# Active Earth Pressure for Sloping Finite Soil While Accounting for Shear Stress and Curved Slip Surface

Authors: Weiwei Wang; Xinxi Liu; Weidong Hu; Hua Luo; Guanghui Wang
Journal: Geofluids
 (2022)

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## Abstract

Calculating the earth pressure of the sloping soil having finite width behind the retaining wall is difficult for stability calculation. Thus, a novel method to calculate the active pressure of cohesionless sloping finite soil behind a retaining wall was developed to investigate. Taking cohesionless soil as the research object and considering the principal stress rotation of soil, the resultant force for active earth pressure, action point position, and earth pressure distribution of sloping finite soil was obtained based on assumptions of the translational mode of the rigid retaining wall and cycloidal slip surface. The accuracy of the proposed method was verified by model tests. The influence of the slope height ratiol/H and slope angle α on earth pressure was analyzed in this study. The result revealed that the horizontal component of the active earth pressure distribution curve for sloping finite soil was linear in area I and nonlinear with a drum shape in area II. There was a noticeable change at the junction of the two areas. The resultant force of earth pressure and the height of action point of resultant force increased and tended to reach a certain value as the aspect ratio l/H increased. When l/H≥0.4, the height of the action point of resultant force tended to be two-fifths the height of the wall. The resultant force and the height of the action point decreased linearly as the slope bottom angle increased.


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## Body

## 1. Introduction

With the continuing development of capital construction projects in recent years, a significant number of retaining structures of high embankment retaining walls and adjacent foundation pits have appeared in the city [1]. Due to the influence of height difference and slope construction, the sloping finite soil forms behind the retaining wall. When the soil mass is destroyed, the sliding fracture surface ends at the slope surface, which contradicts the classical earth pressure theory [2, 3]. The distribution of horizontal earth pressure is assumed to be triangular in the classical earth pressure theory. However, the existing experimental findings [4–11] suggest that the horizontal earth pressure distribution is nonlinear. Moreover, scholars [6, 12–15] also considered the soil arch effect in calculating earth pressure.The active earth pressure of finite soil between the rock face and the rigid retaining wall was studied by centrifugation tests [16, 17] and laboratory model tests [8, 18, 19]. The results revealed that finite soil has lower earth pressure than semi-infinite soil, and the distribution was nonlinear. Some scholars [20–25] have calculated the active earth pressure of finite soil by different methods. These researchers typically used the horizontal flat-element approach, and shear stress between soil layers was ignored in the force calculations. However, the horizontal shear stress between the element layers would be created by the primary stress deflection of the soil element induced by friction on the rear of the retaining wall ([26, 27]). In most research, finite soil’s sliding surface was considered a plane. At the same time, some model experiments have demonstrated that the sliding surface is a curved surface [8, 11]. Based on the curved sliding surface, several researchers [8, 28, 29] developed a theoretical formula for the earth pressure of finite soil between a rock face and a solid retaining wall. However, these researches focused on finite width soil near the building or bedrock. In addition, research on active earth pressure of sloping finite soil behind the wall is uncommon.Using the differential layer approach, scholars have conducted extensive studies on earth pressure distribution on finite soil. But the research on active earth pressure for sloping finite soil while accounting for shear stress and curved slip surface needs further development. Following methodology of Huang et al. [28], it was assumed in this study that the backfill had a cycloidal slip surface of the finite soil in translation mode. The horizontal flat-element method calculates the distribution of active earth pressure while accounting for the shear stress between the horizontal element layers; thus, an active earth pressure calculation model for sloping finite soil is established.


