Research on the influence of width-height ratio and internal friction angle of the TT mode on the trapezoidal sliding surface of backfill behind the retaining wall

2.1. Model test device

The design of the model test device, parameters, etc. are detailed in the previous research results of the research group by Liu [41, 42]. The model pictures are shown in Fig. 2.

The soil filling box is rectangular in shape, with dimensions of 1200 mm in length, 400 mm in width, and 700 mm in height. The front panel of the box is made of transparent glass with a thickness of 16 mm. The right side features a steel plate serving as a movable retaining wall, whereas the left side consists of a steel plate functioning as a fixed retaining wall.

Fig. 2Model testing equipment

2.2. Filler parameters

The fill material used in this model test was sand, which was taken from Dongtongdian Lake in Yueyang, Hunan Province, China. Before the formal model test, a relationship test between the internal friction angle $\phi$ and density $\rho$ of the sand was conducted. The density of the fill soil was obtained by controlling the number of hammer blows on the compaction plate to achieve the compaction degree. The internal friction angle of the fill soil was obtained through direct shear tests, and the results are shown in Table 1. The process of direct shear tests for the compaction degree, density and internal friction angle are detailed in Liu [41, 42].

The data in Table 1 was subjected to linear fitting. The fitting formula is detailed in Eq. (1), with a correlation coefficient $R^2=$0.990:

where $\rho$ is the sand density and $\phi$ is the sand internal friction angle.

Table 1Internal friction angle of different density sandy soils

Impact number $N$	Compaction degree $\lambda$ (%)	Density $\rho$(g/cm-3)	Internal friction angle $\phi$ (°)
0	80.8	1.41	34.9
1	82.6	1.45	36.0
3	83.7	1.47	36.5
5	85.5	1.50	37.3
10	87.2	1.53	38.1
20	92.4	1.62	40.5
30	85.3	1.67	41.9
40	97.1	1.70	42.7
50	98.3	1.72	43.2
60	98.8	1.73	43.5
70	98.8	1.73	43.5

2.3. Model test scheme

To explore the combined influence of width-height ratio and internal friction angle on the active failure mode of finite soil mass. As shown in Table 2, five representative internal friction angles from Eq. (1) were selected, and a total of nine model test schemes were designed. The test operation procedures can be found in the previous work of our research group by Liu [41, 42].

2.4. Introduction to numerical simulation

Numerical simulation is conducted by using PLAXIS^2D software. The model design, basic assumptions and parameters of retaining walls, are detailed in the published results of our research group by Liu [41, 42]. The numerical simulation is based on the following assumptions: The soil behavior is modeled using the Mohr-Coulomb constitutive model, and discretized using 15-node triangular elements. The retaining walls on both sides are simulated using a linear elastic model, with interface elements employed to represent the interaction between the retaining walls and the backfill soil. Rigid retaining walls are modeled using plate elements. Full constraints are applied at the base of the model and on the fixed retaining wall, while the movable retaining wall is constrained only in the $Y$-direction. The parameters of the filler materials are detailed in Table 3.

3.1. The influence of the fill soil width-height ratio on the active sliding surface

In order to better analyze the influence laws of width-height ratio on the morphological characteristics of active sliding surfaces, different width-height ratio active sliding surfaces are plotted on the same figure, the characteristics of soil slip failure surfaces with different width-height ratios are shown in Fig. 3. In Fig. 3, the fitting curves corresponding to width-height ratio of $n=$ 0.2, 0.3, 0.4, 0.5, and 0.6 represent the measured values, whereas the Coulomb curves denote the predicted values.

