1. Introduction
The slip-line field method is a classic theory for calculating earth pressure [
1]. Other methods are the limit equilibrium method [
2], limit analysis [
3], and numerical method [
4,
5]. The limit equilibrium method calculates earth pressure by assuming the shape of a sliding surface [
6,
7]. The limit analysis method is used to obtain the upper and lower bound solutions of earth pressure by constructing a displacement field with mobility permission and a statically permissible stress field [
8,
9,
10,
11]. Among other methods, the slip-line field method is a relatively rigorous method to obtain earth pressure as well as slip-line fields [
12]. Xiong and Wang (2020) [
13] proposed an approach to solve axisymmetric active earth pressure problems by using a rigorous characteristic line theory. Chen et al. (2022) [
14] proposed a slip-line solution to earth pressure of narrow backfill. However, in some cases, stress discontinuity may occur in a slip-line field. How to determine the position of stress discontinuity is crucial to the computation of slip-line fields. Lee et al. (1972) [
15] proposed a theoretical solution for the lateral pressure of a rigid retaining wall considering stress discontinuity, and a cycling process of stress discontinuity line was proposed. Zhu et al. (2001) [
16] proposed the limit condition for the occurrence of stress discontinuity by using the relationship between the wall inclination, the soil–wall interface friction, the backfill surface inclination, and the backfill internal friction for weightless soil. Peng et al. (2002) [
17] found that a folding phenomenon occurs in slip-line fields if the parameters in the retaining wall system meet a certain condition, and they explained that it is the only virtual solution for the boundary value problem in the mathematical sense. Based on the stress discontinuity judgment method proposed by Sokolovskii, Kumar and Chitikela (2002) [
18] studied a method to calculate seismic passive earth pressure coefficients. Liu and Wang (2008) [
19] put forward the occurrence conditions of stress discontinuity and proposed two calculation procedures of active earth pressure for a circular retaining wall. The existence of stress discontinuity was also considered in Keshavarz and Ebrahimi (2017) [
20]’s study, which mainly proposed a method of solving axisymmetric active earth pressure using the method of stress characteristics. Liu et al. (2018) [
21] proposed the conditions of concave failure surface, convex failure surface, and plane failure surface and used the logarithmic spiral method to calculate passive earth pressure; the result for the concave failure surface indicates that there is stress discontinuity in the slip-line field. There was a similar phenomenon in Zhu et al. (2000) [
22] ’s study, which calculated the critical slip field of earth pressure based on the triangular slice method. In addition, Li and Jiang (2022) [
23] proposed a solution to solve the ultimate bearing capacity of strip footings by considering stress discontinuity in slip-line fields. Smith and Gilbert (2022) [
24] established the stress function basis of the upper bound theorem of plasticity by considering a discontinuous slip line. In the author’s previous study, a new method for computing earth pressure slip-line fields was proposed [
25], which did not take stress discontinuity into account.
Therefore, a numerical method for computing slip-line fields with stress discontinuity was proposed in the present study. Based on the triangular slice method, the force and moment equilibrium equations for a typical slice are established. The relationship between the inclination of the interslice force and the inclination of the failure surface is established by considering the Mohr–Coulomb criterion. An iterative procedure for calculating the lateral force of the triangular slices by changing the failure surface inclination in the transition zone is performed until the interslice force satisfies the stress condition of the transition zone boundary, resulting in a stress discontinuity line if the Rankine zone and the transition zone intersect. It should be noted that the equations derived for the presented active case are also applicable to passive cases if the signs of the soil–wall interface friction angle and the backfill internal friction angle are changed, that is δ0 = −δ0 and φ = −φ.
2. Fundamentals
Figure 1 shows a typical stress system of a retaining wall in an active case. The potential failure zone can be divided into the Rankine zone and the transition zone according to the soil plasticity theory. If there is no stress discontinuity, the line OA is the transition ray between the two zones, and it is a slip line, as shown in
Figure 1a. Otherwise, the Rankine zone and the transition zone will intersect, as shown in
Figure 1b, where the line OD is the intersection line of the two zones, and the line OA is the boundary of the Rankine zone. The inclination of the line OA, i.e.,
θR, and the inclination of the Rankine zone failure surface, i.e.,
αR, are drawn from Zhu et al. (2001) [
16], as shown in Equations (1) and (2). Additionally, the two conjugate slip lines within the Rankine zone intersect each other at an angle of 90° −
φ/2.
According to the theory of plasticity, the normal stress and the shear stress in the normal direction along both sides of the stress discontinuity line are equivalent (Equations (3) and (4)). Meanwhile, the stress along the tangential direction of the stress discontinuity line is discontinuous, and the stress discontinuity line must not be a stress slip line, as shown in
Figure 2. Thus, if the stress characteristics along both sides of the intersection line OD are consistent with the stress discontinuity line, the intersection line is the stress discontinuity line.
