applied in slope analysis. The calculation block diagram used in the finite element calculation is shown in Figure 1, and its boundary conditions are often of two types. The first type is displacement boundary conditions. The horizontal and bottom vertical displacements around the perimeter in Figure 1 are equal to zero; that is, when the displacement on the left (point i) is equal to zero, the displacement on the right (point j) must also be equal to zero. According to the definition of the displacement calculation:

$$u_i-u_j={\displaystyle {\int }_i^j{\epsilon }_{i\to j}}dl$$

(1)

where $u_i,u_j$ denotes the displacements at points i and j in the figure, ${\epsilon }_{i\to j}$ denotes the strain in the ij segment, and dl denotes the integration along the i to j segments. From Equation (1), it can be seen that to make the ij segment displacement integral equal to zero, only the integration of strain in the ij segment is equal to zero; however, in the horizontal segment surface immediately below the bottom edge, the site should be dominated by compressive strain, and it is unlikely to produce tensile and compressive strain resulting in the sum of the ij segment displacement integral that is equal to zero.

The second type is the stress boundary condition. The boundary condition for the numerical analysis of the slope is shown in Figure 2. The far-field stress boundary condition is calculated according to the textbook of elastic mechanics. At the perimeter boundary, the expression of the boundary condition is as follows.

$${\left.{\left.{\sigma }_{xx}\right|}_{x=0}={\sigma }_{yy}\right|}_{x=0}=\frac{\upsilon }{1-\upsilon }{\gamma }_{z}z;{\left.{\sigma }_{zz}\right|}_{x=0}={\gamma }_{z}z;{\left.{\tau }_{xy}\right|}_{x=0}={\left.{\tau }_{yz}\right|}_{x=0}={\left.{\tau }_{xz}\right|}_{x=0}=0$$

(2)

where $\upsilon ,{\gamma }_y$ are the Poisson ratio and specific gravity of the geological material and ${\sigma }_{xx},{\sigma }_{yy},{\sigma }_{zz},{\tau }_{xy},{\tau }_{yz},{\tau }_{xz}$are the positive and shear stresses, respectively.

The physical meaning of Equation (2) is.

$${\left.{\left.{\sigma }_{xx}\right|}_{x=0}={\sigma }_{yy}\right|}_{x=0}={\sigma }_3;{\left.{\sigma }_{zz}\right|}_{x=0}={\sigma }_1;{\left.{\tau }_{xy}\right|}_{x=0}={\left.{\tau }_{yz}\right|}_{x=0}={\left.{\tau }_{xz}\right|}_{x=0}=0$$

(3)

where ${\sigma }_i,i=\mathrm{1,2},3$ is the principal stress, and the corresponding y-value magnitude can be calculated by substituting Equation (3) into any current strength criterion. According to the inequality, when the numerical calculation results in a the y-axis value greater than the y-value obtained according to the strength criterion, damage has occurred at the site, i.e., with the increase in the calculated depth, the damage area becomes increasingly large; however, the site becomes increasingly stable with increasing depth, which is not compatible with the site. A correct boundary condition corresponds to a correct solution; if the boundary condition is incorrect, the solution is incorrect.

3. Numerical theoretical solutions for side slopes and retaining walls

3.2. Example of a two-dimensional slope retaining wall

Using the basic ideas presented above, the study of the theoretical solution to the two-dimensional slope plane strain retaining wall problem, as an example, is illustrated as follows:

For ①, based on the precisely measured macroscopic geometric features, the geometric characteristic descriptive equations associated with the object of study are established as in Figure 3, and they can be written in the form of $y=kx+b$ to represent the geometric characteristic descriptive equations for the boundaries of AB, BC, CD, DA and FB, respectively (Note: if the boundaries are in the form of curves, they can be described by the equations of curves).

Figure 3. Calculation model of slope and retaining wall.

Figure 3. Calculation model of slope and retaining wall.

For ②, when analyzing the weight distribution of the study object, its associated weight distribution equation is established as ${\gamma }_{w,x},{\gamma }_{w,y}$.