Fig. 3Characteristics of soil slip failure surfaces with different width-height ratios

As shown in Fig. 3, when width-height ratio $n\le$ 0.4, for a smaller width-height ratio, the initial failure angle of the soil increases, and the sliding surface becomes steeper. When width-height ratio $n\le$ 0.5, for finite width soil, the active failure forms multi-segment folded-line sliding surface. When width-height ratio $n=$ 0.2, there are three segmented folded-line sliding surfaces; when width-height ratio $n=$0.3, 0.4, and 0.5, there are two segmented folded-line sliding surfaces. The fewer the width-height ratio, the more the line segments of the sliding surface. The lower the width-height ratio, the lower the height of the first line segment of sliding surface. When width-height ratio $n=$ 0.5, it is approximately 1/16$H$; when width-height ratio $n=$ 0.4, it is approximately 2/10$H$; when width-height ratio $n=$ 0.3, it is approximately 3/8$H$; and when width-height ratio $n=$ 0.2, it is approximately 9/20$H$. The failure angles of the sliding surfaces for width-height ratio $n=$ 0.5 and width-height ratio $n=$ 0.6 are basically consistent with those of the Coulomb theory sliding surface, and the three sliding surfaces in the figure are basically superimposed. The critical width-height ratio of the finite soil is between 0.5 and 0.6. When width-height ratio $n>$ 0.6, the active sliding surface starts from the wall heel and is a straight line. The shape and failure angle ($ɑ\approx$ 61.25°) are basically consistent with the Coulomb theory (45°+ $\phi$/2 = 62.45°), and it is a semi-infinite soil.

3.2. The influence of the internal friction angle of backfill on the active sliding surface

In order to better analyze the influence law of the internal friction angle on the morphological characteristics of the active sliding surface, different active sliding surfaces with different internal friction angles are plotted on the same graph. The characteristics of sliding failure surfaces of soil with different internal friction angles are shown in Fig. 4. In Fig. 4, the fitting curves corresponding to internal friction angles of $\phi =$ 34.9°, 35.6°, 37.7°, 41.0°, and 43.1° represent the measured values, whereas the Coulomb curve for $\phi =$ 34.9° represents the predicted value.

Fig. 4Characteristics of sliding failure surfaces of soil with different internal friction angles

As shown in Fig. 4, The fracture angles and the forms of the broken lines of the sliding failure surfaces with internal friction angles of $\phi =$ 34.9°and $\phi =$ 35.6° are basically the same, both start from the heel of retaining wall and end at the surface of the fill soil. The failure surfaces consist of three segments, the first segment of the failure surface starts from the heel of the movable retaining wall and ends at about 1/2$H$ of the fixed retaining wall; the second segment starts from about 1/2$H$ of the fixed retaining wall and ends at about 1/8$H$ of the movable retaining wall; the third segment starts from about 1/8$H$ of the movable retaining wall and ends at the surface of the soil mass. The larger the internal friction angle, the greater the fracture angle of the sliding surface, and the steeper the sliding surface. The smaller the internal friction angle, the more the number of broken line segments on the sliding surface; the more obvious the finite soil constraint, the more obvious the failure mode of the sliding surface broken line.

4.1. Define the “length factor” of the sliding fracture surface

The folded-line sliding surface is composed of multiple sliding surfaces that are folded and terminate between the fixed retaining wall and the movable retaining wall. The linear sliding surface is a straight line. To better express the quantitative relationship of the number of sliding surface segments, the definition of “length factor” for sliding surface is introduced. Project each sliding surface onto the horizontal direction and adopt normalisation processing. The ratio of the total length of the projection of each segment of the sliding surface in the horizontal direction to the width of the fill soil is called the “length factor”, represented by $N$:

where $L$ denotes the horizontal projection length of each segment of the sliding surface, while $B$ refers to the width of the backfill.

As illustrated in Fig. 5, using the numerical simulation results obtained with an internal friction angle of 37.7° and a width-to-height ratio ($n$) of 0.5 as an example, the calculation method for the “length factor” of the sliding surface is demonstrated.

Fig. 5Schematic diagram of the length factor of the sliding failure line (φ = 37.7°, n= 0.5)

As shown in Fig. 5, $L_{ab}$ denotes the initial segment of the sliding surface (labeled as segment ab in the figure), which extends from the heel of the movable retaining wall to the fixed retaining wall, with a horizontal projected length of 1.0B; $L_{bc}$ represents the second segment of the sliding surface (denoted as the bc section in the figure), extending from the endpoint of the first segment to the surface of the fill material. The horizontal projection length of $L_{bc}$ is 0.2B; $L$ denotes the total horizontal projection length of all segments of the sliding surface, calculated as $L=L_{ab}+L_{bc}$ = 1.0$B$ + 0.2$B$ = 1.2$B$, therefore, length factor $N=$1.2.