In a retaining wall system,
H is the wall height;
ε is the wall back inclination from a vertical line (
ε is positive when the wall back inclines to the backfill, as shown in
Figure 1);
β is the inclination of the backfill (
β is positive when the slope is higher than the horizontal plane, as shown in
Figure 1);
δ0 is the friction angle of the soil-wall interface, which is positive when the wall is far away from the backfill, as shown in
Figure 1;
γ is the unit weight of the backfill;
φ is the internal friction angle of the backfill; and
Pa is the active earth pressure. Since the backfill is cohesionless and free of surcharge, the earth pressure is proportional to the wall height in a triangular distribution, with an acting point of resultant force located at one third of the wall toe. Therefore, the shapes of slip lines at different heights are similar.
4. Numerical Procedure
For a typical retaining wall, the value of the wall inclination,
ε, the friction angle of the soil–wall interface,
δ0, the internal friction angle of the backfills,
φ, the unit weight of the backfills,
γ, and the backfills’ surface inclination angle,
β, can be determined in advance. Firstly, for a retaining wall with a wall height of
H, whether there will be stress discontinuity needs to be determined according to Equation (28). If there is no stress discontinuity, the slip-line fields can be obtained according to the previous study [
25]. To computing slip-line fields with stress discontinuity, the potential failure zone is divided into the Rankine zone and the transition zone; the Rankine zone can be determined strictly based on the theory of plastic mechanics; and the boundary of the transition zone needs to be determined using an iterative procedure. The lateral pressure coefficient and the interslice force inclination angle of the slice lines within the Rankine zone and the transition zone can be calculated by using an iterative procedure, and then we can determine the location of the stress discontinuity line. Finally, the earth pressure and the slip-line fields with stress discontinuity can be obtained. The main calculation process are as follows:
(1) Divide the Rankine zone into m triangular slices, and the included angle of each slice is Δθ. According to Equations (11) and (16), the corresponding interslice force inclination angle of each slice line, i.e., δRi, and the lateral pressure coefficient, i.e., KRi, can be calculated.
(2) Divide the potential zone into
n triangular slices, and the included angle of each slice is Δ
θ. Assuming that the earth pressure coefficient
Ka(j=1) in the first calculation process is the Coulomb earth pressure coefficient, begin with the wall back and calculate the lateral pressure coefficient, i.e.,
Kk, and the interslice force inclination angle, i.e.,
δk, of each slice line successively according to the recursive equation (Equations (23)~(26)); for the iterative process of
δk,, refer to the authors’ previous study [
25]. If the inclination angle of the interslice force on the
kth slice line is equal to the internal friction angle, that is,
δk = φ, stop the calculation, and the inclination angle of the
kth slice line is
θkφ. Then, the following judgment needs to be made immediately:
(a) If θkφ(j=1) = θR, and the lateral pressure coefficient and its inclination of the kth slice line within the Rankine zone are equal to those within the transition zone, that is, Kk = KR (θkφ = θR) and δk = δR = φ (θkφ = θR), Ka(j=1) is the active earth pressure coefficient Ka, αk(k=1,2,…,k), αR is the inclination of the failure surface within the transition zone and the Rankine zone. The kth slice line is the transition ray between the Rankine zone and the transition zone, and it is a slip line. It indicates that there is no stress discontinuity in the slip-line field.
(b) If θkφ(j=1) < θR, and the transition zone intersects the Rankine zone, it is necessary to judge whether there is an intersection line satisfying the features of stress discontinuity, that is, the lateral pressure coefficient and its inclination angle on the intersection line calculated by the Rankine zone equation (Equations (11) and (16)) are equal to those calculated by the transition zone equation (Equations (23)~(26)). (If Δ ≤ 10−3, and , it is considered to satisfy the condition of
equality, where Δδ is the difference of the interslice force inclination angle of the intersection line calculated by the Rankine zone equation and the transition zone equation, and is the average value. ΔK is the difference of the lateral pressure coefficient value of the intersection line calculated by the Rankine zone equation and the transition zone equation, and is the average value.). If there is an intersection line satisfying the conditions, that is, Δ ≤ 10−3, the intersection line is the stress discontinuity line, which inclination angle is denoted by θray; αk(k=1,2,…,k)
is the inclination of the failure surface in the transition zone; and the value of Ka(j=1) is the active earth pressure coefficient Ka. If Δ > 10−3, change the earth pressure coefficient, Ka(j), and repeat step (2) until the stress features of the intersection line in step 2(a) or step 2(b) is satisfied.
(c) If θkφ(j=1) > θR, there is no intersection line between the transition zone and the Rankine zone. Hence, it is necessary to change the earth pressure coefficient, Ka(j=1) (increase the earth pressure coefficient value in an active case), so that Ka = Ka(j=2), repeat step (2) until θkφ ≥ θR, and perform the judgment of step 2(a) or step 2(b).