For ③, when analyzing the stress characteristics of the research object, the stress equations for the boundary conditions relative to the boundary conditions are established according to the boundary conditions in different cases; for the planar problems related to Figure 3, such as horizontal stress (${p_x}^{AB}$ ) and vertical stress (${p_y}^{AB}$ ) on the AB boundary surface, the expressions are

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

(4)

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

(5)

where $l,m$ is the outer normal direction cosine on the AB boundary and ${{\sigma }_{xx}}^{AB},{{\sigma }_{yy}}^{AB},{{\tau }_{xy}}^{AB}$ is the boundary stress on the AB side. Under the condition of stress continuity, Equations (4) and (5) can describe all the different boundary condition stresses corresponding to Figure 3.

In the case of stress discontinuity, the stress at the discontinuity is not equal; however, the boundary condition stress forces need to be balanced along the horizontal and vertical directions.

For ④, the representation of the stress equation is chosen, and the corresponding constant coefficients are calculated in the case that the object of study satisfies the corresponding equilibrium, stress boundary conditions and coordination equations; in the two-dimensional condition, the stress expression is written (note: it can be changed according to different situations) as follows:

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

(6)

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

(7)

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

(8)

The corresponding specific gravity equation (note: other representations are also possible) is written as follows:

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

(9)

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

(10)

where $a_{1,i},a_{2,i},a_{3,i},a_{4,i},a_{5,i}$ is the constant coefficient term, i is taken as zero and an integer, ${\sigma }_{xx},{\sigma }_{yy},{\tau }_{xy}$ is the stress and shear stress in the X- and Y-axis directions, respectively, and${\gamma }_{w,x},{\gamma }_{w,y}$ is the specific gravity in the X- and Y-axis directions, respectively.

Under gravity conditions, the general solution of the above stresses satisfying the equilibrium equation can be expressed as

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

(11)

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

(12)

Then, the corresponding coefficients are zero, which is a necessary condition for the stress balance equation (note: equations (11, 12) can also be used to study the specific gravity relationship), and the following relationship can be obtained from Equation (11):

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

(13)

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

(14)

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

(15)

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

(16)

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

(17)

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

(18)

$$\cdots \cdots$$

From Equation (12), we have:

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

(19)

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

(20)

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

(21)

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

(22)

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

(23)

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

(24)

$$\cdots \cdots$$

By taking the specific gravity as a constant (${\gamma }_{w,x}=0,{\gamma }_{w,y}\ne 0=\gamma$), the relative satisfaction of the boundary conditions, equilibrium equations, etc., can be solved for all constant coefficients, and when they are substituted into the stress expression, the stress solution of the object of study can be solved by Equations (6~8).

For ⑤, the damage characteristics of the object of study are determined by combining the current strength criterion with the specific analysis of its force characteristics; the corresponding intrinsic structure model is selected, the deformation characteristics are studied, and the behavior associated with them is further clarified by comparison with the field. The analysis process is as follows: first, the calculation process expressed in the paper is used to obtain the stress theory solution and then calculate the corresponding principal stress magnitude and substitute it into the strength criterion (e.g., the Moore-Coulomb criterion, Griffith criterion, etc.) to determine the damage state point, and then it is combined with the current strength theory to further determine the damage direction to determine the stress damage path. When the damage driving force is greater than its strength (i.e., : stress discontinuity), there exists stress discontinuity and displacement discontinuity. While solving the stress discontinuity solution according to the above basic methods (① to ④), the damage path of the study object can be determined. The displacement continuity is solved by using coordinate rotation while considering the stress continuity, calculating the corresponding principal strain with the principal equation, and using the principal strain to obtain the strain at any point of the object of study. To solve the stress‒strain problem under the premise of stress discontinuity, it is necessary to consider the deformation characteristics for calculation. By following the above solution steps, the stress‒strain solution of the object of study during the whole damage process can be solved, and the relevant physical-mechanical parameters in the theory can be corrected (i.e., inverse analysis) by comparing the actual situation in the field.