## 2. Calculation of Active Earth Pressure

A mechanical analysis model of finite soil is established, as shown in Figure1. The retaining wall was positioned on the left side, and the cohesionless sloping finite soil was on the right side. The slope dip angle of finite soil is α, the slope top width is l, the gravity of soil is γ, the internal friction angle is φ, the depth of the retaining wall is H, and the external friction angle is δ. The sloping finite soil is divided into two zones, with the ef regarded as a dividing line. The height of zone I and zone II are h1 and h2, respectively, while the angle between the tangent line of any point on the slip surface and the horizontal direction is ψ.Figure 1
Calculation model of active earth pressure.There are four types of commonly applied curved slip surfaces of soil, including broken line sliding surface [20, 22, 24], logarithmic spiral curve [29], experimental fitting curve [8], and cycloidal curve [19, 30], as illustrated in Figure 2.Figure 2
Slip surface through the slope surface.The equations of a cycloid line are as follows:(1)x=R1θ−sinθ,z=R11−cosθ+h1,where R1 is the radius of the cycloid line and θ is the angle of the cycloid line.Whenz=H, the initial angle of the cycloid line is θc, and the length of ef can be calculated as
(2)ef¯=R1θc−sinθc.It can also be obtained from the geometric relationship in Figure2.
(3)cd¯=l+Htanα,ed¯=R11−cosθc,ef¯=cd¯−ed¯tanα.The radius of the cycloid line may be calculated using Equations (2) and (3).
(4)R1=l+H/tanαθc−sinθc+1−cosθc/tanα.The height of the cycloid slip surface is obtained as follows:(5)h2=ed¯=R11−cosθc,(6)h1=H−h2.The slip surface of soil reaches the top of soil in the finite soil range whenh2≥H, which belongs to semi-infinite soil, where h1=0. According to Equation (1), R1 can be calculated as
(7)R1=H1−cosθc.The rotation angle of the cycloid slip surface at any point is expressed as(8)θ=arccos1−z−h1R1.The slope of any point on the cycloid line is expressed as(9)tanψ=dzdx=tanπ2−θ2.At each point on the cycloid line, the angle between the tangent line and the horizontal direction equals(10)ψ=π2−θ2.Figure3 depicts the horizontal differential layer AB, which is used as the study object and has a length of Lz and a height of dz at a distance of z from the soil surface. AB experiences stress deflection and produces an arc-shaped minor principal stress trajectory when the soil behind the retaining wall reaches the active limiting equilibrium state. The circle’s center is at point O, and the radius is R2. The angle between the horizontal direction and the connecting line from any point D in the arc to the center of the circle is ε, the angle between AO and the horizontal direction is εA, and the angle between BO and the horizontal direction is εB.Figure 3
Stress analysis of soil element.
(a)
Trajectory of minor principal stress(b)
Mohr’s stress circle of soil at pointAWhen the pointD is in failure condition, its horizontal stress σh, vertical stress σv, and shear stress τ can be expressed as
(11)σvσ1=sin2ω+Kacos2ω,σhσ1=cos2ω+Kasin2ω,τσ1=sinφ⋅sin2ω1+sinφ,where σ1 is the major principal stress in a differential element, ω is angle between the direction of major principal stress and horizontal direction at point D, and Ka is the coefficient of passive earth pressure. Then,
(12)Ka=σ3σ1=1−sinφ1+sinφ,where σ3 is the minor principal stress in a differential element.Vertical resultant forcedV at point D can be expressed as
(13)dV=σVdA=σ11−cos2ωsinφ1+sinφR2sinωdω,where εA is the angle between the minor principal stress at A point and the horizontal plane and εB is the angle between the minor principal stress at B point and the horizontal plane.When AB is in zone I, Equation (14) can be obtained:
(14)εA=π2−αA2,εB=π2,where αA=arcsinsinδ/sinφ−δ.When AB is in zone II,εA is consistent with zone I.
(15)εB=π4−φ2+ψ.The lateral active earth pressure coefficient is the ratio ofσh to the average vertical pressure σv¯, as follows:
(16)Kawn=σhσv¯=cos2εA+Kasin2εA1+cos3εA−cos3εBKa−1/3cosεA−cosεB.The shear stress coefficientA can be calculated in the same way as follows:
(17)τ¯=∫εAεBτR2sinεdεLz=2sinφsin3εA−sin3εB31+sinφcosεA−cosεBσ1,(18)Kτ=τ¯σv¯=2sinφsin3εA−sin3εB31+sinφcosεA−cosεB−2sinφcos3εA−cos3εB,where τ¯ is the average shear stress on the principal stress trajectory.As shown in Figure4, a stress analysis is performed on the horizontal differential layer element in zone I.Figure 4
Stresses acting on horizontal elements in zone I.The following formula can be obtained:(19)σh1dz+τ1dzcotα+dτ1l+zcotα=0,where σh1 is the horizontal active earth pressure at the depth z and τ1 is the shear stress on the surface of the horizontal element. Assuming that τ1 is uniformly distributed, it can be stated as follows:
(20)τ1=σv1Kτ.It can be concluded from the horizontal element’s vertical static equilibrium condition that(21)τw1dz+σv1dzcotα+dσv1l+zcotα−dw1=0.where τw1 is the friction shear stress on the interface between the retaining wall and the soil, which can be stated as
(22)τw1=σh1tanδ.In addition,dw1 is the self-weight of the horizontal differential element in zone I, and it is formulated as
(23)dw1=γl+zcotαdz.Considering all the above conditions, the differential equation can be expressed as(24)dσv1dz=γ−σh1tanδ+σv1cotαl+zcotα,σh1=Kτγl+zcotαKτtanδ−1.As shown in Figure5, a stress analysis is performed on zone II’s horizontal differential layer element. It can be obtained from the geometric relationship in Figure 5.
(25)kl¯=l+h1cotα−R1θ−sinθ,mn¯=kl¯−cotψdz=kl¯−tanθ2dz.Figure 5
Stresses acting on horizontal elements in zone II.By omitting the second derivative, the following equation is obtained:(26)σh2dz+τ2kl¯−τ2+dτ2mn¯−rcosπ/2+φ−ψdzsinψ=0,where σh2 is the horizontal active earth pressure, r is the reaction force on the sliding inclined plane, σv2 is the average vertical normal stress on the surface of the horizontal element, and τ2 is the horizontal shear stress on the surface of the horizontal element.
(27)τ2=σv2Kτ.Based on the horizontal element’s vertical static equilibrium condition, it can be determined that(28)dw2−σv2kl¯+σv2+dσv2mn¯−τw2dz−rsinπ/2+φ−ψdzsinψ=0,where τw2 is the friction shear stress on the interface between the retaining wall and the soil mass; it can be stated as follows:
(29)τw2=σh2tanδ,where dw2 is the self-weight of the horizontal layer unit in zone II, which can be stated as
(30)dw2=γ2kl¯+mn¯dz.Formulas (25)–(30) are synthesized as follows:
(31)dσv2dzkl¯1+Kτtanφ+θ2=γkl¯+σv2tanθ2−Kawntanδ−Kawntanφ+θ2+Kτtanθ2tanφ+θ2,(32)dσv2dz=1Aγ+BAkl¯σv2,in which
(33)A=1+Kτtanφ+θ2,B=tanθ2−Kawntanφ+θ2−Kawntanδ+Kτtanθ2tanφ+θ2.The difference method is used to estimate a solution that can be achieved using the finite difference method. The soil mass is divided into several horizontal differential element layers. The thickness of each layer isΔz, as shown in Figure 6.Figure 6
Stress of horizontal differential element layers.Equation (31) can be expressed as
(34)Δσiv2Δz=1Aγ+BAkl¯σiv2,σi+1v=σiv+Δσiv,σi+1h2=Kawnσi+1v2,where σ1v is expected to be q in the condition of Z=0, and the corresponding values of σiv for different soil strips can be calculated according to Equation (24) and Equation (34). Afterwards, the values of σih and τwi for all horizontal layers can be computed sequentially. On this basis, the resultant force and overturning moment of earth pressure are obtained according to the following:
(35)E=∑i=1nσihcosδΔz,M=∑i=1nσihH−ziΔz,where n is the number of horizontal layers, zi is the vertical distance between the center of the ith layer and the top surface of the soil, and ψ is the angle between the direction of the resultant force and the horizontal direction. The vertical distance between the resultant force’s action point and the wall heel is defined as
(36)h=MEcosδ=∑i=1nσihH−zi∑i=1nσih.


