As shown in Figure 1, a schematic diagram is established with finite soil as the research object, with the retaining wall on the left, the bedrock on the right, and non-cohesive soil between them. The width of the finite soil is X. The internal friction angle of the soil is ϕ . The gravity is γ . The buried depth of the retaining wall is Z ₁. The external friction angle of the soil is δ 1 . The distance between the intersection of the bedrock and sliding surface and the ground is Z ₂. The external friction angle of the soil is δ 2 . When the soil reaches the limit equilibrium state, a curve slip surface through the bottom of the wall is formed within the soil. H is the height of the slip surface. ψ is the angle between the tangent of any point of the slip line and the horizontal line. According to the different boundary conditions, the finite soil is divided into I and II zones.

The following assumptions are made to simplify the theoretical derivation:

1) The finite soil behind the wall is a single soil layer, which is homogeneous and non-cohesive.

If the classical earth pressure theory is used to calculate the active earth pressure, one of its assumptions is that the sliding surface is a straight line passing through the bottom of the wall. However, the experiments and theories of some scholars proved that the sliding surface of active earth pressure is not a straight line. A number of nonlinear sliding surface models have been proposed by many scholars, such as cycloidal lines (Cao, 1995; Yang et al., 2017), logarithmic spiral curves (Wang et al., 2011; He et al., 2020), and folding lines (Greco, 2013). In this study, it is assumed that when the retaining wall is in limit equilibrium, the soil in the active zone behind the wall produces a cycloidal line slip surface through the heel of the wall as shown in Figure 2.

The right-angle coordinate system is established as shown in Figure 2. The equation of the cycloidal line can be expressed as: { x = R 1 ( θ − sin ⁡ θ ) ; z = R 1 ( 1 − cos ⁡ θ ) + Z 2 , (1) where R ₁ is the radius of the rotating wheel, and θ is the rotating angle.

When the cycloidal line passes through the wall toe, z = Z 1 , the radius of the rotating wheel can be obtained as: R 1 = X θ c − sin ⁡ θ c , (2) where θ c is the initial rotating angle of the cycloidal line.

Thus, the height of the cycloidal line slip surface can be obtained as Eq. 3: H = Z 1 − Z 2 = R 1 ( 1 − cos ⁡ θ c ) , (3)

If H ≥ Z 1 , it indicates that the sliding surface of soil reaches the ground within the range of finite soil, which is semi-infinite at this time. Take Z 2 = 0 , then R ₁ is calculated by the second equation in Eq. 4: R 1 = Z 1 1 − cos ⁡ θ c , (4)

The rotating angle θ at any point on the slip surface is shown as: θ = arccos ( 1 − z − Z 2 R 1 ) , (5)

The slope of any point on the slip surface tanψ is shown in Eq. 6: tan ⁡ ψ = d z d x = tan ( π 2 − θ 2 ) , (6)

The angle between the tangent and horizontal direction at any point on the slip surface ψ is shown as: ψ = π 2 − θ 2 , (7)

With the lateral displacement of the retaining wall during the active failure of the soil, the soil and the back of the wall produce relative slip. The friction between the wall and the soil deflects principal stress of the soil element. After the soil element is deflected, the curve formed by the principal stress direction is called the principal stress trajectory. The principal stress trajectory is generally a catenary curve. Paik and Salgado (2003) compared the catenary trajectory line with the arc trajectory line. The results show that the difference between the two calculation results is not significant. Meanwhile, the circular arc is simpler than the catenary calculation, which is more convenient for practical application. Therefore, this study adopts the circular arc for stress analysis.

Layer AB of Zone I is shown in Figure 3, and layer AB of Zone II is shown in Figure 4. The length is L _z. When the soil after the retaining wall reaches the active limit equilibrium state, the stress deflection occurs in AB, which forms a circular arc minor principal stress trajectory. The center of the circle is located at point O in the figure. The radius is R ₂. The angle between the connection line of any point D in the arc and the center O in the horizontal direction is ε. The angle between AO and the horizontal direction is ε _A. The angle between BO and the horizontal direction is ε _B.