(3) Take the calculated transition ray or stress discontinuity line as the boundary line, and plot the failure lines within the Rankine zone and the transition zone, respectively. Another conjugate slip line can be constructed according to the condition that the two slip lines intersect each other at an angle of π/2 − φ, and, thus, slip-line fields with stress discontinuity can be obtained.
6. Discussion
There is an ultimate inclination of a retaining wall, which refers to the maximum inclination of a retaining wall when there is no stress discontinuity in the slip-line field; that is, if the inclination of the retaining wall is greater than the ultimate inclination, stress discontinuity will occur in the slip-line field.
Figure 12 indicates that the ultimate inclination decreases with an increase in internal friction angle, in other words, the greater the internal friction angle, the easier stress discontinuity will appear in the slip-line fields. From
Figure 13, it can be seen that the ultimate inclination of a retaining wall increases with an increase in the soil–wall interface friction angle. If the soil–wall interface friction angle is less than 25°, the ultimate inclination is basically consistent with the slope of the increase in the soil–wall interface friction angle. When the soil–wall interface friction angle is greater than 25°, the ultimate inclination of the wall back increases significantly, and the slope is much larger than the small soil–wall interface friction angle; this feature can also be seen in
Figure 12. It can be seen from
Figure 14 that with an increase in the backfills’ inclination, the ultimate inclination of a retaining wall gradually decreases. Moreover, it can be clearly seen that when the soil–wall interface friction angle is equal to the internal friction angle, the ultimate inclination increases significantly. Finally, we can draw the conclusion that the soil–wall interface friction angle has a great impact on stress discontinuity.
Table 2 shows the calculation results of active earth pressure coefficients and position of stress discontinuity with a smooth retaining wall,
δ0 = 0°, and a horizontal backfill surface,
β = 0°. The wall inclination angle and the backfill internal friction angle are variable. It can be seen that there is no stress discontinuity in the slip-line field when the wall is vertical or inclined to a free position, that is,
ε ≤ 0°. With an increase in the wall inclination, the slip-line field appears to have stress discontinuity, and the location of the stress discontinuity line gradually moves to the backfill surface with the increase in the wall inclination.
Table 3 shows the calculation results of active earth pressure coefficients and location of stress discontinuity with
ε = 10°,
β = 10°, and
φ = 30°. It is obvious that with a decrease in the soil–wall interface friction angle, the slip-line field gradually shows stress discontinuity, and the smaller the soil–wall interface friction angle, the smaller the inclination angle of the stress discontinuity line, that is, the closer to the backfill surface.
7. Conclusions
In this paper, a numerical method is presented for computing slip-line fields with stress discontinuity lines in cohesionless backfills, and the corresponding earth pressure coefficients. The position of the stress discontinuity line can be accurately calculated. By dividing the potential failure zone into the Rankine zone and the transition zone, the position of the Rankine zone can be strictly determined by using the plasticity theory, and the transition zone is determined by using the numerical method presented in this paper. The potential failure zone is divided into a series of triangular slices, each of which satisfies the force and moment equilibrium conditions and the Mohr–Coulomb criterion. An iterative procedure is proposed for adjusting the inclination of the failure surface to ensure that the stress boundary conditions of the transition zone are satisfied. If the stress discontinuity line occurs in the slip-line field, there is an intersection line between the Rankine zone and the transition zone. The stress characteristics on both sides of the intersection line satisfy two conditions: (i) the lateral pressure coefficient on both sides of the intersection line is equal, and (ii) the interslice force inclination angle on both sides of the intersection line is equal, but not equal to the internal friction angle. Additionally, the two sides of the intersection line refer to the Rankine zone and the transition zone. Since the stress discontinuity line and the first family of slip lines have been obtained, the inclination of the second family of slip lines can be calculated according to the relationship that two conjugate slip lines intersect at an angle of π/2 − φ in an active case.
The method for computing slip-line fields with stress discontinuity presented herein is applicable to a retaining wall with an inclined cohesionless backfill surface without surcharge. Several examples are provided to demonstrate slip-line fields with stress discontinuity, and the earth pressure coefficients are in fair agreement with those of other numerical methods. In addition, the effects of retaining wall inclination, the soil–wall interface friction angle, the backfill friction angle, and the backfill inclination on stress discontinuity are discussed. The results indicate that the soil–wall interface friction angle has the most obvious effect on stress discontinuity. Since the shape of the failure surface and the interslice force inclination angle are not assumed, the results calculated by the present method are strictly approximate to the solution obtained based on the theory of plastic mechanics. If the internal friction angle, φ, and the friction angle of the soil–wall interface, δ0, are replaced by −φ and −δ0, the equations derived in this paper are also applicable to a passive case.