## 3. Experimental Comparison

The proposed approach’s computation outcomes are compared to laboratory test results to ensure its applicability. Three groups of experiments were performed by using cohesionless sand. The geometric parameters of the tests are shown in Table1. The internal friction angle is φ=39.6°, the gravity is γ=16.1kN/m3, and the outer friction angle is δ=2/3φ.Table 1
List of tests.
Test numberThe height of soil (m)The top width of slope (m)The bottom angle of slope10.5000.05033.7°20.5000.10033.7°30.5000.15033.7°As illustrated in Figure7, the theoretical calculation results of the distribution of active earth pressure were compared with the experimental results.Figure 7
Comparison of the proposed model’s results and the experimental validation of the laboratory test.It can be demonstrated that the proposed approach produces a distribution of active earth pressure that was comparable to the findings of laboratory model experiments. The earth pressure distribution curve was found to be nonlinear and drum-shaped. The active earth pressure determined by the model test on sloping finite soil was relatively close to the proposed calculated value. However, some discrepancies may be attributable to the influence of test boundary conditions and parameters.


## 4. Parameter Analysis

In this section, the influences of geometric slope parameters on the distribution, resultant force, and action point of active earth pressure are evaluated. Assuming that the sloping finite soil behind the retaining wall is cohesionless, the wall’s back is defined as vertical, and its height is defined as 10 m. The soil gravity is 17.0 kN/m3, the internal friction angle is φ=43°, and the external friction angle is δ=2/3φ.