$N$ denotes the ratio of the total horizontal projection length of all line segments along the sliding surface from the starting point to the endpoint, to the width of the fill material. When $N=$ 1, it represents the critical condition for transitioning a semi-infinite soil mass to a finite one. If $N<$1, the soil behind the retaining wall remains a semi-infinite soil mass, and the corresponding sliding surface conforms to the classical Coulomb failure criterion. In this case, earth pressure can be calculated using classical earth pressure theory. However, when $N>$ 1, the soil behind the wall is classified as a finite soil mass, and the sliding surface no longer aligns with the classical Coulomb assumption. Consequently, the mechanical behavior and deformation characteristics of the soil deviate from the assumptions of classical Coulomb theory, and the earth pressure must be calculated using a method specifically designed for finite soil masses.

4.2. Increase the number of statistical samples

As stated by Liu [41, 42], the numerical simulation results are basically consistent with the model test results. The reliability of the numerical simulation for other working conditions is relatively high. To increase the statistical sample size, through numerical simulation, the research on other working conditions is obtained, 16 numerical simulation schemes are designed. The supplementary numerical simulation test schemes are shown in Table 4.

Numerical simulation was conducted on the scheme in Table 4, 16 sets of deformation results of sliding shear surfaces were obtained. The morphologies of sliding shear surfaces from both model tests and numerical simulations were summarized, and the summary of sliding failure surfaces under various working conditions are shown in Fig. 6.

Fig. 6Summary of sliding failure surfaces under various working conditions

a) Simulation result, $\phi =$34.9°

b) Simulation result, $\phi =$35.6

c) Simulation result, $\phi =$37.7°

d) Simulation result, $\phi =$41.0°

e) Simulation result, $\phi =$43.1°

As shown in Fig. 6, the width-height ratio and internal friction angle both have an impact on the morphology of the sliding fracture surface. The morphology of the sliding fracture surface changes from a three-segment line to a two-segment line, and finally becomes a single straight line. The folded-line sliding fracture surface represents the characteristic of finite soil failure, while the single straight line represents the characteristic of semi-infinite soil failure.

The number of broken line segments of the sliding fracture surface in Fig. 6 was normalized, and the corresponding length factors were obtained. The Length factor of slip surface for model tests and numerical simulations were summarized in Table 5 and Table 6.

4.3. Linear fitting of width-height ratio, internal friction angle and length factor of sliding fracture surface

As shown in Table 5 and Table 6, the length factor $N$ of the active failure sliding surface of the finite soil mass is closely related to the width-height ratio n and the internal friction angle. After fitting, the relationship curves as shown in Fig. 7 and Fig. 8 are obtained, the relationship between length factor with width-height ratio and internal friction angle of sliding fracture plane is shown in Eq. (3) and Eq. (4), respectively.

where $N_n$ and $N_{\phi }$ are the length factor of sliding fracture surface, $n$ is the width-height ratio of backfill behind the retaining wall, $\phi$ is the internal friction angle of backfill behind the retaining wall.

From Eq. (3) and Eq. (4), it can be seen that the critical value of the finite soil mass decreases linearly with the increase of the width-height ratio n and the internal friction angle. The correlation coefficients between them are $R^2=$0.983 and $R^2=$0.851, respectively; showing a highly linear correlation.

Fig. 7Relationship between length factor and width-height ratio of sliding fracture plane (φ= 34.9°)

Fig. 8Relationship between length factor and internal friction angle of sliding fracture surface (n= 0.2)

4.4. Surface fitting of width-height ratio, internal friction angle and length factor of sliding fracture surface

In order to better analyze the influence laws of the width-height ratio $n$ and the internal friction angle on the active sliding surface length factor $N$, a study on the influence of the width-height ratio n and the internal friction angle on the sliding surface length factor $N$ have been conducted.