When active failure occurs at point D, the horizontal σ h and vertical stresses σ v can be expressed as (Zhu and Zhao, 2014): { σ v σ 1 = sin 2 ω + K a cos 2 ω ; σ h σ 1 = cos 2 ω + K a sin 2 ω ; (8) K a = σ 3 / σ 1 = ( 1 − sin ⁡ φ ) / ( 1 + sin ⁡ φ ) , (9) where ω is the angle between major principal stress and horizontal direction.

The vertical force of point D, i.e., d V is shown as: d V = σ V d A = σ 1 [ 1 − cos ( 2 ω ) sin ⁡ φ ] 1 + sin ⁡ φ R 2 ⁡ sin ⁡ ω d ω . (10)

The relationship between the radius of small principal stress traces R ₂ and the distance L _z between the two points AB is as follows: R 2 = L z / ( cos ⁡ ε B − cos ⁡ ε A ) , (11)

It can be seen from Figure 5 that the angle between the minor principal stress at point A and the horizontal direction is ε _A. The angle between the minor principal stress at point B and the right tangent is ε _B.

When AB is located in Zone I: ε A = π / 2 − α A / 2 , (12) α A = arc sin ( sin ⁡ δ 1 sin ⁡ φ ) − δ 1 , (13) ε B = π / 2 + α B / 2 , (14) α B = arc sin ( sin ⁡ δ 1 sin ⁡ φ ) − δ 2 , (15)

When AB is located in Zone II:

ε _A is the same as in Zone I, ε _B is equal to the sum of the angle between the minor principal stress at point B and the tangential direction of the slip surface, and the angle between the tangential direction of the slip surface and the horizontal direction, namely, ε B = π / 4 − φ / 2 + ψ , (16)

In the calculation of earth pressure on retaining walls by the horizontal differential layer method, the active lateral earth pressure coefficient K a w n is the ratio between σ h and vertical average stress σ v ¯ .

The following equation can be deduced: K a w n = σ h σ v ¯ = σ 1 ( cos 2 ε A + K a ⁡ sin 2 ε A ) ∫ l A B d V L z = cos 2 ε A + K a ⁡ sin 2 ε A 1 + ( cos 3 ε A − cos 3 ε B ) ( K a − 1 ) 3 ( cos ⁡ ε A − cos ⁡ ε B ) , (17)

It can be seen from Eq. 17 that when the horizontal differential layer is located in Zone I, K a w n is a fixed value; when the horizontal differential layer is located in Zone II, it changes with the slope of the slip line, namely, K a w n changes with the rotation angle of the cycloidal line.

A horizontal differential element layer at z from the ground is taken, and the equilibrium equations are established for analysis according to the different stresses on the soil in Zone I and Zone II. Assuming that there is no relative slip between the horizontal differential layers of soil, namely, the shear stress between layers is not considered.

The mechanical model is shown in Figure 6 when the horizontal differential layer is located in the soil of Zone I, z < Z 2 . σ v 1 is the average vertical stress acting on the upper surface of the differential element, σ v 1 + d σ v 1 is the average vertical stress acting on the lower surface, σ h 1 is the horizontal stress of the retaining wall structure side, and σ n is the horizontal stress of the bedrock side. The thickness of the differential horizontal element is d z , and the volume is d V .

According to the balance of forces in the horizontal direction, Eq. 18 can be obtained: σ h 1 d z − σ n d z = 0, (18)

According to the balance of stresses in the vertical direction, Eq. 19 can be obtained: X d σ v 1 + σ h 1 ⁡ tan ⁡ δ 1 d z + σ n ⁡ tan ⁡ δ 2 dz = γ d V , (19)

Combining Eqs 17–19: X d σ v 1 + K a w n σ v 1 ⁡ tan ⁡ δ 1 d z + K a w n σ v 1 ⁡ tan ⁡ δ 2 dz = X γ d z . (20)

Differential Eq. 21 can be obtained: d σ v 1 dz = γ − K a w n σ v 1 ( ⁡ tan ⁡ δ 1 + tan ⁡ δ 2 ) X . (21)

The horizontal earth pressure is: σ h 1 = K a w n σ v 1 , (22)

When the horizontal differential layer is located in Zone II, Z 2 < z < Z 1 . The mechanical model is shown in Figure 7. σ v 2 is the average vertical stress acting on the upper surface of the differential element. σ v 2 + d σ v 2 is the average vertical stress acting on the lower surface. σ h 2 is the horizontal stress acting on the side of the retaining wall. r is the reaction of soil to the differential element. φ is the internal friction angle of the soil. ψ is the angle between the tangent of the slip surface at the differential unit and the horizontal direction. The thickness of the differential unit is d z . The volume is d V .