### 4.1. Analysis of the Distribution of Earth Pressure

Figure8 illustrates the influence of different l/H on the distribution of active earth pressure with α=40°. The distribution of active earth pressure in zone I is linear; however, it is nonlinear with a drum shape in zone II.Figure 8
Distribution of active earth pressure under differentl/H (α=40).A sharp change at the junction of the two zones was observed. The horizontal component of active earth pressure increased gradually with an increase in the aspect ratiol/H. When l/H>0.4, it tended to be stable and could be regarded as semi-infinite soil. Figure 9 demonstrates the influence of different α on the distribution of the horizontal component of active earth pressure with l/H=0.1. The horizontal component of active earth pressure gradually declined as α increased, particularly in zone I.Figure 9
Distribution of active earth pressure under differentα (l/H=0.1).


### 4.2. Analysis of Resultant Force and Earth Pressure

Figure10 depicts the impact of various l/H on the resultant earth pressure and its action point height. The resultant force of finite earth pressure had an action point slightly greater than that calculated by Coulomb’s theory. In the case of l/H<0.4, the resultant force and its action point of active earth pressure increased nonlinearly with an increase in the aspect ratio. On the other hand, for l/H≥0.4, the resultant force and its action point of active earth pressure tended to be stable, and slope finite soil could be regarded as semi-infinite soil.Figure 10
Effect ofl/H on active earth pressure.
(a)
Effect ofl/H on the resultant force(b)
Effect ofl/H on the position of resultant forceFigure11 shows the influence of different α on the resultant earth pressure and its action point height. The resultant force of active earth pressure declined linearly as α increased, and the acting point height of the resultant force of earth pressure similarly decreased, which was approximately two-fifths the height of the wall.Figure 11
Effect ofα on active earth pressure.
(a)
Effect ofα on the resultant force(b)
Effect ofα on the position of resultant force


## 4.1. Analysis of the Distribution of Earth Pressure

Figure8 illustrates the influence of different l/H on the distribution of active earth pressure with α=40°. The distribution of active earth pressure in zone I is linear; however, it is nonlinear with a drum shape in zone II.Figure 8
Distribution of active earth pressure under differentl/H (α=40).A sharp change at the junction of the two zones was observed. The horizontal component of active earth pressure increased gradually with an increase in the aspect ratiol/H. When l/H>0.4, it tended to be stable and could be regarded as semi-infinite soil. Figure 9 demonstrates the influence of different α on the distribution of the horizontal component of active earth pressure with l/H=0.1. The horizontal component of active earth pressure gradually declined as α increased, particularly in zone I.Figure 9
Distribution of active earth pressure under differentα (l/H=0.1).


## 4.2. Analysis of Resultant Force and Earth Pressure

Figure10 depicts the impact of various l/H on the resultant earth pressure and its action point height. The resultant force of finite earth pressure had an action point slightly greater than that calculated by Coulomb’s theory. In the case of l/H<0.4, the resultant force and its action point of active earth pressure increased nonlinearly with an increase in the aspect ratio. On the other hand, for l/H≥0.4, the resultant force and its action point of active earth pressure tended to be stable, and slope finite soil could be regarded as semi-infinite soil.Figure 10
Effect ofl/H on active earth pressure.
(a)
Effect ofl/H on the resultant force(b)
Effect ofl/H on the position of resultant forceFigure11 shows the influence of different α on the resultant earth pressure and its action point height. The resultant force of active earth pressure declined linearly as α increased, and the acting point height of the resultant force of earth pressure similarly decreased, which was approximately two-fifths the height of the wall.Figure 11
Effect ofα on active earth pressure.
(a)
Effect ofα on the resultant force(b)
Effect ofα on the position of resultant force


## 5. Conclusions

In the current study, a calculation method was developed for the soil pressure of a sloping finite soil, considering the shear stress and the curved slip surface. To validate the rationality of these formulas, the earth pressure distribution was compared to current test outcomes.With the increase in aspect ratio, the horizontal component of the active earth pressure distribution curve was linear in zone I and nonlinear with a drum shape in zone II. Besides, a noticeable change at the junction of the two zones was observed. The change became weaker as the aspect ratiol/H decreased and stronger as α increased.The resultant force of earth pressure and the position of the resultant force action point increased and tended to achieve a specific value as the aspect ratiol/H increased. With an increase in α, the resultant force and the height of the resultant force acting point were reduced linearly.



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