Based on the data in Table 5 and Table 6, according to the test range and the verification results, the function relationship of the sliding surface length factor $N$ with the width-height ratio n and the internal friction angle $\phi$ was fitted. The final fitting formula is shown in Eq. (5):

where $N$ is the length factor of sliding fracture surface, $n$ is the width-height ratio of backfill behind the retaining wall, $\phi$ is the internal friction angle of backfill behind the retaining wall.

The statistical correlation coefficients associated with the fitting Eq. (5) are presented in Table 7.

From Eq. (5), it can be seen that the length factor $N$ of the sliding surface of the finite soil mass decreases with the increase of the width-height ratio $n$ and the internal friction angle. The correlation coefficient $R^2=$ 0.938, indicating a highly linear correlation.

The length factor $N$ of the sliding surface in the oblique view and bottom view varies with the width-height ratio n and the internal friction angle $\phi$ are shown in Fig. 9 and Fig. 10, respectively.

As shown in Fig. 9, the length factor $N$ of the sliding fracture surface has a nonlinear relationship with the width-height ratio $n$ and the internal friction angle $\phi$, and is distributed in a curved surface form. As shown in Fig. 10, the contour lines of the length factor $N$ of the sliding surface present an S shape distribution. The blue distribution area indicates that the critical width-height ratio ($n_{cr}$) of the finite soil mass is between 0.55-0.60; when the width-height ratio $n<$0.3, the number of the broken line segments of the sliding surface morphology increases sharply, and the characteristics of the finite soil mass change significantly. The red area is located in the upper left corner. The smaller the internal friction angle and the smaller the width-height ratio, the more broken line segments of the sliding surface morphology there are, and the more obvious the characteristics of the finite soil mass are. The width-height ratio has a significant impact on the finite soil mass, while the internal friction angle has a relatively smaller impact on the finite soil mass.

Fig. 9The oblique view on the length factor of sliding fracture surface varies with the width-height ratio n and the internal friction angle

Fig. 10The bottom view on the length factor of sliding fracture surface varies with the width-height ratio n and the internal friction angle

4.5. Verification of the formula for surface function

In order to verify the reliability of the fitting Eq. (5), the results on length factor of the sliding fracture surface N by model test in this paper and the calculation results of the fitting Eq. (5) are presented in Table 8.

In order to further verify the reliability of the fitting Eq. (5), the test results of the TT mode in Wang [10] were compared with the calculation results of the fitting Eq. (5). The test device in this paper and that of Wang [10] are of the same model, but the fillers used are different. The summary results are shown in Table 9.

From Table 8 and Table 9, it can be seen that the difference values between the model test results and the calculated results of the fitting Eq. (5) are less than 12.86 % except for one of 18.75 %, which are basically consistent. Through indoor model tests, the Eq. (5) has been verified to be reliable and worthy of promotion. The reasons for the relatively large individual differences are analyzed as follows: The numerical simulation simplifies the interaction in soil mass, ignores the internal action of sliding surface, and simplifies the contact relationship. Due to the limitations of the available verified data and statistical sources, the applicable range of the fitted Eq. (5) is defined as: 0.2 $\le n\le$ 0.6 and 30° $\le \phi \le$ 43.1°.

The fitting Eq. (5) can be applied to the determination of the failure mode of soil mass. By comparing the calculation results of Eq. (5) with the critical value “1” of the finite soil mass, the critical state is identified when $N=$ 1; when $N>$ 1, it indicates a finite soil mass with a multi-segmental line sliding surface morphology, and the earth pressure is calculated using the finite soil mass theory; when $N<$ 1, it represents a semi-infinite soil mass with a straight-line sliding surface, and the earth pressure is calculated using the classical Coulomb theory. The width-height ratio n and the internal friction angle $\phi$ are two common parameters in engineering practice. The fitting Eq. (5) has the advantages of simple parameters, easy calculation, and simple determination in the application process of engineering design and construction guidance.