3.4 Computational Analysis

3.4.1 Example of a waste transfer station project

The Shennong Creek Area Garbage Transfer Station Project is located in Fengjia Dagou, Group 1, Wulidui Village, Shennong Creek Area, Guandukou Town, Badong County, Hubei Province, and National Highway 209 crosses the west side of the site. The project covers an area of 1443.06 m2 , and the basic features of the garbage transfer station are as follows: the platform elevation is 283 m, the bottom elevation of the retaining wall is 274.8~283 m, and the slope height is 0~8.2 m (see Figure 4 and Figure 5). The surface layer of the waste transfer station is red clay and was formed according to the I-I section (see Figure 5). The lower part of the transfer station is composed of T b22 sandstone with high strength, uniaxial compressive strength 40~60 MPa, and rock inclination 260 ~300, and its retaining wall foundation is located above the weathered sandstone.

3.4.2. Calculating the Model Dimensions

According to the I-I profile of the Guandukou waste transfer station, the specific gravity of the backfill clay of the waste transfer station is set to $19.6\mathrm{kN}/{\mathrm{m}}^3$, the friction angle is 18°, and the basic dimensions of the ABCD area of the slope are AB=6.8 m, BC=10.48 m, CD=6.14 m, and DA=2.12 m. The retaining wall ABEF is C25 plain concrete, and the basic dimensions are AB=6.8 m, BF=4.2 m, EF=5.8 m, AE=1.62 m , EF = 5.8 m, and AE = 1.62 m (see Figure 3). The specific gravity is set to $25\mathrm{kN}/{\mathrm{m}}^3$

3.4.3. Analysis of the calculation results

3.4.3.1. Stress calculation results of the slope and retaining wall

According to the calculation of the slope retaining wall model, the coordinates corresponding to each point in the model are determined, the geometric boundary descriptive equations are established, the conditions associated with each boundary are determined, and the correlation coefficients under different discount factors ($f=\mathrm{1.00,1.50,2.00}$) are solved. The correlation coefficients are solved under different discount factors. The obtained coefficients are substituted into (6~8) to obtain the distribution of ${\sigma }_{xx}、{\sigma }_{yy}、{\tau }_{xy}$ (see Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14) and the distribution of the principal stress ${\sigma }_1、{\sigma }_3$ (see Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20) at any point according to the corresponding coordinate points. Assuming that the peak stress of the retaining wall satisfies the Moore Coulomb criterion, its friction angle is set to $\phi =40^0$. According to the stress distribution characteristics of the retaining wall, the corresponding distribution of the cohesive force $C$ values can be calculated, as shown in Figure 21, Figure 22 and Figure 23. According to the magnitude of the $C$ values, it can be determined where the retaining wall will be damaged first.

3.4.3.2. Slope strain calculation results

Using the intrinsic structure relationship, the strain of the clay slope body is obtained according to the test, and it satisfies the Duncan-Zhang intrinsic structure model, whose basic equation is

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

(34)

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

(35)

where ${\epsilon }_1,{\epsilon }_3$ represents the first and third principal strains, respectively. According to the test results, $a_1$ is 0.0002, $a_2$ is 0.00012099, $b_1$ is -0.000056, and $b_2$ is 0.0002099. The distribution characteristics of the principal strains of the slope under different reduction factors can be obtained (see Figure 24, Figure 25, Figure 26, Figure 27, Figure 28 and Figure 29)

For a two-dimensional problem, the expression for the strain (${\epsilon }_{ij}$) in either direction of the rotation (rotation angle $\varphi$) is

$${\epsilon }_{xx}={\epsilon }_{1}co{s}^2\text{}\varphi +{\epsilon }_{3}si{n}^2\text{}\varphi$$

(36)

$${\epsilon }_{yy}={\epsilon }_{1}si{n}^2\text{}\varphi +{\epsilon }_{3}co{s}^2\text{}\varphi$$

(37)

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

(38)

where ${\epsilon }_{xx},{\epsilon }_{yy}，{\gamma }_{\mathrm{xy}}$ indicates the strain and $\varphi$ indicates the rotation angle.