The top width of the differential element can be calculated as: B 1 = X ( 1 − θ − sin θ θ c − sin ⁡ θ c ) . (23)

The bottom width of the microelement can be calculated as: B 2 = B 1 − cot ⁡ ψ d z = B 1 − tan θ 2 d z , (24)

The microelement weight can be calculated as: γ d V = γ 2 ( B 1 + B 2 ) d z , (25)

Omitting the higher order differential, we get: γ d V = γ B 1 d z , (26)

According to the balance of stresses in the horizontal direction, the following equations can be obtained: σ h 2 d z − r ⁡ cos ( π 2 + φ − ψ ) d z/sin ψ = 0 , (27) σ h 2 = r ⁡ cos ⁡ ( φ + θ 2 ) /cos θ 2 , (28)

According to the balance of stresses in the vertical direction, the following equations can be obtained: σ v 2 B 1 + γ d V − σ h 2 ⁡ tan ⁡ δ 1 d z − r ⁡ sin ( π 2 + φ − ψ ) d z /sin ψ − ( σ v 2 + d σ v 2 ) B 2 = 0 , (29) d σ v 2 d z = γ + σ v 2 ⁡ tan θ 2 − σ h 2 ⁡ tan ⁡ δ 1 − r ⁡ sin ( φ + θ 2 ) /cos θ 2 B 1 , (30)

The Eq. 31 can be obtained by combining Eq. 17. d σ v 2 d z = γ + σ v 2 [ tan θ 2 − K a w n ⁡ tan ⁡ δ 1 − K a w n ⁡ tan ( φ + θ 2 ) ] B 1 , (31)

The horizontal earth pressure is shown as follows: σ h 2 = K a w n σ v 2 , (32)

For a given soil and retaining wall, parameters δ , φ , γ , and H are known. θ c is unknown. However, due to a large number of parameters, it is difficult to obtain the analytical solution in the aforementioned derivation process of active earth pressure. This study adopts MATLAB software to calculate by the numerical method, and the specific calculation process is as follows:

1) Assuming an initial rupture angle, according to Eq. 1.

Geotechnical engineering is concerned with determining the critical width of finite soil. The width calculated by the Coulomb earth pressure theory is commonly used as the critical value by most researchers. However, the critical width of the soil is not accurate because the Coulomb earth pressure assumes that the sliding surface behind the wall is straight.

Based on this, after deriving θ c by the aforementioned method, the slip crack surface width X ₀ can be deduced as: X 0 = R 1 ( θ c − sin ⁡ θ c ) , (33)

This study selects the retaining wall height of 10 m, the filling weight of 14.6 kN/m³, and the filling surface without load as examples to investigate the influence of various parameters on the critical width of finite soil.

The wall–soil friction angle is taken as a fixed value δ 1 = 20 ° . The internal friction angle is varied for analysis. The critical slip surface of the finite soil and semi-infinite soil is shown in Figure 10. The corresponding critical widths of the finite soil are 5.41, 4.63, 4.08, 3.56, and 3.11 m. At this time, the critical widths calculated according to the Coulomb earth pressure theory are 6.75, 5.92, 5.18, 4.52, and 3.92 m. It can be seen that when the internal friction angle increases, the critical width of the finite soil decreases gradually. The change rate also decreases gradually. The critical width value obtained by this method is obviously smaller than the calculated value of Coulomb earth pressure.

The internal friction angle is taken as φ = 40 ° and the wall–soil friction angle is taken as δ 1 = 10 ° , 20 ° , 30 ° , 40 ° for analysis. The critical slip surface of the finite soil and semi-infinite soil is shown in Figure 11. The corresponding critical widths of the finite soil are 3.89, 4.08, 4.42, and 5.05 m, respectively. At this time, the critical widths calculated according to the Coulomb soil pressure theory are 4.94, 5.18, 5.42, and 5.67 m. The results show that when the wall–soil friction angle increases, the critical width of the finite soil gradually increases, and the change rate gradually increases. The critical width of finite soil is smaller than the Coulomb theoretical value under different wall–soil friction angles.