The rotation angle $\varphi$ is determined by the following equation:

$$tan\text{}2\varphi =\frac{-2{\tau }_{xy}}{{\sigma }_x-{\sigma }_y}\Rightarrow \varphi =\frac{1}{2}arctan\text{}\left(\frac{-2{\tau }_{xy}}{{\sigma }_x-{\sigma }_y}\right)$$

(39)

The calculated strain distribution obtained for each point of the sliding body is shown in Figure 30, Figure 31, Figure 32, Figure 33, Figure 34, Figure 35, Figure 36, Figure 37 and Figure 38.

3.4.3.3. Results of the strain calculation for the retaining wall

Calculating the retaining wall strain can be considered a plane strain problem when ${\epsilon }_z=0$ and ${\sigma }_z\ne 0$. The general form according to Hooke's law is expressed as follows:

$${\epsilon }_{xx}=\frac{1}{E}[{\sigma }_{\mathrm{xx}}-\mu ({\sigma }_{\mathrm{yy}}+{\sigma }_{\mathrm{zz}})]$$

(40)

$${\epsilon }_{zz}=\frac{1}{E}[{\sigma }_{\mathrm{zz}}-\mu ({\sigma }_{\mathrm{xx}}+{\sigma }_{\mathrm{yy}})]$$

(41)

$${\gamma }_{xy}=\frac{{\tau }_{xy}}{G},{\gamma }_{yz}=\frac{{\tau }_{yz}}{G},{\gamma }_{xz}=\frac{{\tau }_{xz}}{G}$$

(42)

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

E denotes the modulus of elasticity and is set to 300 MPa, G expresses the shear modulus, and $\mu$ denotes the Poisson ratio, which is 0.11.

According to the above process, the strain distribution solutions of the retaining wall can be obtained, as shown in Figure 39, Figure 40, Figure 41, Figure 42, Figure 43, Figure 44, Figure 45, Figure 46 and Figure 47. The corresponding principal strains are shown in Figure 48, Figure 49, Figure 50, Figure 51, Figure 52 and Figure 53.

3.4.3.4. Retaining wall stability analysis

From the calculation results of the retaining wall, it can be seen that the maximum tensile stress occurs at the corner of the wall and the slip body reduction factor becomes increasingly large (f=1.00,473.17 kPa; f=1.50,500.82 kPa; and f=2.00,530.71 kPa), but its value falls within the strength range of C25 plain concrete; the absolute value of the compressive stress of the retaining wall increases with the slip body reduction factor and becomes increasingly larger (-537.74 kPa, -600.51 kPa, and -640.35 kPa), but it also falls within the strength range of plain concrete and foundation rock; for the C25 plain concrete retaining wall, the absolute value of the compressive stress increases with the slip body reduction factor. The absolute value of the compressive stress of the retaining wall increases with the discount factor of the sliding body (-537.74 kPa, -600.51 kPa, and -640.35 kPa), but it also falls within the strength range of plain concrete and foundation rock; for the C25 plain concrete retaining wall, the maximum value of the cohesion of the back calculated points is 323.76 kPa under the condition that the friction angle is equal to 40 degrees, which also falls within the strength range of C25 plain concrete. The maximum value of cohesion is 323.76 kPa, which also falls within the cohesion value range of the C25 plain concrete strength, and the corresponding principal strain of the retaining wall is 10^-3 level, which is within the peak strain range. Through the stress and deformation analysis of the retaining wall, the stress and strain at the point of the retaining wall are within the strength range, which means that the conditions under which damage occurs do not exist for the retaining wall. For the above problem, the finite unit method and Ansys software were used to calculate the error. The deviation in the stress‒strain calculation for the backfill clay at the transfer station was less than 12%, and the deviation in the retaining wall result was less than 5%. Based on the analysis of the regulated retaining wall and the analysis results presented in this paper, the whole retaining wall is in a stable state, which is consistent with the results of many years of field operation.

4. Conclusion

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